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Higher-order sensitivity analysis of finite element method by automatic differentiation. (English) Zbl 0845.73071

The authors study a computerization of finite element techniques used in optimization of structures. Need for higher order sensitivity analysis in design of structures or mechanical systems was recognized by several authors (R. T. Haftka, E. J. Haug, Gene Hou, and others). Programs have been developed capable of computing higher order Fréchet derivatives and permitting reasonable accuracy in determining sensitivity of structure to changes in parameters.
The authors describe the basic principles of automatic differentiation for computing high order derivatives. Use of non-standard analysis approach and, particularly, the fact that non-Archimedean analysis must be used in treating very small quantities, permits replacement of finite “small” increments by appropriate derivatives applied to a collection of ordered monomials. Examples are offered of computing first order and higher order derivatives by purely algebraic techniques.
The methodology suggested by the authors gives much freedom in changing such parameters as Young’s modulus, as well as geometric parameters, and offers a wide choice of cost functionals, such as angular deflections at a particular node, as well as more commonly used criteria such as total weight, or stiffness of the entire structure. From engineers’ point of view such developments are very important. They will eventually provide efficient machine oriented design changes and optimal design of projects that are too complex to be analyzed by structural or mechanical engineers who are “expert designers”.
Reviewer: V.Komkov (Roswell)

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74P99 Optimization problems in solid mechanics

Software:

Cosy; DAFOR
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Full Text: DOI

References:

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