Maximization of Minimal Eigenvalue of Structures by using Sequential Semidefinite Programming

3. Introduction In this paper, we investigate the design optimization of structures to maximize the lowest eigenvalue of free vibration, which has been studied extensively [20, 22]. In designing civil, mechanical and aerospace structures, the eigenvalues of free vibration, as well as the linear buckling load factor, have been used widely for decades as a performance measure of structures. It was shown that the design optimization of truss structures under the frequency constraints can be formulated as the semidefinite programming (SDP) problem [18]. We formulate the maximization problem of the lowest eigenvalue as a nonlinear programming problem, in which we attempt to minimize the linear function over the constraint such that a symmetric matrix Z(y) defined as a polynomial function of the design variables y should be positive semidefinite. This problem class is referred to as the polynomial semidefinite program (polynomial SDP), because it includes the SDP [6] as a particular case in which Z(y) is supposed to be an affine function of y. It is known that the SDP problem is convex, while the polynomial SDP problem is nonconvex in general. It is well known that optimum designs for maximization problem of the fundamental eigenvalue often have multiple (repeated) eigenvalues [16, 19, 22]. It has been shown that the multiple eigenvalues are not differentiable in the ordinary sense, and only directional derivatives with respect to the design variables may be calculated [3]. Therefore, it is very difficult to obtain the optimal design related to eigenvalue optimization by using a gradient-based nonlinear programming algorithm for a large structure, especially for the topology optimization in which we allow some elements of the structure to vanish. Several computational approaches have been developed for sensitivity analysis of multiple eigenvalues of finite dimensional structures [22]. Khot [12] presented an optimality criteria approach for optimum design of trusses with multiple frequency constraints. Rodorigues et al. [21] developed necessary conditions for optimality for problems under constraints on the linear buckling load factor based on Clarke’s generalized gradient. An optimal topology may be obtained based on the conventional ground structure method, in which the locations of structural elements are fixed and the optimal topology is obtained by removing the elements with vanishing design variables. In the authors’ previous paper [18], it has been shown that topology optimization of trusses under frequency constraints can be formulated as SDP, and an algorithm has been proposed based on the primal-dual interior-point method [14], which is applicable to cases with any multiplicity of the lowest eigenvalues. As a natural extension, we consider not only truss structures but also general structures discretized into finite elements in this paper. We show that the maximization problem of the minimal eigenvalue of a structure is formulated as a polynomial SDP problem. Recently, in continuation of the interest in SDP, several extended models of SDP have been proposed, which are called the nonlinear SDP problems. For distinction, the conventional SDP is sometimes called the linear SDP, in spite of the fact that the linear SDP is a nonlinear optimization. It is known that the nonlinear SDP has the application in the control theory [15, 17] as well as the structural engineering [2, 8, 10]. The structural optimization over the lower bound constraint for the linear buckling load factor can be formulated as a nonlinear SDP problem [2, 8]. The first author showed that the robust optimal design of a structure subjected to uncertain loads can be formulated as a nonlinear SDP problem [10]. The interior-point methods for the linear SDP have been extended to the nonlinear SDP [7, 15]. In a manner similar to the case of the sequential linear programming method for the differentiable nonlinear optimization, the sequential