Stochastic Elastic-Plastic Finite Element Method

Transcription

1 Stochastic Elastic-Plastic Finite Element Method Boris In collaboration with Kallol Sett, You-Chen Chao and Levent Kavvas CEE Dept., University of California, Davis and ESD, Lawrence Berkeley National Laboratory Intel Corporation Webinar Series, October 2010

2 Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

3 Stochastic Systems: Historical Perspectives Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

4 Stochastic Systems: Historical Perspectives History Probabilistic fish counting Williams DEM simulations, differential displacement vortexes SFEM round table Kavvas probabilistic hydrology

5 Stochastic Systems: Historical Perspectives Types of Uncertainties Epistemic uncertainty - due to lack of knowledge Can be reduced by collecting more data Mathematical tools are not well developed trade-off with aleatory uncertainty Aleatory uncertainty - inherent variation of physical system Can not be reduced Has highly developed mathematical tools Deterministic Epistemic uncertainty Aleatory uncertainty Heisenberg principle

6 Stochastic Systems: Historical Perspectives Ergodicity Exchange ensemble averages for time averages Is elasto-plastic-damage response ergodic? Can material elastic-plastic-damage statistical properties be obtained by temporal averaging? Will elastic-plastic-damage statistical properties "renew" at each occurrence? Are material elastic-plastic-damage statistical properties statistically independent? Important assumptions must be made and justified

7 Stochastic Systems: Historical Perspectives Historical Overview Brownian motion, Langevin equation PDF governed by simple diffusion Eq. (Einstein 1905) With external forces Fokker-Planck-Kolmogorov (FPK) for the PDF (Kolmogorov 1941) Approach for random forcing relationship between the autocorrelation function and spectral density function (Wiener 1930) Approach for random coefficient Functional integration approach (Hopf 1952), Averaged equation approach (Bharrucha-Reid 1968), Numerical approaches, Monte Carlo method

8 Uncertainties in Material Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

9 Uncertainties in Material Material Behavior Inherently Uncertainties Spatial variability Point-wise uncertainty, testing error, transformation error (Mayne et al. (2000)

10 Uncertainties in Material Soil Uncertainties and Quantification Natural variability of material (for soil, Fenton 1999) Function of material formation process Testing error (for soil, Stokoe et al. 2004) Imperfection of instruments Error in methods to register quantities Transformation error (for soil, Phoon and Kulhawy 1999) Correlation by empirical data fitting (for soil, CPT data friction angle etc.)

11 Uncertainties in Material Probabilistic material (Soil Site) Characterization Ideal: complete probabilistic site characterization Large (physically large but not statistically) amount of data Site specific mean and coefficient of variation (COV) Covariance structure from similar sites (e.g. Fenton 1999) Moderate amount of data Bayesian updating (e.g. Phoon and Kulhawy 1999, Baecher and Christian 2003) Minimal data: general guidelines for typical sites and test methods (Phoon and Kulhawy (1999)) COVs and covariance structures of inherent variability COVs of testing errors and transformation uncertainties.

12 Uncertainties in Material Recent State-of-the-Art Governing equation Dynamic problems Mü + Cü + Ku = φ Static problems Ku = φ Existing solution methods Random r.h.s (external force random) FPK equation approach Use of fragility curves with deterministic FEM (DFEM) Random l.h.s (material properties random) Monte Carlo approach with DFEM CPU expensive Perturbation method a linearized expansion! Error increases as a function of COV Spectral method developed for elastic materials so far New developments for elasto-plastic-damage applications

13 PEP Formulations Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

14 PEP Formulations Uncertainty Propagation through Constitutive Eq. Incremental el pl constitutive equation dσ ij dt D ijkl = D el ijkl Dijkl el Del ijmn m mnn pq Dpqkl el n rs Drstu el m tu ξ r = D ijkl dɛ kl dt for elastic for elastic-plastic

15 PEP Formulations Previous Work Linear algebraic or differential equations Analytical solution: Variable Transf. Method (Montgomery and Runger 2003) Cumulant Expansion Method (Gardiner 2004) Nonlinear differential equations (elasto-plastic/viscoelastic-viscoplastic): Monte Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah et al. 2000, Griffiths et al ) accurate, very costly Perturbation Method (Anders and Hori 2000, Kleiber and Hien 1992, Matthies et al. 1997) first and second order Taylor series expansion about mean - limited to problems with small C.O.V. and inherits "closure problem"

16 PEP Formulations Problem Statement Incremental 3D elastic-plastic-damage stress strain: { dσ ij = Dijkl el Del ijmn m } mnn pq Dpqkl el dɛ kl dt n rs Drstu el m tu ξ r dt Focus on 1D a nonlinear ODE with random coefficient (material) and random forcing (ɛ) dσ(x, t) dt = β(σ(x, t), D el dɛ(x, t) (x), q(x), r(x); x, t) dt = η(σ, D el, q, r, ɛ; x, t) with initial condition σ(0) = σ 0

17 PEP Formulations Solution to Probabilistic Elastic-Plastic Problem Use of stochastic continuity (Liouiville) equation (Kubo 1963) With cumulant expansion method (Kavvas and Karakas 1996) To obtain ensemble average form of Liouville Equation Which, with van Kampen s Lemma (van Kampen 1976): ensemble average of phase density is the probability density Yields Eulerian-Lagrangian form of the Fokker-Planck-Kolmogorov equation

18 PEP Formulations Eulerian Lagrangian FPK Equation P(σ ij (x t, t), t) t + Z t 0 =»jfi fl η mn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)) σ mn dτcov 0» ηmn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)) σ ab ; ff η ab (σ ab (x t τ, t τ), D abcd (x t τ ), ɛ cd (x t τ, t τ) P(σ ij (x t, t), t) +»jz 2 t dτcov 0»η mn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)); σ mn σ ab 0 ff η ab (σ ab (x t τ, t τ), D abcd (x t τ ), ɛ cd (x t τ, t τ)) P(σ ij (x t, t), t)

19 PEP Formulations Eulerian Lagrangian FPK Equation Advection-diffusion equation P(σ ij, t) t = [ N σ (1) P(σ ij, t) { N(2) P(σ ij, t) } ] mn σ ab Complete probabilistic description of response Solution PDF is second-order exact to covariance of time (exact mean and variance) Deterministic equation in probability density space Linear PDE in probability density space simplifies the numerical solution process Applicable to any elastic-plastic-damage material model (only coefficients N (1) and N (2) differ)

20 PEP Formulations 3D FPK Equation P(σ ij (x t, t), t) t + Z t 0 =»jfi fl η mn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)) σ mn dτcov 0» ηmn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)) σ ab ; ff η ab (σ ab (x t τ, t τ), D abcd (x t τ ), ɛ cd (x t τ, t τ) P(σ ij (x t, t), t) +»jz 2 t dτcov 0»η mn(σ mn(x t, t), D mnrs(x t), ɛ rs(x t, t)); σ mn σ ab 0 ff η ab (σ ab (x t τ, t τ), D abcd (x t τ ), ɛ cd (x t τ, t τ)) P(σ ij (x t, t), t) 6 equations, 3D probabilistic stress-strain response 3 equations, 3D in principal stress-strain space 2 equations, 2D in principal stress-strain space

21 Probabilistic Elastic Plastic Response Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

22 Probabilistic Elastic Plastic Response Elastic Response with Random G General form of elastic constitutive rate equation dσ 12 dt = 2G dɛ 12 dt = η(g, ɛ 12 ; t) Advection and diffusion coefficients of FPK equation N (1) = 2 dɛ 12 dt < G > N (2) = 4t ( dɛ12 dt ) 2 Var[G]

23 Probabilistic Elastic Plastic Response Elastic Response with Random G Prob Density Strain (%) Time (Sec) Stress (MPa) < G > = 2.5 MPa; Std. Deviation[G] = 0.5 MPa

24 Probabilistic Elastic Plastic Response Verification Variable Transformation Method 0 Strain (%) Stress (MPa) Std. Deviation Lines (Variable Transformation) Std. Deviation Lines (Fokker Planck) Mean Line (Variable Transformation) Mean Line (Fokker Planck) Time (Sec)

25 Probabilistic Elastic Plastic Response Drucker-Prager Linear Hardening with Random G dσ 12 dt = G ep dɛ 12 dt = η(σ 12, G, K, α, α, ɛ 12 ; t) Advection and diffusion coefficients of FPK equation N (2) = t N (1) = dɛ 12 dt ( dɛ12 dt ) 2 Var 2G G 2 G + 9K α I 1 α 2G G 2 G + 9K α I 1 α 3

26 Probabilistic Elastic Plastic Response Verification of D P E P Response - Monte Carlo 0 Strain (%) Std. Deviation Lines (Fokker Planck) Stress (MPa) Std. Deviation Lines (Actual) Mean Line (Actual) Mean Line (Fokker Planck) Time (Sec)

27 Probabilistic Elastic Plastic Response Modified Cam Clay Constitutive Model η = 2G dσ 12 dt = G ep dɛ 12 dt (1 + e 0 )p(2p p 0 ) 2 + κ = η(σ 12, G, M, e 0, p 0, λ, κ, ɛ 12 ; t) ) (36 G2 M 4 σ 2 12 (18 GM 4 ) σ e 0 λ κ pp 0(2p p 0 ) Advection and diffusion coefficients of FPK equation [ ] t N (i) (1) η = (i) η (i) (t) (t) + dτcov ; η (i) (t τ) 0 t N (i) (2) = t 0 dτcov [ ] η (i) (t); η (i) (t τ)

28 Probabilistic Elastic Plastic Response Low OCR Cam Clay with Random G, M and p 0 Non-symmetry in probability distribution Difference between mean, mode and deterministic Divergence at critical state because M is Stress (MPa) Strain (%) 1.62 Std. Deviation Lines Deterministic Line Mean Line 0 uncertain Time (s) 30 Mode Line 10 50

29 Probabilistic Elastic Plastic Response Comparison of Low OCR Cam Clay at ɛ = 1.62 % PDF of Shear Stress Random G Random G, p0 Random G, M Random G, M, p0 Det. Value = MPa Shear Stress (MPa) None coincides with deterministic Some very uncertain, some very certain Either on safe or unsafe side

30 Probabilistic Elastic Plastic Response High OCR Cam Clay with Random G and M Large non-symmetry in probability distribution Significant differences in mean, mode, and deterministic Stress (MPa) Strain (%) 2.0 Mean Line Std. Deviation Lines 50 Divergence at Deterministic Line 30 Mode Line critical state, 0 M is uncertain Time (s)

31 Probabilistic Elastic Plastic Response Probabilistic Yielding Weighted elastic and elastic-plastic Solution ) ) P(σ, t)/ t = (N(1) (N w P(σ, t) (2) w P(σ, t) / σ / σ Weighted advection and diffusion coefficients are then N(1,2) w (σ) = (1 P[Σ y σ])n(1) el + P[Σ y σ]n el pl (1) Cumulative Probability Density function (CDF) of the yield function F y P y Σ y (b) (a) Σ y MPa

32 Probabilistic Elastic Plastic Response Transformation of a Bi Linear (von Mises) Response Mode Stress (MPa) Mean Std. Deviations Deterministic Solution Strain (%)

33 SEPFEM Formulations Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

34 SEPFEM Formulations Governing Equations & Discretization Scheme Governing equations of mechanics: Aσ = φ(t); Bu = ɛ; σ = Dɛ Discretization (spatial and stochastic) schemes Input random field material properties (D) Karhunen Loève (KL) expansion, optimal expansion, error minimizing property Unknown solution random field (u) Polynomial Chaos (PC) expansion Deterministic spatial differential operators (A & B) Regular shape function method with Galerkin scheme

35 SEPFEM Formulations Truncated Karhunen Loève (KL) expansion Representation of input random fields in eigen-modes of covariance kernel Error minimizing property q T (x, θ) = q T (x) + M n=1 λn ξ n (θ)f n (x) D C(x 1, x 2 )f (x 2 )dx 2 = λf (x 1 ) ξ i (θ) = 1 [q T (x, θ) q T (x)] f i (x)dx λi Optimal expansion minimization of number of stochastic dimensions D

36 SEPFEM Formulations Polynomial Chaos (PC) Expansion Covariance kernel is not known a priori L u(x, θ) = e j χ j (θ)b j (x) j=1 Can be expressed as functional of known random variables and unknown deterministic function u(x, θ) = ζ[ξ i (θ), x] Need a basis of known random variables PC expansion u(x, θ) = L P j=1 i=0 χ j (θ) = P i=0 γ (j) i ψ i [{ξ r }] γ (j) i ψ i [{ξ r }]e j b j (x) = P ψ i [{ξ r }]d i (x) i=0

37 SEPFEM Formulations Spectral Stochastic Elastic Plastic FEM Minimizing norm of error of finite representation using Galerkin technique (Ghanem and Spanos 2003): N K mn d ni + n=1 N P M d nj n=1 j=0 k=1 C ijk K mnk = F mψ i [{ξ r }] K mn = B n DB m dv K mnk = B n λk h k B m dv D D C ijk = ξ k (θ)ψ i [{ξ r }]ψ j [{ξ r }] F m = φn m dv D

38 SEPFEM Formulations Inside SEPFEM Explicit stochastic elastic-plastic-damage finite element computations FPK probabilistic constitutive integration at Gauss integration points Increase in (stochastic) dimensions (KL and PC) of the problem (parallelism) Development of the probabilistic elastic-plastic-damage stiffness tensor

39 SEPFEM Verification Example Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

40 SEPFEM Verification Example 1 D Static Pushover Test Example Linear elastic model: < G >= 2.5 kpa, Var[G] = 0.15 kpa 2, correlation length for G = 0.3 m. F Elastic plastic material model, von Mises, linear hardening, < G >= 2.5 kpa, Var[G] = 0.15 kpa 2, correlation length for G = 0.3 m, C u = 5 kpa, C u = 2 kpa. L

41 SEPFEM Verification Example Linear Elastic FEM Verification Mean (MC) 0.1 Mean (SFEM) Load (N) Standard Deviations (MC) Standard Deviations (SFEM) Displacement at Top Node (mm) Mean and standard deviations of displacement at the top node, linear elastic material model, KL-dimension=2, order of PC=2.

42 SEPFEM Verification Example SEPFEM verification Load (N) Mean (SFEM) Mean (MC) Standard Deviations (MC) Standard Deviations (SFEM) Displacement at Top Node (mm) Mean and standard deviations of displacement at the top node, von Mises elastic-plastic linear hardening material model, KL-dimension=2, order of PC=2.

43 Probabilistic Shear Response Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

44 Probabilistic Shear Response Probabilistic Material Response Shear Stress (MPa) Mode Mean 0.1 Mean ± Standard Deviation Shear Strain (%)

45 Probabilistic Shear Response Elasticity and Strength, Mean and Std.Dev Depth (m) Depth (m) Depth (m) Depth (m) Mean of Standard Deviation of Shear Modulus (MPa) Shear Modulus (MPa) Mean of Shear Strength (MPa) 12 Standard Deviation of Shear Strength (MPa)

46 Probabilistic Shear Response Displacement Results at the Top 0.30 Load (kn) Mean Mean ± Standard Deviation Displacement (m)

47 Probabilistic Shear Response Statistics of the Tangent Stiffness: Mean and λ for the 1st and 2nd KL-spaces Mean of Tangent Modulus (MPa) Load (kn) λ at 1st KL space at 2nd KL space Load (kn)

48 Probabilistic Shear Response Mean Stiffness Evolution, Cor.Len. 0.1m, 1.0m 0 0 Depth (m) Load = 0 kn Depth (m) Load = 0 kn Mean of Tangent Modulus (MPa) Mean of Tangent Modulus (MPa)

49 Probabilistic Shear Response Mean Prob. Stiffness ( λ 1 f 1 ) Evolution, Cor.Len. 0.1m, 1.0m 0 0 Depth (m) Load = 0 kn λf at 1st KL space Depth (m) λf at 1st KL space Load = 0 kn

50 Probabilistic Shear Response Mean Prob. Stiffness ( λ 2 f 2 ) Evolution, Cor.Len. 0.1m, 1.0m Depth (m) Load = 0 kn λf at 2nd KL space Depth (m) Load = 0 kn λ f at 2nd KL space

51 Probabilistic Shear Response Std.Dev Stiffness Evolution, Cor.Len. 0.1m, 1.0m 0 0 Depth (m) Load = 0 kn Depth (m) Load = 0 kn Standard Deviation of Tangent Modulus (MPa) Standard Deviation of Tangent Modulus (MPa)

52 Probabilistic Shear Response Probability for Softening (?), Cor.Len. 1.0m Depth (m) Load = 0 kn Mean of Tangent Modulus (MPa) Depth (m) Load = 0 kn Standard Deviation of Tangent Modulus (MPa)

53 Seismic Wave Propagation Through Uncertain Soils Outline Motivation Stochastic Systems: Historical Perspectives Uncertainties in Material Probabilistic Elasto Plasticity PEP Formulations Probabilistic Elastic Plastic Response Stochastic Elastic Plastic Finite Element Method SEPFEM Formulations SEPFEM Verification Example Uncertain Response of Solids Probabilistic Shear Response Seismic Wave Propagation Through Uncertain Soils Summary

54 Seismic Wave Propagation Through Uncertain Soils Uncertain Dynamics Stochastic elastic plastic simulations of soils and structures Probabilistic inverse problems Geotechnical site characterization design Optimal material design

55 Seismic Wave Propagation Through Uncertain Soils Decision About Site (Material) Characterization Do an inadequate site characterization (rely on experience): conservative guess for soil data, COV = 225%, large correlation length (length of a model). Do a good site characterization: COV = 103%, correlation length calculated (= 0.61m) Do an excellent (much improved) site characterization if probabilities of exceedance are unacceptable!

56 Seismic Wave Propagation Through Uncertain Soils Random Field Parameters from Site Data Maximum likelihood estimates Finite Scale Typical CPT q T Fractal

57 Seismic Wave Propagation Through Uncertain Soils "Uniform" CPT Site Data (Courtesy of USGS)

58 Seismic Wave Propagation Through Uncertain Soils Statistics of Stochastic Soil Profile(s) Soil as 12.5m deep 1 D soil column (von Mises material) Properties (including testing uncertainty) obtained through random field modeling of CPT q T q T = 4.99 MPa; Var[q T ] = MPa 2 ; Cor. Length [q T ] = 0.61 m; Testing Error = 2.78 MPa 2 q T was transformed to obtain G: Assumed transformation uncertainty = 5% G = 11.57MPa; Var[G] = MPa 2 Cor. Length [G] = 0.61m G/(1 ν) = 2.9q T Input motions: modified 1938 Imperial Valley

59 Seismic Wave Propagation Through Uncertain Soils Evolution of Mean ± SD for Guess Case 600 mean ± standard deviation Displacement (mm) mean Time (sec)

60 Seismic Wave Propagation Through Uncertain Soils Evolution of Mean ± SD for Real Data Case Displacement (mm) mean 5 10 mean ± standard deviation Time (sec)

61 Seismic Wave Propagation Through Uncertain Soils Full PDFs for Real Data Case 0.06 PDF Displacement (mm) Time (sec)

62 Seismic Wave Propagation Through Uncertain Soils Example: PDF at 6 s PDF Real Soil Data Conservative Guess Displacement (mm)

63 Seismic Wave Propagation Through Uncertain Soils Example: CDF at 6 s CDF Real Soil Data Conservative Guess Displacement (mm)

64 Motivation Probabilistic Elasto Plasticity SEPFEM Uncertain Response of Solids Seismic Wave Propagation Through Uncertain Soils Probability of Exceedance of 50 cm (%) Probability of Unacceptable Deformation (50cm) Conservative Guess 40 Real Site 20 Excellently Characterized Site Jeremic Time (s) Summary

65 Seismic Wave Propagation Through Uncertain Soils Risk Informed Decision Process Probability of Exceedance (%) Conservative Guess Real Site Excellently Characterized Site Displacement (cm)

66 Summary Second-order (mean and variance) exact, method for simulations of solids with probabilistic elastic-plastic-damage material and stochastic loading Input: elastic-plastic-damage point-wise uncertainty (PDFs of material paramaters, LHS), spatial uncertainty (KL eigen modes, LHS) and uncertain loading (RHS) Output: accurate PDFs and CDFs of displacements, velocities, accelerations, stress, strain... Probably numerous applications

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