Finite Element Systems for Vector Bundles: Elasticity and Curvature
We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. W...
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| Veröffentlicht in: | Foundations of computational mathematics 2023-04, Vol.23 (2), p.545-596 |
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| creator | Christiansen, Snorre H. Hu, Kaibo |
| description | We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes. |
| doi_str_mv | 10.1007/s10208-022-09555-x |
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| subjects | Applications of Mathematics Computer Science Curvature Economics Elasticity Finite element method Homology Linear and Multilinear Algebras Math Applications in Computer Science Mathematical analysis Mathematics Mathematics and Statistics Matrix Theory Methods Numerical Analysis Tensors |
| title | Finite Element Systems for Vector Bundles: Elasticity and Curvature |
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