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An engineering interpretation of Nesterov’s convex minimization algorithm and time integration: application to optimal fiber orientation

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Abstract

Nesterov’s 1983 first-order minimization algorithm 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.

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Acknowledgements

The first author acknowledges the support of FCT, through IDMEC, under LAETA, Project UIDB/50022/2020. The first author would like to thank Professor Leonel Fernandes at IST for the outstanding insight concerning time integration algorithms.

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Authors and Affiliations

  1. DEM - Mechanical Engineering Department, Instituto Superior Técnico, University of Lisbon, Avenida Rovisco Pais, 1049-001, Lisbon, Portugal

    P. Areias

  2. IDMEC, Lisbon, Portugal

    P. Areias

  3. Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, 99423, Weimar, Germany

    T. Rabczuk

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  1. P. Areias
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  2. T. Rabczuk
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Correspondence to P. Areias.

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Appendix

Appendix

See Table 4.

Table 4 First order amplification matrices (using Hughes terminology [24]) for four ODE explicit integrators
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Areias, P., Rabczuk, T. An engineering interpretation of Nesterov’s convex minimization algorithm and time integration: application to optimal fiber orientation. Comput Mech 68, 211–227 (2021). https://doi.org/10.1007/s00466-021-02027-z

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