Optimal design versus maximal Monge-Kantorovich metrics
| Autor: | Karol Bołbotowski, Guy Bouchitté |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN) |
| Rok vydání: | 2021 |
| Předmět: |
90B06
02 engineering and technology 01 natural sciences peusdo-metric [SPI]Engineering Sciences [physics] Mathematics (miscellaneous) monotone maps FOS: Mathematics Monge-Kantorovich distance [MATH]Mathematics [math] 0101 mathematics Mathematics - Optimization and Control 49K20 Minimal compliance 49J20 74P05 pre-stressed membrane 28A50 Mechanical Engineering 010102 general mathematics 021001 nanoscience & nanotechnology duality and saddle point 2010 Mathematics Subject Classification: 49J45 Optimization and Control (math.OC) 2010 MSC-class: 49J45 49K20 49J20 90B06 28A50 74P05 0210 nano-technology Analysis geodesics |
| DOI: | 10.48550/arxiv.2104.04894 |
| Popis: | A remarkable connection between optimal design and Monge transport was initiated in the years 1997 in the context of the minimal elastic compliance problem and where the euclidean metric cost was naturally involved. In this paper we present different variants in optimal design of mechanical structures, in particular focusing on the optimal pre-stressed elastic membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to existence and (in a smooth case) to PDE optimality conditions, we present a general geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. These two aspects are enlightened by several explicit examples and also by numerical solutions in which optimal structures very often turn out to be truss-like i.e supported by piecewise affine geodesics. In case of a discrete load, we are able to relate the existence of such truss-like solutions to an extension property of maximal monotone maps which is of independent interest and that we propose here as a conjecture. Comment: 54 pages, 26 figures |
| Databáze: | OpenAIRE |
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