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7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 143
Mechanical Engineering CAE Research Lab1
An Introduction to the
Boundary Element Method (BEM) andIts Applications in Engineering
Yijun LiuProfessor of Mechanical Engineering University of Cincinnati
Cincinnati Ohio 45221-0072 USA
E-mail YijunLiuuceduWeb wwwyijunliucom
(Updated November 8 2013)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 243
Mechanical Engineering CAE Research Lab2
Boundary Element Method (BEM)
n
n
n
bull Boundary element method applies surfaceelements on a 3-D domain and line elementson a 2-D domain Number of elements is O(n2)as compared to O(n3) for other domain basedmethods ( n = number of elementsdimension)
bull BEM is good for problems with complicatedgeometries stress concentration problems
infinite domain problems wave propagation problems and many othersbull F inite element method can now solve a model
with 1 million DOFs on a PC with 1GB RAM
bull F ast mul tipole BEM can also solve a modelwith 1 million DOFs on a PC with 1 GB RAMHowever these DOFs are on the boundary ofthe model only which would require 1 billion
DOFs for a corresponding domain modelANSYS
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 343
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 443
Mechanical Engineering CAE Research Lab4
A Brief History of the BEM
BEM emerged in 1980rsquos hellip
Integral equations(Fredholm 1903)
Modern numericalsolutions of BIEs
(in early 1960rsquos)
Jaswon and Symm (1963)
ndash 2D Potential Problems
F J Rizzo (Dissertation in 1964at TAM UIUC paper in 1967)
ndash 2D Elasticity Problems
T A Cruse and F J Rizzo (1968) ndash 2D elastodynamics
P K Banerjee (1975) ndash Coined the name ldquoboundary element methodrdquo
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 543
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 243
Mechanical Engineering CAE Research Lab2
Boundary Element Method (BEM)
n
n
n
bull Boundary element method applies surfaceelements on a 3-D domain and line elementson a 2-D domain Number of elements is O(n2)as compared to O(n3) for other domain basedmethods ( n = number of elementsdimension)
bull BEM is good for problems with complicatedgeometries stress concentration problems
infinite domain problems wave propagation problems and many othersbull F inite element method can now solve a model
with 1 million DOFs on a PC with 1GB RAM
bull F ast mul tipole BEM can also solve a modelwith 1 million DOFs on a PC with 1 GB RAMHowever these DOFs are on the boundary ofthe model only which would require 1 billion
DOFs for a corresponding domain modelANSYS
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 343
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 443
Mechanical Engineering CAE Research Lab4
A Brief History of the BEM
BEM emerged in 1980rsquos hellip
Integral equations(Fredholm 1903)
Modern numericalsolutions of BIEs
(in early 1960rsquos)
Jaswon and Symm (1963)
ndash 2D Potential Problems
F J Rizzo (Dissertation in 1964at TAM UIUC paper in 1967)
ndash 2D Elasticity Problems
T A Cruse and F J Rizzo (1968) ndash 2D elastodynamics
P K Banerjee (1975) ndash Coined the name ldquoboundary element methodrdquo
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 543
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 343
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 443
Mechanical Engineering CAE Research Lab4
A Brief History of the BEM
BEM emerged in 1980rsquos hellip
Integral equations(Fredholm 1903)
Modern numericalsolutions of BIEs
(in early 1960rsquos)
Jaswon and Symm (1963)
ndash 2D Potential Problems
F J Rizzo (Dissertation in 1964at TAM UIUC paper in 1967)
ndash 2D Elasticity Problems
T A Cruse and F J Rizzo (1968) ndash 2D elastodynamics
P K Banerjee (1975) ndash Coined the name ldquoboundary element methodrdquo
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 543
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 443
Mechanical Engineering CAE Research Lab4
A Brief History of the BEM
BEM emerged in 1980rsquos hellip
Integral equations(Fredholm 1903)
Modern numericalsolutions of BIEs
(in early 1960rsquos)
Jaswon and Symm (1963)
ndash 2D Potential Problems
F J Rizzo (Dissertation in 1964at TAM UIUC paper in 1967)
ndash 2D Elasticity Problems
T A Cruse and F J Rizzo (1968) ndash 2D elastodynamics
P K Banerjee (1975) ndash Coined the name ldquoboundary element methodrdquo
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 543
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 543
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 643
Mechanical Engineering CAE Research Lab6
FEMResults
(50 min)
BEMResults
(16 min)
A Comparison of the FEM and BEMwith An Engine Block Model (Cont)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 743
Mechanical Engineering CAE Research Lab7
Formulation The Potential Problembull Governing Equation
with given boundary conditions on S bull The Greenrsquos function for potential problem
bull Boundary integral equation formulation
wherebull Comments The BIE is exact due to the use of the Greenrsquos function
Note the singularity of the Greenrsquos function G(x y)
0)(2 V u = xx
[ ] or )()()()()()()( S V dS u F qGuC S
minus= int xyyyxyyxxx
2Din1
ln21
)(
=r
Gπ
yx
nG F nu q part part part part ==
r
S x
y
n
V
3Din4
1)(
r G
π =yx
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 843
Mechanical Engineering CAE Research Lab8
Formulation The Potential Problem (Cont)bull Discretize boundary S using
N boundary elementsline elements for 2D problemssurface elements for 3D problems
bull The BIE yields the following BEM equation
bull Apply the boundary conditions to obtain
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
b
b
x
x
x
aaa
aaa
aaa
=
N NN N N
N
N
N NN N N
N
N
u
u
u
g g g
g g g
g g g
q
q
q
f f f
f f f
f f f
2
1
21
22221
11211
2
1
21
22221
11211
r
i(x)
yn
V
Each nodeelementinteracts with all other
nodeelement directlyThe number of
operations is of orderO (N 2)
Storage is also of order
O (N 2)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 943
Mechanical Engineering
Advantages and Disadvantages of the BEM
Advantages bull Accuracy ndash due to the semi-analytical nature and use of integrals
bull More efficient in modeling due to the reduction of dimensions
bull Good for stress concentration and infinite domain problems
bull Good for modeling thin shell-like structuresmaterials
bull Neat hellip (integration superposition boundary solutions for BVPs)
Disadvantages
bull Conventional BEM matrices are dense and nonsymmetricalbull Solution time is long and memory size is large (Both are O( N 2))
bull Limited to solving small-scale models (Not any more with new fast
solution methods)CAE Research Lab9
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1143
Mechanical Engineering CAE Research Lab11
The Simple Idea
Conventional BEM approach ( O( N 2
)) FMM BEM approach ( O( N ) for large N )
Apply iterative solver (GMRES) and accelerate matrix-vector multiplications by replacing element-element interactions with cell-cell interactions
bAx == or2
1
2
1
21
22221
11211
N N NN N N
N
N
b
bb
x
x x
aaa
aaaaaa
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1243
Mechanical Engineering
Adaptive Cross Approximation (ACA)
bull Hierarchical decomposition of a BEM matrix
(from Rjasanow and Steinbach 2007)
bull A lower-rank submatrix A away from the main diagonal can berepresented by a few selected columns ( u ) and rows ( vT ) (crosses) basedon error estimates
bull The process is independent of the kernels (or 2-D3-D)
bull Can be integrated with iterative solvers (GMRES)12
)()()(with11
i j jik T
k AvAuAvuA ===asymp sum=
γ γ α
α α α
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1343
Mechanical Engineering CAE Research Lab13
Some Applications of the Fast Multipole
Boundary Element Methodbull 2-D3-D potential problemsbull 2-D3-D elasticity problemsbull 2-D3-D Stokes flow problemsbull 2-D3-D acoustics problemsbull Applications in modeling porous materials fiber-reinforced composites and
microelectromechanical systems (MEMS)
bull All software packages used here can be downloaded from wwwyijunliucom
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1443
Mechanical Engineering CAE Research Lab14
2-D Potential Accuracy and Efficiency of the
Fast Multipole BEM
N
FMM BEM ConventionalBEM
36 -401771619 -401771546
72 -400400634 -400400662
360 -400014881 -400014803
720 -400003468 -400003629
1440 -400000695 -400000533
2400 -400001929 -400000612
4800 -400001557 -400000561
7200 -399997329 -399998183
9600 -399997657 -399996874
AnalyticalSolution -4000
aqa b
O
V
S b
S a
Results for a simple potential problem in an annular region V
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1543
Mechanical Engineering CAE Research Lab15
3-D Potential Modeling of Fuel Cells
Thermal Analysis of FuelCell (SOFC) Stacks
There are 9000 smallside holes in this model
Total DOFs = 530230solved on a desktop PCwith 1 GB RAM)
ANSYS can only modelone cell on the same PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1643
Mechanical Engineering CAE Research Lab16Computed charge density
3-D Electrostatic Analysis
Applied potential ( plusmn5)
X Y
Z
One BEM mesh
bull 11 conducting spheres
bull Forces can be found with the charge density
bull Largest model has 118800 DOFs
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1743
Mechanical Engineering CAE Research Lab17
3-D Electrostatic Analysis (Cont)
Applications
in MEMS
A comb drive
bull Beams are applied with +- voltages
bull Forces can be found with the charge density
bull Model shown has 55 beams (179300 DOFs)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1843
Mechanical Engineering CAE Research Lab18
2-D Elasticity Modeling of Perforated Plates
Computed effective Youngrsquos modulusfor the perforated plate ( x E )
No Holes DOFsUniformly
Distributed Holes
Randomly Distributed
Holes
2x2 3680 0697892 0678698
4x4 13120 0711998 0682582
6x6 28320 0715846 0659881
8x8 49280 0717643 0651026
12x12 108480 0719345 0672084
20x20 296000 0720634 0676350
30x30 660000 0721255 0676757
40x40 1168000 0721558 0675261
A BEM model of a perforated plate(with 1600 holes)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 1943
Mechanical Engineering CAE Research Lab19
3-D Elasticity Modeling of Scaffold Materials
(Hollister et al 2002)
PreliminaryBEMmodels and
results
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2043
Mechanical Engineering CAE Research Lab20
2-D Stokes Flow Multiple Cylinders
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2143
Mechanical Engineering CAE Research Lab21
3-D Stokes Flow Modeling of RBCs
Drag force in the flow direction
An exterior Stokes flow problem
Total DOFs = 900 K Solved on alaptop PC
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2243
Mechanical Engineering
3-D Stokes Flow MEMS Analysis
bull BEM model with362662 elements(1087986 totalDOFs)
bull An angular velocityis applied
bull Drag forces arecomputed
bull Solved on a desktopPC
CAE Research Lab22
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2343
Mechanical Engineering CAE Research Lab23
Modeling CNT Composites
CNT fibersFiber (Linear
elastic anisotropic)
Cohesive interface(Linearnonlinear)
Matrix (Linearelastic isotropic)
(a) An RVE with many CNT fibers (to
be solved by the fast multipole BEM)
(b) Models for the CNTs andinterfaces (to be extracted from
MD simulations)
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2443
Mechanical Engineering CAE Research Lab24
A Multiscale Model for CNT Composites
bull A rigid-inclusion model is applied to represent the CNT fibers in polymer matrixbull The cohesive model from MD study is applied for the CNTpolymer interfaces
bull The fast multipole BEM is applied to solve the large BEM systemsbull This approach is a first step toward the more general multiscale model withcontinuum BEM for matrix and nanoscale MD for CNTs and interfaces
InterfaceCNT (rigid inclusion)
Matrix (elastic)u
u (CNT ))(
α S CNT =minus yCtuu
A cohesive interface model
with C being the compliancematrix ( determined by MD )
α S
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2543
Mechanical Engineering CAE Research Lab25
A Typical RVE Using the BEM
A model containing 2197 short CNT fibers with the total DOF = 3018678
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2643
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2743
Mechanical Engineering CAE Research Lab27
Modeling of CNT Composites (Cont)Effects of the Cohesive Interface
Computed effective moduli of CNTpolymer composites(same CNT and RVE dimensions as used in the previous perfect bonding case)
Case 1 C11=C22=C33=0(perfect bonding)
Case 2 C11=C22=C33=Cr =002157 (large stiffness)
Case 3 C11=C22=C33=Cz =3506 (small stiffness)
Cr Cz are interfacecompliance ratios in theradial and longitudinaldirection of the fiberrespectively and aredetermined from theMD simulations
Closer to
experimental data
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2843
Mechanical Engineering
Acoustic Wave Problems
bull Helmholtz equation
bull - acoustic pressure - wavenumber bull BEM for solving 3-D full-half-space interiorexterior
radiationscattering problems
k cω =φ
2 2
( ) 0Qk Q E φ φ δ + + = x x x
CAE Research Lab28
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 2943
Mechanical Engineering
Examples A Radiating Sphere
CAE Research Lab29
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3043
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3143
Mechanical Engineering
Windmill Turbine Analysis
Plot of the SPL on the field due to 5 windmills (with 557470 DOFs)31 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3243
Mechanical Engineering
FEMBEM Coupled Analysis (Freq Response)
32 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3343
Mechanical Engineering
Noise Prediction in Airplane LandingTaking Off
Noise propagation on the ground
during the landing of an airplane BEMmodel with 539722 elements andsolved with the FMM BEM in 8940 sec
on a PC ( ka = 615 or f = 90 Hz)
CAE Research Lab33
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3443
Mechanical Engineering
Acoustic Noise During Launch of
A Space Vehicle
34
bull Jet flow was modeled using CFD by NASA
bull Acoustic field was modeled using ouracoustic fast BEM code
bull FFT used to compute the time domainsolutions
bull The BEM model with 300K elements wassolved on a laptop PC
CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3543
Mechanical Engineering
Bio-Medical Applications
Pressure plots at 11 kHz
with a plane wave in ndashx direction
A human head model
with 90000 elements35 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3643
Mechanical Engineering
Bio-Medical Applications (Cont)
CAE Research Lab36
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3743
Mechanical Engineering
Applications in Computer Animation
Work done by the Group of Professor Doug James at CornellUniversity Using the FastBEM Acoustics code
(Click on the images to play the YouTube video and hear the computed sound)
37 CAE Research Lab
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3843
Mechanical Engineering
Fast Multipole Boundary Element Method ( FastBEM ) Softwarefor Education Research and Further Development
(httpurbanamieuceduyliuSoftware )
CAE Research Lab38
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 3943
Mechanical Engineering
Summary
bull BEM is very efficient for solving large-scale problems withcomplicated geometries or in infinite domains
bull Fast multipole method has re-energized the BEM research anddramatically expanded its range of applications
bull More large-scale realistic engineering problems can be and should be solved by the fast multipole BEM
bull Other developments in fast multipole BEM fracture mechanicselastodynamic and electromagnetic wave propagation problems time-domain problems black-box fast multipole method (bbFMM)
coupled field and nonlinear problemsbull Other fast solution methods for solving BIEBEM equations include
adaptive cross approximation (ACA) method precorrected FFTmethod wavelet method and others
CAE Research Lab39
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4043
Mechanical Engineering
A Bigger Picture of the CM ndash A Numerical Toolbox
FEM Large-scale structural nonlinearand transient problems
BEM Large-scale continuum linearand steady state (wave) problems
Meshfree Large deformation fractureand moving boundary problems ldquoIf the only tool
you have is ahammer thenevery problem you
can solve lookslike a nailrdquo
CAE Research Lab40
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4143
Mechanical Engineering
References
1 L F Greengard The Rapid Evaluation of Potential Fields in Particle Systems (The MIT PressCambridge 1988)
2 N Nishimura ldquoFast multipole accelerated boundaryintegral equation methodsrdquo Applied Mechanics Reviews 55 No 4 (July) 299-324 (2002)
3 Y J Liu Fast Multipole Boundary Element Method -Theory and Applications in Engineering (CambridgeUniversity Press Cambridge 2009)
4 Y J Liu and N Nishimura ldquoThe fast multipole boundary element method for potential problems atutorialrdquo Engineering Analysis with Boundary
Elements 30 No 5 371-381 (2006)5 Y J Liu Fast Multipole Boundary Element Method
( FastBEM ) Software for Education Research andFurther Development (1997-2010)httpurbanamieuceduyliuSoftware(or Google search ldquo fast multipole BEM rdquo)
CAE Research Lab41
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4243
Mechanical Engineering CAE Research Lab42
Acknowledgments
bull The US National Science Foundation
bull NASAbull Prof Subrata Mukherjee at Cornell University
bull Prof Naoshi Nishimura at Kyoto University (Japan)
bull Prof Dong Qian at the University of Cincinnati
bull Students at the University of Cincinnati and Kyoto University
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
7212019 BEM Introduction
httpslidepdfcomreaderfullbem-introduction-56da2cdc1f990 4343
Mechanical Engineering CAE Research Lab43
Contact
Dr Yijun LiuMechanical EngineeringPO Box 210072University of CincinnatiCincinnati Ohio 45221-0072
USA
E-mail YijunLiuuceduWebsite wwwyijunliucom Phone 1 (513) 556-4607
