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Primal-dual Newton Methods in Structural Optimization

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Computing and Visualization in Science

Abstract

We consider the numerical solution of optimization problems for systems of partial differential equations with constraints on the state and design variables as they arise in the optimal design of the shape and the topology of continuum mechanical structures. After discretization the resulting nonlinear programming problems are solved by an “all-at-once” approach featuring the numerical solution of the state equations as an integral part of the optimization routine. In particular, we focus on primal-dual Newton methods combined with interior-point techniques for an appropriate handling of the inequality constraints. Special emphasis is given on the efficient solution of the primal-dual system that results from the application of Newton’s method to the Karush–Kuhn–Tucker conditions where we take advantage of the special block structure of the primal-dual Hessian. Applications include structural optimization of microcellular biomorphic ceramics by homogenization modeling, the shape optimization of electrorheological devices, and the topology optimization of high power electromotors.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Ronald H. W. Hoppe

  2. Institute of Mathematics, University of Augsburg, 86159, Augsburg, Germany

    Christopher Linsenmann

  3. CLLP, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    Svetozara I. Petrova

Authors
  1. Ronald H. W. Hoppe
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  2. Christopher Linsenmann
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  3. Svetozara I. Petrova
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Correspondence to Ronald H. W. Hoppe.

Additional information

Communicated by G. Wittum

Dedicated to Peter Deuflhard on the occasion of his 60th birthday

The first and the third author have been supported by the DFG within the Collaborative Research Center SFB 438 and within the Priority Program SPP 1095 under the Grants Ho 877/5-1 and Ho 877/5-2. The first author acknowledges further support by the BMBF under Grant 03HOM3A1

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Hoppe, R.H.W., Linsenmann, C. & Petrova, S.I. Primal-dual Newton Methods in Structural Optimization. Comput. Visual Sci. 9, 71–87 (2006). https://doi.org/10.1007/s00791-006-0018-9

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