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XFEM as an Alternative for the Classicalh-refinement

bySafdar Abbas & Thomas-Peter Fries

USNCCMJuly 17, 2009

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Enrichment Functions for High Gradient Solutions

• Motivation: Standard FEM (No Stabilization)

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Enrichment Functions for High Gradient Solutions

• Motivation: Stabilized FEM (SUPG Stabilization)

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Enrichment Functions for High Gradient Solutions

• XFEM approximation.

– Standard finite element approximation.

– Enrichment.

W

W

s

• Instead of stabilization and/or refinement we want to enrich the

approximation space.

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Enrichment Functions for High Gradient Solutions

• Enrichment Functions

– Weak discontinuity Abs-Enrichment

– Strong discontinuity Sign/Heaviside Enrichment

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Enrichment Functions for High Gradient Solutions

• Enrichment functions–Regularized sign function [Patzak and Jirasek, 2003] ( C4 continuous at ).

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Optimal Set of Enrichment Functions

Can be integrated

Can be captured

by FEM

Interpolation Problem:Interpolated function fInterpolating functions Ψ = [ψ1, ψ2, ψ3]

Find ∫ ωuh = ∫ ωf, for ω ∈ Ψ, where uh = ∑ ψiui= ΨΤuminimize the error |f – uh|

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Enrichment Functions for High Gradient Solutions

• An optimal set of 7 enrichment functions.

• Enrichment functions are relative to the element size.

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Enrichment Functions for High Gradient Solutions

• An optimal set of 7 enrichment functions.

• Enrichment functions are relative to the element size.

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Stationary High Gradient Developing Over Time

• First test case: In-stationary Burger’s Equation with stationary high gradientthat develops over time.

– Time-Stepping for the temporal discretization.

– Non-linear term is linearized using Newton-Raphson iterations.

– Diffusion coefficient is very small.

– No stabilization is used.

– Position of the highest gradient is known and stationary.

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Stationary High Gradient Developing Over Time

XFEM Results

(No Stabilization)

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Stationary High Gradient Developing Over Time

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Moving High Gradient (position known a priori)

• Second test case: In-stationary linear advection equation with moving high

gradient specified as an initial condition.

– Time stepping is not fully appropriate.

– Equation is discretized using Space-Time discretization with

Discontinuous-Galerkin in time.

– No stabilization is used.

– Position of the highest gradient known a priori at each time.

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Moving High Gradient (position known a priori)

XFEM Results

(No stabilization)

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Moving High Gradient (position not known)

• Third test case: In-stationary Burger’s Equation with unknown position ofthe highest gradient.

– Level-set function is transported using transport equation for the level-set.

– Equation is discretized using Space-Time discretization withDiscontinuous-Galerkin in time.

– Non-linear term is linearized using Newton-Raphson iterations.– Diffusion coefficient is very small.– No stabilization is used.– Position of the highest gradient in each time step is found iteratively by

a strong coupling loop.

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Moving High Gradient (position not known)

Strong Coupling

Solution ofBurger’s Equation

Solution ofTransportEquation

Solution ofBurger’s Equation

Solution ofTransportEquation

tn tn+1

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Moving High Gradient (position not known)

XFEMResultsSpace-Timeview

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Moving High Gradient (position not known)

FEMResults

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Moving High Gradient (position not known)

XFEMResults

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori).

• Conclusions.

• Future outlook.

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Moving High Gradient in 2D (position known a priori).

• Fourth test case: A high gradient scalar function transported in a circular

velocity field.

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Moving High Gradient in 2D (position known a priori).

2d Advectionequation

(No stabilization)

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions.

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Conclusions

• A complete range of gradients is captured using an optimal set of high

gradient enrichment functions.

• No oscillations are observed near the high gradient.

• Solution quality is better than that achieved from stabilization without

refining the mesh.

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The XFEM as an Alternative for h-Adaptivity

• Enrichment functions for high gradient solutions.

• Optimal set of enrichment functions

• Stationary high gradient developing over time.

• Moving high gradient (position known a priori).

• Moving high gradient (position not known).

• Moving high gradient in 2D (position known a priori)..

• Conclusions.

• Future outlook.

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Future Outlook

• Using the optimal set of enrichment functions to simulate the cohesive

cracks in quasi-brittle materials.

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Financial support from the DeutscheForschungsgemeinschaft (German ResearchAssociation) through grant GSC 111 is gratefullyacknowledged.

Acknowledgements

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Thanks for your Attention