ELSEVIER Finite Elements in Analysis and Design 26 (1997) 1-19 FINITE ELEMENTS IN ANALYSIS AND DESIGN Optimization of finite element grids using shape sensitivity analysis in terms of nodal positions Koo Tae Kang, Byung Man Kwak* Korea Advanced Insti~te of Science and Technology, 373-1, Kusong-Dong, Yusong-Ku, Taejon 305-701, South Korea Abstract An approach of finite element grid optimization is proposed as an application of the shape design sensitivity analysis. Change of the mesh is described by design velocity fields that can be simply obtained by a piecewise linear interpolation from the nodal positions. For a given topology of finite elements mesh, the strain energy is maximized for static problems and the eigenvalues are minimized for eigenvalue problems with respect to the nodal positions. Numerical examples for the Timoshenko beams and the Mindlin plates are obtained and the proposed approach is shown to be a feasible method that can be used for shape or configuration designs where large distortion of meshes is often involved. Keywords: Finite element grids; Optimization; Shape design sensitivity; Design velocity field; Eigenvalue; Timoshenko beam; Reissner-Mindlin plate I. Introduction In the 1970s and early 1980s, there were many studies on optimal grids and suggestions in selecting the best finite element grids based on these studies [1, 2]. While the numerical or the analytical results for simple examples were impressive, applications were very limited due to complexity of optimization process and insufficiency of computer facilities, The studies of improv- ing the accuracy of finite element analysis are mainly focused on the h-adaptive mesh I-3,4-1 and hp-adaptive mesh [:5] based on the local error estimation. However, these approaches are not only unsuitable for global domain problems with eigenvalues, for example, but also limited in applica- tions when refinement is not possible in practice. In recent years, shape optimal design problems have attracted much attention and become a new area which requires remeshing after an optimization step. Although the theory and analytical formulas for shape design sensitivity analysis have been well developed, they sometimes do not * Corresponding author. 0168-874X/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved PII S0 1 68-874X(96)00068-6