1. University of Texas at San Antonio Complex Variable Finite Element Methods for Fracture Mechanics Analysis Harry Millwater, Ph.D. David Wagner, MS Jose Garza, MS Andrew Baines, BS Kayla Lovelady, BS Carolina Dubinsky, MS Thomas Ross, MS Sivirt Student Seminary, April 2013

2. University of Texas at San Antonio Overview  Development of the complex variable finite element method (ZFEM)  Derivatives wrt shape, material prop, load and others  Implemented into the Abaqus finite element code  Verification against known solutions  Applications to fracture mechanics  Future efforts

3. University of Texas at San Antonio Re Im h Forward Differencing Finite Difference Method Finite Differencing Perturb along the imaginary axis Determining h is problematic

4. University of Texas at San Antonio Complex Taylor Series Expansion Im Re h Perturb along the imaginary axis Subset of Fourier Differentiation h can be “very” small ~ 10 -30

5. University of Texas at San Antonio Example: No subtraction of nearly equal numbers h can be as small as desired Exact derivative can be recovered as h becomes small Finite Difference CTSE

6. University of Texas at San Antonio Finite element Implementation

7. University of Texas at San Antonio Finite Element Analysis Shape sensitivities “difficult” to compute using finite difference method h must be kept small Mesh must be perturbed enough to “see” the perturbation but not too large to distort elements

8. University of Texas at San Antonio Complex Variable Advantage Perturb nodal coordinates in imaginary axis only! Minimal mesh distortion issues!

9. University of Texas at San Antonio Computational Issues Standard finite element EXCEPT complex nodal coordinates Stiffness matrix is now complex->COMPLEX solver required Apply an imaginary displacement to the nodal coordinates to represent a shape (domain) change Shape change is arbitrary Sensitivities for entire displacement/strain/stress available Imaginary displacement

10. University of Texas at San Antonio Example: 1 st Der. of σ θ w.r.t. R i CTSEAnalytic Solution UTSA Matlab code A. Voorhees, H.R. Millwater, R.L. Bagley, “Complex Variable Methods for Shape Sensitivity of Finite Element Models,” Finite Elem. Anal. Des., 47 (2011) 1146–1156, doi:10.1016/j.finel.2011.05.003

11. University of Texas at San Antonio L2 Norm – Error controlled by Mesh L2 norm reduces as mesh is refined

12. University of Texas at San Antonio Current Effort – Implementation into Abaqus  Abaqus user element implementation (uel)  6 dof/node (3 real, 3 imag)  Abaqus cannot solve complex stiffness matrix, represent as real n x n complex matrix solved as 2n x 2n real matrix

13. University of Texas at San Antonio Abaqus Example Real nodes (physical coordinates) Imaginary nodes (perturbation in physical coordinates Abaqus UEL (user element) Plane stress/strain Element has 8 nodes (4 real, 4 imag) ½ P Plane Strain

14. University of Texas at San Antonio COMPLEX(KIND=r8),DIMENSION(MCRD,NNODE/2 ) :: zCOORDS COMPLEX(KIND=r8),DIMENSION(NNODE,NNODE)::zAMATRX !Complex Stiffness Matrix !Construct complex nodal coordinate array zCOORDS = CMPLX(COORDS(1:2,1:3), COORDS(1:2,NNODE/2+1:NNODE)*h) f1 = zCOORDS(1,2)*zCOORDS(2,3) - zCOORDS(1,3)*zCOORDS(2,2) f2 = zCOORDS(1,3)*zCOORDS(2,1) - zCOORDS(1,1)*zCOORDS(2,3) f3 = zCOORDS(1,1)*zCOORDS(2,2) - zCOORDS(1,2)*zCOORDS(2,1) b1 = zCOORDS(2,2) - zCOORDS(2,3) b2 = zCOORDS(2,3) - zCOORDS(2,1) b3 = zCOORDS(2,1) - zCOORDS(2,2) c1 = zCOORDS(1,3) - zCOORDS(1,2) c2 = zCOORDS(1,1) - zCOORDS(1,3) c3 = zCOORDS(1,2) - zCOORDS(1,1) Area = half * (f1 + f2 + f3) zAMATRX = MATMUL(Bt, MATMUL(Cthick,zB_ip) ) * Area Abaqus UEL Example Code

15. University of Texas at San Antonio Obtaining Derivatives  Shape sensitivity – input imaginary coordinates to represent shape change. (All Imag nodes not perturbed have a value of zero) ½ P 1 2 3 4 5 6 8 7 h = 10 -10 L Nodal Coordinates 1 (0,0) 2 (1,0) 3 (0,1) 4 (1,1) 5 (0,0) 6 (0,0) 7 (0,1)h 8( 0,1)h

16. University of Texas at San Antonio Obtaining Derivatives  Load sensitivity – Apply perturbation in loading to IM nodes. ½ P 1 2 3 4 5 6 8 7 h = 10 -10 Applied Loads 3 (0,1/2 P) 4 (0,1/2 P) 7 (0,1/2)h 8( 0,1/2)h Imaginary coordinates all zero

17. University of Texas at San Antonio Obtaining Derivatives  Material sensitivity – Apply perturbation to constitutive matrix. ½ P 1 2 3 4 5 6 8 7 h = 10 -10 Imaginary coordinates all zero

18. University of Texas at San Antonio Abaqus Verification Results Exact SolutionsABAQUS UEL Solutions Exact SolutionsABAQUS UEL Solutions

19. University of Texas at San Antonio Abaqus Verification Results SDVsStress Values SDV1 SDV2 SDV3 SDV4 SDV5 SDV6 SDVsStrain Values SDV7 SDV8 SDV9 SDV10 SDV11 SDV12 SDVsStrain Energy Values SDV13 SDV14 SDVsDisplacement Values SDV15 SDV16 SDV17 SDV18 SDV19 SDV20

20. University of Texas at San Antonio Abaqus 2D Verification AnalyticalAbaqus

21. University of Texas at San Antonio Applications to Fracture Mechanics

22. University of Texas at San Antonio Weight Function Development Calculation of the partial derivative of the crack opening displacement with respect to crack length required Use multiple reference solutions to solve for Mi Standard research approaches: Assume 3-4 term approximations to du/da or approximate the weight fn. directly.

23. University of Texas at San Antonio Perturbation of Crack Length Crack tip element can be perturbed in the imaginary domain – no perturbation of real mesh Perturb a no. of elements around the crack tip Perturbation of Crack Length in Imaginary domain

24. University of Texas at San Antonio Accuracy: Infinite Array of Cracks Accurate calculation of the crack opening displacement with respect to crack length all along the crack line Crack Opening Displacement 67 – 181 nodes on crack line

25. University of Texas at San Antonio Infinite Array Weight Function 3-term [reference] 3-term [WCTSE] 7-term [WCTSE] Accuracy controlled by the user: high order expansions possible

26. University of Texas at San Antonio Accuracy: Crack from a Hole WCTSE ±0.15% WCTSE ±0.06% WCTSE weight function consistently better than published weight fns. D. Wagner, and H. Millwater, “2D Weight Function Development using a Complex Taylor Series Expansion Method,” Engng Fract Mech 86 (2012), 23-37

27. University of Texas at San Antonio Strain Energy Release Rate Double Cantilever Beam (DBC) Energy release rate (Exact) = 0.800 20 Contours842

28. University of Texas at San Antonio Strain Energy Release Rate Double Cantilever Beam (DBC) Exact = 0.8 Exact = 0.800 # of ContoursPerturbation Method Abaqus J Integral Abaqus CTSE 20Tip only0.79120.7914 8Tip only0.79030.7918 4Tip only0.7900, 0.7841, 0.7832 0.7914 2Tip only0.9026, 0.77640.9244 204 Contours0.79120.7914 84 Contours0.79030.7897 21 Contour + Quarterpoints 0.9026, 0.77640.7764 2Crack tip+Quarterpoints 0.9026, 0.77640.9026

29. University of Texas at San Antonio 3D Applications

30. University of Texas at San Antonio 3D Applications

31. University of Texas at San Antonio Implementation into Abaqus **[…] ** **===================== Real Mesh ======================= *NODE, NSET=rnodes 1, 0.0d0, 0.0d0 2, 2.0d0, 0.0d0 3, 1.2d0, 1.0d0 4, 0.2d0, 1.0d0 5, 1.0d0, 0.0d0 6, 1.6d0, 0.5d0 7, 0.6d0, 1.0d0 8, 0.1d0, 0.5d0 ** *ELEMENT, TYPE=CPE8, ELSET=relems 1, 1, 2, 3, 4, 5, 6, 7, 8 ** **===================== Material ======================== *SOLID SECTION, ELSET=relems, MATERIAL=aluminum 1d0 *MATERIAL, NAME=aluminum *ELASTIC 100d9, 0.3d0 **======== Node Sets, Element Sets, and Surfaces ======== *NSET, NSET=rbotedge 1, 5, 2 ** **================ Boundary Conditions ================== *BOUNDARY rbotedge, 2 1, 1 ** **==================== Load Step ======================== *STEP, NAME=Step-1 *STATIC, DIRECT 1d0, 1d0, 1d-05, 1d0 **[…] *USER ELEMENT, TYPE=U28, NODES=16, COORDINATES=2, UNSYMM, PROPERTIES=5, IPROPERTIES=2, VARIABLES=108 1, 2 **===================== Real Mesh ======================= *NODE, NSET=rnodes 1, 0.0d0, 0.0d0 2, 2.0d0, 0.0d0 3, 1.2d0, 1.0d0 4, 0.2d0, 1.0d0 5, 1.0d0, 0.0d0 6, 1.6d0, 0.5d0 7, 0.6d0, 1.0d0 8, 0.1d0, 0.5d0 **==================== Complex Mesh ===================== *NODE, NSET=inodes 101, 0.0d0, 0.0d0 102, 0.0d0, 0.0d0 103, 1.0d0, 0.0d0 104, 0.0d0, 0.0d0 105, 0.0d0, 0.0d0 106, 0.5d0, 0.0d0 107, 0.5d0, 0.0d0 108, 0.0d0, 0.0d0 *ELEMENT, TYPE=U28, ELSET=zelems 101, 1, 2, 3, 4, 5, 6, 7, 8, 101,102,103,104,105,106,107,108 **===================== Material ======================== ** *UEL PROPERTY, ELSET=zelems 100d9, 0.3d0, 1d-9, 1d0, 1d0, 3, 0 **======== Node Sets, Element Sets, and Surfaces ======== *NSET, NSET=rbotedge 1, 5, 2 *NSET, NSET=ibotedge 101,105,102 **================ Boundary Conditions ================== *BOUNDARY rbotedge, 2 1, 1 ibotedge, 2 101, 1 **==================== Load Step ======================== *STEP, NAME=Step-1, UNSYMM=YES *STATIC, DIRECT 1d0, 1d0, 1d-5, 1d0 **[…] Abaqus Input File Changes Needed Imaginary nodes UEL definition UEL properties

32. University of Texas at San Antonio Multicomplex Mathematics Bi-complex numbers representation : Matrix representation of bi-complex numbers: Tri-complex numbers representation :

33. University of Texas at San Antonio Multicomplex Mathematics Example

34. University of Texas at San Antonio Bicomplex Analysis Higher Order Sensitivities Thick walled cylinder

35. University of Texas at San Antonio Progressive Fracture  Construct a 3 rd order Taylor series of the strain energy using tricomplex elements in front of the crack tip.  Predict the crack path along the max energy release  Progress the crack, remesh, repeat

36. University of Texas at San Antonio Progressive Fracture  Construct a 3 rd order Taylor series of the strain energy using tricomplex elements. Predict the crack path along the max energy release  Progress the crack, remesh, repeat

37. University of Texas at San Antonio Progressive Fracture Similar results to Franc2D with far fewer steps

38. University of Texas at San Antonio Progressive Fracture

39. University of Texas at San Antonio ZFEM User Element Features “Easy” to Use Only a Single Input File Needed Single User Subroutine File No Abaqus Configuration Needed Analytic Extensions of Built-in Elements Identical Response of Real values 2D (6,8 noded), 3D (8, 15, 20) Modular Fortran 95 Implementation Extensible Very Good Performance abaqus job=mycomplexmodel user=zfem

40. University of Texas at San Antonio Future Interests  High order sensitivities:  Multicomplex mathematics element library (complete for linear elastic statics)  Extension to non-linear materials  Plasticity enhancement (in progress)  Composites, anisotropic materials  3D progressive fracture (in progress)  Structural dynamics (in progress)  Thermoelastic analysis  Expanded element library  Plates and shells

41. University of Texas at San Antonio Acknowledgements  Efficient Sensitivity Methods for Probabilistic Lifing and Engine Prognostics, Pat Golden, AFRL/RXLMN, Aug. 2007-Sep. 2010  Efficient Finite Element-based 3D Fracture Mechanics Crack Growth Analysis using Complex Variable Sensitivity Methods, DoD PETTT, Sep. 2010 - Aug. 2011  Implementation of Complex Variable Finite Element Methods in Abaqus, DOD PETTT, Sep. 2011- Aug. 2012  Enhanced Fracture Mechanics Crack Growth Analysis using Complex Variable Sensitivity Methods, AFOSR (David Stargel), May 2011-2014