SBFEM_Demo.pdf (PDF, 2.5 MB)

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

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Motivation I

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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

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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 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:

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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

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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)

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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 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

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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

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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

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courtesy of Kostas Agathos

xFEM DEMO

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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

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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

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

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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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