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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
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
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14