Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations

Transcription

1 Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations Boris Jeremić 1 Kallol Sett 2 1 University of California, Davis 2 University of Akron, Ohio ICASP Zürich, Switzerland August 2011

2 Outline Motivation Probabilistic Elasto Plasticity PEP Formulations Stochastic Elastic Plastic Finite Element Method Applications to Risk Analysis Seismic Wave Propagation Through Uncertain Soils Probabilistic Analysis for Decision Making Summary

3 Determining Risk for Civil Engineering Object Behavior Risk: inherent, intrinsic, constitutive part of civil engineering Uncertain loads (!) Uncertain materials (!!) Uncertain human factor (!) Mü + C u + Ku = F

4 Material Behavior Inherently Uncertain Spatial variability Point-wise uncertainty testing error transformation error (Mayne et al. (2000)

5 PEP Formulations Outline Motivation Probabilistic Elasto Plasticity PEP Formulations Stochastic Elastic Plastic Finite Element Method Applications to Risk Analysis Seismic Wave Propagation Through Uncertain Soils Probabilistic Analysis for Decision Making Summary

6 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 What if all (any) material parameters are uncertain Since material is inherently spatially variable and uncertain at the point, PEP and SEPFEM methods were developed

7 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 Forward Kolmogorov (Fokker-Planck-Kolmogorov) equation

8 PEP Formulations Eulerian Lagrangian FPK Equation P(σ ij (x t, t), t) t + t 0 = [{ 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 ; ]} 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) + [{ 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 ] } ] 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)

9 PEP Formulations Eulerian Lagrangian FPK Equation Advection-diffusion equation P(σ ij, t) t = [ N (1) σ ab P(σ ij, t) { } ] N (2) ab σ abcd P(σ ij, t) cd 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) ab and N(2) abcd differ)

10 P[Shear Stress] PEP Formulations Probabilistic Elastic-Plastic Response Shear Strain (%) 0.75 Shear Stress (MPa)

11 SEPFEM Outline Motivation Probabilistic Elasto Plasticity PEP Formulations Stochastic Elastic Plastic Finite Element Method Applications to Risk Analysis Seismic Wave Propagation Through Uncertain Soils Probabilistic Analysis for Decision Making Summary

12 SEPFEM Governing Equations & Discretization Scheme Governing equations: 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

13 SEPFEM 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

14 SEPFEM Inside SEPFEM Stochastic elastic plastic (explicit) finite element computations FPK probabilistic constitutive integration at Gauss integration points Increase in (stochastic) dimensions (KL and PC) of the problem Development of the probabilistic elastic plastic stiffness tensor

15 Seismic Wave Propagation Through Uncertain Soils Outline Motivation Probabilistic Elasto Plasticity PEP Formulations Stochastic Elastic Plastic Finite Element Method Applications to Risk Analysis Seismic Wave Propagation Through Uncertain Soils Probabilistic Analysis for Decision Making Summary

16 Seismic Wave Propagation Through Uncertain Soils Risk Assessment Applications Any problem (Mü + C u + Ku = F) with known PDFs of material parameters, PDFs of loading can be analyzed using PEP and SEPFEM to obtain PDFs of DOFs, stress, strain... PEP solution is second order accurate (exact mean and standard deviation) SEPFEM solution (PDFs) can be made as accurate as need be Tails of PDFs can than be used to develop accurate risk Application to a realistic case of seismic wave propagation

17 Seismic Wave Propagation Through Uncertain Soils "Uniform" CPT Site Data

18 Seismic Wave Propagation Through Uncertain Soils Seismic Wave Propagation through Stochastic Soil Soil as 12.5 m 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

19 Seismic Wave Propagation Through Uncertain Soils Seismic Wave Propagation through Stochastic Soil Displacement mm Time s Mean± Standard Deviation

20 Seismic Wave Propagation Through Uncertain Soils Full PDFs All Results 0.06 PDF Displacement (mm) Time (sec)

21 Probabilistic Analysis for Decision Making Outline Motivation Probabilistic Elasto Plasticity PEP Formulations Stochastic Elastic Plastic Finite Element Method Applications to Risk Analysis Seismic Wave Propagation Through Uncertain Soils Probabilistic Analysis for Decision Making Summary

22 Probabilistic Analysis for Decision Making Example: Three Approaches to Modeling Do nothing about site (material) characterization (rely on experience): conservative guess for soil data, COV = 225%, correlation length = 12m. Do better than standard site (material) characterization: COV = 103%, correlation length = 0.61m) Do the best site (material) characterization to reduce probabilities of exceedance

23 Probabilistic Analysis for Decision Making Example: PDF at 6 s PDF Real Soil Data Conservative Guess Displacement (mm)

24 Probabilistic Analysis for Decision Making Example: CDF (Non-Exceedance) at 6 s CDF Real Soil Data Conservative Guess Displacement (mm)

25 Motivation Probabilistic Elasto Plasticity Applications to Risk Analysis Probabilistic Analysis for Decision Making Probability of Exceedance of 20 cm (%) Probability of Exceedance of 20cm Jeremic and Sett Conservative Guess 80 Real Site 60 Excellently Characterized Site Displacement (cm) Summary

26 Motivation Probabilistic Elasto Plasticity Applications to Risk Analysis Summary Probabilistic Analysis for Decision Making Probability of Exceedance of 50 cm (%) Probability of Exceedance of 50cm Conservative Guess 40 Real Site Jeremic and Sett Excellently Characterized Site Displacement (cm) 25

27 Probabilistic Analysis for Decision Making Risk of Unacceptable Deformation Probability of Exceedance (%) Conservative Guess Real Site Excellently Characterized Site Displacement (cm)

28 Summary Behavior of all civil engineering objects (structures, soils...) is probably probabilistic! Presented methodology (PEP and SEPFEM) allows for (very) accurate numerical simulation of PDFs of DOFs (and stress, strain) from known (given) PDFs of material properties and PDFs of loads. Human nature: how much do you want to know about potential problem?