1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 31

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 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. 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.

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 Figure 31.1

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4 The General Approach Discretization/ First step is dividing the solution domain into finite elements. Figure 31.2

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5 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;

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6 Approximation or shape function Interpolation functions

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 7 Figure 31.3

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 8 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.

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9 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.

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10 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}

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11 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).

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 12 Postprocessing/ Upon obtaining solution, it can be displayed in tabular form or graphically.

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13 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.

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14 Figure 31.9

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 15 Figure 31.10