Anisotropic multi-element polynomial chaos expansion for high-dimensional non-linear structural problems

Basmati, A. A.; Fau, Amélie; Urrea-Quintero, J. H.; Dannert, Mona M.; Voelsen, Esther; Nackenhorst, Udo

doi:10.1016/j.probengmech.2022.103366

Cited by 12 publications

(4 citation statements)

References 40 publications

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“…PC expansions and stochastic collocation methods are used to discretize the original stochastic problem and nonlinear convex programming is adopted for the numerical implementation. Also, since the multi‐element PC expansion 21 can be used to well capture sharp and discontinuous stochastic solutions, it has been applied to stochastic elastoplastic problems in Reference 22. There are also other effective methods for solving nonsmooth stochastic solutions of nonlinear stochastic problems, for example, the stochastic perturbation method, 23,24 the stochastic collocation methods, 25,26 the polynomial/spline dimensional decomposition methods 27‐29 …”

Section: Introductionmentioning

confidence: 99%

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

Zheng

Nackenhorst

2023

Numerical Meth Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

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

Zheng

Nackenhorst

2023

Numerical Meth Engineering

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“…In addition, the decomposition of the input space was applied in [28] where the different behaviours of the response surface are locally approximated. The multi-element approaches were applied in [29][30][31] where the parametric space is discretised into non-overlapping elements. Thereafter, the surrogate model is constructed element-wise which weakens the non-smoothness influence of the response within each element.…”

Section: Introductionmentioning

confidence: 99%

Globally supported surrogate model based on support vector regression for nonlinear structural engineering applications

2022

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This work presents a global surrogate modelling of mechanical systems with elasto-plastic material behaviour based on support vector regression (SVR). In general, the main challenge in surrogate modelling is to construct an approximation model with the ability to capture the non-smooth behaviour of the system under interest. This paper investigates the ability of the SVR to deal with discontinuous and high non-smooth outputs. Two different kernel functions, namely the Gaussian and Matèrn 5/2 kernel functions, are examined and compared through one-dimensional, purely phenomenological elasto-plastic case. Thereafter, an essential part of this paper is addressed towards the application of the SVR for the two-dimensional elasto-plastic case preceded by a finite element method. In this study, the SVR computational cost is reduced by using anisotropic training grid where the number of points are only increased in the direction of the most important input parameters. Finally, the SVR accuracy is improved by smoothing the response surface based on the linear regression. The SVR is constructed using an in-house MATLAB code, while Abaqus is used as a finite element solver.

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“…On the other hand, stochastic FEM problems are more difficult to substitute by an efficient surrogate approach (e.g. [42,[73][74][75][76][77][78]). Besides the aforementioned application of SVR, in [42] Gaussian process regression (GPR) is applied to approximate the FEM-based displacement, stresses and strains under plastic deformation.…”

Section: State Of the Artmentioning

confidence: 99%

“…In [42] Gaussian process regression is found to fails to accurately approximate plastic deformation, indicating limited applicability in highly nonlinear problems that are not smooth. Furthermore, in [74][75][76][77] a polynomial chaos expansion (PCE) [79][80][81][82][83] is employed to model nonlinear behavior obtained from a structural finite element response. In polynomial chaos expansion the uncertain input parameters (i.e.…”

Section: State Of the Artmentioning

confidence: 99%

Towards an interactive crash simulation

van de Weg

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Anisotropic multi-element polynomial chaos expansion for high-dimensional non-linear structural problems

Cited by 12 publications

References 40 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

Globally supported surrogate model based on support vector regression for nonlinear structural engineering applications

Towards an interactive crash simulation

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