Sensitivity analysis of the dynamic response of a frame. Part I: Direct differentiation method

Olga Dąbrowska,

Henryk Ciurej

Abstrakt

This paper presents a sensitivity analysis related to the solution of a stationary, linear system of second order equations of motion obtained by the Finite Element Method. The Direct Differentiation Method was presented in this paper. The essence of this method is the explicit differentiation of the system of equations with respect to parameters. As a result, derivatives of vectors and matrices are obtained. The necessary material derivatives of vectors and matrices associated with the simplest 3D beam element are presented. Sensitivity analysis consists in searching for changes in physical quantities in relation to selected parameters. Ultimately, the sensitivity analysis comes down to calculating derivatives of specific functions with respect to parameters. Real and continuous design variables are considered.

Keywords: sensitivity analysis, direct differentiation method, explicit differentiation

Streszczenie
Analiza wrażliwości polega na poszukiwaniu zmian wielkości fizycznych względem wybranych parametrów. Sprowadza się ona do obliczania pochodnych określonych funkcji. W pracy przedstawiono analizę wrażliwości rozwiązania stacjonarnego, liniowego układu równań ruchu drugiego rzędu otrzymanego metodą elementów skończonych. Przedstawiono metodę bezpośrednią analizy wrażliwości. Polega ona na bezpośrednim zróżniczkowaniu równań względem parametrów. W rezultacie uzyskano pochodne wektorów i macierzy. Przedstawiono niezbędne pochodne materialne wektorów i macierzy związanych z najprostszym elementem belki 3D. Rozpatrywano rzeczywiste i ciągłe  zmienne projektowe.

Słowa kluczowe: analiza wrażliwości, bezpośrednia analiza wrażliwości, pochodne równania ruchu
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