lecture42 - Lecture 42 Determining Internal Node Values As...

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Unformatted text preview: Lecture 42 Determining Internal Node Values As seen in the previous section, a finite element solution of a boundary value problem boils down to finding the best values of the constants { C j } n j =1 , which are the values of the solution at the nodes. The interior nodes values are determined by variational principles . Variational principles usually amount to minimizing internal energy . It is a physical principle that systems seek to be in a state of minimal energy and this principle is used to find the internal node values. Variational Principles For the differential equations that describe many physical systems, the internal energy of the system is an integral. For instance, for the steady state heat equation u xx + u yy = g ( x, y ) (42.1) the internal energy is the integral I [ u ] = integraldisplayintegraldisplay R u 2 x + u 2 y + 2 g ( x, y ) u ( x, y ) dA, (42.2) where R is the region on which we are working. It can be shown that u ( x, u ) is a solution of (42.1) if and only if it is minimizer of I [ u ] in (42.2). The finite element solution Recall that a finite element solution is a linear combination of finite element functions: U ( x, y ) = n summationdisplay j =1 C j Φ j ( x, y ) , where n is the number of nodes. To obtain the values at the internal nodes, we will plug U ( x, y ) into the energy integral and minimize. That is, we find the minimum of I [ U ] for all choices of { C j } m j =1 , where m is the number of internal nodes. In this as with any other minimization problem, the way to find a possible minimum is to differentiate the quantity with respect to the variables and set the results to zero. In this case the free variables are { C j } m j =1 . Thus to find the minimizer we should try to solve ∂I [ U ] ∂C j = 0 for 1 ≤ j ≤ m. (42.3) 144 145 We call this set of equations the internal node equations . At this point we should ask whether the internal node equations can be solved, and if so, is the solution actually a minimizer (and not a maximizer). The following two facts answer these questions. These facts make the finite elementa maximizer)....
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lecture42 - Lecture 42 Determining Internal Node Values As...

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