Mac-based mode-tracking in structural topology optimization Tae Soo Kim, Yoon Young Kim* School of Mechanical and Aerospace Engineering, Seoul National University Shinlim-Dong, San 56-1, Kwanak-Gu Seoul 151-742, South Korea Received 25 September 1998; accepted 07 January 1999 Abstract An ecient mode-tracking method based on the modal assurance criterion (MAC) is formulated for the structural topology optimization of maximizing the eigenfrequencies of desired modes. The present method is very eective in tracing each of the desired modes correctly even when the structural topology and shape, change substantially from the initial con®guration. The use of MAC eliminates the need for mass orthogonalization and the use of selected nodal data enhances numerical eciency greatly. The penalized density function approach for topology optimization allows an interesting interpretation in the sensitivity analysis. The use of an explicit penalty function in the objective function is proposed, which helps suppress undesirable intermediate densities. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Topology; Optimization; MAC; Structure; Vibration; Eigenvalue; Eigenvector 1. Introduction Structural topology optimization has been a subject of numerous investigations since Bendsùe and Kikuchi [1±4] introduced a homogenization method for the op- timal material layout problems. Successful applications of this technique to various problems have been reported in recent years. In the ®nite element-based topology optimization, the element densities are used as design parameters. An optimal density distribution is used to determine the ®nal shape and topology of an optimal structure, and it is desirable to have regions of material or no material. The rigorous homogenization technique may be used to model an element with a hole where the density of a holey element is used as the design variable. For more ecient sensitivity analysis, the element density may be penalized between 0 (no material) and 1 (material) by an arti®cial function. The advantage of using this penalized density function over the rigorous homogenization-based technique is that, only the ex- pressions for the element strain and kinetic energies are needed for sensitivity analysis. Furthermore, this technique allows the direct application of any commer- cial ®nite element package in topology optimization problems. The comparison of the density function and the homogenization-based technique is provided by Yang [10,11]. Bendsùe [12] examines various design parametrization techniques which can be used eec- tively for topology optimization problems. Although many practical applications of the top- ology optimization technique have been made in static problems, successful applications to the vibration pro- Computers and Structures 74 (2000) 375±383 0045-7949/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0045-7949(99)00056-5 www.elsevier.com/locate/compstruc * Corresponding author. Tel.: +82-2-880-7154; fax: +82-2- 883-1513. E-mail address: yykim@snu.ac.kr (Y.Y. Kim)