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Adrian Egger
Applications of the
Scaled Boundary Finite Element Method in
Linear Elastic Fracture Mechanics
Contents
Motivation
What is SBFEM?
SBFEM Theory Derivation via Virtual work approach
Solution process and interpretation
Calculating Stress intensity factors
Error estimators
Applications in linear elastic fracture mechanics ABAQUS demo
xFEM demo
SBFEM demo
Conclusion
Questions
12/8/2016 2Adrian Egger | FEM II | HS 2015
Motivation I
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FEM has been extended
to 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
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#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
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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 (PVV 1)
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SBFEM (PVV 2)
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SBFEM (PVV 3)
Geometry transformation
Jacobian
Differential unit volumen
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SBFEM (PVV 4)
The linear differential operator L may thus be written as:
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with
SBFEM (PVV 5)
Assuming an analytical solution in radial direction:
And therefore the strains and stresses become:
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SBFEM (PVV 6)
Setting up the virtual work formulation:
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SBFEM (PVV 7)
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SBFEM (PVV 8)
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SBFEM (PVV 9)
Introducing some substitutions
Leads to some significant simplifications
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SBFEM (PVV 10)
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SBFEM (PVV 11)
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SBFEM (Solution 1)
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SBFEM (Solution 2)
Notice the similarity to the mode superposition method
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18
Adrian Egger | FEM II | HS 2015
http://web.sbe.hw.ac.uk/acme2011/Handout_Scaled_boundary_methods_CA.pdf
SBFEM (Solution 3)
A quadratic eigenproblem results from substituting the
general solution into the scaled boundary finite element
equation in displacements:
The boundary forces are now expressed as in an equivalent
modal formulation. Conceptually, they represent the nodal
force modes {q} required to balance the corresponding
displacement modes {ϕ}.
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SBFEM (Solution 4)
Linearizing the quadratic eigenproblem is obtained by
rearringing the force mode equation and substituting into
the scaled boundary finite element equation in
displacements.
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SBFEM (Solution 5)
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SBFEM (Solution 6)
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SBFEM (Solution 7)
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SBFEM implementation
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Stress Intensity factors (SIFs)
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http://solidmechanics.org/Text/Chapter9_3/Chapter9_3.php
Analog to stress concentration factors in i.e. tunneling
In fracture mechanics predicts stress distribution
near crack tip and is useful for providing a failure criterion:
𝜎𝑖𝑗 𝑟, 𝜃 =𝐾
2𝜋𝑟𝑓𝑖𝑗 𝜃 + higher order terms
K = stress intensity factor
𝑓𝑖𝑗 = function of load and geometry
SIFs by SBFEM (1)
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Strain:
Stress:
Take limit as ξ→0; Singularity for eigenvalues -1 < λs < 0
SIFs by SBFEM (2)
Take limit as ξ→0; Singularity for eigenvalues -1 < λs < 0
By matching expressions with the exact solution:
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where:
Stress recovery for SIFs (1)
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Stress recovery for SIFs (2)
Accelerate procedure
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Optimal placement of scaling center
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What is being compared?
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ABAQUS xFEM SBFEM
Countour Integral Local enrichment
(Interaction Integral)
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 on
tractions and displacements
Requires information about
crack propagation direction
Cannot predict
how a crack will propagate
12/8/2016 32Adrian Egger | FEM II | HS 2015
http://imechanica.org/files/Modeling%20fracture%20mechanics.pdf
P.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
12/8/2016 35Adrian 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 solved
by 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
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courtesy of Kostas Agathos
XFEM DEMO
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SBFEM DEMO
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Numerical experiments:
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1 2 3
Numerical example 1 (SBFEM)
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Numerical example 1 (SBFEM)
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Numerical example 2
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Numerical example 2
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Numerical example 2
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Numerical example 3
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Numerical example 3
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Numerical example 3
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Conclusion
SBFEM combines many of the desireable characteristics of
FEM and BEM into one method with additional benefits of
ist 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 of various orders of singularity)
Dense and fully populated matrices:
Higher order elements don’t (noticeably) impact performance
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Questions
12/8/2016 50Adrian Egger | FEM II | HS 2015