A stochastic 3-scale method to predict the thermo-elastic behaviors of polycrystalline structures

Transcription

1 Computational & Multiscale Mechanics of Materials CM3 A stochastic 3-scale method to predict the thermo-elastic behaviors of polycrystalline structures Wu Ling, Lucas Vincent, Golinval Jean-Claude, Paquay Stéphane, Noels Ludovic 3SMVIB: The research has been funded by the Walloon Region under the agreement no (CT-INT ) in the context of the ERA-NET MNT framework. Experimental measurements provided by IMT Bucharest (Voicu Rodica, Baracu Angela, Muller Raluca) CM3 CMCS November 2017, Paris, France

2 MEMS structures Are not several orders larger than their micro-structure size The problem Parameters-dependent manufacturing process Low Pressure Chemical Vapor Deposition (LPCVD) Properties depend on the temperature, time process, and flow gas conditions Scatter in the structural properties Due to the fabrication process (photolithography, etching ) Due to uncertainties of the material The objective of this work is to estimate this scatter CM3 CMCS November 2017, Paris, France - 2

3 The problem MEMS structures Are not several orders larger than their micro-structure size Parameters-dependent manufacturing process Low Pressure Chemical Vapor Deposition (LPCVD) Properties depend on the temperature, time process, and flow gas conditions Scatter in the structural properties Due to the fabrication process (photolithography, etching ) Due to uncertainties of the material The objective of this work is to estimate this scatter Application example Poly-silicon resonators Quantities of interest Eigen frequency Quality factor due to thermoelastic damping Q ~ W/ W Thermoelastic damping is a source of intrinsic material damping present in almost all materials τ T τ T τ ~T isothermal process adiabatic process Q CM3 CMCS November 2017, Paris, France - 3

4 The problem Material structure: grain size distribution SEM Measurements (Scanning Electron Microscope) Grain size dependent on the LPCVD temperature process 2 µm-thick poly-silicon films 1 μm 1 μm Deposition temperature: 580 o C Deposition temperature: 650 o C Deposition temperature [ o C] Average grain diameter [µm] SEM images provided by IMT Bucharest, Rodica Voicu, Angela Baracu, Raluca Muller CM3 CMCS November 2017, Paris, France - 4

5 The problem Material structure: grain orientation distribution Grain orientation by XRD (X-ray Diffraction) measurements on 2 µm-thick poly-silicon films I [cps] Si(111) Si(220) Si(311) PolySi, T=580 o C Si(331) Si(511)/ Si(333) Si(422) I [cps] Si(111) Si(220) Si(311) Si(400) PolySi, T=650 o C Si(331) Si(422) Si(511)/ Si(333) Si(400) [ o ] Deposition temperature: 580 o C [ o ] Deposition temperature: 630 o C Deposition temperature [ o C] <111> [%] <220> [%] <311> [%] <400> [%] <331> [%] <422> [%] XRD images provided by IMT Bucharest, Rodica Voicu, Angela Baracu, Raluca Muller CM3 CMCS November 2017, Paris, France - 5

6 Monte-Carlo for a fully modelled beam The first mode frequency distribution can be obtained with A 3D beam with each grain modelled Grains distribution according to experimental measurements Monte-Carlo simulations Considering each grain is expensive and time consuming Motivation for stochastic multi-scale methods CM3 CMCS November 2017, Paris, France - 6

7 Motivations Multi-scale modelling 2 problems are solved concurrently The macro-scale problem Material response Macro-scale Extraction of a mesoscale Volume Element The meso-scale problem (on a meso-scale Volume Element) BVP Length-scales separation L macro >>L VE >>L micro For accuracy: Size of the mesoscale volume element smaller than the characteristic length of the macro-scale loading To be statistically representative: Size of the meso-scale volume element larger than the characteristic length of the microstructure CM3 CMCS November 2017, Paris, France - 7

8 Motivations For structures not several orders larger than the micro-structure size L macro >>L VE >~L micro For accuracy: Size of the mesoscale volume element smaller than the characteristic length of the macro-scale loading Meso-scale volume element no longer statistically representative: Stochastic Volume Elements* Possibility to propagate the uncertainties from the micro-scale to the macro-scale *M Ostoja-Starzewski, X Wang, 1999 P Trovalusci, M Ostoja-Starzewski, M L De Bellis, A Murrali, 2015 X. Yin, W. Chen, A. To, C. McVeigh, 2008 J. Guilleminot, A. Noshadravan, C. Soize, R. Ghanem, CM3 CMCS November 2017, Paris, France - 8

9 A 3-scale process Grain-scale or micro-scale Meso-scale Macro-scale Samples of the microstructure (volume elements) are generated Each grain has a random orientation Intermediate scale The distribution of the material property P(C) is defined Uncertainty quantification of the macro-scale quantity E.g. the first mode frequency P f 1 /Quality factor P Q Mean value of material property Stochastic Homogenization SVE size SFEM Probability density Variance of material property SVE size Quantity of interest CM3 CMCS November 2017, Paris, France - 9

10 From the micro-scale to the meso-scale Thermo-mechanical homogenization Definition of Stochastic Volume Elements (SVEs) & Stochastic homogenization Need for a meso-scale random field The meso-scale random field Definition of the thermo-mechanical meso-scale random field Stochastic model of the random field: Spectral generator & non-gaussian mapping From the meso-scale to the macro-scale 3-Scale approach verification Application to extract the quality factor Accounting for roughness effect Content From the micro-scale to the meso-scale The meso-scale random field From the meso-scale to the macro-scale CM3 CMCS November 2017, Paris, France - 10

11 From the micro-scale to the meso-scale Thermo-mechanical homogenization Definition of Stochastic Volume Elements (SVEs) & Stochastic homogenization Need for a meso-scale random field The meso-scale random field Definition of the thermo-mechanical meso-scale random field Stochastic model of the random field: Spectral generator & non-gaussian mapping From the meso-scale to the macro-scale 3-Scale approach verification Application to extract the quality factor Accounting for roughness effect Content From the micro-scale to the meso-scale The meso-scale random field From the meso-scale to the macro-scale CM3 CMCS November 2017, Paris, France - 15

12 From the micro-scale to the meso-scale Definition of Stochastic Volume Elements (SVEs) Poisson Voronoï tessellation realizations SVE realization ω j Each grain ω i is assigned material properties Elasticity tensor C m i; Heat conductivity tensor κ m i ; Thermal expansion tensors α m i. Defined from silicon crystal properties Each set C m i, κ m i, α m i is assigned a random orientation Following XRD distributions Stochastic homogenization Several SVE realizations C m i, κ m i, α m i C m i, κ m i, α m i ω j = i ω i C m i, κ m i, α m i C m i, κ m i, α m i C m i, κ m i, α m i For each SVE ω j = i ω i C m i, κ m i, α m i i Computational homogenization C M j, κ M j, α M j Samples of the mesoscale homogenized elasticity tensors Homogenized material tensors not unique as statistical representativeness is lost* * C. Huet, 1990 CM3 CMCS November 2017, Paris, France - 16

13 From the micro-scale to the meso-scale Thermo-mechanical homogenization Down-scaling ε M = 1 V ω ω ε m dω ε M, M θ M, θ M x σ M, q M, ρ M C vm C M, κ M,α M C M,. M θ M = 1 V ω θ M = 1 V ω ω m θ m dω ρ m C vm θ ω ρ M C m dω vm x Meso-scale BVP resolution x Up-scaling ω = i ω i σ M = 1 V ω q M = 1 V ω ω σ m dω ω q m dω ρ M C vm = 1 V ω ρ mc vm dv σ M C M = u M M κ M = q M M θ M & α M : C M = σ M θ M Consistency Satisfied by periodic boundary conditions CM3 CMCS November 2017, Paris, France - 17

14 From the micro-scale to the meso-scale Distribution of the apparent mesoscale elasticity tensor C M COV = Variance mean 100% For large SVEs, the apparent tensor tends to the effective (and unique) one The bounds do not depend on the SVE size but on the silicon elasticity tensor However, the larger the SVE, the lower the probability to be close to the bounds CM3 CMCS November 2017, Paris, France - 19

15 From the micro-scale to the meso-scale Use of the meso-scale distribution with macro-scale finite elements Beam macro-scale finite elements C M 1 C M 2 C M 3 Use of the meso-scale distribution as a random variable Monte-Carlo simulations COV = Variance mean 100% First bending mode of a 3.2 μm-long beam Coarse macro-mesh Fine macro- mesh CM3 CMCS November 2017, Paris, France - 20

16 From the micro-scale to the meso-scale Use of the meso-scale distribution with macro-scale finite elements Beam macro-scale finite elements C M 1 C M 2 C M 3 Use of the meso-scale distribution as a random variable Monte-Carlo simulations COV = Variance mean 100% First bending mode of a 3.2 μm-long beam Convergence Coarse macro-mesh Fine macro-mesh No convergence: the macro-scale distribution (first resonance frequency) depends on SVE and mesh sizes CM3 CMCS November 2017, Paris, France - 21

17 From the micro-scale to the meso-scale Need for a meso-scale random field Introduction of the (meso-scale) spatial correlation Define large tessellations SVE (x, y+l) SVE (x+l, y+l) SVEs extracted at different distances in each tessellation Evaluate the spatial correlation between the components of the meso-scale material operators For example, in 1D-elasticity Young s modulus correlation E y t SVE (x, y ) SVE (x, y) SVE (x+l, y) E x Young s modulus correlation R Ex τ = E E x x E E x E x x + τ E E x E E x E E x 2 Correlation length L Ex = REx τ dτ R Ex 0 CM3 CMCS November 2017, Paris, France - 23

18 From the micro-scale to the meso-scale Need for a meso-scale random field (2) The meso-scale random field is characterized by the correlation length L Ex The correlation length L Ex depends on the SVE size Random field with different SVEs sizes l SVE = 0. 1 μm l SVE = 0. 4 μm CM3 CMCS November 2017, Paris, France - 24

19 From the micro-scale to the meso-scale Thermo-mechanical homogenization Definition of Stochastic Volume Elements (SVEs) & Stochastic homogenization Need for a meso-scale random field The meso-scale random field Definition of the thermo-mechanical meso-scale random field Stochastic model of the random field: Spectral generator & non-gaussian mapping From the meso-scale to the macro-scale 3-Scale approach verification Application to extract the quality factor Accounting for roughness effect Content From the micro-scale to the meso-scale The meso-scale random field From the meso-scale to the macro-scale CM3 CMCS November 2017, Paris, France - 27

20 The meso-scale random field Use of the meso-scale distribution with stochastic (macro-scale) finite elements Use of the meso-scale random field Monte-Carlo simulations at the macro-scale BUT we do not want to evaluate the random field from the stochastic homogenization for each simulation Meso-scale random field from a generator Need for a stochastic model of meso-scale elasticity tensors Stochastic model C M (x + τ, θ) C M (x, θ) C M (x + 2τ, θ) CM3 CMCS November 2017, Paris, France - 28

21 The meso-scale random field Definition of the thermo-mechanical meso-scale random field Elasticity tensor C M (x, θ) (matrix form C M ) & thermal conductivity κ M are bounded Ensure existence of their inverse Define lower bounds C L and κ L such that ε: C M C L : ε > 0 ε θ κ M κ L θ > 0 θ Use a Cholesky decomposition when semi-positive definite matrices are required C M x, θ = C L + A + A x, θ T A + A x, θ κ M x, θ = κ L + B + B x, θ T B + B x, θ α Mij x, θ = V (t) + V (t) x, θ We define the homogenous zero-mean random field V x, θ, with as entries Elasticity tensor A x, θ V (1) V (21), Heat conductivity tensor B x, θ V (22) V (27) Thermal expansion tensors V (t) V (28) V (33) CM3 CMCS November 2017, Paris, France - 29

22 The meso-scale random field Characterization of the meso-scale random field Generate large tessellation realizations For each tessellation realization SVE (x, y+l) SVE (x+l, y+l) Extract SVEs centered on x + τ For each SVE evaluate C M (x + τ), κ M (x + τ), α M x + τ From the set of realizations C M x, θ, κ M x, θ, α M x, θ E y t SVE (x, y ) Evaluate the bounds C L and κ L Apply the Cholesky decomposition A x, θ, B x, θ SVE (x, y) SVE (x+l, y) E x Fill the 33 entries of the zero-mean homogenous field V x, θ Compute the auto-/cross-correlation matrix R V rs τ = E V r x V s x + τ E V r 2 E V s 2 Generate zero-mean random field V x, θ Spectral generator & non-gaussian mapping CM3 CMCS November 2017, Paris, France - 30

23 The meso-scale random field Polysilicon film deposited at 610 ºC SVE size of 0.5 x 0.5 μm 2 Comparison between micro-samples and generated field PDFs C M11 [GPa] κ M33 [W/(m K)] CM3 CMCS November 2017, Paris, France - 33

24 The meso-scale random field Polysilicon film deposited at 610 ºC (2) Comparison between micro-samples and generated field cross-correlations Micro-Samples Generator CM3 CMCS November 2017, Paris, France - 34

25 The meso-scale random field Polysilicon film deposited at 610 ºC (3) Comparison between micro-samples and generated random field realizations Micro-Samples Generator CM3 CMCS November 2017, Paris, France - 35

26 From the micro-scale to the meso-scale Thermo-mechanical homogenization Definition of Stochastic Volume Elements (SVEs) & Stochastic homogenization Need for a meso-scale random field The meso-scale random field Definition of the thermo-mechanical meso-scale random field Stochastic model of the random field: Spectral generator & non-gaussian mapping From the meso-scale to the macro-scale 3-Scale approach verification Application to extract the quality factor Accounting for roughness effect Content From the micro-scale to the meso-scale The meso-scale random field From the meso-scale to the macro-scale CM3 CMCS November 2017, Paris, France - 36

27 From the meso-scale to the macro-scale 3-Scale approach verification with direct Monte-Carlo simulations Use of the meso-scale random field Monte-Carlo simulations at the macro-scale Macro-scale beam elements of size l mesh Convergence in terms of α = l Ex l mesh C M 1(x) C M 1(x + τ) COV = Variance mean 100% First bending mode of a 3.2 μm-long beam Coarse macro-mesh Fine macro-mesh CM3 CMCS November 2017, Paris, France - 37

28 From the meso-scale to the macro-scale 3-Scale approach verification (α~2) with direct Monte-Carlo simulations First bending mode Eigen frequency First bending mode of a 3.2 μm-long beam Second bending mode Eigen frequency Second bending mode of a 3.2 μm-long beam CM3 CMCS November 2017, Paris, France - 38

29 From the meso-scale to the macro-scale Quality factor Micro-resonators Temperature changes with compression/traction Energy dissipation Eigen values problem Governing equations M 0 u 0 0 θ u D uϑ (θ) D ϑϑ θ + K uu(θ) 0 Free vibrating problem u(t) θ(t) = u 0 θ 0 e iωt K uu (θ) K uθ (θ) 0 0 K ϑϑ (θ) I Quality factor From the dissipated energy per cycle u θ u K uϑ (θ) K ϑϑ (θ) = iω u θ = F u F ϑ 0 0 M D ϑu (θ) D ϑϑ 0 I 0 0 u θ u Q 1 = 2 Iω Iω 2 + Rω 2 CM3 CMCS November 2017, Paris, France - 39

30 From the meso-scale to the macro-scale Application of the 3-Scale method to extract the quality factor distribution 3D models readily available The effect of the anchor can be studied w support t t L w L support g L w 15 x 3 x 2 μm 3 -beam, deposited at 610 ºC 15 x 3 x 2 μm 3 -beam & anchor, deposited at 610 ºC CM3 CMCS November 2017, Paris, France - 42

31 From the micro-scale to the meso-scale Thermo-mechanical homogenization Definition of Stochastic Volume Elements (SVEs) & Stochastic homogenization Need for a meso-scale random field The meso-scale random field Definition of the thermo-mechanical meso-scale random field Stochastic model of the random field: Spectral generator & non-gaussian mapping From the meso-scale to the macro-scale 3-Scale approach verification Application to extract the quality factor Accounting for roughness effect Content From the micro-scale to the meso-scale The meso-scale random field From the meso-scale to the macro-scale CM3 CMCS November 2017, Paris, France - 43

32 Surface topology: asperity distribution Accounting for roughness effect Upper surface topology by AFM (Atomic Force Microscope) measurements on 2 µmthick poly-silicon films Deposition temperature [ o C] Std deviation [nm] AFM data provided by IMT Bucharest, Rodica Voicu, Angela Baracu, Raluca Muller CM3 CMCS November 2017, Paris, France - 44

33 Accounting for roughness effect From the micro-scale to the meso-scale Second-order homogenization ε M, κ M n M = C M1 : ε M + C M2 : κ M x n M, C M1, C M2 m M, C M3, C M4 m M = C M3 : ε M + C M4 : κ M A h/2 Meso-scale BVP resolution Stochastic homogenization Several SVE realizations ω = i ω i For each SVE ω j = i ω i The density per unit area is now non-constant ω j = i ω i C m i C m i C m i C m i i Computational homogenization C M1 j,c M2 j, C M3 j, C M4 j ρ M j U M j Samples of the mesoscale homogenized elasticity matrix U M & density ρ M CM3 CMCS November 2017, Paris, France - 45

34 The meso-scale random field Accounting for roughness effect Generate large tessellation realizations For each tessellation realization C m i C m i Extract SVEs centred at x + τ For each SVE evaluate U M (x + τ), ρ M (x + τ) ω j = i ω i C m i From the set of realizations U M x, θ, ρ M x, θ, Evaluate the bounds U L and ρ L Apply the Cholesky decomposition A x, θ Fill the 22 entries of the zero-mean homogenous field V x, θ Compute the auto-/cross-correlation matrix R V rs τ = E V r x V s x + τ E V r 2 E V s 2 CM3 CMCS November 2017, Paris, France - 46

35 Accounting for roughness effect From the meso-scale to the macro-scale Cantilever of 8 x 3 x t μm 3 deposited at 610 ºC Flat SVEs (no roughness) - F Rough SVEs ( Polysilicon film deposited at 610 ºC ) - R Grain orientation following XRD measurements Si pref Grain orientation uniformly distributed Si uni Reference isotropic material Iso L t w Roughness effect is the most important for 8 x 3 x 0.5 μm 3 cantilevers Roughness effect is of same importance as orientation for 8 x 3 x 2 μm 3 cantilevers CM3 CMCS November 2017, Paris, France - 47

36 Efficient stochastic multi-scale method Conclusions & Perspectives Micro-structure based on experimental measurements Computational efficiency relies on the meso-scale random field generator Used to study probabilistic behaviors Perspectives Other material systems Non-linear behaviors Non-homogenous random fields CM3 CMCS November 2017, Paris, France - 48

37 Thank you for your attention! CM3 CMCS November 2017, Paris, France - 49

38 Thermo-mechanical problem Governing equations Thermo-mechanics Linear balance ρ u σ ρb = 0 Clausius-Duhem inequality in terms of volume entropy rate Helmholtz free energy F ε, T = F 0 T ε: 2 ψ ε ε : α T T 0 + ψ ε S = q T σ = F S = F 2 F 0 ε, & T T ε T T = ρc v Strong form in terms of the displacements u and temperature change θ (linear elasticity) ρ u C: ε C: αθ ρb = 0 ρc v θ + T 0 α: C: ε κ θ = 0 Finite element discretization M(ρ) u θ D ϑu (α, C) D ϑϑ (ρc v ) u θ + K uu(c) K uϑ (α, C) 0 K ϑϑ (κ) u θ = F u F ϑ CM3 CMCS November 2017, Paris, France - 50

39 The meso-scale random field Stochastic model of the meso-scale random field: Spectral generator* Start from the auto-/cross-covariance matrix R V rs rs τ = σ V r σ V s R V τ = E V r x E V r V s x + τ E V s Evaluate the spectral density matrix from the periodized zero-padded matrix R P V τ rs S V ω (m) = P (rs) τ n e 2πiτ n ω (m) & S V ω (m) = H V ω (m) H V R V ω (m) n ω gathers the discrete frequencies τ gathers the discrete spatial locations Generate a Gaussian random field V x, θ V (r) x, θ = 2Δω R (rs) H V ω (m) η (s,m) e 2πi x ω m +θ (s,m) s m η and θ are independent random variables Quid if a non-gaussian distribution is sought? *Shinozuka, M., Deodatis, G., 1988 CM3 CMCS November 2017, Paris, France - 51

40 The meso-scale random field Stochastic model of the meso-scale random field: non-gaussian mapping* Start from micro-sampling of the stochastic homogenization The continuous form of the targeted PSD function S T rs rs (ω) = ΔτS V ω (m) = Δτ P R (rs) V τ n e 2πiτ n ω (m) n Iterate The targeted marginal distribution density function F NG(r) of the random variable V (r) A marginal Gaussian distribution F G(r) of zero-mean and targeted variance σ V (r) Generate Gaussian random vector V G (x) from S rs ω S rs ω S rs ω ST rr ω S T ss ω S NG rr ω S NG ss ω Map V G (x) to a non-gaussian field: V NG(r) x = F NG r 1 F G r V G(r) (x) Evaluate the PSD S NG rs (ω) of V NG x NG rs S No S T rs? * Deodatis, G., Micaletti, R., 2001 CM3 CMCS November 2017, Paris, France - 52