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Adrian Egger
Applications of the Scaled Boundary Finite Element Method in Linear Elastic Fracture Mechanics
Contents
Motivation
SBFEM computational steps
SBFEM vs. ABAQUS: A displacement based comparison
Crack related phenomena ABAQUS demo
xFEM demo
SBFEM demo
Numerical comparisons Edge crack under uniform tension
Edge crack under uniform shear
Slant crack under uniform tension
Conclusion
Questions12/10/2015 2Adrian Egger | FEM II | HS 2015
Motivation I
12/10/2015 3Adrian Egger | FEM II | HS 2015
FEM has been extendedto handle a multitude of problems:
Non-linearity Geometric
Material
Dynamics
Contact problems
Crack problems
Optimization problems
Inverse problems
http://gem-innovation.com/services/mesh-independent-fem/
Motivation II
12/10/2015 4Adrian Egger | FEM II | HS 2015
#DOF▲
Non linear
Dynamic
Cracks / Contact
Only partially alleviated by:
• Vectorization and parallelization of code
• Multiscale schemes, sub-structuring, …
• Fast multipole boundary element method (FMBEM)
• Extended finite element method (xFEM)
• Etc.
Conceptual Comparison of FEM, BEM and SBFEM
12/10/2015 5Adrian Egger | FEM II | HS 2015
FEM:
- High amount of DOF
- Crack surface discretized
- Discretization error in all
directions
BEM:
- Discretization on boundary only
- Crack surface discretized
- non-symmetric dense matrices
SBFEM bounded domain:
- Introduction of a scaling center
- Discretization on boundary only
- Crack surface not discretized
- Analytical solution in radial
direction
SBFEM unbounded domain:
- Introduction of a scaling center
- Discretization on boundary only
- Analytical solution in radial
direction
SBFEM Fundamentals I:
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SBFEM Fundamentals II:
Applying the principle of virtual work* yields 2 equations:
General solution for displacements is assumed of form:
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Boundary:
Domain:
where
* Derivation at end of presentation
SBFEM Fundamentals III:
Solution assumed as a power series: Notice the similarity to the mode superposition method!
8
Adrian Egger | FEM II | HS 2015
http://web.sbe.hw.ac.uk/acme2011/Handout_Scaled_boundary_methods_CA.pdf
12/10/2015
Leads to a Hamiltonian eigenvalue problem of [Z]: Introducing a new variable leads to a first order differential equation
Substituting the general solution into the obtained equation:
Having calculated the decomposition
SBFEM Fundamentals IV:
12/10/2015 9Adrian Egger | FEM II | HS 2015
with:
Displacements:
Forces:
boundeddomain
unboundeddomain
�������� = + �� ����
�������� = − Φ�� Φ����
Basic Analysis Procedure
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+
Reduction of problem dimension by 1
Semi-analytical solution in radial direction
Bounded and unbounded domains
-
Assembly of multiple coefficient matrices
Schur/Eigen decomposition
Fully populated stiffness matrix
Comparison to ABAQUS reference solution
12/10/2015 11Adrian Egger | FEM II | HS 2015
ABAQUS:#DOF = 50’000+
Real time = 11s
Time savings:
102x @ ~0.1‰
0.8 s
-0.1%
0.1%
0.2%
-0.1%
0.1%
0.2%
0.8 s
Cost of using higher order elements
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#DOF
tim
e0.8 seconds
360 DOF
Stress Intensity factors (SIFs)
12/10/2015 13Adrian Egger | FEM II | HS 2015
http://solidmechanics.org/Text/Chapter9_3/Chapter9_3.php
Analog to stress concentration factors in i.e. tunneling
In fracture mechanics predicts stress distributionnear crack tip and is useful for providing a failure criterion:
��� �, � =�
������ � + higher order terms
K = stress intensity factor
��� = function of load and geometry
What is being compared?
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ABAQUS xFEM SBFEM
Countour Integral Local enrichment Analytical limit in radial direction
Compiled (Fortran?) Scripted (Matlab) Scripted (Matlab)
Vectorized Non-vectorized Non-vectorized
Highly optimized Proof of concept code Proof of concept code
Total CPU time Matlab tic - toc Matlab tic - toc
ABAQUS: Contour Integral
Integral based ontractions and displacements
Requires information aboutcrack propagation direction
Cannot predict how a crack will propagate
12/10/2015 15Adrian Egger | FEM II | HS 2015
http://imechanica.org/files/Modeling%20fracture%20mechanics.pdfP.H. Wen, M.H. Aliabadi, D.P. Rooke, A contour integral for the evaluation of stress intensity factors, Applied Mathematical Modelling, Volume 19, Issue 8, August 1995, Pages 450-455, ISSN 0307-904X, http://dx.doi.org/10.1016/0307-904X(95)00009-9
ABAQUS DEMO
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Extended finite element method (xFEM) I
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FEM XFEM FEM XFEM
Goal: Separate geometry from mesh XFEM achieves this by locally enriching the FE approximation with
local partitions of unity enrichment functions
Extended finite element method (xFEM) II
xFEM aims to overcome the shortcomings of FEM
Does so by introducing two kinds of enrichment Jump enrichment
Tip enrichment
Achieves: Higher accuracy for
stresses at crack tip
Less remeshing required
Level set method used to efficiently track cracks
12/10/2015 18Adrian Egger | FEM II | HS 2015
courtesy of Kostas Agathos
xFEM: Jump enrichment
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Heaviside step function
courtesy of Kostas Agathos
xFEM: Tip enrichment
Analytical solution for the crack problem solvedby Westergaard (1939) using a complex Airy stress function
These can be spanned by the following basis, which are used as enrichment functions for the crack tip
12/10/2015 20Adrian Egger | FEM II | HS 2015
courtesy of Kostas Agathos
xFEM DEMO
12/10/2015 21Adrian Egger | FEM II | HS 2015
SBFEM: Analytical limit in radial direction
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Strain:
Stress:
Take limit as ξ→0; Singularity for eigenvalues -1 < λ < 0
By matching expressions with the exact solution:
where:
Effects of Stress Smoothing
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Exact solution
Error Estimator for Stress Intensity Factors
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Raw SIF
Recovered SIF
SIF error estimator
Calc
ula
ted
Err
or
in S
IF [
%]
SBFEM DEMO
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Numerical experiments: Stress intensity factors
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1 2 3
Numerical Experiment 1
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1.0%
1.0%
0.5%
0.1%
0.5%
0.1%
DOF106
SIF
K1
Numerical Experiment 1
12/10/2015 28Adrian Egger | FEM II | HS 2015
1.0%
1.0%
0.5%
0.1%
0.5%
0.1%
seconds102
SIF
K1
Numerical Experiment 2
12/10/2015 29Adrian Egger | FEM II | HS 2015
1.0%
1.0%
0.5%
0.1%
0.5%
0.1%
seconds102
SIF
K1
Numerical Experiment 2
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1.0%
1.0%0.5%0.1%
0.5%0.1%
seconds102
SIF
K2
Numerical Experiment 3
12/10/2015 31Adrian Egger | FEM II | HS 2015
SIF
K2
SIF
K1
seconds102 seconds
102
0.5%
2.0%
Observations
Incredibly fast convergence of SBFEM using few DOF
Exeptional speed considering non-optimized code
Stress recovery methods dramatically improve accuracy atvirtually no additional computational cost
Always choose at least two elements per side
Three elements per side are better
Do not use two node elements if possible
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Conclusion
SBFEM combines many of the desireable characteristics ofFEM and BEM into one method with additional benefits ofist own: Analytical solution in radial direction:
Higher accuracy per DOF
permits elegant and efficient calculation of stress intensity factors
Stress recovery enhances results greatly Must only be performed on the boundary
Large workload can be performed in advance
No change necessary to solution process to extract crack related phenomena (i.e. SIFs and T-stress of various orders of singularity)
Dense and fully populated matrices: Higher order elements don’t (noticeably) impact performance
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Questions
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SBFEM derivation I
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SBFEM derivation II
Geometry transformation
Jacobian
Differential unit volumen
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SBFEM derivation III
The linear differential operator L may thus be written as:
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with
SBFEM derivation IV
Assuming an analytical solution in radial direction:
And therefore the strains and stresses become:
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SBFEM derivation V
Setting up the virtual work formulation:
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SBFEM derivation VI
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SBFEM derivation VII
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SBFEM derivation VIII
Introducing some substitutions
Leads to some significant simplifications
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SBFEM derivation IX
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SBFEM derivation X
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