This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.

Multiple eigenvalues in structural optimization problems

A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher order, or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and several illustrative results are presented.

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Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps

Structural optimization with frequency constraints - A review

In this paper, we apply asymptotic–numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Pade approximants play an important role in the determination of the size of the domain of convergence.

Asymptotic-numerical methods and pade approximants for non-linear elastic structures

Dynamic stability of columns subjected to follower loads : A survey

Optimal structural design under multiple eigenvalue constraints

Non-stationary optimality conditions in structural design☆

The application of standard stationarity conditions to the optimal design of a structure against buckling (or other eigenvalue constraints) may actually lead to a reduction of its real strength in the sense that the “optimized” structure has been weakened with respect to higher modes to the extent that these have now become fundamental. In this case a true optimum occurs when the magnitudes of several eigenvalues coalesce, and the corresponding optimality condition is then no longer stationary. This singular condition is derived and applied to several examples of major practical significance, together with a method of obtaining an approximate solution to the resulting set of nonlinear equations.

E.F. Masur

Papers

Optimal structural design under multiple eigenvalue constraints

Non-stationary optimality conditions in structural design☆

Singular Solutions in Structural Optimization Problems

On the use of the complementary energy in the solution of buckling problems

Some additional comments on optimal structural design under multiple eigenvalue constraints