Keywords: design sensitivity analysis, semi-analytical method, coupled problem, displacement load method, thermal-structure problem, electromagnetic-structure problem.

Design sensitivity analysis is widely desired in various engineering fields such as the optimal design process, reliability analysis and the determination of relative importance of design variables. A finite element package is placed as a daily design tool for engineering designers in order to analyze and improve the design. However, finite element analysis cannot provide enough information to improve the design such as design sensitivity coefficients. Therefore, design sensitivity analysis with the finite element package is necessary for practical application in an industrial setting.

Design sensitivity analysis using finite element program can be implemented inside or outside of the finite element program. Design sensitivity analysis inside the finite element program requires a profound knowledge of source code and information of element shape function. Design sensitivity analysis implemented outside of finite element program, however, must be performed without access to source code of finite element program and has been used for some structural problems [1,2,3]. Recently, many design problems are often included multi-physics that interacts among several disciplines and its design sensitivity analysis focused on continuum approach or using the internal matrix information of finite element program [4,5,6].

In this paper, the thermal-structure coupled problem and the static electromagnetic-structure coupled problem are considered for design sensitivity analysis. The semi-analytical method is used for the development of the design sensitivity module outside of the commercial finite element package of ANSYS. An effective computational technique named displacement load method is proposed for semi-analytical design sensitivity analysis of coupled systems. The calculation of pseudo-load of coupled system that requires partial design derivatives of coupled load is the most important in implementation of semi-analytical design sensitivity analysis. With the displacement load method, the displacement of the original design becomes the input load for a perturbed design. It is proposed to numerically evaluate partial derivatives of the coupled load. Using the displacement load method, we can calculate the pseudo-load outside a commercial finite element program robustly. Numerical examples are used to show the accuracy of the developed method and to provide insights for implementation of the suggested method into other commercial finite element packages.

Design sensitivity equation for coupled problem is derived analytically from the discrete finite element equations. The displacement load method is proposed to implement the developed design sensitivity equation for the coupled system into a commercial finite element program without access to the internal information of the finite element package. The design sensitivity analysis module outside of the commercial finite element package, ANSYS, is developed with the basis of displacement load method. Numerical examples such as a thermal-structure coupled problem and a electromagnetic-structure coupled problem are illustrated to verify the accuracy of the method developed. Based on the numerical study, the developed method provides reasonably accurate design sensitivities compared with those obtained by finite difference method.

1

W.H. Zhang and M. Domaszewski, "Recent Developments of Sizing Optimization with ABAQUS Code", AIAA-98-4710, pp. 190-197, 1998.

2

T.H. Lee, J.H. Yoo and M.U. Lee, "Refined Semi-analytical Design Sensitivity Analysis with Commercial Finite Element Package", AIAA-2004-4534, 2004.

3

T. Chen, "Sensitivity Analysis and Shape Optimization of Axisymmetric Structures", International Journal for Numerical Methods in Engineering, 28, 95-108, 1989. doi:10.1002/nme.1620280108

4

H.L. Thomas and Y.K. Shyy, "Optimization of Structure with Static and Thermal Loads", Proceedings of the AIAA/ASME/ASCE/AHS/ASC 34th Structure, Structural Dynamics and Materials Conference, 1438-1442, 1993.

5

H. Shin, H. Moon and S. Wang, "3D Topology Optimization of Magneto-Thermal Systems", ISEF2005-MTT-SA.28, 2005.

6

T.H. Lee, J.H. Yoo and H.S. Kim, "Displacement-Load Method for Semi-Analytical Design Sensitivity Analysis", Transaction of KSME A., 28 (10), 1590-1597, 2004.

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