Chapter 7

Finite element programming

May 17, 2011

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.

7.1 Input data

I. Node numbers II. 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 = 9Total of elements: NELMT = 8

I. Node numbers II. 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

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:

20/ onnu

3

1

),(3

),(j

jjeyxg

Sdxdyyxg Given :),( yxg

e i

ei dxdyyxLyxff ),(),()(

(7.1)

(7.2)

(7.3)

(7.4)

ijjjjjijjjj yxfyxLyxfyxg ),(),(),(),(

3/),()(ii

ei yxSff

7.3Creating 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

1 32

1 54

Table 7.2

7.4Output 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.

7.5 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)

INPUT ASSEM

ECM

F

SOLVE OUTPUT

Fig.7.2 Relationship of each function

7.6 Examples of program use (1/3)

• Using the given program, let’s solve several problems.

Figures 7.3 and 7.4

7.6 Examples of program use (2/3)

1 u 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

7.6 Examples of program use (3/3)Table 7.3 Results of Example 7.5

Fig.7.7 and distributions along the centerline lineu

u

Exact solution

Fig.7.6 Error estimates Exact solution

even number

odd number

gradient -2 line

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 .Am Am

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.

Appendix B

• Source of a C language program example is given as follows:

http://www.s.kyushu-u.ac.jp/~z7kh03in/full_FEM.c