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# Having symmetry in mind we will select functions u

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Unformatted text preview: u and v that produce bounded values of A(u u) = Z1 p(u )2 + qu2]dx: 0 0 Actually, since p and q are smooth functions, it su ces for u and v to have bounded values of Z 1 (u )2 + u2]dx: 0 0 (1.2.10) Functions where (1.2.10) exists are said to be elements of the Sobolev space H 1. We've also required that u and v satisfy the boundary conditions (1.2.1b). We identify those functions in H 1 that also satisfy (1.2.1b) as being elements of H01. Thus, in summary, the variational problem consists of determining u 2 H01 such that A(v u) = (v f ) 1 8v 2 H0 : (1.2.11) The bilinear form A(v u) is called the strain energy. In mechanical systems it frequently corresponds to the stored or internal energy in the system. We obtain approximate solutions of (1.2.11) in the manner described earlier for the more general method of weighted residuals. Thus, we replace u and v by their approximations U and V according to (1.2.3). Both U and V are regarded as belonging to the N same nite-dimensional subspace S0 of H01 and j , j = 1 2 : : : N , forms a basis for S0N . Thus, U is determined as the solution of A( V U ) = ( V f ) N 8V...
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