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Semi-analytical sensitivity analysis for nonlinear transient problems

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Abstract

Efficient analytical sensitivity computations are essential elements of gradient-based optimization schemes; unfortunately, they can be difficult to implement. This implementation issue is often resolved by adopting the semi-analytical method which exhibits the efficiency of the analytical methods and the ease of implementation of the finite difference method. However, care must be taken as semi-analytical sensitivities may exhibit errors due to truncation and round-off. Additional errors are introduced if the convergence tolerance of the primal analysis is not sufficiently small. This paper gives a general overview and some new developments of the analytical and semi-analytical sensitivity analyses for nonlinear steady-state, transient, and dynamic problems. We discuss the restrictive assumptions, accuracy, and consistency of these methods. Both adjoint and direct differentiation methods are studied. Numerical examples are provided.

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Notes

  1. For α = 0, 1/2, or 1, we recover the forward Euler, Crank-Nicolson, and backward Euler strategies respectively.

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Authors and Affiliations

  1. Department of Mechanical Sciences and Engineering, University of Illinois at Urbana-Champaign, Champaign, IL, USA

    Felipe Fernandez & Daniel A. Tortorelli

  2. Center for Design and Optimization, Lawrence Livermore National Laboratory, Livermore, CA, USA

    Daniel A. Tortorelli

Authors
  1. Felipe Fernandez
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  2. Daniel A. Tortorelli
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Correspondence to Felipe Fernandez.

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Responsible Editor: Hai Huang

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This work was partially performed under the auspices of the US Department of Energy by Lawrence Livermore Laboratory under contract DE-AC52-07NA27344, cf. ref number LLNLCONF-717640.

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Fernandez, F., Tortorelli, D.A. Semi-analytical sensitivity analysis for nonlinear transient problems. Struct Multidisc Optim 58, 2387–2410 (2018). https://doi.org/10.1007/s00158-018-2096-y

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