Post on 26-Jan-2021
transcript
Formulation of a discontinuous Galerkin method for
unstructured causal grids in spacetime and linear dispersive
electromagnetic mediaReza Abedi
Mechanical, Aerospace & Biomedical Engineering
University of Tennessee Space Institute (UTSI) / Knoxville (UTK)
Saba Mudaliar
Air Force Research Laboratory
Comparison of DG and CFEM methods
Consider FEs for a scalar field and polynomial order p = 2
p dofDG/CFEM
1 4
2 2.25
3 1.78
4 1.56
5 1.44
High order polynomial DG competitive
Disjoint basis
1. Balance laws at the element level
2. More flexible h-, hp-adaptivity
3. Less communication between
elements
Better parallel performance
CFEM: transition element DG
Comparison of DG and CFEM methods:
Dynamic problems
Discontinuities are preserved or generated from smooth initial conditions!
t = 0,
smooth
solutio
nt > 0,
shock has
formed
Burger’s equation (nonlinear)
Global numerical oscillations COMSOL
2. Hyperbolic problems: resolving shocks / discontinuities
1. Parabolic & Hyperbolic problems: O(N) solution complexity
CFEM DG
O(N) O(N1.5) d = 2
O(N2) d = 3
Explicit solver
Example:
10x finer mesh (1000x elements in 3D)
Cost:
DG: 103x
CFEM: 106-107x
Spacetime Discontinuous Galerkin Finite Element method:
1. Discontinuous Galerkin Method
2. Direct discretization of spacetime
3. Solution of hyperbolic PDEs
4. Use of patch-wise causal meshes
A local, O(N), asynchronous solution scheme
Discontinuous Galerkin (DG) Finite
Element Methods
Weakly enforce conservation jump conditions
(e.g., Rankine–Hugoniot)
Can recover balance properties at the element level (vs global domain)
Support for nonconforming meshes and
Arbitrary changes in element polynomial orderno transition elements
needed
Arbitrary
change in size
and polynomial
order
Superior performance for resolving
discontinuities (discrete solution space better
resembles the continuum solution space)
Sample DG solutions
with no evident
numerical
artifacts numerical artifacts generally
spoil continuous FE solutions
in the presence of shocks
In contrast to conventional finite element methods, DG
methods use discontinuous basis functions and
Direct discretization of spacetime
Replaces a separate time integration;
no global time step constraint
Unstructured meshes in spacetime
No tangling in moving boundaries
Arbitrarily high and local order of
accuracy in time
Unambiguous numerical framework
for boundary conditions
shock capturing more
expensive, less accurate
Shock tracking in spacetime:
more accurate and efficient
Results by Scott Miller
http://www.personal.psu.edu/stm134/
Spacetime Discontinuous Galerkin (SDG)
Finite Element Method
Local solution property
O(N) complexity (solution cost scales linearly vs. number of elements N)
Asynchronous patch-by-patch solver
DG + spacetime meshing + causal meshes for hyperbolic problems:
Elements labeled 1 can be solved in
parallel from initial conditions; elements
2 can be solved from their inflow
element 1 solutions and so forth.
SDG
Time marching
Time marching or the use of extruded
meshes imposes a global coupling that
is not intrinsic to a hyperbolic problem
- incoming characteristics on red boundaries
- outgoing characteristics on green boundaries
- The element can be solved as soon as inflow
data on red boundary is obtained
- partial ordering & local solution property
- elements of the same level can be solved
in parallel
Tent Pitcher:
Causal spacetime meshing
causality constraint
tent–pitching sequence
Given a space mesh, Tent Pitcher
constructs a spacetime mesh such
that the slope of every facet on a
sequence of advancing fronts is
bounded by a causality constraint
Similar to CFL condition, except
entirely local and not related to
stability (required for scalability)
time
Tent Pitcher:
Patch–by–patch meshing
meshing and solution are interleaved
patches (‘tents’) of tetrahedra are solve immediately O(N) property
rich parallel structure: patches can be created and solved in parallel
tent–pitching sequence
Advantages of Spacetime discontinuous Galerkin
(SDG) Finite element method
1. Arbitrarily high temporal order of accuracy
• Achieving high temporal orders in semi-discrete methods (CFEMs and
DGs) is very challenging as the solution is only given at discrete times.
• Perhaps the most successful method for achieving high order of accuracy
in semi-discrete methods is the Taylor series of solution in time and
subsequent use of Cauchy-Kovalewski or Lax-Wendroff procedure (FEM
space derivatives time derivatives). However, this method becomes
increasingly challenging particularly for nonlinear problems.
• High temporal order adversely affect stable time step size for explicit DG
methods (e.g. or worse for RKDG and ADER-DG methods).
• Spacetime (CFEM and DG) methods, on the other hand can achieve
arbitrarily high temporal order of accuracy as the solution in time is
directly discretized by FEM.
2. Asynchronous / no global time step
• Geometry-induced stiffness results from simulating domains with
drastically varying geometric features. Causes are:
• Multiscale geometric features
• Transition and boundary layers
• Poor element quality (e.g. slivers)
• Adaptive meshes driven by FEM discretization errors.
Time step is limited by smallest elements for explicit methods
Explicit: Efficient / stability concerns
Implicit: Unconditionally stable
Time-marching methods
Improvements:
Implicit-Explicit (IMEX) methods increase the time step by geometry
splitting (implicit method for small elements) or operator splitting.
Local time-stepping (LTS): subcycling for smaller elements enables
using larger global time steps
SDGFEM Small elements locally have smaller progress in
time (no global time step constrains)
None of the complicated “improvements” of time
marching methods needed
SDGFEM graciously and
efficiently handles highly
multiscale domains
A. Taube, M. Dumbser, C.D. Munz
and R. Schneider, A high-order
discontinuous Galerkin method with
time accurate
local time stepping for the Maxwell
equations, Int. J. Numer. Model.
2009; 22:77–103
2. Asynchronous / no global time step
tim
e
3. Spacetime grids and Moving interfaces
• Problems with moving interfaces:
* Solid-fluid interaction * Non-linear free surface water waves
* helicopter rotors /forward fight * flaps and slats on wings and piston engines
• Derivation of a conservative scheme is very challenging:
• Even Arbitrary Lagrangian Eulerian (ALE) methods do not automatically
satisfy certain geometric conservation laws.
• Spacetime mesh adaptive operations
Enable mesh smoothing and adaptive
operations Without projection errors of
semi-discrete methods.
4. Adaptive mesh operations
• Local-effect adaptivity: no need for reanalysis of the entire domain
• Arbitrary order and size in time:
• Adaptive operations in spacetime:
Example:
LTS by
Dumbser,
Munz, Toro,
Lorcher, et. al.
SDGADER-DG with LTS
- Front-tracking better than shock capturing
- hp-adaptivity better than h-adaptivity
Sod’s shock tube problem
Results by Scott Miller
Shock capturing: 473K elements Shock tracking: 446 elements
http://www.personal.psu.edu/stm134/
4. Adaptive mesh operationshighly multiscale grids in spacetime
These meshes for a crack-tip wave scattering problem are generated by
adaptive operations. Refinement ratio smaller than 10-4.
click to play movie click to play movie
Color: log(strain energy); Height: velocity Time in up direction
Shock visualization link
cracktip-soln-top.mp4movies/cracktip-spacetime-side.mp4https://www.youtube.com/watch?v=hqvGWd0S_rwhttps://www.youtube.com/watch?v=6kh4fp5fJt0shockVisualization_crack1e10.mov
5. Riemann-solution free scheme
• Reimann solutions are often complicated, expensive, and even difficult to
derive particularly for nonlinear and anisotropic materials.
Example: Simple linear elastodynamic problem
( regions III and IV)
• Riemann solutions required for inter-element noncausal
boundaries
• If interior facets are eliminated we obtain a Riemann-
solver free method
• Riemann-solution free scheme can also be more efficient
active elements
predecessor
elements
active element =
list of int. cells
predecessor
elements
integration
cells
Single-element
patch
Outstanding base properties of serial mode
O(N) Complexity
Favors highest polynomial order
Favors multi-field over single-field FEMs
Asynchronous
Nested hierarchical structure for HPC:
1.patches, 2.elements/cells, 3.quadrature points
4,5.rows & columns of matrices
Domain decomposition at patch level:
Near perfect scaling for non-adaptive case
95% scaling for strong adaptive refinement
Diffusion-like asynchronous load balancing
Multi-threading & Vectorization UIUC
Multi-threading
LU solve
Number of OpenMP ThreadsW
all
Tim
e (
s)
assembly
physics
more efficient
CG: BatheSDG
6. parallel computing (asynchronous structure)
http://mechanical.illinois.edu/directory/faculty/r-haber
Other Applications
Multiscale & Probabilistic Fracture
Probabilistic crack nucleation.
Exact tracking of crack interfaces.
Structural Health Monitoring
Multiscale and noise free
solution of scattering enables
detection of defects at
unprecedented resolutions.
Dynamic Contact/Fracture
SDGFEM eliminates common
artifacts at contact transitions
High resolution slip-stick-separation waves
Click here to play movie
Click here to play movie
Click here to play movie
Click here to play movie
movie
YouTube link
YouTube link
YouTube link
YouTube linkYouTube link
movie
Fluid mechanics: Euler’s equation Hyperbolic thermal model Solid mechanics
movies/Multiscale_ForwardAnalysis.mp4movies/brake2e3_tf_420_720_HS500_cm3e6_DS_400.mov.mp4movies/CircularCrack_Path_Damage.mp4movies/SolutionDependentCrackPath.mp4movies/shockVisualization_crack1e10.mp4https://www.youtube.com/watch?v=EX2Vsw50ygMhttps://www.youtube.com/watch?v=IxoS7_bBpHchttps://www.youtube.com/watch?v=sTYTqe0trLUhttps://www.youtube.com/watch?v=NxGapy31rOQhttps://www.youtube.com/watch?v=X5_fRZlBhSwMachReflection.movmovies/MachReflection.mp4
Time Domain Electromagnetics:
Discontinuous Galerkin Methods
• Time Domain (TD) vs. Frequency Domain (FD) solvers:
• Transient problems TD
• Steady state and frequency response:
• Small problem size FD
• Large problem size TD (TDDG ≈ O(N)FD = O (Na), a ≥ 1.5
• Material nonlinearities are better modeled in TD
• Entire spectrum obtained by one TD simulation of
broadband signalBusch:2009
Electromagnetics formulation in spacetime:
Fields, fluxes and sources
• Electromagnetic fields
• Electromagnetic (total) flux densities
• Constitutive equation
,
Inductive Conductive Dispersive
Spacetime electromagnetic flux
Spacetime electromagnetic flux density
Acting on time-like (“vertical”) boundaries
Spacetime flux
Acting on space-like (“horizontal”) boundaries
Comparison with d-form fluxes in spacetime
(d – 1)-form fluxes in spacetime for EM problem
Comparison with other differential form
formulation of EM problem
Source: Peter Russer, Exterior Differential Forms in Teaching Electromagnetics, 2004.
Our electromagnetic fields have an extra dt
Balance laws in spacetime
Source terms: current and charge densities
Integral form of Maxwell equations
Spacetime electromagnetic flux density
(f is a 1-form)
Stokes’ theorem
Strong form of Maxwell equations
(Provides PDEs)
(Provides Boundary conditions and
interface jump conditions)
Diffuse part
Jump part
Strong Form to weak form and FEM
formulation
a. Maxwell’s balance laws (diffuse part of the strong form)
b. Jump equations (BCs, interfaces, etc.)
- Weighted residual statement (WRS)
- Weak statement (WKS)
Stokes’ theorem
Auxiliary Differential Equations:
Elimination of convolutions in TD
• Electromagnetic equations in FD
• Frequency domain constitutive relation for an isotropic media:
• Pull-back in time domain involves a convolution:
• Solution: Elimination of convolution integral by introducing additional fields.
Auxiliary Differential Equations:
• Assume electrical permittivity uses a Debye dispersion model,
• By the introduction of Auxiliary Field P we get,
• That is convolution term is eliminated by the addition of the field P(k).
Auxiliary Differential Equations:
Examples
Time Domain Electromagnetics SDG:
Formulation of PML for dispersive media
• Perfectly matched layer for bi-anisotropic dispersive mediaPML stretching dispersive relations
2 Levels of ADEs needed in TD
ConductiveInductive DispersiveLevel
Base
L1
L2
P
M
L
Constitutive model
The recursive formulation of ADEs enables automatic formation of PML
equations for dispersive media in TD
Teixeira, Chew, General closed-form PML constitutive
tensors to match arbitrary bianisotropic and dispersive
linear media, IEEE Microw. Guid. Wave Lett., 1998
Time Domain Electromagnetics SDG:
Balance of energy / proof of numerical stability
• Energy stability proof (dissipative method)
• Sketch of the proof:
• Bilinear form from the weighted residual statement:
• By showing
and manipulation of bilinear form we can show:
(interchanging * and ^ on
fields and flux densities)
Time Domain Electromagnetics SDG:
Balance of energy / proof of numerical stability
That is
But,
= 0 outflow
> 0 inflow
> 0 non-causal
(using Riemann
values, etc.)
Numerical results
Convergence studies:
Smooth solution (Elastodynamics)
• Smooth analytical solution
(harmonic function)
• 1D problem
• D: Elastodynamic numerical energy dissipation.
High convergence rates are achieved due to high spatial and temporal orders of
elements.
Convergence plot
2D scattering problem / Nonadaptive meshing
𝐻𝑧 = cos(𝜋
2
𝑥
𝑑𝑥)cos(
𝜋
2
𝑦
𝑑𝑦)
Electric permittivity:
𝜀𝑖 = 1𝜀𝑜 = 10
Magnetic permeability:
𝜇 = 1
Initial condition:
Transverse electric formulation
movies/hybrid_2016_10_10_NA3Sq001_p3.movmovies/hybrid_2016_10_10_NA3Sq001_p3_x264.mp4
2D scattering problem adaptive meshing
t = 0.18 t = 0.50t = 0.70
t = 0.95 t = 1.30 t = 2.35
movies/sq3_tol1e10.mp4movies/sq3_tol1e10.mov
Comparison of Adaptive / nonadaptive schemes
Nonadaptive Adaptive
Initial spatial
Mesh
25K elements 46 elements
movies/EM_CompAdaptNonAdapt.png
Comparison of Adaptive / nonadaptive schemes
Nonadaptive Adaptive
93 Hours
Numerical dissipation 8x10-472 Hours
Numerical dissipation 1x10-4
Initial Mesh:
83K elements
Even finer nonadaptive
simulation
> 500 Hours
Numerical dissipation 2x10-4
(still larger than adaptive one)
Characterization of
Dispersive media
S-parameters for a slab
Material Properties
Transmission / reflection coefficients
Unit cell to dispersive response
(S-parameters) retrieval method
1. (Computationally) solve for t, r:
2. Inverse solution for Z, c
(non-uniqueness)s
Computation of S-parameters
Frequency Domain (FD)
• For each frequency w one
FD simulation is done
• For high frequencies very
fine meshes are required.
• Solution is globally coupled
(Elliptic PDE)
Large problem size FD = O (Na), a ≥ 1.5
Computation of S-parameters
Time Domain (TD)
Gaussian pulse
Properties:
• Ultrashort duration pulses
Broadband frequency content
• (Almost) zero solutions for
initial and final times
facilitates Fourier analysis
Time Domain vs. Frequency Domain
• Time Domain (TD) vs. Frequency Domain (FD) solvers:
• Transient problems TD
• Steady state and frequency response:
• Small problem size FD
• Large problem size TD (TDDG ≈ O(N) FD = O (Na), a ≥ 1.5
• Material nonlinearities are better modeled in TD
• Entire spectrum obtained by one TD simulation of
broadband signalBusch:2009
highly multiscale grids in spacetime
These meshes for a crack-tip wave scattering problem are generated by
adaptive operations. Refinement ratio smaller than 10-4.
click to play movie click to play movie
Color: log(strain energy); Height: velocity Time in up direction
Shock visualization link
Motivation:
Adaptive SDG method to model dispersive mediatim
e
movies/cracktip-soln-top.mp4movies/cracktip-spacetime-side.mp4https://www.youtube.com/watch?v=hqvGWd0S_rwhttps://www.youtube.com/watch?v=6kh4fp5fJt0shockVisualization_crack1e10.mov
Test problem:
Retrieval method for solid slab
𝐸 = 1, = 1
Initial space mesh
𝐸 = 1, = 1𝐸 = 10, = 1
Movie: Solution
movies/inc_Test_x264.mp4movies/inc_Test.movmovies/inc_TestSln.mov
Test problem:
Retrieval method for electromagnetics problem
Transmission and reflection coefficients
Conductive mid-layer
Test problem:
Retrieval method for electromagnetics problem
Conductive mid-layer
Test problem:
Retrieval method for electromagnetics problem
3-layer set-up
Test problem:
Retrieval method for electromagnetics problem
3-layer set-up
Retrieval method for 2D unit cells
Ying Wu, Yun Lai, and Zhao-Qing Zhang. Effective
medium theory for elastic metamaterials in two
dimensions. Physical Review B, 2007.
lead
rubber
epoxy
Spatial mesh for the SDG method
Retrieval method for 2D unit cells
Movie: Solution, Mesh
movies/inc_LRE_mesh_n2.mp4movies/inc_LRE_mesh_n2.movmovies/inc_LRE_sln_n2.mp4movies/inc_LRE_sln_n2.movmovies/inc_LRE_sln_n2.movmovies/inc_LRE_mesh_n2.mov