xfem as an alternative for the classical h-refinement€¦ · 2 the xfem as an alternative for...
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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.
![Page 4: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/4.jpg)
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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.
![Page 10: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/10.jpg)
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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.
![Page 13: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/13.jpg)
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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.
![Page 14: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/14.jpg)
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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.
![Page 18: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/18.jpg)
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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.
![Page 21: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/21.jpg)
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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.
![Page 27: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/27.jpg)
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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.
![Page 30: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/30.jpg)
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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.
![Page 32: XFEM as an Alternative for the Classical h-refinement€¦ · 2 The XFEM as an Alternative for h-Adaptivity •Enrichment functions for high gradient solutions. •Optimal set of](https://reader033.vdocuments.mx/reader033/viewer/2022043014/5fb19c75afd94e6fa17bd5e8/html5/thumbnails/32.jpg)
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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