Mathematical Physics - Volume II - Numerical Methods

3.6 Program for solving of elliptical problems

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Figure 3.20: Isoparametric finite elements.

Base function for a rectangular element in parametric plane are: ψ 1 = 1 4 ( ξ 2 − ξ )( η 2 − η ) ψ 5 = 1 2 ( 1 − ξ

2 )( η 2 − η ) 2 + ξ )( 1 − η 2 ) 2 )( η 2 + η ) 2 − ξ )( 1 − η 2 )

1 4 ( ξ 1 4 ( ξ 1 4 ( ξ

1 2 ( ξ

2 + ξ )( η 2 − η ) ψ 2 + ξ )( η 2 + η ) ψ 2 − ξ )( η 2 + η ) ψ

ψ 2 = ψ 3 = ψ 4 =

6 = 7 = 8 =

1 2 ( 1 − ξ 1 2 ( ξ

ψ 9 = ( 1 − ξ 2 )( 1 − η 2 ) Base function for a triangular element in parametric plane are:

ψ 1 = 2 ζ ( ζ − 0 . 5 ) ψ 4 = 4 ζ ξ ψ 2 = 2 ξ ( ξ − 0 . 5 ) ψ 5 = 4 ξη ψ 3 = 2 η ( η − 0 . 5 ) ψ 6 = 4 ηζ ζ = 1 − ξ − η

3.6.2 Subprogram structure The program consists of three basic parts: the preprocessor, processor and postprocessor. Subpro grams which make up the preprocessor read input data (from an input file named “fem.in”), which define the problem and are used as the base for generating of all of the necessary quantities. The processor then forms a system matrix and system vector and finally solves the obtained system of equations. The postprocessor displays the obtained results. All relevant input and output data are written in the output database, “fem.out”. Let us now list some of the most important variables in the program: • MAXN – maximum number of nodes allowed by the program. • MAXE – maximum number of elements. • MAXM – Maximum number of subregions into which region Ω can be divided into. • MAXBCE – maximum number of points in which function ˆ u is defined. • MAXBCN – maximum number of element sides at the boundary of region Ω where function u variation is defined in the normal direction. • MAXPT – Maximum number of points at which f has a δ function type singularity. • MAXBAND - maksimalna poluširina trake oko dijagonale matrice sistema u kojoj su skon centrisane vrednosti razlicˇite od nule. ??????????????????? All of these quantities need to be defined in the program (depending on the available working memory of the computer), before compiling the program. Let us briefly describe all subprograms which are a part of this program. • SETINT defines all quantities necessary for numerical integration via Gauss method along a surface or a line. • PREP is the preprocessor. Every series of input data for a given problem starts with a short text than identified it. If this text only contains the word “end”, the program stops there.