Biomimetic approach to compliance optimization and multiple load cases.

*(English)*Zbl 1432.74184Summary: The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a known heuristic algorithm. The speed method of shape optimization is applied to obtain necessary optimality conditions for representative test examples. The algorithm originates from the biomimetic approach to compliance optimization. The trabecular bone adapts its form to mechanical loads and is able to form structures that are lightweight and very stiff at the same time. In this sense, it is a problem pertaining to both the nature or living entities which is similar to structural optimization, especially topology optimization. The paper presents the biomimetic approach, based on the trabecular bone remodeling phenomenon, with the aim of minimizing the compliance in multiple load cases. The method employed aims at minimizing the energy and combines structural evolution inspired by trabecular bone remodeling and the shape gradient framework, with strict analysis based on functionals in the 3-dimensional elasticity model. The method is enhanced to handle the problem of structural optimization under multiple loads. The new biomimetic approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Structural evolution is based on shape gradient approximation by the speed method, and it is separated from the finite element method of the model solution. Numerical examples confirm that the heuristic algorithm for structural optimization is efficient.

##### MSC:

74P10 | Optimization of other properties in solid mechanics |

35C20 | Asymptotic expansions of solutions to PDEs |

35J15 | Second-order elliptic equations |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

49J40 | Variational inequalities |

49Q12 | Sensitivity analysis for optimization problems on manifolds |

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\textit{M. Nowak} et al., J. Optim. Theory Appl. 184, No. 1, 210--225 (2020; Zbl 1432.74184)

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