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Chapter 31. Finite-Element Method Chapter 31. Finite element method provides an alternative to finite-difference methods, especially for systems with irregular geometry, unusual boundary conditions, or heterogeneous composition.
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Finite-Element MethodChapter 31 • Finite element method provides an alternative to finite-difference methods, especially for systems with irregular geometry, unusual boundary conditions, or heterogeneous composition. • This method divides the solution domain into simply shaped regions or elements. An approximate solution for the PDE can be developed for each element. • The total solution is generated by linking together, or “assembling,” the individual solutions taking care to ensure continuity at the interelement boundaries.
The General Approach Figure 31.2 Discretization/ • First step is dividing the solution domain into finite elements.
Element Equations/ • Next step is develop equations to approximate the solution for each element. • Must choose an appropriate function with unknown coefficients that will be used to approximate the solution. • Evaluation of the coefficients so that the function approximates the solution in an optimal fashion. Choice of Approximation Functions: For one dimensional case the simplest case is a first-order polynomial;
Approximation or shape function Interpolation functions
The fact that we are dealing with linear equations facilitates operations such as differentiation and integration: Obtaining an Optimal Fit of the Function to the Solution: • Most common approaches are the direct approach, the method of weighted residuals, and the variational approach.
Mathematically, the resulting element equations will often consists of a set of linear algebraic equations that can be expressed in matrix form: [k]=an element property or stiffness matrix {u}=a column vector of unknowns at the nodes {F}=a column vector reflecting the effect of any external influences applied at the nodes.
Assembly/ • The assembly process is governed by the concept of continuity. • The solutions for contiguous elements are matched so that the unknown values (and sometimes the derivatives) at their common nodes are equivalent. • When all the individual versions of the matrix equation are finally assembled: [K] = assemblage property matrix {u´} and {F´}= assemblage of the vectors {u} and {F}
Boundary Conditions/ • Matrix equation when modified to account for system’s boundary conditions: Solution/ • In many cases the elements can be configured so that the resulting equations are banded. Highly efficient solution schemes are available for such systems (Part Three).
Postprocessing/ • Upon obtaining solution, it can be displayed in tabular form or graphically.
Two-dimensional Problems • Although the mathematical “bookkeeping” increases significantly, the extension of the finite element approach to two dimensions is conceptually similar to one-dimensional applications. It follows the same steps.
