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Higher-Order XFEM for
Arbitrary Weak Discontinuities
Kwok-Wah Cheng and Thomas-Peter Fries
RWTH Aachen University, Germany
Columbus, Ohio
July 17, 2009
� Motivation
� Interpolation of interface
� Sub-cell integration
� Higher-order XFEM formulations
� Standard XFEM
� Modified abs-enrichment (Möes et. al., 2003)
� Corrected XFEM (Fries, 2007)
� Numerical results
� Conclusion
7/14/2009
K.W. Cheng, Higher-Order XFEM
2
Contents
Motivation
7/14/2009 K.W. Cheng, Higher-Order XFEM 3
� High-order XFEM: what has already been done?
� Higher-order convergence rates for strong and weak, straight
discontinuities in 2-D have been achieved by previous workers (e.g.
Laborde et. al., 2005; Legay et. al., 2006).
� Higher-order convergence rates for strong, curved discontinuities in
2-D have been achieved (e.g. Dréau et. al., 2008).
� This study considers higher-order XFEM for weak, curved
discontinuities in 2-D. Both quadratic and cubic
approximations will be investigated.
�Motivation
� Interpolation of interface
� Sub-cell integration
� Higher-order XFEM formulations
� Standard XFEM
� Modified abs-enrichment (Möes et. al., 2003)
� Corrected XFEM (Fries, 2007)
� Numerical results
� Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 4
Contents
Interpolation of interface
7/14/2009 K.W. Cheng, Higher-Order XFEM 5
The position of the interface is defined by the level-set method:
� The level-set function is a signed distance function
which stores the shortest distance to the discontinuity.
� The zero-level of the level-set function is the discontinuity.
discontinuity
1D 2D
Interpolation of interface
7/14/2009 K.W. Cheng, Higher-Order XFEM 6
� Interpolation using finite element shape functions.
Bilinear Quadratic Cubic
-0.4 -0.2 0 0.2 0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.4 -0.2 0 0.2 0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.4 -0.2 0 0.2 0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
�Motivation
� Interpolation of interface
� Sub-cell integration
� Higher-order XFEM formulations
� Standard XFEM
� Modified abs-enrichment (Möes et. al., 2003)
� Corrected XFEM (Fries, 2007)
� Numerical results
� Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 7
Contents
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 8
� In the XFEM, integration of element matrices needs to take into account different material properties across interfaces.
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Material A
Material B
Interface
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 9
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
� Projection of integration points
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 10
� Projection of integration points
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Triangular sub-cells
Quadrilateral sub-cell
Always end up with triangular and quadrilateral sub-cellswith at most one curved side.
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 11
� Projection of integration pointsReference
elements
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 12
� Projection of integration pointsReference
elements
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 13
� Objective: Locate the 4 points on the interface
Exact interfaceBilinear interpolation
interpolated interface
-0.4 -0.3 -0.2 -0.1 0
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Sub-cell integration
7/14/2009 K.W. Cheng, Higher-Order XFEM 14
Reference
elements
� Projection of integration points
�Motivation
� Interpolation of interface
� Sub-cell integration
� Higher-order XFEM formulations
� Standard XFEM
� Modified abs-enrichment (Möes et. al., 2003)
� Corrected XFEM (Fries, 2007)
� Numerical results
� Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 15
Contents
Standard XFEM formulation
7/14/2009 K.W. Cheng, Higher-Order XFEM 16
� Standard XFEM approximation:
Standard XFEM formulation
7/14/2009 K.W. Cheng, Higher-Order XFEM 17
� Standard XFEM approximation:
� For weak discontinuities
Problems in blending elements
7/14/2009 K.W. Cheng, Higher-Order XFEM 18
� Using the standard XFEM formulation with the abs-enrichment leads to problems in blending elements (e.g. Sukumar et. al., 2001; Chessa et. al., 2003).
� Remedies for such problems� Modified abs-enrichment (Möes et. al., 2003)
� Corrected XFEM (Fries, 2007)
Problems in blending elements
7/14/2009 K.W. Cheng, Higher-Order XFEM 19
� Partition-of-unity functions do not build a partition-of-unity over the blending elements
Problems in blending elements
7/14/2009 K.W. Cheng, Higher-Order XFEM 20
� Introduces unwanted parasitic terms into the approximation space of the blending elements.
� Degrades both accuracy and convergence rates.
� Affects the abs-enrichment
Modified abs-enrichment
7/14/2009 K.W. Cheng, Higher-Order XFEM 21
� Modified abs-enrichment (Möes et. al., 2003)
Modified abs-enrichment
7/14/2009 K.W. Cheng, Higher-Order XFEM 22
� Modified abs-enrichment (Möes et. al., 2003)
Modified abs-enrichment
7/14/2009 K.W. Cheng, Higher-Order XFEM 23
� Modified abs-enrichment (Möes et. al., 2003)
Modified abs-enrichment
7/14/2009 K.W. Cheng, Higher-Order XFEM 24
� Modified abs-enrichment (Möes et. al., 2003)
Corrected XFEM formulation
7/14/2009 K.W. Cheng, Higher-Order XFEM 25
� Corrected XFEM (Fries, 2008)
Corrected XFEM formulation
7/14/2009 K.W. Cheng, Higher-Order XFEM 26
� Corrected XFEM (Fries, 2008)
J*
J*
�Motivation
� Interpolation of interface
� Sub-cell integration
�Higher-order XFEM formulations
� Standard XFEM
�Modified abs-enrichment (Möes et. al., 2003)
�Corrected XFEM (Fries, 2007)
� Numerical results
� Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 27
Contents
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 28
� Bi-material with a circular inclusion (Sukumar et. al., 2001; Legay et. al., 2005; Fries, 2008)
Material 1Material 2
Plane-strain conditions
25.0,1
3.0,10
22
11
==
==
vE
vE
Computational Domain
Material Properties
Loading results from constant radial displacement
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 29
� Computational domain
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Reproducing elements
Blending elements
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 30
� Computational domain
Reproducing elements
Blending elements
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
U = 0
U = 0
v = 0 v = 0
Dirichlet boundary conditions
0)0,1(
0)1,0(
=±
=±
v
u
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 31
� Computational domain
Reproducing elements
Blending elements
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
U = 0
U = 0
v = 0 v = 0
Dirichlet boundary conditions
0)0,1(
0)1,0(
=±
=±
v
u
Neumann boundary conditions
yyyyxxy
xyxyxxx
tnn
tnn
=+
=+
σσ
σσ
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 32
� Quadratic elements (i.e. order of =2)
Mesh sizes
10 X 10
20 X 20
40X 40
80 X 80
160 X 160
320 X 320
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 33
� Cubic elements (i.e. order of =3)
120 X 120
Numerical results
7/14/2009 K.W. Cheng, Higher-Order XFEM 34
� Comparison of convergence rates
� Standard XFEM 2.5/3.0 3.4/4.0
(order = 1)
� Modified abs-enrichment 2.4/3.0 2.5/3.0
� Corrected XFEM 2.8/3.0 3.7/4.0
(order = order )
• Corrected XFEM has higher accuracy as well.
Quadratic Cubic
�Motivation
� Interpolation of interface
� Sub-cell integration
�Higher-order XFEM formulations
� Standard XFEM
�Modified abs-enrichment (Möes et. al., 2003)
�Corrected XFEM (Fries, 2007)
�Numerical results
� Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 35
Contents
Conclusion
7/14/2009 K.W. Cheng, Higher-Order XFEM 36
� Standard XFEM yields best results when first order partition-of-unities are used; however, convergence rates are still suboptimal.
� Modified abs-enrichment yields suboptimalconvergence rates for all orders of
� Corrected XFEM yields close-to-optimalconvergence rates when order of = order of .
� Higher-order XFEM for curved weak discontinuities is possible.
Thank you!
Kwok-Wah Cheng and Thomas-Peter Fries