MATHEMATICAL ENGINEERING TECHNICAL REPORTS Eigenvalue Optimization of Structures via Polynomial Semidefinite Programming
نویسندگان
چکیده
This paper presents a sequential semidefinite programming (SDP) approach to maximize the minimal eigenvalue of the generalized eigenvalue problem, in which the two symmetric matrices defining the eigenvalue problem are supposed to be the polynomials in terms of the variables. An important application of this problem is found in the structural optimization which attempts to maximize the minimal eigenvalue of the free vibration. It is shown that the maximization of minimal eigenvalue of a structure can be formulated as the linear optimization over a polynomial matrix inequality (polynomial SDP). We propose a bisection method for the polynomial SDP, at each iteration of which we solve a maximization problem of a convex function over a linear matrix inequality. A sequential SDP method is proposed for the subproblem based on the DC (difference of convex functions) algorithm. Optimal topologies are computed for various framed structures to demonstrate that the algorithm presented can converge to optimal solutions with multiple lowest eigenvalues without any difficulty.
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