1. Chapter 7 Finite element programming May 17, 2011

2. Brief introduction Until Chapter 6, there are FEM (Finite Element Method) and solution of simultaneous linear equations. Chapter 7 focuses to explain how to make basic FEM programs. Some easy Fortran technique is needed in some program examples. In Appendix B, a C language program example appears.

3. 7.1 Input data I. Node numbersII. Element numbers First, consider the element division. Node numbers and node coordinates are necessary. Don’t confuse global node numbers with element node numbers. Total of nodes: NNODE = 9 Total of elements: NELMT = 8 I. Node numbersII. Element numbers 1 2 3 4 5 6 7 8 9 [1] [2] [3] [4] [5] [6] [7] [8] [1] [2] [3] [4] [5] [6] [7] [8] Fig. 7.1 FEM mesh division

4. 7.2 Element coefficient matrix creating Consider the boundary value problem of the 2-dimensional Poisson equation in Chapter 5. The natural boundary condition is From (5.29) and (5.30), the element rhs vector is as follows: Given : (7.1) (7.2) (7.3) (7.4)

5. 7.3 Creating the whole coefficient matrix and solving linear equations The direct stiffness method produces all coefficient matrixes and the rhs vectors. As explained in Chapter 5, element node numbers and global node numbers must be consistent. Elem. Global 132 154 Table 7.2

6. 7.4 Output and important points Don’t forget that input data have close relation to output data. Element numbers, node numbers, boundary conditions, and node coordinates are indispensable for input data checking.

7. 7.5 Program examples Program’s structure: 1.Main program: the whole subprogram call 2.Input : input data’s reading 3.Assembling : getting the global matrix and the vector, and collecting element stiffness matrixes and element vectors 4.ECM: element matrix and element vector calculation 5.Solve: solving the linear equations by the Gauss elimination method 6.Output : showing obtained values. 7.Function: preparation of function f(x, y) (MAIN) INPUTASSEM ECM F SOLVEOUTPUT Fig.7.2 Relationship of each function

8. 7.6 Examples of program use (1/3) Using the given program, let’s solve several problems. Figures 7.3 and 7.4

9. 7.6 Examples of program use (2/3) number of nodes 36 number of nodes 121 m =5 m =10 number of elements 50 number of elements 200 The domain Ω is a unit square (0<x, y<1). The Poisson equation is given as follows: Fig.7.5 A mesh division example in Ω on Γ

10. 7.6 Examples of program use (3/3) Table 7.3 Results of Example 7.5 Fig.7.7 and distributions along the centerline line Exact solution Fig.7.6 Error estimates Exact solution even number odd number gradient -2 line

11. 7.7 An example program using a symmetric band matrix The big change in a symmetric band matrix is subroutines SOLVE and BAND. Some modification is needed in ASSEM For half band, input data should be + 1.

12. 7.8 Ending The things described in above sections have told us how to use and understand FEM problems using an easy model. This program has been separated to several parts, good for beginners.

13. Appendix B Source of a C language program example is given as follows: http://www.s.kyushu-u.ac.jp/~z7kh03in/full_FEM.c