Programs in Physics & Physical Chemistry
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|Manuscript Title: A finite element program package for axisymmetric scalar field problems.|
|Authors: A. Konrad, P. Silvester|
|Program title: AXISYMM-SCALAR-HELMHOLTZ-FINTEL4|
|Catalogue identifier: ACSB_v1_0|
Distribution format: gz
|Journal reference: Comput. Phys. Commun. 5(1973)437|
|Programming language: Fortran.|
|Computer: IBM 360/75.|
|Operating system: OS/360 HASP II.|
|RAM: 25K words|
|Word size: 32|
|Keywords: Plasma physics, Laplace, Helmholtz, Acoustics, Poisson, Axisymmetric, Finite element, Heat transfer, Collisionless plasma, Scalar field, Electrostatics.|
|Classification: 10, 19.3.|
Nature of problem:
Electric field problems requiring solution to Laplace's or Poisson's equations in cylindrical coordinates with no azimuthal variation (e.g. coaxial cable discontinuities, electron guns, electrostatic lenses). Acqustics problems requiring solution to the scalar Helmholtz equation in cylindrical coordinates (e.g. Helmholtz resonators); heat-flow problems, plasma simulations, fluid-flow problems.
Laplace's, Poisson's or Helmholtz's equations subject to Dirichlet or homogeneous Neumann boundary conditions are solved by the high-order polynomial triangular finite-element method. The method does not involve iterative procedures of any kind. The solution region is subdivided into triangular subregions (finite elements). Over each element the solution is expressed as a linear combination of a complete set of Nth order interpolation polynomials. The variationally station- ary energy functional associated with the Helmholtz equation is approximated in this way by a matrix quadratic form. Extremization with respect to the unknown coefficients yields a simple, relatively low order matrix equation or matrix eigenvalue equation. The latter are solved by standard methods.
AXISYMM-SCALAR-HELMHOLTZ-FINTEL 4 accepts up to and including 9 different equipotentials (Dirichlet boundary conditions) and the homogeneous Neumann boundary condition. It provides up to and including 4th order polynomial approxiamtion. Mixed order polynomial approximat- ions and mixed material properties can be accomodated, provided that material discontinuities coincide with interelement boundaries. It is intended for problems of matrix size approximately 100 by 100.
Data in the BLOCK DATA segment of the program are punched in Z format, a feature of the IBM FORTRAN IV. The BLOCK DATA can be generated in other formats using the program GENERATE which must be suitably modified. It is described in this paper.
The execution times are heavily problem-dependent. Using an IBM 360/75, for example, a coaxial cable discontinuity problem which involved the solution of Laplace's equation using 10 fourth-order elements (overall matrix order 67) required 5 s of arithmetic execution time. Aspace charge problem which involved the solution of Poisson's equation using 16 second-order and 10 fourth-order elements (matrix order 123) required 12 s. More than 50 % of the arithmetic running times are spent on the solution of the matrix equations. Total timings may therefore be estimated from the known matrix algebra timings for any computer.
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