0 2nd Column negative 73 slash 15 3rd Column 158 slash 15 4th Column negative 73 slash 15 4th Row 1st Column 0 2nd Column 0 3rd Column negative 73 slash 15 4th Column 158 slash 15 EndMatrix Start 4 By 1 Matrix 1st Row a 2 2nd Row a 3 3rd Row a 4 4th Row a 5 EndMatrix equals Start 4 By 1 Matrix 1st Row 73 slash 13 2nd Row 0 3rd Row 0 4th Row 146 slash 15 EndMatrix"/>
alternatively:
where the first and sixth equations are placeholders for the boundary conditions
Exercise 1.14 Solve the problem in Example 1.7 with the boundary conditions
Exercise 1.15 Solve the problem in Example 1.7 with the boundary conditions
1.4 Post‐solution operations
Following assembly of the coefficient matrix and enforcement of the essential boundary conditions (when applicable) the resulting system of simultaneous equations is solved by one of several methods designed to exploit the symmetry and sparsity of the coefficient matrix. The solvers are classified into two broad categories; direct and iterative solvers. Optimal choice of a solver in a particular application is based on consideration of the size of the problem and the available computational resources.
At the end of the solution process the finite element solution is available in the form
(1.81)
where the indices reference the global numbering and Nu is the number of degrees of freedom plus the number of Dirichlet conditions.
The basis functions are decomposed into their constituent shape functions and the element‐level solution records are created in the local numbering convention. Therefore the finite element solution on the kth element is available in the following form:
(1.82)
1.4.1 Computation of the quantities of interest
The computation of typical engineering quantities of interest (QoI) by direct and indirect methods is outlined in this section.
Computation of uFE(x0)
Direct computation of
and
(1.84)
Direct computation of
Direct computation of
(1.85)
