The Choice of Representative Volumes for Random Materials

Transcription

1 The Choice of Representative Volumes for Random Materials Julian Fischer IST Austria

2 Effective properties of random materials Material with random microstructure Effective large-scale description? Stochastic homogenization Effective material laws? Given by infinite-volume cell formula Method of representative volumes

3 Effective properties of random materials Material with random microstructure Effective large-scale description? Stochastic homogenization Effective material laws? Given by infinite-volume cell formula Method of representative volumes

4 The Choice of Representative Volumes Requirements on representative volume: Should capture statistical properties of material Large material sample desirable But: Computational cost Better accuracy for fixed size of the material sample? Material science: Special quasirandom structures, Zunger et al. 90 Mechanics: Statistically similar representative volume elements, Schröder, Balzani, Brands 11 Mathematics: Selection approach for representative volumes, Le Bris, Legoll, Minvielle 15

5 A Selection Approach for Representative Volumes Le Bris, Legoll, Minvielle 15 Choose material sample which captures statistical properties exceptionally well Consider large number of samples of the random material Evaluate one or more statistical quantities on samples Choose the sample which is most representative Statistical quantities close to mean Simplest statistical quantity for composite material: Volume fraction

6 A Selection Approach for Representative Volumes Le Bris, Legoll, Minvielle 15

7 A Selection Approach for Representative Volumes Le Bris, Legoll, Minvielle 15

8 Stochastic Homogenization Linear elliptic PDE (a u) = f Random coefficient field a Stationary distribution Uniform ellipticity Length of dependence ε Examples: Random spherical inclusions (Poisson point process) Functions of Gaussian random fields: ξ (ρ ε W) with W white noise, ρ smooth kernel with diameter ε

9 Stochastic Homogenization Homogenized limit: constant coefficient a hom (a hom u hom ) = f Quantitative error estimates { C (a) f u u hom L 2 L 2ε logε for d = 2 C (a) f L 2ε for d 3 with E[exp(C (a) 2 δ )] C(δ) Gloria, Neukamm, Otto 14, Gloria, Otto 16, Armstrong, Kuusi, Mourrat 16

10 Computation of Effective Coefficient Representative volume method: Choose sample [0,Lε] d Compute homogenization corrector φi (a(e i + φ i )) = 0 Compute average flux a hom e i a RVE e i := a(e i + φ i ) [0,Lε] d Error estimate? Expected fluctuations of a RVE : fluct. of average of L d iid random variables Standard deviation of a RVE L d/2

11 Excursus: Boundary layers in the RVE method Problem: Boundary layers limit order of convergence Solution: Periodization of probability distribution Alternative: Screening of boundary Add massive term in PDE for corrector Average flux only in interior

12 Error analysis RVE approximation a RVE e i := a(e i + φ i ) [0,Lε] d Fluctuations: Var a RVE L d/2 Gloria, Otto 11 Systematic error: E[a RVE ] a hom L d Gloria, Otto 11 Reduction of variance would increases accuracy

13 A Selection Approach for Representative Volumes Le Bris, Legoll, Minvielle 15 Choose only realizations of a which capture statistical properties exceptionally well Simplest case: Spatial average Expect by CLT [ ] a dx E [0,Lε] d a dx [0,Lε] d L d/2 Choose only realizations with [ ] [0,Lε] a dx E a dx δ L d/2 d [0,Lε] d Numerical observation: variance reduction by factor 10

14 A Selection Approach for Representative Volumes Le Bris, Legoll, Minvielle 15 Improved method: Keep F 1 (a) := a dx [0,Lε] d Additional random variable F 2 (a) := a v i dx [0,Lε] d with v i = (ae i ) Require F i (a) E[F i (a)] δ VarF i (a) for i = 1,2 Observation: Variance reduction factor up to 50

15 Analysis of the Selection Approach Rigorous results: one-dimensional case qualitative convergence towards effective coefficient Le Bris, Legoll, Minvielle 15 General case: open Theorem: Fischer 18 Var a sel-rve Var a RVE 1 (1 δ 2 ) ρ 2 C + δ N (1 ρ 2 ) 3/2 L d/2 log L C with ρ := Cov[aRVE,F (a)] Var a RVE VarF(a)

16 Analysis of the Selection Approach Theorem: Fischer 18 Systematic error: E[a sel-rve ] a hom L d log L C δ N (1 ρ 2 ) 3/2 Selection approach also reduces (moderate) tails: a P[ sel-rve a hom t Var ] a sel-rve (1 + Cδ ) N L β P [ N (0,1) t L β ] + C exp ( cl β ).

17 Underlying Principle of the Selection Approach a sel-rve obtained from a RVE by conditional probability Key observation: Joint distribution of a RVE and F (a) Multivariate Gaussian Error quantified in suitable distance Fischer 18 Known for every component of a RVE in functional inequality case Nolen 14, Gloria, Nolen 15, Duerinckx, Gloria, Otto 16

18 Underlying Principle of the Selection Approach

19 Underlying Principle of the Selection Approach

20 Underlying Principle of the Selection Approach

21 Underlying Principle of the Selection Approach

22 Underlying Principle of the Selection Approach

23 Underlying Principle of the Selection Approach

24 Failure of the Variance Reduction Methods Is the covariance Cov[F (a),a RVE ] nonzero? In general: No! Even for the average we may have [ ] Cov a(x)dx,a RVE = 0. [0,Lε] d

25 Failure of the Variance Reduction Methods Idea: Have [ ] Cov a(x)dx,a RVE > 0 [0,Lε] d for random checkerboard Need to construct probability distribution with [ ] Cov a(x)dx,a RVE < 0 [0,Lε] d

26 Failure of the Variance Reduction Methods Idea: Use tiles with microstructure Have [ ] Cov a eff (x)dx,a RVE > 0 [0,Lε] d Exploit a eff (Tile) Tile a(x)dx to get [ ] Cov a(x)dx,a RVE < 0 [0,Lε] d

27 Success of the Variance Reduction Methods Consider F (a) := [0,Lε] d a dx Suppose a is monotone and local function of iid random variables Then Cov[a RVE, [0,Lε] d a dx] L d Variance reduction by a factor 1 c Examples: Random iid checkerboard Monotone functions of Gaussian random fields: ξ (ρ ε W) with W white noise, ρ 0 smooth kernel with diameter ε, ξ monotone: ξ (x) ξ (y) c(x y)id for all x y

28 Normal Approximation for the Effective Coefficient Fischer 18 Stein s method Need to write a RVE as sum of random variables X i with local dependence structure Impossible But: Approximation by sum of X i with multilevel local dependence structure possible

29 Normal Approximation for the Effective Coefficient Fischer 18 Use corrector decomposition φ i (x) = 0 u i(x,s)ds with d ds u i(,s) = (a u i (,s) ), u i (,0) = (ae i ). Dependence of u i (,s) approximately local on scale s Decay of u i (,s) s 1 d/2 May write n a RVE e i e i = ae i e i dx u i (x,s) 2 dx ds [0,L] d 0 [0,L] d k=1

30 Normal Approximation for the Effective Coefficient Fischer 18 Use corrector decomposition φ i (x) = 0 u i(x,s)ds with d ds u i(,s) = (a u i (,s) ), u i (,0) = (ae i ). Dependence of u i (,s) approximately local on scale s Decay of u i (,s) s 1 d/2 May write a RVE 4 n e i e i = ae i e i dx u i (x,s) 2 dx ds [0,L] d 4 n 1 [0,L] d k=1

31 Normal Approximation for the Effective Coefficient a RVE e i e i =... 4 n L d k=1 4 n 1 [0,L] d u i (x,s) 2 dx ds Xk 4 := L d B 4 k 0 u i (x,s) 2 dx ds Xk 3 := L d B 3 k 1 u i (x,s) 2 dx ds Xk 2 := L d B 2 k 2 u i (x,s) 2 dx ds Xk 1 := L d B 1 k 3 u i (x,s) 2 dx ds X0 4 X0 3 X1 3 X0 2 X1 2 X2 2 X3 2 X0 1 X1 1 X2 1 X3 1 X4 1 X5 1 X6 1 X7 1 X 0 k := L d B 4 k u i (x,s) 2 dx ds X0 0 X 1 0 X 2 0 X 3 0 X 4 0 X 5 0 X 6 0 X 7 0 X 8 0 X 9 0 X 10 0 X 11 0 X 12 0 X 13 0 X 14 0 X 15 0 Decay of u i (,s) s 1 d/2 Gloria, Otto 16 All Xk n satisfy bounds L d Suitable adjusted Stein s method succeeds

32 Thank you for your interest Literature: C. Le Bris, F. Legoll, W. Minvielle Special Quasirandom Structures: a selection approach for stochastic homogenization, Monte Carlo Methods Appl., 2016 X. Blanc, C. Le Bris, F. Legoll Some variance reduction methods for numerical stochastic homogenization, Philos. Trans. A, 2016 J.F., The choice of representative volumes in the approximation of effective properties of random materials, in preparation, 2018 Questions?