A complete methodology for the implementation of XFEM models. Team: - Carlos Hernán Villanueva - Kai Yu

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1 A complete methodology for the implementation of XFEM models Team: - Carlos Hernán Villanueva - Kai Yu

2 Problem Statement QUAD4 DISCONTINUITY

geometry Changes in topology or shape Require

3 Problem Statement Traditional FEM Mesh generates discrete representation of potentially complex geometry Changes in topology or shape Require new mesh Discontinuities must align with element edges

4 Problem Statement GOAL OBJECTIVE Develop a numerical framework Solving problems of complex geometry in 2D and 3D Using XFEM and LSM Decompose a 3D element Arbitrary complex geometry Homogeneous domains capable of integration

5 Background: XFEM Numerical technique Provides local enrichment For solutions to differential equations with discontinuous functions Avoids need to re-mesh Solve shortcomings of FEM

6 Background: Discontinuity A discontinuity can be defined as a rapid change in a field variable (a) Cracks, (b) Shear bands, (c) Interface of two fluids Image from: oduction/figures/fig1.png

Background: Level-set function Numerical Scheme Discontinuity of interest is represented as the zero level-set function XFEM XFEM places discontinuity at the boundary layer

7 Background: Level-set function Numerical Scheme Discontinuity of interest is represented as the zero level-set function XFEM XFEM places discontinuity at the boundary layer dividing the grid into negative and positive phases Zero level-set Positive phase Negative phase Image from: res/fig3.png

8 Background: XFEM and LSM LSM XFEM LSM: - Model discontinuity - Update its motion at each calculation XFEM: - Solves the problem - Determines direction of discontinuity

9 Background: Delaunay triangulation 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 Image from : ommons/c/c9/delaunay_circumcircles. png

10 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

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

12 Methodology: Overview

13 Methodology: Point-to-cell list Point Id Number of cells Cell Ids connected , ,2,3, , , ,

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

15 Methodology: Point-to-edge list Global Edge Id Point Ids Cell Ids Local Edge Num

16 Methodology: Edges Edge representation in a QUAD4 element. Green edges represent the local edge index at the element.

17 Methodology: Intersection points Mapping of intersection point to element.

18 Delaunay triangulation Phase 2 Phase 1 Combined

19 Enrichment algorithm Sub-phase (element level) Enrichment (nodal level) Assign degrees of freedom (nodal level)

20 Sub-phase algorithm

21 Enrichment Levels Phase 1 Phase 2 TEMP TEMP_e8 TEMP_e1 TEMP_e9 TEMP_e2 TEMP_e10 TEMP_e3 TEMP_e11 TEMP_e4 TEMP_e12 TEMP_e5 TEMP_e13 TEMP_e6 TEMP_e14

22 Sub-phase algorithm With 27 enrichment levels Triangle number Main Phase Sub-Phase

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

24 Enrichment algorithm Nodes/Ele ments C C C C C

25 Enrichment algorithm Nodes/Elem ents 1 U 15 2 C C C U 16 6 C U 17 8 C U 18

26 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

27 Results: Gauss points Gauss points on volume: Phase 1, tetrahedron 1

28 Results: Gauss points on interface and Normal vector Gauss points on interface and normal vector to surface: Phase 1, tetrahedron 1

29 Corroborate results Compare different interface contribution formulations (Stabilized Lagrangian, Nietsche, etc.) and declare a winner 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 Develop a protocol e i e (u + u ) 2 dγ i i dγ i (u 1 u 2 ) 2 dγ (u 2 ) 2 dγ

30 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

31 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 Mean Error Maximum Minimum Std. Dev

Results Stabilization factor sweep Mesh size: 30x30 Conductivity ratios: k2/k1=10 Stabilization factor: variable Interface formulation: stabilized Lagrange Solver: UMFPACK Pre-conditioner: Spatial

32 Results Stabilization factor sweep Mesh size: 30x30 Conductivity ratios: k2/k1=10 Stabilization factor: variable Interface formulation: stabilized Lagrange Solver: UMFPACK Pre-conditioner: Spatial Derivatives Shape Functions Maximum Mean Interface and L-2 Errors of Temperature over Stabilization Factor Levels Refinement Mean Error Maximum Minimum Std. Dev

Results Conductivity ratio sweep Mesh size: 30x30 Conductivity ratios: k2/k1=variable Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: UMFPACK Pre-conditioner: Spatial

33 Results Conductivity ratio sweep Mesh size: 30x30 Conductivity ratios: k2/k1=variable Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: UMFPACK Pre-conditioner: Spatial Derivatives Shape Functions Maximum Mean Interface and L-2 Errors of Temperature over Conductivity Ratio Levels Refinement Mean Error E E E E E E E E-04 Maximum E E E E E E E E-04 Minimum E E E E E E E E-04 Std. Dev E E E E E E E E-05

34 Results UMFPACK/GMRES sweep 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 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 E E E E E E+05 Maximum E E E E E E+07 Minimum E E E E E E+00 Std. Dev E E E E E E+06

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

36 Results UMFPACK/GMRES sweep Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: variable Pre-conditioner: Spatial Derivatives Shape Functions Maximum L2 ERROR

formulation: stabilized Lagrange Solver: variable

37 Results UMFPACK/GMRES sweep 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

38 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Shape Functions Maximum Differences in L2 errors Matlab UMPFACK, Shape Functions T scaling femdoc UMFPACK, Shape Functions T scaling DSOL L2 Error Interface Error L2 Error Condition # Enrichment Interface Error L2 Error Condition # Enrichment Mean Error Maximum Minimum Std. Dev E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05

39 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Shape Functions Maximum

40 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Shape Functions Maximum

41 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Spatial Derivatives Shape Functions Maximum Differences in L2 errors Time Matlab UMFPACK, Spatial Derivatives Shape Functions T scaling femdoc UMFPACK, Spatial Derivatives Shape Functions T scaling DSOL L2 Error Interface Error L2 Error Condition # Enrichment Interface Error L2 Error Condition # Enrichment Mean Error E E E E E E E E E E-05 Maximum Minimum Std. Dev E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05

42 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Spatial Derivatives Shape Functions Maximum

43 Results Matlab comparison Mesh size: 30x30 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct Pre-conditioner: Spatial Derivatives Shape Functions Maximum

44 Results 2.5D Cylindrical inclusion 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

45 Results 2.5D Cylindrical inclusion 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

Interface formulation: stabilized Lagrange Solver: GMRES

46 Results 3D Cylindrical inclusion Mesh size: 50x50 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: GMRES Aztec with ILU pre-conditioner Pre-conditioner: No pre-conditioner

47 Results 3D Cylindrical inclusion Mesh size: 50x50 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: GMRES Aztec with ILU pre-conditioner Pre-conditioner: No pre-conditioner

48 Results 3D Cylindrical inclusion Mesh size: 50x50 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: GMRES Aztec with ILU pre-conditioner Pre-conditioner: No pre-conditioner

49 Results: 2D niv50 thermal

50 Results: 2D niv50 thermal

51 Results: 2D niv50 thermal

52 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

53 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

54 Results

55 Results

56 Results: 3D niv50 thermal

57 Partial Results Stabilization factor sweep across mesh refinements Mesh size: 20x20, 30x30, 40x40, 50x50 Conductivity ratios: k2/k1 = 10 Stabilization factor: 10 Interface formulation: stabilized Lagrange Solver: Direct UMFPACK Pre-conditioner: Spatial Derivatives Shape Functions Maximum

58 Partial Results Stabilization factor sweep across mesh refinements

59 Questions?

60 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 , [3] Fries, T.P., The extended Finite Element Method. < Accessed on September 21 st, [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 , [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 , [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 , [7] Joe, B., GEOMPACK3 Computational Geometry in 2D, 3D, ND. < Accessed on November 10 th, [8] Joe, B., Geompack3_prb.f90, a sample problem. < Accessed on November 10 th, [9] Flood Fill. Wikipedia, the free encyclopedia. < Accessed on November 29 th, Images used in this presentation:

A complete methodology for the implementation of XFEM inclusive models

University of Colorado Boulder A complete methodology for the implementation of XFEM inclusive models Author: Carlos Hernan Villanueva Kai Yu Center for Aerospace Structures Department of Mechanical Engineering