## Details

Original language | English |
---|---|

Pages (from-to) | 667-684 |

Number of pages | 18 |

Journal | Computational mechanics |

Volume | 73 |

Issue number | 3 |

Early online date | 29 Aug 2023 |

Publication status | Published - Mar 2024 |

## Abstract

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

## Keywords

- Curse of dimensionality, Polynomial Chaos expansion, Stochastic virtual element method, Uncertainty quantification, Weakly intrusive approximation

## ASJC Scopus subject areas

- Engineering(all)
**Computational Mechanics**- Engineering(all)
**Ocean Engineering**- Engineering(all)
**Mechanical Engineering**- Computer Science(all)
**Computational Theory and Mathematics**- Mathematics(all)
**Computational Mathematics**- Mathematics(all)
**Applied Mathematics**

## Cite this

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- Harvard
- Apa
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- RIS

**Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity.**/ Zheng, Zhibao; Nackenhorst, Udo.

In: Computational mechanics, Vol. 73, No. 3, 03.2024, p. 667-684.

Research output: Contribution to journal › Article › Research › peer review

*Computational mechanics*, vol. 73, no. 3, pp. 667-684. https://doi.org/10.1007/s00466-023-02384-x

}

TY - JOUR

T1 - Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity

AU - Zheng, Zhibao

AU - Nackenhorst, Udo

N1 - Funding information: The authors are grateful to the Alexander von Humboldt Foundation and the International Research Training Group 2657 (IRTG 2657) funded by the German Research Foundation (DFG) (Grant number 433082294).

PY - 2024/3

Y1 - 2024/3

N2 - This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

AB - This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

KW - Curse of dimensionality

KW - Polynomial Chaos expansion

KW - Stochastic virtual element method

KW - Uncertainty quantification

KW - Weakly intrusive approximation

UR - http://www.scopus.com/inward/record.url?scp=85168877414&partnerID=8YFLogxK

U2 - 10.1007/s00466-023-02384-x

DO - 10.1007/s00466-023-02384-x

M3 - Article

AN - SCOPUS:85168877414

VL - 73

SP - 667

EP - 684

JO - Computational mechanics

JF - Computational mechanics

SN - 0178-7675

IS - 3

ER -