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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.15410 |
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| _version_ | 1866915165380804608 |
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| author | Millar, Cameron Schulze, Bernd Theran, Louis |
| author_facet | Millar, Cameron Schulze, Bernd Theran, Louis |
| contents | For a bar-joint framework $(G,p)$, a subgroup $Γ$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $Γ$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equilibrium stresses in frameworks via symmetric averaging Millar, Cameron Schulze, Bernd Theran, Louis Metric Geometry 52C25, 74K25, 20C35 For a bar-joint framework $(G,p)$, a subgroup $Γ$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $Γ$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs. |
| title | Equilibrium stresses in frameworks via symmetric averaging |
| topic | Metric Geometry 52C25, 74K25, 20C35 |
| url | https://arxiv.org/abs/2502.15410 |