A complete methodology for the implementation of XFEM · PDF fileA complete methodology for...

A complete methodology for the implementation of XFEM

models

Team: - Carlos Hernán Villanueva

- Kai Yu

WHAT IS XFEM?

It stands for the eXtended finite element method

WHY XFEM?

Why do we need to extend the classical FEM?

A discontinuity is a rapid

change in a field variable

(a) Cracks, (b) Shear bands, (c) Interface

of two fluids

Discontinuity in our element

QUAD4 DISCONTINUITY

WHY NOT A RE-MESHING TOOL?

Re-meshing is computationally more expensive that XFEM

HOW DOES XFEM WORK?

The eXtended finite element method adds enrichment degrees of freedom to interpolate sub-

domains of discontinuity

Regular FE temperature element

QUAD4

Node 4: TEMP

Node 1: TEMP

Node 3: TEMP

Node 2: TEMP

eXtended FE temperature element

QUAD4

Node 4: TEMP

TEMP enriched

Node 1: TEMP

TEMP enriched

Node 3: TEMP TEMP enriched

Node 2: TEMP TEMP enriched

Material Phase II Material Phase I

To summarize: WHAT IS AND WHY DO WE NEED XFEM?

Traditional FEM

• Mesh generates discrete representation of potentially complex geometry

Why XFEM?

Changes in topology or

shape • Require new

mesh • Discontinuities

must align with element edges

Why XFEM?

The solution: eXtended Finite Element Method

Numerical technique

Provides local enrichment

For solutions to differential equations with discontinuous functions

Avoids need to re-mesh

Solve shortcomings of FEM

What is XFEM?

HOW DO WE MODEL DISCONTINUITIES?

A level-set function can model discontinuities in a fixed mesh

Positive phase

Negative phase

Zero level-set

Discontinuity of interest is represented as the zero level-set function

Numerical Scheme XFEM places

discontinuity at the boundary layer dividing the grid into negative and positive phases

XFEM

Background: Level-set function

FEM

XFEM

Background: XFEM and LSM

LSM

XFEM

LSM: - Model

discontinuity XFEM:

- Solves the problem

Natural coupling XFEM and LSM

+ - + + + -

+

+

- + - + - +

-

+ -

+

+ -

- - -

-

How do level-set functions work? Compute nodal level-set values based on

desired topology

HOW ARE ENRICHMENTS APPLIED?

Enrichment functions are applied to interpolate discontinuities

• n-dimensional domain: Ω ϵ R • nel elements, numbered from 1 to nel. • I is the set of all nodes in the domain • Iel

k are the nodes of element k ϵ {1,…, nel}.

How does XFEM provide local enrichment?

• Standard extended finite element method approximation of u(x)

• uh(x): approximated function • Ni(x): standard FE function of node i • ui: unknown of the standard FE part at node i • Mi(x): local enrichment function of node i • ai: unknown of the enrichment at node i • I*: nodal subset of the enrichment, built by the nodes of all

elements that are cut by the discontinuity


• The local enrichment functions have the form:

• Where Ni*(x) are the partition of unity functions and Ψ(x) are the global enrichment functions

• These Ni*(x) build a partition of unity in elements whose nodes are all in the nodal subset I*


HOW DO WE IMPLEMENT XFEM?

Let’s begin by the triangulation of our element for integration

Sub-division of geometric objects into integrable domains such as triangles and tetrahedra

Circumcircle of any triangle does not contain the vertices of other triangle

Delaunay triangulation

• Straightforward for 2D • Only 6 cases

+ - + + + -

+

+

- + - + - +

-

+ -

+

+ -

- - -

-

Delaunay triangulation – 2D

• More complicated for 3D • Multiple tetrahedra per interface • Needed a tool to triangulate the intersections for us

Phase 1

Combined

Phase 2

Delaunay triangulation – 3D

• Fortran90 Library for Computational Geometry in 2D, 3D, ND • Delaunay triangulation in 3D routine (dtris3)

Geompack3

Phase 1

Combined

Phase 2

Delaunay triangulation – 3D • A wrapper was used to interface with code in

C++

WHICH DEGREES OF FREEDOM TO USE?

Enrichment functions needs to continuous across nodal clusters

• A level-set function is used to represent a discontinuity

• Initially we don’t know how many enrichment functions we will need –We can estimate a number, not 100% accurate

• We can assign a maximum: Regular degree of freedom + 27 enrichment levels –First 14 number correspond to degrees of freedom of

material phase I –Second half corresponds to degrees of freedom of

material phase II

How to apply enrichments

• Consider the following 6 element model:

I

I

I

I

I

I

II II

II II

II II


• Important to obtain enrichment values when using XFEM.

I

I

I

I

I

I

II II

II II

II II This node will affect the behavior of 4 elements.

This node will affect the behavior of 2 elements.


• Important to obtain enrichment values when using XFEM.

I

I

I

I II II

II II

All the phase 1 components will receive the same enrichment

value because they are all connected.

I I II II

This two elements will receive different enrichments values

because they do not share an interface.


METHODOLOGY OVERVIEW

Test example

Methodology: Setup

4-element 2D mesh. Red areas: phase 2, positive level-set value; blue areas: phase 1 negative level-set value at the node

Methodology: Discretization

4-element 2D mesh. Red numbers represent the Global Element Id, blue numbers represent the Global Node Id.

Methodology: Edges

Edge representation in a QUAD4 element. Green edges represent the local edge index at the element.

Methodology: Intersection points

Mapping of intersection point to element.

Phase 1

Combined

Phase 2

Delaunay triangulation

Enrichment algorithm

Sub-phase (element level)

Enrichment (nodal level)

Assign degrees of freedom (nodal level)

Sub-phase algorithm

Sub-phase algorithm

Triangle number

Main Phase Sub-Phase

1 1 0

2 1 0

3 1 0

4 1 0

5 2 14

6 2 15

• With 27 enrichment levels

Methodology: Enrichment algorithm

4-element 2D mesh. Red areas: phase 2, positive level-set value; blue areas: phase 1 negative level-set value at the node


Nodes/Elements

1 2 3 4

1 14 2 “C” 0 0 3 “C” 15 14 14 15 4 “C” 0 0 5 15 6 “C” 0 0 7 15 8 “C” 0 0 9 14


Nodes/Elements

1 2 3 4

1 “U” 15 2 “C” 0 0 3 “C” 14 14 14 14 4 “C” 0 0 5 “U” 16 6 “C” 0 0 7 “U” 17 8 “C” 0 0 9 “U” 18

Results: Gauss points

Gauss points on volume: Phase 1,

tetrahedron 1

Results: Gauss points on interface and Normal vector

Gauss points on interface and

normal vector to surface: Phase 1, tetrahedron 1

Pre-conditioner

• Calculate the integral of the shape functions for each

enrichment level • Clip degrees of freedom that have a high integral • Apply pre-conditioner matrix with integral values to Jacobian

and solution vector • 4 pre-conditioner types implemented:

• Maximum value of integrals of shape functions per enrichment level

• Sum of values of integrals of shape functions per enrichment level

• Maximum value of integrals of the derivatives of the shape functions per enrichment level

• Sum of values of integrals of the derivatives of the shape functions per enrichment level

Corroborate results

• By using a combination of different mesh

refinements, constraint parameters and property ratios

• Check how variation of these methods and values affect the results of XFEM

Thermal Tests • Mesh size: 20x20,30x30,40x40,50x50

• Conductivity ratios: 0.01,0.1,1,10,100,1000

• Stabilization factor: 1,10,100,1000

Nominal values are in bold face Vary values of one parameter (mesh size, conductivity ratio, etc)

while setting the other parameters to the nominal values Sweep radius of circular inclusion between 2 and 6 using 500

increments

Results – Mesh refinement sweep • Mesh size: variable • Conductivity ratios: k2/k1=10 • Stabilization factor: 10 • Interface formulation: stabilized Lagrange • Solver: UMFPACK • Pre-conditioner: No scaling

Mean Interface and L-2 Errors over Mesh-Refinement Levels Refinement 50 40 30 20 Mean Error 0.0113027 0.0001866 0.0177428 0.0002690 0.0314534 0.0004533 0.0713550 0.0011360 Maximum 0.0229412 0.0027586 0.0350934 0.0017806 0.0598878 0.0017406 0.1756957 0.0045631 Minimum 0.0072671 0.0001382 0.0110625 0.0002151 0.0195454 0.0003847 0.0361082 0.0008666 Std. Dev. 0.0030040 0.0002624 0.0050765 0.0002105 0.0092470 0.0001821 0.0261411 0.0006217

Results – Conductivity ratio sweep

Mean Interface and L-2 Errors of Temperature over Conductivity Ratio Levels Refinement 0.1 1 10 100 1000

Mean Error 2.749304722E-02 1.237790051E-14 3.145337711E-02 4.246747531E-04 3.916329049E-02 2.865812389E-03 4.029740022E-02 5.008780772E-04

Maximum 5.994371110E-02 2.073944120E-14 5.988784790E-02 5.352580837E-04 7.514461690E-02 6.970382421E-03 7.912210350E-02 5.848057930E-04

Minimum 1.257535240E-02 8.022076050E-15 1.954542120E-02 3.844238522E-04 2.502952140E-02 1.113303684E-03 2.570895700E-02 4.485954533E-04

Std. Dev. 1.164141257E-02 2.028024133E-15 9.247029226E-03 2.833694953E-05 1.108102236E-02 1.299709537E-03 1.154149447E-02 2.455657070E-05

• Mesh size: 30x30 • Conductivity ratios: k2/k1=variable • Stabilization factor: 10 • Interface formulation: stabilized Lagrange • Solver: UMFPACK • Pre-conditioner: Spatial Derivatives Shape Functions Maximum

• Mesh size: 30x30 • Conductivity ratios: k2/k1 = 10 • Stabilization factor: 10 • Interface formulation: stabilized Lagrange • Solver: variable • Pre-conditioner: Spatial Derivatives Shape Functions Maximum

• NOTE: UMFPACK and GMRES yielded same condition number results. GMRES

pre-conditioner not dumped to Matlab

Results – UMFPACK/GMRES sweep

Time

UMPFACK, no T scaling

UMFPACK, T scaling GMRES, ILU, no T

scaling GMRES, ILU, T scaling

Condition # Condition # Enrichment Condition # Condition # Enrichment

Mean Error 3.965580161E+15 1.304228344E+04 1.203674188E+05 3.965580161E+15 1.304228344E+04 1.203674188E+05

Maximum 1.952676140E+18 1.391935730E+04 3.868249460E+07 1.952676140E+18 1.391935730E+04 3.868249460E+07

Minimum 1.050431090E+04 1.213103390E+04 9.472486604E+00 1.050431090E+04 1.213103390E+04 9.472486604E+00

Std. Dev. 8.724494218E+16 3.112311742E+02 1.787898160E+06 8.724494218E+16 3.112311742E+02 1.787898160E+06

• Mesh size: 30x30 • Conductivity ratios: k2/k1 = 10 • Stabilization factor: 10 • Interface formulation: stabilized Lagrange • Solver: variable • Pre-conditioner: Spatial Derivatives Shape Functions Maximum

CONDITION NUMBER

Results – UMFPACK/GMRES sweep

• Mesh size: 30x30 • Conductivity ratios: k2/k1 = 10 • Stabilization factor: 10 • Interface formulation: stabilized Lagrange • Solver: GMRES Aztec with ILU pre-conditioner • Pre-conditioner: No pre-conditioner

Results – 2.5D Cylindrical inclusion


Results – 2.5D Cylindrical inclusion


Results – 3D Cylindrical inclusion





Results: 2D niv50 thermal



Results: 2D niv50 thermal • Symmetry in interface errors • EXTREME VALUES (these extremes appeared in the old enrichment algorithm

also): • At iterations ~194 and ~305 (194 and 305 are identical) • Location of double intersections

Results: 2D niv50 thermal • Due to the method for computing double intersections and their main phase, the

following case occurs: • Small phase 1 diamond in center of intersections

Results

Results


FUTURE WORK

Parallelization Different interface formulations

Questions?

Bibliography • [1] Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z., “Finite Element Method - Its Basis and Fundamentals (6th

Edition).” • [2] Abdelaziz, Y., Hamouine, A., A survey of the extended finite element, Computers & Structures, Volume

86, Issues 11-12, Pages 1141-1151, 2008. • [3] Fries, T.P., “The eXtended Finite Element Method.” <http://www.xfem.rwth-aachen.de/index.php>

Accessed on September 21st, 2011. • [4] Stolarska M., Chopp D.L., Moës N., Belytschko T., Modeling crack growth by level sets in the

extended finite element method, International Journal for Numerical Methods in Engineering, Volume 51, Issue 8, Pages 943-960, 2001.

• [5] Lee, D.T., Schachter, B.J., Two Algorithms for Constructing a Delaunay Triangulation, International Journal of Computer & Information Sciences, Volume 9, Issue 3, Pages 219-42, 1980.

• [6] Hansbo, P., Hansbo, A., A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Computer Methods in Applied Mechanics and Engineering, Volume 193, Issue 33-35, Pages 3523-3540, 2004.

• [7] Joe, B., “GEOMPACK3 Computational Geometry in 2D, 3D, ND.” <http://people.sc.fsu.edu/~jburkardt/f_src/geompack3/geompack3.html> Accessed on November 10th, 2011.

• [8] Joe, B., “Geompack3_prb.f90, a sample problem.” <http://people.sc.fsu.edu/~jburkardt/f_src/geompack3/geompack3_prb.f90> Accessed on November 10th, 2011.

• [9] Flood Fill. Wikipedia, the free encyclopedia. <http://en.wikipedia.org/wiki/Flood_fill> Accessed on November 29th, 2011.

• Images used in this presentation: – http://www.xfem.rwth-aachen.de/Background/Introduction/Figures/Fig4.png – http://upload.wikimedia.org/wikipedia/commons/8/80/Example_of_2D_mesh.png – http://www.mathresources.com/products/mathresource/dictionary/maple_images/volterras_integral_equation_1.png – http://www.sigma.ms/stream/images/trommeln/fem_trommel_stage_b_prop.jpg – http://image.tutorvista.com/content/feed/u364/discontin.GIF

htt // l d iki di / iki di / / / 9/D l i i l

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