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An engineering interpretation of Nesterov’s convex minimization algorithm and time integration: application to optimal fiber orientation. (English) Zbl 1480.74250

Summary: Yu. E. Nesterov’s 1983 first-order minimization algorithm [Sov. Math., Dokl. 27, 372–376 (1983; Zbl 0535.90071); translation from Dokl. Akad. Nauk SSSR 269, 543–547 (1983)] is equivalent to the numerical solution of a second-order ODE with non-constant damping. It is known that the algorithm can be obtained by time discretization with asynchronous damping, a long-standing technique in computational explicit dynamics. We extend the solution of the ODE to other time discretization algorithms, analyze their properties and provide an engineering interpretation of the process as well as a prototype implementation, addressing the estimation of the relevant parameters in the computational mechanics context. The main result is that a standard Newmark-type time-integration finite element code can be adopted to perform classical optimization in mechanics. Standard FE analysis, sensitivity analysis and optimization are performed in sequence using a typical FE framework. Geometric nonlinearities in the conservative case are addressed by the adjoint-variable method and examples of optimal fiber orientation are shown, exhibiting remarkable advantages with respect to more traditional optimization algorithms.

MSC:

74P10 Optimization of other properties in solid mechanics
74E30 Composite and mixture properties
74S05 Finite element methods applied to problems in solid mechanics

Citations:

Zbl 0535.90071
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Full Text: DOI

References:

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