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Accelerated proximal gradient method for elastoplastic analysis with von Mises yield criterion

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Abstract

It is known that, under certain conditions, the quasi-static incremental analysis problem of elastoplastic structures with the von Mises yield criterion can be formulated as a second-order cone programming (SOCP) problem, which can be solved with a primal-dual interior-point method. Alternatively, this paper proposes to solve an equivalent unconstrained nonsmooth convex optimization problem, which has a form similar to a class of regularized least-square problems, known as group LASSO. We propose an accelerated proximal gradient method with an adaptive restart scheme for solving this unconstrained optimization problem. The algorithm is easy to implement, and free from numerical solution of linear equations unlike conventional methods in computational mechanics. Numerical experiments suggest that the presented algorithm outperforms a standard solver that implements a primal-dual interior-point method for conic optimization.

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Notes

  1. Most of them use interior-point methods.

  2. This condition follows from (5) and \(\mathop {\mathrm {tr}}\nolimits ({{\mathrm{dev}}}(\varvec{\sigma }))=0\).

  3. The use of the same notation \(\varvec{u}\) to denote the displacement field in the continuum and the nodal displacement vector in the finite-element model should not cause any confusion. Hereinafter \(\varvec{u}\) always denotes the latter.

  4. Reduction of 19 into SOCP is not unique.

  5. This part of the numerical experiments was carried out on a \(2.6\,\mathrm {GHz}\) Intel Core i5 processor with \(8\,\mathrm {GB}\) RAM.

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Acknowledgements

The work of the second author is partially supported by JSPS KAKENHI 26420545 and 17K06633.

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Correspondence to Yoshihiro Kanno.

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Shimizu, W., Kanno, Y. Accelerated proximal gradient method for elastoplastic analysis with von Mises yield criterion. Japan J. Indust. Appl. Math. 35, 1–32 (2018). https://doi.org/10.1007/s13160-017-0280-x

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