Applications of computer algebra to finite element software construction over the past two decades are reviewed. Its use in element matrix evaluation is well established, as is the exploration of matrix symmetries and group properties. Automatic generation of complete finite element software systems is a more recent but possibly stillborn development. Other applications include creation of material models appropriate to finite element analysis and development of postprocessing techniques. Aspects of algebra systems important to users include algebraic expression display and manipulation, code generation, programming style, and linkages to other software. The publication of symbolic programs can and should replace publication of derivation details or mere results.
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References
1.
CalmetJ. and van HulzenJ.A., Computer algebra applications, in: Computer algebra: symbolic and algebraic computation (Computing, Supplementum 4) (Springer, Berlin, 1982) pp. 245–258.
2.
CohenH.I. and FitchJ.P., Uses made of computer algebra in physics, J. Symb. Comp.11 (1991) 291–305.
3.
SilvesterP.P., Symbolic computation as a basis for numerical methods, Int. Conf. on Computational Electromagnetics (IEE Conference Publication No. 350), London, 1991, pp. 1–5.
4.
RentiaN. and ViraN., Why symbolic computation in robotics?Comput. Ind.17 (1991) 49–62.
SilvesterP.P. and FerrariR.L., in: Finite elements for electrical engineers, 2nd ed. (Cambridge University Press, Cambridge, 1990).
7.
LuftW., RoessetJ.M. and ConnorJ.J., Automatic generation of finite element matrices, J. Struct. Div. ASCE97 (1971) 349–362.
8.
GundersonR.H. and CetinerA., Element stiffness matrix generator, J. Struct. Div. ASCE97 (1971) 363–375.
9.
CecchiM.M. and LamiC., Automatic generation of stiffness matrices for finite element analysis, Int. J. Numer. Methods Engr.11 (1977) 396–400.
10.
NagabhushanamJ., SrinivasC.J. and GaonkarG.H., Symbolic generation of elemental matrices for finite element analysis, Comput. Structures42 (1992) 375–380.
11.
KonradA. and SilvesterP., Triangular finite elements for the generalized Bessel equation of order m, Int. J. Numer. Methods Engr.7 (1973) 43–55.
12.
ChepurnyiN., Evaluation of high-order polynomial triangular finite elements using FORMAC, Proc. 2nd Symp. on Symbolic Algebraic Manipulation (1971) pp. 365–371.
13.
SilvesterP., Universal finite element matrices for tetrahedra, Int. J. Numer. Methods Engr.18 (1981) 1055–1061.
14.
PedersenP., On computer-aided analytic element analysis and the similarities of tetrahedron elements, Int. J. Numer. Methods Engr.11 (1977) 611–22.
15.
NoorA.K. and AndersenC.M., Mixed isoparametric elements for the Saint-Venant problem, Comput. Meth. Appl. Mech. Engng, 6 (1975) 195–218.
16.
NoorA.K. and AndersenC.M., Mixed isoparametric finite element models of laminated composite shells, Comput. Meth. Appl. Mech. Engng.11 (1977) 255–280.
17.
AndersenC.M., Evaluation of integrals for a ten-node isoparametric tetrahedral finite element, Comput. Math. Applic. (1978).
18.
KorncoffA.R. and FenvesS.J., Symbolic generation of finite element stiffness matrices, Comput. Structures10 (1979) 119–124.
19.
SaleebA.F. and ChangT.Y., An efficient quadrilateral element for plate bending analysis, Int. J. Numer. Methods Engr.24 (1987) 1123–1155.
20.
KikuchiM., Application of symbolic mathematics system of the finite element program, Computational Mechanics5 (1989) 41–48.
21.
ChangT.Y., TanH.Q., ZhengD. and YuanM.W., Application of symbolic method to hybrid/mixed finite elements and computer implementation, Comput. Structures35 (1990) 293–299.
22.
NoorA.K. and AndersenC.M., Computerized symbolic manipulation in structural mechanics-progress and potential, Comput. Structures10 (1979) 95–118.
23.
NoorA.K. and AndersenC.M., Computerized symbolic manipulation in nonlinear finite element analysis, Comput. Structures13 (1981) 379–403.
24.
WangP.S., Taking advantage of symmetry in the automatic generation of numerical programs for finite element analysis, Proc. EUROCAL 1985, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 572–582.
25.
BardellN.S., The application of symbolic computing to the hierarchical finite element method, Int. J. Numer. Methods Engrg.28 (1989) 1181–1204.
26.
IoakimidisN.I., The location of discontinuity intervals of sectionally analytic functions: application to the interface crack problem, Computers and Mathematics with Applications21 (1991) 69–74.
27.
IoakimidisN.I., Orders of singularity at wedge apices: the computer algebra approach, Eng. Fract. Mech.38 (1991) 349–352.
28.
StoutemyerD.R., Automatic error analysis using computer algebraic manipulation, ACM Transactions on Mathematical Software3 (1977) 26–43.
29.
StoutemyerD.R., Computing with impure numbers: automatic consistency checking and units conversion using computer algebra, IEEE Trans. Software Eng.SE-3 (1977) 478–80.
30.
FerrariM., HardingJ.H. and MarcheseM., Computer simulation of plasma-sprayed coatings. II. Effective bulk properties and thermal stress calculations, Surface Coat. Technol.48 (1991) 147–154.
31.
SilvesterP.P., Computational models for laminated magnetic materials, Int. J. Numer. Model.5 (1992) 83–92.
32.
WeggelC.F. and SchwartzD.P., New analytical formulas for calculating magnetic field, IEEE Trans. on Magnetics24 (1988) 1544–1547.
33.
SchenckJ.F. and HussainM.A., Formulation of design rules for NMR imaging coil by using symbolic manipulation, Proc. SYMSAC'81, WangP.S, ed. (ACM, New York, 1981) pp. 85–93.
34.
WantP.S., ChangT.Y. and van HulzenJ.A., Code generation and optimization for finite element analysis, Proc. EUROSAM'84 (Springer, Berlin, 1984) pp. 237–247.
35.
WangP.S., Taking advantage of symmetry in the automatic generation of numerical programs for finite element analysis, Proc. EUROCAL'85, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 572–582.
36.
WangP.S., FINGER: A symbolic system for automatic generation of numerical programs in finite element analysis, J. Symb. Comput.2 (1986) 305–316.
37.
FitchJ. and HallR., Symbolic computation and the finite element method, Proc. EUROCAL'87 (Springer, Berlin, 1989) pp. 95–96.
38.
SharmaN. and WangP.S., Symbolic derivation and automatic generation of parallel routines for finite element analysis, Proc. ISSAC'88 (Springer, Berlin, 1989) pp. 33–56.
39.
SilvesterP. and HaslamC.R.S., Magnetotelluric modelling by the finite element method, Geophys. Prospect.20 (1972) 872–891.
40.
SilvesterP.P., Differentiation of finite element approximations to harmonic functions, IEEE Trans. on Magnetics27 (1991) 3774–3779.
41.
SilvesterP.P., Extended Poisson kernels for differentiation of harmonic functions, Electrosoft2 (1991) 284–300.
42.
FosterK.R. and BauH.H., Symbolic manipulation programs for the personal computer, Science243 (1989) 679–684.
43.
AvitzurR., Suggestions for a friendlier user interface, Design and Implementation of Symbolic Computation Systems, Proc. DISCO'90, MiolaA., ed. (Springer, Berlin, 1990) pp. 282–283.
44.
KajlerN., Building graphic user interfaces for computer algebra systems, Design and Implementation of Symbolic Computation Systems, Proc. DISCO'90, MiolaA., ed. (Springer, Berlin, 1990) pp. 235–244.
45.
DavenportJ.H., Current problems in computer algebra systems design, Design and Implementation of Symbolic Computation Systems, Proc. DISCO'90, MiolaA., ed. (Springer, Berlin, 1990) pp. 1–9.
46.
CalmetJ. and LugiezD., A knowledge-based system for computer algebra, SIGSAM Bulletin21(1) (1987) 7–13.
47.
GielenG., WalschartsH. and SansenW., ISAAC: A symbolic system for analog integrated circuits, IEEE J. Solid-State CircuitsSC-24 (1989) 1587–1597.
48.
SaitoO., KannoM. and AbeK., Symbolic manipulation CAD of control engineering by using REDUCE, Int. J. Control48 (1988) 781–790.
49.
LelerW. and SoifferN., An interactive graphical interface for REDUCE, SIGSAM Bulletin19(3) (1985) 17–23.
50.
AbdaliS.K. and CherryG., Visual environments for symbolic computation, Proc. EUROCAL'85, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 166–167.
51.
van den HeuvelP., HulshofB.J.A. and van HulzenJ.A., Some simple pretty-print facilities for REDUCE, SIGSAM Bulletin21(1) (1987) 14–17.
52.
FatemanR.J. and LaiD.T., A simple display package for polynomials and rational functions in common Lisp, SIGSAM Bulletin25(4) (1991) 1–3.
53.
MarzinkewitschR., Operating computer algebra systems by handprinted input, Proc. ISSAC'91, WattS.M, ed. (Springer, Berlin, 1991).
54.
YoungD.A. and WangP.S., GI/S: A graphical user interface for symbolic computation systems, J. Symb. Comput.4 (1988) 365–380.
55.
HearnA.C., Structure: The key to improved algebraic computation, Second RIKEN International Symposium on Symbolic and Algebraic Computation by Computers, InadaN. and SomaT., eds. (World Scientific, Singapore, 1985) pp. 215–230.
56.
HulshofB.J.A. and van HulzenJ.A., An expression compression package for REDUCE based on factorization and controlled expansion, Proc. EUROCAL'85, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 315–316.
57.
SteinbergS. and RoacheP., Using VAXIMA to write FORTRAN code, in: Applications of Computer Algebra, PavelleR., ed. (Kluwer, Boston, 1985) pp. 74–94.
58.
GatesB.L., GENTRAN: An automatic code generation facility for REDUCE, SIGSAM Bulletin19(3) (1985) 24–42.
59.
GatesB.L. and van HulzenJ.A., Automatic generation of optimized programs, Proc. EUROCAL'85, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 583–584.
60.
LaureP. and DemayY., Symbolic computation and equation on the center manifold: application to the Couette-Taylor problem, Comput. Fluids16 (1988) 229–238.
61.
LuppiJ. and PajunenP., Symbolic computation and automatic FORTRAN code generation for eigenvalue determination by phase integral method, J. Comput. Phys.88 (1990) 15–30.
62.
FatemanR.J., TEX output from MACSYMA-like systems, SIGSAM Bulletin21(4) (1987) 1–5.
63.
AntweilerW., StrotmannA. and WinkelmannV., A TEX-REDUCE-interface, SIGSAM Bulletin, 23(2) (1989) 26–31.
64.
DrskaL., LiskaR. and SinorM., Two practical packages for computational physics-GCPM, RLFI, Comp. Phys. Comm.61 (1990) 225–230.
65.
YoungD.A. and WangP.S., An improved plotting package for Vaxima, Proc. EUROCAL'85, Vol. 2, CavinessB.F, ed. (Springer, Berlin, 1985) pp. 431–432.
66.
WangP.S., A system independent graphing package for mathematical functions, Design and Implementation of Symbolic Computation Systems, Proc. DISCO'90, MiolaA., ed. (Springer, Berlin, 1990) pp. 245–254.
67.
DolehY. and WangP.S., SUI: a system independent user interface for an integrated scientific computing environment, Proc. ISSAC'90 (Springer, Berlin, 1990).
68.
MartiJ.B., The role of explanation in symbolic computation, Second RIKEN Int. Symposium on Symbolic and Algebraic Computation by Computers, InadaN. and SomaT., eds. (World Scientific, Singapore, 1985) pp. 13–34.
69.
dos SantosR.P. and RoqueW.L., On the design of an expert help system for computer algebra systems, SIGSAM Bulletin24(4) (1990) 22–25.
70.
ClarksonM., Praxis: a rule-based expert system for MACSYMA, Design and Implementation of Symbolic Computation Systems, Proc. DISCO'90, MiolaA., ed. (Springer, Berlin, 1990) pp. 264–265.
71.
WolframS., Mathematica: a system for doing mathematics by computer, 2nd ed. (Addison-Wesley, Redwood City, 1991).
72.
BarronD., Thoroughly modern misses, Times Higher Education Supplement, 2 November 1990.
