By Zhang Q.

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Higher order nite di erences as central di erences are not used. Such a method has been developed and published by Olho , Rasmussen & Lund (1992) and Lund & Olho (1993a, 1993b), and the remainder of this chapter is devoted to a presentation of this new method of S-A design sensitivity analysis. 3. 3 \Exact" Numerical Di erentiation of Special Element Functions The foundation for the new S-A method using \exact" numerical di erentiation of nite element matrices is the observation, that element functions g, used in the de nition of element matrices for large groups of nite elements, have the same mathematical form.

19) where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains (typically 20C). It is not possible to add these thermally induced strains directly to the mechanical strains in order to calculate the induced stress eld. , a structure is free of thermally induced stresses if its supports do not inhibit thermal expansion or contraction. 20) where is the domain of the nite element and ne is the number of nite elements.

1 where the unperturbed mesh also is shown. The active element strategy implies that in the assembly of element matrix derivatives, only active elements contribute, and the pseudo load vector in Eq. 6 can thereby be calculated at the element level in the following computationally ecient way @K D + @F = X @ai @ai nae ! 9) where nae is the number of active nite elements, ka is the element sti ness matrix, da is the element displacement vector, and f a is the element load vector for an active element.

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A BMO estimate for the multilinear singular integral operator by Zhang Q.

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