Higher-Order XFEM for Arbitrary Weak Discontinuities · Motivation 7/14/2009 K.W. Cheng, Higher-Order XFEM 3 High-order XFEM: what has already been done? Higher-order convergence

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




� Standard XFEM




� Conclusion


Contents

Interpolation of interface


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


� 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




� Standard XFEM




� Conclusion


Contents

Sub-cell integration


� 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



-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




-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.



� 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



� 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



Reference

elements


�Motivation




� Standard XFEM




� Conclusion


Contents

Standard XFEM formulation


� Standard XFEM approximation:

Standard XFEM formulation


� Standard XFEM approximation:

� For weak discontinuities

Problems in blending elements


� 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)




� Partition-of-unity functions do not build a partition-of-unity over the blending elements



� 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



Corrected XFEM formulation



Corrected XFEM formulation



J*

J*

�Motivation



�Higher-order XFEM formulations

� Standard XFEM

�Modified abs-enrichment (Möes et. al., 2003)

�Corrected XFEM (Fries, 2007)


� Conclusion


Contents

Numerical results


� 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


� 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




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




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


� 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


� Cubic elements (i.e. order of =3)

120 X 120

Numerical results


� 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



�Higher-order XFEM formulations

� Standard XFEM

�Modified abs-enrichment (Möes et. al., 2003)

�Corrected XFEM (Fries, 2007)

�Numerical results

� Conclusion


Contents

Conclusion


� 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

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