Keywords: optimization, eigenfrequencies, two-materials, 2D scalar case.

One strategy for passive vibration control of mechanical structures is to design the structures so that eigenfrequencies lie as far away as possible from the excitation frequency. This paper exploits the possibility for using finite element analysis and the method of topology optimization to maximize the separation of two adjacent eigenfrequencies in structures with two material components. This study is restricted to 2D structures where the vibrations are governed by the scalar wave equation. Results related to the 1D scalar case can be found in [1].

When we optimize a 2D domain with respect to maximizing the gap between eigenfrequencies there are a number of extra difficulties we must deal with. The primary source of the difficulties is the possibility of multiple eigenfrequencies. The multiple eigenfrequencies can be calculated using e.g. the subspace iteration method [2].

The objective for the optimization is given by

where the gap between the eigenfrequencies of order and is maximized.

If the eigenfrequencies of order and are both distinct, with squared eigenfrequencies and and corresponding eigenvectors and , no problems arise and we use the objective (81) directly since the sensitivities of a squared eigenfrequency with respect to a design parameter are given by

where is it assumed that the eigenvector have been normalized so that . In the case of multiple eigenvalues we cannot use (82) to find the sensitivities. Instead we can use an extended method presented in [3] and used more recently in [4] and [1].

It is possible to find the sensitivities of multiple eigenfrequencies. There is, however, still a problem because the sensitivities are given for specific eigenvectors that vary for each design parameter. It is therefore difficult to solve the optimization problem as formulated in (81). As an alternative formulation we propose to use a double bound formulation.

We apply topology optimization with a material interpolation scheme where constant material properties is assigned to each finite element and associate these material properties with continuous design variables [5]. We choose one design variable per element and let it vary continuously between 0 and 1. We let the material properties in each element be a specified interpolation function of this design variable.

Since the design variable vary continuously between 0 and 1 we may expect that in the optimal design we can end up with material properties that do not correspond to either of the two materials, but instead to some intermediate values. In order to ensure a well defined distribution of material 1 and 2 in the structure, we can manipulate the interpolation functions [6]. Especially when dealing with eigenvalue problems the choice of the interpolation function is important [7]. In the present study we can not use a simple linear interpolation of mass and stiffness, therefore a more involved interpolation is needed, and results obtained with a new interpolation function are shown.

The results of the optimization of a square design domain with free-free boundary conditions is shown. The results show a clear separation of the two materials in the domain.

1

J.S. Jensen, N.L. Pedersen, "On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases", Submitted, 2003. doi:10.1016/j.jsv.2005.03.028

2

K.J. Bathe, "Finite Element Procedures, 2nd Ed.", Prentice-Hall, 1996.

3

A.P. Seyranian, E. Lund, N. Olhoff, "Multiple eigenvalues in structural optimization problems", Structural Optimization 8(4): 207-227, 1994. doi:10.1007/BF01742705

4

N.L. Pedersen, A.K. Nielsen, "Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses and buckling", Structural and Multidisciplinary Optimization 25(5-6): 436-445, 2003. doi:10.1007/s00158-003-0294-7

5

M.P. Bendsøe, O. Sigmund, "Topology Optimization - Theory, Methods and Applications", Springer Verlag, Berlin Heidelberg, 2003.

6

M.P. Bendsøe, O. Sigmund, "Material interpolations in topology optimization", Archive of Applied Mechanics 69: 635-654, 1999. doi:10.1007/s004190050248

7

N.L. Pedersen, "Maximization of eigenvalues using topology optimization" Structural and Multidisciplinary Optimization 20: 2-11, 2000.

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