Details
| Original language | English |
|---|---|
| Pages (from-to) | 1187-1202 |
| Number of pages | 16 |
| Journal | Computational mechanics |
| Volume | 73 |
| Issue number | 5 |
| Early online date | 3 Nov 2023 |
| Publication status | Published - May 2024 |
Abstract
In the present article, a stability- and consistency-preserving integration scheme for polynomial Galerkin approaches of arbitrary order is presented. The basis is formed by Taylor series expansions of the stresses with respect to the strains, which in turn are expanded towards the spatial directions. With a split of the material and geometric nonlinearities and the assumption of a material behavior linearly variable within an element, the strain energy in elements of arbitrary shape and polynomial order can be evaluated exactly. Therefore, geometric moments have to be calculated in preprocessing, requiring only evaluations of derivatives at a single integration point during the analysis. The moments can be effectively integrated analytically over the boundary of the elements. As one of the manifold applications, the use in the context of second order virtual elements is elaborated for which the assembly time can be significantly reduced. The combination with the automatic differentiation and expression optimization software AceGen provides performant element routines. In the numerical examples, the integration scheme shows promising accuracy and makes the application in more complex material models up to computational homogenization attractive.
Keywords
- Integration split, Moment integration, Quadrature by differentiation, Virtual Element Method
ASJC Scopus subject areas
- Mathematics(all)
- Computational Mathematics
- Engineering(all)
- Mechanical Engineering
- Engineering(all)
- Ocean Engineering
- Mathematics(all)
- Applied Mathematics
- Engineering(all)
- Computational Mechanics
- Computer Science(all)
- Computational Theory and Mathematics
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In: Computational mechanics, Vol. 73, No. 5, 05.2024, p. 1187-1202.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - One-point quadrature of higher-order finite and virtual elements in nonlinear analysis
AU - Bode, Tobias
N1 - Funding Information: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy within the Cluster of Excellence
PY - 2024/5
Y1 - 2024/5
N2 - In the present article, a stability- and consistency-preserving integration scheme for polynomial Galerkin approaches of arbitrary order is presented. The basis is formed by Taylor series expansions of the stresses with respect to the strains, which in turn are expanded towards the spatial directions. With a split of the material and geometric nonlinearities and the assumption of a material behavior linearly variable within an element, the strain energy in elements of arbitrary shape and polynomial order can be evaluated exactly. Therefore, geometric moments have to be calculated in preprocessing, requiring only evaluations of derivatives at a single integration point during the analysis. The moments can be effectively integrated analytically over the boundary of the elements. As one of the manifold applications, the use in the context of second order virtual elements is elaborated for which the assembly time can be significantly reduced. The combination with the automatic differentiation and expression optimization software AceGen provides performant element routines. In the numerical examples, the integration scheme shows promising accuracy and makes the application in more complex material models up to computational homogenization attractive.
AB - In the present article, a stability- and consistency-preserving integration scheme for polynomial Galerkin approaches of arbitrary order is presented. The basis is formed by Taylor series expansions of the stresses with respect to the strains, which in turn are expanded towards the spatial directions. With a split of the material and geometric nonlinearities and the assumption of a material behavior linearly variable within an element, the strain energy in elements of arbitrary shape and polynomial order can be evaluated exactly. Therefore, geometric moments have to be calculated in preprocessing, requiring only evaluations of derivatives at a single integration point during the analysis. The moments can be effectively integrated analytically over the boundary of the elements. As one of the manifold applications, the use in the context of second order virtual elements is elaborated for which the assembly time can be significantly reduced. The combination with the automatic differentiation and expression optimization software AceGen provides performant element routines. In the numerical examples, the integration scheme shows promising accuracy and makes the application in more complex material models up to computational homogenization attractive.
KW - Integration split
KW - Moment integration
KW - Quadrature by differentiation
KW - Virtual Element Method
UR - http://www.scopus.com/inward/record.url?scp=85175658219&partnerID=8YFLogxK
U2 - 10.1007/s00466-023-02406-8
DO - 10.1007/s00466-023-02406-8
M3 - Article
VL - 73
SP - 1187
EP - 1202
JO - Computational mechanics
JF - Computational mechanics
SN - 0178-7675
IS - 5
ER -