Fokker–Planck linearization for non-Gaussian stochastic elastoplastic finite elements

Karapiperis, Konstantinos; Sett, Kallol; Kavvas, M. L.; Jeremić, Boris

doi:10.1016/j.cma.2016.05.001

Cited by 14 publications

(6 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and the matrix assembly of K T (u k−1 (t i , 𝜃) , 𝜃) (or 𝜕𝒩 (u) ∕𝜕u) is implemented by using the tangent tensor  T in (19). In this way, the stochastic solution u k (t i , 𝜃) of the kth iteration at the time step t i in ( 21) is approximated as…”

Section: Stochastic Newton Linearizationmentioning

confidence: 99%

“…References 16 and 17 introduce two fictitious bounding bodies to approximate stochastic elastoplastic constitutive equations and then PC‐based expansions are adopted to solve the corresponding stochastic finite element equations. Stochastic elastoplastic problems are solved by coupling the spectral SFEM to the Fokker Planck Kolmogorov (FPK) equation approach in References 18 and 19. The randomness of the elastoplastic constitutive model is propagated through FPK equations and the unknown stochastic solution is solved via the spectral SFEM.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Zheng

Nackenhorst

2023

Numerical Meth Engineering

View full text Add to dashboard Cite

This article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history‐dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time‐independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non‐intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.

show abstract

Section: Stochastic Newton Linearizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Zheng

Nackenhorst

2023

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…FPK approach to probabilistic elasto‐plasticity has been used to simulate both monotonic and cyclic hardening and softening type uncertain material behaviors . It has also been successfully integrated with the spectral approach of the stochastic finite element (FE) method to probabilistically solve stochastic elastic‐plastic boundary value problems …”

Section: Introductionmentioning

confidence: 99%

“…8,9 It has also been successfully integrated with the spectral approach of the stochastic finite element (FE) method to probabilistically solve stochastic elastic-plastic boundary value problems. 10,11 Traditionally, in mechanics, an FPK equation is solved numerically using the finite difference or FE technique. [12][13][14] Finite difference and FE methods are based on local representations of functions such as low-order polynomials.…”

Section: Introductionmentioning

confidence: 99%

Efficient solution algorithms for multiaxial probabilistic elasto‐plastic constitutive simulations of soils

Sadrinezhad

Sett

Hariharan

2017

Num Anal Meth Geomechanics

Self Cite

View full text Add to dashboard Cite

Summary A Fokker‐Planck‐Kolmogorov (FPK) equation approach has recently been developed to probabilistically solve any elastic‐plastic constitutive equation with uncertain material parameters by transforming the nonlinear, stochastic constitutive rate equation into a linear, deterministic partial differential equation (PDE) and thereby simplifying the numerical solution process. For an uniaxial problem, conventional numerical techniques, such as the finite difference or finite element methods, may be used to solve the resulting univariate FPK PDE. However, for a multiaxial problem, an efficient algorithm is necessary for tractability of the numerical solution of the multivariate FPK PDE. In this paper, computationally efficient algorithms, based on a Fourier spectral approach, are presented for solving FPK PDEs in (stress) space and (pseudo) time, having space‐independent but time‐dependent coefficients and both space‐ and time‐dependent coefficients, that commonly arise in probabilistic elasto‐plasticity. The algorithms are illustrated by probabilistically simulating 2 common laboratory constitutive experiments in geotechnical engineering, namely, the unconfined compression test and the unconsolidated undrained triaxial compression test.

show abstract

“…In the field of solid mechanics, stochastic Galerkin approaches are so far employed mainly to solve static problems -both linear (elastic) and nonlinear (elastic-plastic as well as geometric) -with uncertain material parameters [21,23,2,26,55,3,31]. Solutions of dynamic problems are also attempted, but mostly in the frequency domain [24,25], thereby restricting the use of the algorithms only to linear problems.…”

Section: Introductionmentioning

confidence: 99%

Time-domain stochastic finite element simulation of uncertain seismic wave propagation through uncertain heterogeneous solids

Wang

Sett

2016

Soil Dynamics and Earthquake Engineering

Self Cite

View full text Add to dashboard Cite

This paper presents an efficient numerical methodology in probabilistically solving the governing partial differential equation of solid mechanics with uncertainties in both the material parameter and forcing function in the time domain using the stochastic Galerkin approach. The methodology hypothesizes the input forcing function and the elastic modulus of the solid to be a nonstationary random process and a heterogeneous random field, respectively, and efficiently represents them in terms of multidimensional Hermite polynomial chaos -orthogonal and uncorrelated polynomials of zero-mean, unit variance Gaussian random variables -by taking advantage of the optimality of the Kosambi-Karhunen-Loève theorem. The methodology allows for any non-Gaussian marginal distributions and any arbitrary correlation structures for the input process and field. The solution random processes (displacement, velocity, and acceleration) are also represented in

show abstract

Fokker–Planck linearization for non-Gaussian stochastic elastoplastic finite elements

Cited by 14 publications

References 38 publications

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Efficient solution algorithms for multiaxial probabilistic elasto‐plastic constitutive simulations of soils

Time-domain stochastic finite element simulation of uncertain seismic wave propagation through uncertain heterogeneous solids

Contact Info

Product

Resources

About