Harry Millwater, Ph.D. David Wagner, MS Jose Garza, MS Andrew Baines, BS Kayla Lovelady, BS Carolina Dubinsky, MS Thomas Ross, MS

Transcription

1 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 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 Finite Difference Method Im Perturb along the imaginary axis Finite Differencing Re Forward Differencing h df (x o ) dx» f (x o + h) - f (x o ) h Determining h is problematic

4 Complex Taylor Series Expansion Perturb along the imaginary axis Im Subset of Fourier Differentiation h Re F(x + ih) = F(x) + ih df dx - h 2 2! - ih 3 3! d 3 F dx 3 + d 2 F dx 2 df (x o ) dx» Im( f (x o + ih)) h h can be very small ~ 10-30

5 Example: dexp[x] dx = Exp[x] Exp[(x + h)] - Exp[x] h Finite Difference Im[Exp[x +ih]]/h = Exp[x]sin(h)/h CTSE No subtraction of nearly equal numbers h can be as small as desired Exact derivative can be recovered as h becomes small

6 Finite element Implementation ì Pü í ý î 0 þ = EI é 12 6L ù ì L 3 ê ë 6L 4L 2 ú d ü í ý û î fþ ì Pü í ý î 0 þ = EI é 12 6(L + ih) ù ì (L + ih) 3 ê ë 6(L + ih) 4(L + ih) 2 ú d ü í ý û î fþ d = ì PL3 3EI - PLh 2 í î EI ü ý þ + i ì PL 2 h í î EI - Ph 3 3EI ü ý þ d L = Im[d]/h

7 Finite Element Analysis Shape sensitivities difficult to compute using finite difference method Mesh must be perturbed enough to see the perturbation but not too large to distort elements h must be kept small

8 Complex Variable Advantage Perturb nodal coordinates in imaginary axis only! Minimal mesh distortion issues!

9 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 u a Imaginary displacement

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

11 L2 Norm Error controlled by Mesh L2 norm reduces as mesh is refined

12 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 { } é Re P ê ê ë Im P { } ù = é Re K ú ê ú û ë ê Im K [ ] -Im[ K] [ ] Re[ K] ù é Re U ú ê û ú ê ë Im U n x n complex matrix solved as 2n x 2n real matrix { } { } ù ú ú û

13 Abaqus Example Plane Strain ½ P ½ P 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)

14 Abaqus UEL Example Code 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

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

16 Obtaining Derivatives Load sensitivity Apply perturbation in loading to IM nodes. ½ P ½ P 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 h = 10-10

17 Obtaining Derivatives Material sensitivity Apply perturbation to constitutive matrix. ½ P ½ P E (1+n)(1-2n) é 1-n n 0 ù ê ú ê n 1-n 0 ú ë ê 0 0 1/2(1-2n) û ú E = E+ ih n =n + ih h = Imaginary coordinates all zero

18 Abaqus Verification Results Exact Solutions ABAQUS UEL Solutions Exact Solutions ABAQUS UEL Solutions u L = P EA = U L = 1 P 2 2 EA = u P = L EA = U P = PL EA = u L = U L = u P = U P = s L = 0 e L = 0 s P = 1 A = e P = 1 EA = s L = 0 e L = 0 s P = e P = u E = - PL E 2 A = U E = -1 P 2 L 2 E 2 A = u E = U E = s E = 0 e E = -P E 2 A = s E = 0 e E = u A = - PL EA = U A = -1 P 2 L 2 EA = u A = U A = s A = -P A = e A = -P EA = s A = e A =

19 Abaqus Verification Results SDVs Stress Values SDV 1 SDV 2 SDV 3 SDV 4 SDV 5 SDV 6 s xx s yy s xy s xx q s yy q s xy q SDVs SDV7 SDV8 SDV9 SDV1 0 SDV11 SDV1 2 Strain Values e xx e yy e xy e xx q e yy q e xy q SDVs SDV13 SDV14 Strain Energy Values U U q SDVs SDV1 5 SDV1 6 SDV1 7 SDV1 8 SDV1 9 SDV2 0 Displacement Values u x1 q u y1 q u x2 q u y2 q u x3 q u y3 q

20 Abaqus 2D Verification Analytical Abaqus s q r i

21 Applications to Fracture Mechanics

22 Weight Function Development Calculation of the partial derivative of the crack opening displacement with respect to crack length required m(a,x) = E u 2K A a Standard research approaches: m(x,a) = Assume 3-4 term approximations to du/da or approximate the weight fn. directly. 2 2p(a - x) 1+ M 1 (1- x /a)1/ 2 + M 2 (1- x /a) M n (1- x /a) n / 2 [ ] Use multiple reference solutions to solve for Mi

23 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 Accuracy: Infinite Array of Cracks Crack Opening Displacement nodes on crack line u A a Accurate calculation of the crack opening displacement with respect to crack length all along the crack line

25 Infinite Array Weight Function Accuracy controlled by the user: high order expansions possible 7-term [WCTSE] 3-term [reference] 3-term [WCTSE]

26 ComparisontoDomainIntegral (K/K16) ComparisontoDomainIntegral (K/K16) Accuracy: Crack from a Hole HolewithaSingleCrack StresInensityFactor (R/W=0.5, H/W=2) HolewithaSingleCrack StresIntensityFactor (R/W=0.2, H/W=2) WCTSE ±0.15% WCTSE ±0.06% TensionRef.[Wu] TensionWCTSE Tension[Wu] TensionBEM BendingWCTSE Bending[Wu] BendingBEM CrackLength(a/B) 0.98 TensionRef.[Wu] TensionWCTSE Tension[Wu] TensionBEM BendingWCTSE Bending[Wu] BendingBEM CrackLength(a/B) WCTSE weight function consistently better than published weight fns. University Series Expansion of Method, Texas Engng at Fract San Mech 86 (2012), Antonio D. Wagner, and H. Millwater, 2D Weight Function Development using a Complex Taylor

27 Strain Energy Release Rate Double Cantilever Beam (DBC) Energy release rate (Exact) = Contours 8 4 2

28 Strain Energy Release Rate # of Contours Perturbation Method Double Cantilever Beam (DBC) Exact = 0.8 Exact = Abaqus J Integral Abaqus CTSE 20 Tip only Tip only Tip only , , Tip only , Contours Contours Contour + Quarterpoints 2 Crack tip+quarterpoints , ,

29 3D Applications u L

30 3D Applications s R 2

31 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 [ ] Abaqus Input File Changes Needed [ ] *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 [ ] Imaginary nodes UEL definition UEL properties

32 Multicomplex Mathematics Bi-complex numbers representation : Matrix representation of bi-complex numbers: Tri-complex numbers representation : 2) X «x 1 -x 2 -x 3 x 4 x 2 x 1 -x 4 -x 3 x 3 -x 4 x 1 -x 2 x 4 x 3 x 2 x 1 é ë ê ê ê ê ê ù û ú ú ú ú ú 1) Z «z 1 -z 2 z 2 z 1 é ë ê ê ù û ú ú 2)X = x 1 +i 1 x 2 +i 2 x 3 +i 1 i 2 x 4 ;x 1,, x 4 Î R, i 1 2 = i 2 2 = -1, i 1 2 i 2 2 = i 2 2 i 1 2 1) Z = z 1 +i 2 z 2 ; z 1, z 2 Î C 1, i 2 2 = -1 1,,, )]; ( ) [( )] ( ) [( 1) i i i R x x x i x i x i x i x i x i x i x X ( 2 ) C ( 3 ) C 1,, ); ( ) ( 2) i C z z z i z i z i z Z

33 Multicomplex Mathematics Example f ( x +i 1 h +i 2 h) = 2( x+i 1 h+i 2 h) 3 = 2x 3 +(6i 1 h+6i 2 h)x 2 +(12i 1 i 2 h 2-12h 2 )x +2i 1 3 h 3-6i 1 h 3-6i 2 h 3 +2i 2 3 h 3. f ( x +i 1 h +i 2 h) = 2x 3-12h 2 x+i 1 h( 6x 2-8h 2 )+i 2 h( 6x 2-8h 2 )+12i 12 h 2 x Re[ f ] = 2x 3 +O( h 2 ) f x» 1 h Im 1 f [ ] = 6x 2 +O h 2 ( ) 2 f x 2» 1 h 2 Im 12 f [ ] =12x +O h 2 ( )

34 Bicomplex Analysis Higher Order Sensitivities Thick walled cylinder s r r o s q r o 2 s r r o 2 2 s q r o 2 2 s r p r o 2 s q p r o

35 Progressive Fracture (2) Compute the curved crack path to the desired distance (Dd). (1) Perturb the crack tip in x and y. (3) Add curved crack faces, remesh, and repeat. 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 Progressive Fracture (2) Compute the curved crack path to the desired distance (Dd). (1) Perturb the crack tip in x and y. (3) Add curved crack faces, remesh, and repeat. 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 y/w Progressive Fracture 1 Crack Path through a Cruciform Specimen, P 2 /P 1 =1/2, C Dd/W=0.08, 12 Steps [Franc2D] Dd/W=0.04, 25 Steps [Franc2D] Dd/W=0.02, 50 Steps [Franc2D] Dd/W=0.01,100 Steps [Franc2D] Dd/W=0.24, 3 Steps [ZFEM] Dd/W=0.16, 6 Steps [ZFEM] Dd/W=0.08, 12 Steps [ZFEM] Dd/W=0.04, 24 Steps [ZFEM] x/w Similar results to Franc2D with far fewer steps

38 y/w y/w Progressive Fracture Crack Path toward a Hole, Three-point Bending, C 3 Dd/W=0.04, 8 Steps [Franc2D] Dd/W=0.02, 16 Steps [Franc2D] Dd/W=0.005, 66 Steps [Franc2D] Dd/W= 0.1, 3 Steps [ZFEM] Dd/W=0.08, 4 Steps [ZFEM] Dd/W=0.04, 8 Steps [ZFEM] Dd/W=0.02, 16 Steps [ZFEM] x/w Crack Path toward a Hole, Three-point Bending, C 3 Dd/W=0.04, 8 Steps [Franc2D] Dd/W=0.02, 16 Steps [Franc2D] Dd/W=0.005, 66 Steps [Franc2D] Dd/W= 0.1, 3 Steps [ZFEM] Dd/W=0.08, 4 Steps [ZFEM] Dd/W=0.04, 8 Steps [ZFEM] Dd/W=0.02, 16 Steps [ZFEM] x/w

39 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 High order sensitivities: Future Interests 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 Acknowledgements Efficient Sensitivity Methods for Probabilistic Lifing and Engine Prognostics, Pat Golden, AFRL/RXLMN, Aug Sep Efficient Finite Element-based 3D Fracture Mechanics Crack Growth Analysis using Complex Variable Sensitivity Methods, DoD PETTT, Sep Aug Implementation of Complex Variable Finite Element Methods in Abaqus, DOD PETTT, Sep Aug Enhanced Fracture Mechanics Crack Growth Analysis using Complex Variable Sensitivity Methods, AFOSR (David Stargel), May