a comparison of various enrichment functions for weak discontinuities … · 2011-07-05 · a...

A Comparison of Various Enrichment

Functions for Weak Discontinuities

with respect to Higher-Order

XFEM Approximations

Kwok-Wah Cheng and Thomas-Peter Fries

RWTH Aachen University, Germany

Hannover, Germany

September 21, 2009

2

Overview

21/09/2009 2K.W. Cheng, Higher-Order XFEM

Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order XFEM

formulation

Standard XFEM

Modified

abs-enrichment

(Möes et. al., 2003)

Corrected XFEM

(Fries, 2007)

Numerical

Results

Conclusions

Motivation

3

� High-order XFEM: what has already been done?

� Higher-order convergence rates for strong and weak, straight

discontinuities have been achieved by previous workers (e.g. Laborde

et al., 2005; Legay et al., 2006).

� Higher-order convergence rates for strong, curved discontinuities

have been achieved (e.g. Dréau et al., 2008).

� This study considers higher-order XFEM for weak, curved

discontinuities.

� Has been investigated previously but only suboptimal convergence

rates were achieved (e.g. Legay et al., 2006)

� Both quadratic and cubic approximations will be investigated.


4

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

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 2D21/09/2009 5K.W. Cheng, Higher-Order XFEM

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


7

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

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-cells

with at most one curved

side.



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



� 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



13

� Objective: Locate the 4 points on the interface

Exact interfaceBilinear interpolation

interpolated interface


unknown

unknown

-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


14

Reference

elements



15

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order

XFEM

formulation

Standard XFEM

Standard XFEM formulation

� Standard XFEM approximation:

standard enrichment

partition-of-unity

functions


Standard XFEM formulation

� Standard XFEM approximation:

� For weak discontinuities


Problems in blending elements

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).

PU functions do not build

a partition-of-unity

over the blending elements


Problems in blending elements

19

� Introduces unwanted parasitic terms into the approximation space of the blending elements.

� Degrades both accuracy and convergence rates.

� Affects the abs-enrichment

Remedies

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

� Corrected XFEM (Fries, 2007)


20

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order XFEM

formulation

Standard XFEM

Modified

abs-enrichment


Modified abs-enrichment

21




22




23




24



25

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order XFEM

formulation

Standard XFEM

Modified

abs-enrichment


Corrected XFEM

(Fries, 2007)

Corrected XFEM formulation

26


Expanded enriched nodal set J*


Corrected XFEM formulation

27


J*

J*


28

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order XFEM

formulation

Standard XFEM

Modified

abs-enrichment


Corrected XFEM

(Fries, 2007)

Numerical

Results

Numerical results

29

� 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

30

� 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

31



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

32



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

33

� 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

34

� Cubic elements (i.e. order of =3)

120 X 120


Numerical results

35

� 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


36

Overview


Higher-order

XFEM

Higher-order

representation

Of geometry

Interpolation of

interface

Sub-cell

integration

Higher-order XFEM

formulation

Standard XFEM

Modified

abs-enrichment


Corrected XFEM

(Fries, 2007)

Numerical

Results

Conclusions

Conclusions

37

� 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


a comparison of various enrichment functions for weak discontinuities … · 2011-07-05 · a...

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