Resolution of the Poisson's equation with the WRM using global Shape Functions
Revision as of 09:08, 4 November 2009 by JMora
- The residual formulation is based on the Weighted Residual Method (WRM). The differential equation is converted in an integral equation with certain weighting functions applied to each equation.
- The residual form of the Poisson's equation, that is, by using an approach solution , is written:
- Therefore, the integral equation to be solved, including the weightings ( and ), become:
- that can be written as:
- with K is a coefficients matrix that depends on the geometrical and physical properties of the problem, a is the vector with the n unknowns () to be obtained and f is a vector that depends on the source values and boundary conditions.
- Using the same basis functions for the whole domain, and if they are selected to naturally accomplish the boundary conditions, the integral form can be written:
- and therefore:
- For simplicity reasons, we select as basis functions the Fourier Series , that automatically accomplish the boundary conditions and . If the material is constant, we can write:
- Depending on the kind of weighting functions used, different weighted residual methods (WRM) can be considered, as follows in the next sections.
- The matrix components are:
- If the material k is constant and the source Q is also constant over a part of the domain, and we still use Fourier series as basis functions, the matrix form becomes:
- that is, the Subdomain Method is equivalent to be the residual integral equal to zero for each subdomain (note that now the integral is applied just to the subdomain.
- If the material k is constant and the source Q is also constant over a part of the domain, and Fourier series are used too as basis functions, the matrix form becomes:
- If the material k is constant and the source Q is also constant over a part of the domain, and Fourier series are used as basis functions again, the matrix form becomes:
Least Squares Method
- The expression to minimize is:
- Therefore, the minimization process can be understood as a form of the weighting residues with:
- If the material k is constant and the source Q is also constant over a part of the domain, and Fourier series are used as basis functions , the matrix form becomes:
- Note that, in this case and without generality, due to the selected basis functions, the obtained solution is the same that the obtained one using the Galerkin method.
- In order to compare the different methods, check some specific examples.