Structural optimization with parameter-transfer finite element (Q1905469)

The authors discuss some disadvantages of the classical applications of finite elements method (FEM). They observe that in analyzing the response of a structure to various types of loads (dynamic loads, thermal loads, etc...) one has to assemble mass, stiffness and load matrices. Optimization is usually replaced by computation of sensitivity of these matrices to changes of parameters affecting properties of these matrices. It is known that the FEM replaced the techniques based on finding the transfer functions because of the difficulties encountered in computing transfer functions (or transfer matrices for discretized systems), or near impossibility of computing them, except in the cases of extremely simple structures. The parameter transfer-finite element method (PTFM) is a compromise between these two approaches. Let a body be discretized so that its state is described by a finite dimensional vector of state variables \(\{Y\}\). The control is exerted by a bounded finite-dimensional vector \(\{U\}\), \((U_{\min} < U < U_{\max})\). The problem of natural vibration, or of the loss of stability, is assumed to be reduced to an eigenvalue problem: \([R^* (U, \Lambda)] \{Y\} = 0\). Here \(\{\Lambda\} = \{\Lambda^*\}\) is vector of the squares of natural frequencies. The authors give an example of vibrating Euler-Bernoulli beam. The cost function \(F\) is the total weight of the beam, control variables are the geometric dimensions, such as thickness of each element. Since matrix \(R^*\) is singular, \(\text{det} [R^*] = 0\). Direct differentiation applied to each component of \(U\) results in a matrix-polynomial formula of the type \(\partial R/ \partial u_\ell=\sum \lambda^m D^m (U)\), which permits a direct computation of eigenvalue sensitivity \(\partial \lambda/ \partial u_\ell\). The authors compare their results and the results obtained by the standard use of finite element methodology. However, for cases like the beam equation there exist effective techniques of distributed parameter optimization. In more recent times there has been rapid advance based on functional analytic methods. Basic ideas of such approaches are outlined in several books, for example [\textit{E. J. Haug}, \textit{K. K. Choi} and \textit{V. Komkov}, Design sensitivity analysis of structural systems (1986; Zbl 0618.73106)]. It may be of interest to compare authors' technique with these methods. In particular, the sensitivity of eigenvalues for a vibrating Euler-Bernoulli beam can be readily compared.