By Lund E.
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Additional info for Finite Element Based Design Sensitivity Analysis and Optimization
Works by Courant & Hilbert (1953), Wittrick (1962), and Lancaster (1964) have provided a basis for calculating the sensitivities of multiple eigenvalues. It is shown that the design sensitivities of multiple eigenvalues can be found by formulation and solution of a subeigenvalue problem. Let us rst consider a small change "ai of a single, arbitrarily chosen design parameter ai where " is a small positive parameter. 34) K + " @@aK ai and M + " @@aM ai; i = 1; : : : ; I i i Chapter 4. 36) where j and j are unknown eigenvalue and eigenvector sensitivities, respectively, and o(") represents higher order terms.
For this purpose the direct approach seems much easier to implement and to be just as applicable to solve the problem as the continuum approach. , Modulef, see Kibsgaard (1991). So, the direct approach to design sensitivity analysis has been chosen due to its ease of implementation, and it will be shown in the following that this method is very ecient. The direct approach will be only used in the discrete version as the adjoint method is not suited for the way optimization problems are formulated in ODESSY, cf.
4, and consider the formulas in Eq. 2 for f = g and k = rj = 0; 1; 2; : : :. 3. \Exact" Numerical Dierentiation of Special Element Functions 3 1 2 3 1 + j + j + j 2 4 1 c4 = ::: 0 k1 ! 11 X 1 k ck = @ k j(k 1) pA p p=0 where the latter expression for ck is given in terms of binomial coecients. 1: Illustration of relationsship between the analytical derivative @g=@aj and its rst order nite dierence approximation g=aj . In Fig. 1 this relationship of proportionality between the analytical derivative @g=@aj of an element function g and its rst order nite dierence approximation g=aj is illustrated.
Finite Element Based Design Sensitivity Analysis and Optimization by Lund E.