Computational Homogenization and Multiscale Modeling

Transcription

1 Computational Homogenization and Multiscale Modeling Kenneth Runesson and Fredrik Larsson Chalmers University of Technology, Department of Applied Mechanics Runesson/Larsson, Geilo p.1/23

2 Lecture 2: Contents Nested macro-micro computations (basis for FE 2 ) Variationally consistent homogenization Generalized macrohomogeneity (Hill-Mandel) condition Prolongation conditions - nonlinear SVE-problems Boundary conditions versus macroscale strain or stress control Overview (table) Displacement Boundary Condition (nonlinear DBC-problem) Traction Displacement Boundary Condition (nonlinear TBC-problem) Weakly Periodic Boundary Condition (nonlinear PBC-problem) generalization of classical strong displacement periodicity Runesson/Larsson, Geilo p.2/23

3 Model-based homogenization Two separate scales a priori within each SVE. Local u( X;X) U, global ū( X) Ū u( X;X) = u M ( X;X)+u s ( X;X) Macroscale prolongation toγ. For given X Ω: u M (X) = u M ( H (0) def = ū, H (1) def = H, H (2),..., H (K) ;X) with "higher order" gradients (K = order of homogenization): (k) def H = ū... }{{} k 1st order (conventional): u M (X) = ū+ H [X X] 2nd order: u M (X) = ū+ H [X X]+ 1 2 [X X] H (2) [X X] Note: H(2) 3rd order tensor (such as curvature), e.g. KUZNETSOVA ET AL Runesson/Larsson, Geilo p.3/23

4 Model-based homogenization Prolongation condition defines (implicitly) how the fluctuation depends on the macro-solution: u s {u M }, u M U M ū( X) SVE u M ( X,X) u s {u M ( X,X)} u{ū( X),X} Homogenized problem: y( X) y ( X), X Ω Note: Global boundary data assumed to vary "slowly" y( X) ȳ( X) onγ Variational problem: Findu M U M s. t. R(u{u M };δu) def = l(δu) a(u{u M };δu) = 0 δu M U M,0. where a(u;δu) def = Ω P : δh dv, l(δu) def = f δu dv + p δūds Ω Γ N t Runesson/Larsson, Geilo p.4/23

5 VMS vs. Homogenization Single-scale problem: Find u U such that R(u;δu) def = l(δu) a(u;δu) = 0 δu U 0 Variational Multiscale Method (HUGHES 1995)(u M,u s ) U M U s R(u M +u s ;δu M ) = 0 δu M U M,0 U 0 R(u M +u s ;δu s ) = 0 δu s u s {u M } Localized problem for u s We seek solutionsũ ũ Ũ def = {U u = u M +u s {u M }, u M U M } Issue: Due to approximations/restrictions R(ũ;δu) 0 In general not even for δu = δu M +u s {δu M } Runesson/Larsson, Geilo p.5/23

6 Two possible choices for weighted residuals I Classical assumption (in VMS and Homogenization) R(u M +u s ;δu M ) = 0 δu M U M,0 Straightforward choice (presuming globally exact u s { }, i.e. U s = U U M ) δu M "homogenizer" Balance of homogenized fluxes/stresses II Generalized (non-linear) Galerkin-type R(u } M {{ +u } s ;δu M +(u s ) {u M ;δu M }) = 0 δu M U M,0 }{{} ũ δũ Test space: tangent space to approximation space δũ Ũ0 (u M ) def = {U δu = δu M +(u s ) {u M ;δu M }, δu M U M } }{{} sensitivity Variation of energy (if ) Preserves symmetry for R (u;δu,δu) Runesson/Larsson, Geilo p.6/23

7 Variationally Consistent Macrohomogeneity Condition (VCMC) By construction of local problems, i.e. by definingu s { } R(u M +u s ;δu M ) }{{} I R(u M +u s ;δu M +(u s ) {u M ;δu M }) }{{} II = R(u M +u s ;(u s ) {u M ;δu M }) }{{} Fluctuation residual Solve I and get II "for free" Energy equivalence (Hill-Mandel condition) Symmetric VMS-problem = 0 δu M U M,0 Runesson/Larsson, Geilo p.7/23

8 Macrohomogeneity condition "Local" version of VCMC in terms of SVE-residual R(u;δu s ) def = R (u;δu s )dv Ω R (u;δu s ) def = f δu s P : δh s = 1 Ω = 1 Ω Γ t δu s ds Ω [f δu s P : δh s ] dv Sufficent condition for VCMC R ( u{u M };(u s ) {u M ;δu M } ) = 0, δu M U M,0. Note: No need to solve for sensitivity fields Runesson/Larsson, Geilo p.8/23

9 VCMC - 1st order homogenization VCMC HMC (classical Hill-Mandel macrohomogeneity condition) u M (X) = ū+ H [X X] δu M (X) = δū+δ H [X X], δh M (X) = δ H P : δh = P : δh = P : δ H, δu U Note: HMC valid only if the condition δh = δ H is satisfied: automatically by the choice of test functions imposed as part of the SVE-problem Note: Higher order homogenization "stresses" and "strains" on macro-level will appear in case of higher order homogenization, e.g. GEERS ET AL. Runesson/Larsson, Geilo p.9/23

10 Explicit homogenization results Homogenized variational forms a(u{u M };δu M ) = l(δu M ) = Ω Ω P : δh M d V, f δu M d V + Γ N t p δūd S SVE-forms: P : δh = P{ H} : δ H, f δu M = f δū+ f (2) δ H }{{} ignored Macroscale problem: Findū Ū s. t. R{ū;δū} def = l{δū} ā{ū;δū} = 0 δū Ū0. ā{ū;δū} def = Ω P{ H} : δ Hd V, l{δū} def = Ω f δūd V + Γ N t p δūd S Note: ā may represent higher order continuum model Runesson/Larsson, Geilo p.10/23

11 Nested FE 2 -algorithm Solution via Newton iterations based on the tangent form ā {ū;δū, ū} def = = Ω Ω a (u{u M };δu M, u M +(u s ) {u M ; u M })d V δ H : L AT : Hd V Macroscale Algorithmic Tangent (AT) stiffness L AT defined from d P{ H} = L AT { H} : d H LAT obtained from linearized SVE-problem (sensitivity problem ) Sensitivity problem formulated for each choice of prolongation condition Runesson/Larsson, Geilo p.11/23

12 Nested FE 2 -algorithm Macroscale H, P Subscale (RVE) H,P ū, H Prolongation u,h P Homogenization P 1. Assume (nonequilibrium) macroscale stress P, in the macroscale iteration, corresponding to given ū (and H). 2. Prolongation-homogenization: For given H = H( X i ) in each macroscale (Gauss) quadrature point X i, solve the nonlinear RVE-problem (with chosen prolongation conditions) for subscale u( X i,x) and homogenized (macroscale) stress P( X i ). 3. Check macroscale residual. If convergence, R < TOL, then stop, else compute a new (updated) ū (while using L AT ) and return to 1. Runesson/Larsson, Geilo p.12/23

13 DBC Macroscale strain control Appropriate subscale spaces U D = {u U Ĥ R3 3 s.t. u = Ĥ [X X] onγ } U D,0 = {u U u = 0 onγ } SVE-problem: For given value of H R 3 3, findu s U D,0 that solves a (u M ( H)+u s ;δu s ) = 0 δu s U D,0. withu M ( H) = H [X X ]. Solved by Newton iterations in standard fashion Postprocessing: P = P Runesson/Larsson, Geilo p.13/23

14 DBC Macroscale strain control: AT stiffness Macroscale Algorithmic Tangent (AT) stiffness tensor: Compute sensitivity in u s for change in H: d H du = du M + du s = du M +(u s ) {u M ; du M } Ansatz: du s = i,j ûs(ij) d H ij "Unit fluctuation fields" (sensitivity fields)û s(kl) for k,l = 1,2,...,NDIM (a ) ( ;δu s,û s(kl) ) = (a ) ( ;δu s,û M(kl) ) δu s U D,s Primal approach for computing AT-tensor L AT = L + NDIM k,l=1 L : Ĥ s(kl) e k e l Dual approach for computing AT-tensor: Useful method in conjunction with goal-oriented error computation for subscale FE-discretization, LARSSON AND RUNESSON 2007, "power of duality"! Runesson/Larsson, Geilo p.14/23

15 TBC Macroscale strain control Appropriate subscale spaces U N = {u U s.t. u( X) = 0} U N,s = {u U : H = 0} SVE-problem: For given value of H R 3 3, findu U N and P R 3 3 that solve a (u;δu) c (H) (δu; P) = 0 δu U N c (H) (u;δ P) = δ P : H δ P R 3 3 where c (H) (u; P) def = H : P Note: Term c (H) (δu; P) follows from model assumption t = P N = P N in Γ No postprocessing of P = P, "built-in" property in variational format Runesson/Larsson, Geilo p.15/23

16 TBC Macroscale strain control: AT stiffness Macroscale Algorithmic Tangent (AT) stiffness tensor: Compute sensitivity in u and P for change in H: d H du, d P Ansatz: du s = i,j ûs(ij) d H ij, d P = i,j ˆ P(ij) d H ij "Unit fluctuation fields" (sensitivity fields)û s(kl) and ˆ P (ij) for k,l = 1,2,...,NDIM (a ) ( ;δu,û (kl) ) c (H) (δu, ˆ P(kl) ) = 0 δu U N c (H) (û (kl),δ P) = (δ P) kl δ P R 3 3 AT-tensor L AT = NDIM k,l=1 ˆ P (kl) e k e l Note: MIEHE AND KOCH 2002 in the FE-discrete setting, LARSSON AND RUNESSON, 2007 in the continuous setting Runesson/Larsson, Geilo p.16/23

17 WBC (weak periodicity) Macroscale strain control t Γ Γ + t Periodic boundary conditions using "mirroring" (X : Γ + Γ ), Γ = Γ + Γ + Periodic displacements and anti-periodic tractions [u] def = u(x) u(x (X)) = H [X] onγ + t def = t(x) = t(x (X)) onγ + Runesson/Larsson, Geilo p.17/23

18 Appropriate subscale spaces WBC cont d U = U N = {u U s.t. u( X) = 0} T = {t "suff. regular" } Note: Built-in self-equilibration Weak micro-periodicity formulation for SVE: For given value of H, find u,t U T s.t. a (u;δu) d (t,δu) = 0 δu U d (δt,u) = d (δt, H [X X] δt T where d (t,u) def = 1 Ω Γ + t [u]dγ Remark: Variational framework guarantees Hill-Mandel condition for continuous or discrete U and T. ATS-tensor LARSSON ET AL CMAME 2011 Runesson/Larsson, Geilo p.18/23

19 WPBC Mixed FE-approximation Adopted FE: P.w. linear u and p.w. linear (continuous) t in 2D Construction of traction mesh No need for periodic mesh displacement Advantageous for adaptive meshing! Runesson/Larsson, Geilo p.19/23

20 WPBC Mixed FE, cont d Remarks: Stable combination (in the sense of LBB-condition) for "complete" t-mesh, follows from analysis of contact problems, cf. EL-ABBASI & BATHE 2001, WRIGGERS Stable for arbitrarily coarsened t-mesh; however, dim(t,h ) can not be too large Special choice: Constant t on each side Standard TBC (Neumann b.c) Possible to use p.w. constant t from regularity viewpoint Γ +, t 1 TYXB t 1 t 2 TXYB t 2 A Weakly-periodic B Neumann Runesson/Larsson, Geilo p.20/23

21 Weak format of periodicity WPBC Mixed FE, cont d d (δt h,u s h) = 0 δt h T,h Special case: Strong periodicity, [ u s h ] = 0, X Γ+ Increase dim(t,h ) indefinitely. Not feasable in practice due to instability unlessu,h is enlarged indefinitely Introduce strictly periodic mesh, i. e. trace(u,h ) onγ + and trace(u,h ) on Γ are identical. Choose T,h = trace(u,h ) Γ + = trace(u,h ) Γ Settingδt h = [ u s h ] T,h [ u s h ] = 0, i. e. strong periodicity! Note: Strong periodicity does not require any particular arrangement or precautions. It is merely considered as a special choice out of many possible meshes. Runesson/Larsson, Geilo p.21/23

22 WPBC Typical results Snapshots of deformed RVEs for single realization, from CMAME, in print L L ref = 1.25 Dirichlet Weakly periodic Neumann L = 5.00 P N L ref Dirichlet D D Weakly periodic P Neumann N Remark: Non-periodic mesh for weakly periodic bc! Suitable for adaptive meshing. Runesson/Larsson, Geilo p.22/23

23 SVE-problems Displacement b.c. (DBC), Traction b.c. (TBC), Weak periodicity b.c. (WPBC) Macroscale data (input or output) for SVE-problem: Macrostrain H R 3 3 or Macrostress P R 3 3 Subscale variables (fields) Displacement u U D,U P,U N Tractions t T Classical versions: DBC with strain control, TBC with stress control DBC WPBC TBC Strain control H u u,t u, P P from post-proc P from post-proc Stress control P u, H u,t, H u H from post-proc Runesson/Larsson, Geilo p.23/23