A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917518346551296 |
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| author | Boom, Pieter D. Kosmas, Odysseas Margetts, Lee Jivkov, Andrey |
| author_facet | Boom, Pieter D. Kosmas, Odysseas Margetts, Lee Jivkov, Andrey |
| contents | A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector-valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_06000 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus Boom, Pieter D. Kosmas, Odysseas Margetts, Lee Jivkov, Andrey Mathematical Physics A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector-valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes. |
| title | A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2104.06000 |