By Eschenburg J. H., Tribuzy R.

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An element function g as dened by Eq. , the formula for k = 1 in Eq. 3) j j j aj aj = 1a (pj + qj (aj + aj )rj pj qj (aj )rj ) ; j = 1; : : : ; J j Then, in Eq. e. 5) Substitute now g from Eq. 1 into the right hand side of Eq. 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.

20) where is the domain of the nite element and ne is the number of nite elements. This nodal force vector due to thermally induced strains is added to the mechanical nodal force vector in Eq. 2 when solving the static equilibrium equations. This results in a displacement vector D used to calculate the element strains ", see Eq. 9. Design Sensitivity Analysis of Eigenvalues where E is the constitutive matrix, B is the strain-displacement matrix and d is the element displacement vector. In this way boundary conditions are considered when calculating thermal stresses.

39) jk s @a @a k j sk = 0; s = 1; : : : ; N i i k=1 Ts where the M-orthonormalization, Eq. 26, has been used. A nontrivial solution to these equations only exists if the determinant of the system is equal to zero ! 40) i i This is the main equation for determining the coecients j ; j = 1; : : : ; N , of the power series in Eq. 35 which represent the sensitivities of the multiple eigenvalue ~ with respect to changes ai of a single design parameter ai . As in the case of simple eigenvalues, the derivatives of the K and M matrix, respectively, must be calculated rst, and then the subeigenvalue problem of Eq.

### (1,1)-geodesic maps into grassmann manifolds by Eschenburg J. H., Tribuzy R.

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