Investigations on the restrictions of stochastic collocation methods for high dimensional and nonlinear engineering applications

Dannert, Mona M.; Bensel, Fynn; Fau, Amélie; Fleury, Rodolfo M.N.; Nackenhorst, Udo

doi:10.1016/j.probengmech.2022.103299

Cited by 11 publications

(7 citation statements)

References 36 publications

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“…[34][35][36] However, these classical expansions are not applicable to the stochastic elastoplastic problem. In this article, we develop an efficient and accurate numerical algorithm to solve stochastic elastoplastic problems by using the weakly intrusive approximation (25) and a dedicated iteration.…”

Section: Approximation Of Stochastic Solutionsmentioning

confidence: 99%

“…Combining (24) and (25) we rewrite (25) as an incremental format similar to (24). To this end, the previous approximation u k−1 (t i , 𝜃) and the stochastic increment Δu k (t i , 𝜃) in ( 24) are given by…”

Section: Approximation Of Stochastic Solutionsmentioning

confidence: 99%

“…On the other hand, the recalculated Equation ( 40) only requires few effort for assembling the reduced-order matrix and vector in ( 41) and ( 42) and its solution by Algorithm 2 is fully non-intrusive. In these senses, we implement the intrusive approximation (25) of the stochastic solution only in weakly intrusive ways and most existing codes can be reused as-is or slightly modified.…”

Section: Algorithm Schemementioning

confidence: 99%

“…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][28][29] As discussed above, developing efficient and accurate methods for solving stochastic elastoplastic problems remains an attractive topic.…”

Section: Introductionmentioning

confidence: 99%

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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: Approximation Of Stochastic Solutionsmentioning

confidence: 99%

Section: Approximation Of Stochastic Solutionsmentioning

confidence: 99%

Section: Algorithm Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Zheng

Nackenhorst

2023

Numerical Meth Engineering

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“…Consequently, approximating the non-smooth response of the underlying physical problem by a smooth global surrogate model might lead to large errors and poor prediction capability. This issue has been addressed in the literature in [24][25][26]. To overcome this problem, several attempts have been proposed in the literature based on data mining techniques such as clustering [27] which automatically determines the input domain of similar responses.…”

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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Efficient and accurate uncertainty quantification in engineering simulations using time-separated stochastic mechanics

Geisler,

Junker

2024

Arch Appl Mech

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A robust method for uncertainty quantification is undeniably leading to a greater certainty in simulation results and more sustainable designs. The inherent uncertainties of the world around us render everything stochastic, from material parameters, over geometries, up to forces. Consequently, the results of engineering simulations should reflect this randomness. Many methods have been developed for uncertainty quantification for linear elastic material behavior. However, real-life structure often exhibit inelastic material behavior such as visco-plasticity. Inelastic material behavior is described by additional internal variables with accompanying differential equations. This increases the complexity for the computation of stochastic quantities, e.g., expectation and standard deviation, drastically. The time-separated stochastic mechanics is a novel method for the uncertainty quantification of inelastic materials. It is based on a separation of all fields into a sum of products of time-dependent but deterministic and stochastic but time-independent terms. Only a low number of deterministic finite element simulations are then required to track the effect of (in)homogeneous material fluctuations on stress and internal variables. Despite the low computational effort the results are often indistinguishable from reference Monte Carlo simulations for a variety of boundary conditions and loading scenarios.

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Investigations on the restrictions of stochastic collocation methods for high dimensional and nonlinear engineering applications

Cited by 11 publications

References 36 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

Efficient and accurate uncertainty quantification in engineering simulations using time-separated stochastic mechanics

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