Finite element approximation for optimal shape design : theory and applications

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書誌事項

Finite element approximation for optimal shape design : theory and applications

J. Haslinger and P. Neittaanmäki

Wiley, c1988

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注記

Bibliography: p. 311-331

内容説明・目次

内容説明

The goal in optimal shape design (structural organization, or redesign) is to computerize the design process and therefore shorten the time it takes to design or improve the existing design. In an optimal shape design process one wishes to optimize a criteria involving a solution of a partial differential equation with respect to its domain of definition. The detailed study of this subject is at the interface of four fields - optimal control, partial differential equations, numerical analysis and optimization. This work is devoted to the mathematical basis of optimal shape design as well as to finite element approximation and to numerical realization by applying optimization techniques. The main aim is to see the optimal design problems via the optimal control theory when the state systems are governed by variational equalities. The most characteristic property of variational inequalities is the fact that their solution does not depend smoothly, in general, on the control. Hence the shape sensitivity analysis is a crucial question, especially in the design of the solution procedures, as the objective functional may not be smooth.

目次

  • Preliminaries - Green's formula
  • abstract setting of optimal shape design problem and its approximation
  • shape optimization of systems governed by unilateral boundary value state problem - scalar case
  • approximation of the optimal shape design problems by finite elements - scalar case
  • numerical realization
  • shape optimization in unilateral boundary value problems with "flux" cost functional
  • optimal shape design in contact problems - elastic case
  • shape optimization of elastic/perfectly plastic bodies in contact
  • on the design of the optimal covering supported by an obstacle
  • state constrained optimal control problems and their approximations
  • FE-grid optimization
  • concluding remarks on references on optimal shape design and related topics. Appendices: algorithms for FEM
  • on the differentation of stiffness and mass matrices and force vectors
  • subgradient method for convex linearly constrained optimization
  • description of the sequential quadratic programming (SQP) algorithm
  • on the differentiability of a projection on a convex set in Hilbert space.

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