A Brief History of Finite Element Method and Its Applications to Computational Electromagnetics
DOI:
https://doi.org/10.13052/2022.ACES.J.370501Keywords:
Finite elements, history of computation, numerical methodsAbstract
The development of the finite element method is traced, from its deepest roots, reaching back to the birth of calculus of variations in the 17th century, to its earliest steps, in parallel with the advent of computers, up to its applications in electromagnetics and its flourishing as one of the most versatile numerical methods in the field. A survey on papers published on finite elements, and on ACES Journal in particular, is also included.
Downloads
References
R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc., vol. 49, no. 1, pp. 1-23,1943.
R. Courant, K. O. Friedrichs, and H. Lewy, “Über die partiellen differenzengleichungen der mathematischenphysic [About the partial difference equations of the mathematical physic],” Math. Ann. (in German), vol. 100, pp. 32-74, 1928.
J. Bernoulli, “Problema novum ad cujus solutionem mathematici invitantur [A new problem to whose solution mathematicians are invited],” ActaEruditorum (in Latin), vol. 18, p. 269, 1696.
G. Pelosi, “The finite-element method, Part I: R.L. Courant,” IEEE Antennas Propagat. Mag., vol. 49, pp. 180-182, 2007.
G. Pelosi and S. Selleri, “A prelude to finite elements: The fruitful problem of the brachistochrone,” URSI Radio Science Bullettin, no. 367, pp. 10-14, 2018.
G. Leibnitz, “Nova Methodus pro Maximi set Minimis [A new method for maxima and minima],” ActaEroditorum (in Latin), vol. III, pp. 439-473, 1684.
I. Newton, Philosophiæ Naturalis Principia Mathematica [Mathematical principles of natural philosophy], J. Steater (in Latin), London UK, 1687.
K. Schellbach, “Probleme der variationsrechnung [Problems of the calculation of variations],” Journal für die reine und angewandte Mathematik (in German), no. 41, pp. 293-363, 1851.
J. W. Strutt and L. Rayleigh, The Theory of Sound, MacMillan & Co., London, UK, 1894 (vol. I) - 1896 (vol. II).
W. Ritz, “Über eine neue methode zur lösung gewisser variationsprobleme der mathematischenphysic [About a new method for solving certain variation problems in mathematical physics],” Journal für die reine und angewandte Mathematik (in German), vol. CXXXV, pp. 1-6, 1908.
B. G. Galerkin, “Series occurring in various questions concerning the elastic equilibrium of rods and plates,” Engineers Bulletin (Vestnik Inzhenerov - In Russian), vol. 19, pp. 897-908, 1915.
S. Faedo, “Un nuovo metodo per l’analisi esistenziale e quantitativa dei problemi di propagazione [A new method for the existential and quantitative analysis of propagation problems],” Ann. Scuola Norm. Sup Pisa Cl. Sci. (in Italian), vol. 1, no. 3, pp. 1-40, 1949.
G. Kron, Tensor Analysis of Networks, John Wiley & Sons, New York, NY, 1939.
A. Hrennikoff, “Solution of problems in elasticity by the framework method,” J. Appl. Mech., vol. 8, pp. 619-715, 1941.
D. McHenry, “A lattice analogy for the solution of plane stress problems,” J. Inst. Civ. Eng, vol. 21, pp. 59-82, 1949.
J. H. Argyris, “Energy theorem and structural analysis,” Aircraft Engineering, vol. 26, pp. 347-358 and 383-387, 394, 1954.
S. Levy, “Structural analysis and influence coefficients for delta wings,” J. Aeronautical Sc., vol. 20, pp. 449-454, 1953.
M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp, “Stiffness and deflection analysis of complex structures,” J. Aeronautical Sc., vol. 23, pp. 805-823, 854, 1956.
R. W. Clough, “The finite element method in plane stress analysis,” Proc. 2nd
ASCE Conf. on Electronic Comput., Pittsburg, PA, 1960.
J. T. Oden, “A general theory of finite elements; I topological considerations,” Int. J. Num. Meth. Eng., vol. 1, pp. 205-221, 1969.
J. T. Oden, “A general theory of finite elements; II applications,” Int. J. Num. Meth. Eng., vol. 1, pp. 247-259, 1969.
J. T. Oden, “A finite elements analogue of the Navier-Stokes equations,” J. Eng. Mech. Div. ASCE, vol. 96, pp. 529-534, 1970.
O. C. Zienkiewicz and Y. K. Cheung, The Finite Element in Structural and Continuum Mechanics, London, UK: McGraw-Hill, 1967.
O. C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, London, UK, 1971.
P. P. Silvester, “Finite elements solution of homogeneous waveguide problems,” Alta Frequenza, vol. 38, pp. 313-317, 1969.
R. Ferrari, “The Finite-Element Method, Part 2: P.P. Silvester, an Innovator in Electromagnetic Numerical Modeling,” IEEE Antennas Propagat. Mag., vol. 49, pp. 216-234, 2007.
P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Cambridge University Press, Cambridge, UK, 1983.
F. Kang, “A difference formulation based on the variational principle,” Appl. Math. Comp. Math. (in Chinese), vol. 28, pp. 963-971, 1965.
M. W. Johnson and R. W. McLay, “Convergence of the finite element method in the theory of elasticity,” J. Appl. Mech., vol. 35, pp. 274-278, 1968.
S. L. Sobolev, “Méthode nouvelle à résoudre le problème de Cauchy pour les équations linèaires hyperboliques normales [New method to solve the Cauchy problem for normal hyperbolic linear equations],” Matematicheskii Sbornik - Rec. Math. Moscou (in French), vol. 1, pp. 39-71, 1936.
L. Schwartz, “Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques [Generalization of the notion of function, derivation, Fourier transformation and their mathematical and physical applications],”Annales Univ. Grenoble (in French), vol. 21, pp. 57-74, 1945.
L. Schwartz, Théorie des Distributione [Distribution Theory], Hermann (in French), Paris, F, vol. I, 1950, vol. II, 1951.
M. Zlámal, “On the finite element method,” Numerische Mathematik, vol. 12, pp. 394-409, 1968.
I. Babusˇ
ka and A. K. Aziz, “Survey lectures on the mathematical foundation of the finite element method,” in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz, Academic Press, New York, NY, pp. 5-359, 1972.
Z. J. Cendes and P. P. Silvester, “Numerical solution of dielectric loaded waveguides: I-Finite element analysis,” IEEE Trans. Microw. Theory Tech, vol. 18, pp. 1124-1131, 1970.
D. G. Corr and J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microw. Theory Tech., vol. 20, pp. 669-678, 1972.
A. Konrad, “Triangular finite elements for vector fields in electromagnetics,” Ph.D. Thesis, McGill Univ. 1974.
A. Konrad, “Vector variational formulation of electromagnetic fields in anisotropic media,” IEEE Trans. Microw. Theory Tech., vol. 24, pp. 553-559, 1976.
B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech., vol. 32, pp. 922-928, 1984.
K. D. Paulsen and D. R. Lynch, “Elimination of vector parasites in finite element Maxwell solutions,” IEEE Trans. Microw. Theory Tech., vol. 39, pp. 395-404, 1991.
D. R. Lynch and K. D. Paulsen, “Origin of vector parasites in numerical Maxwell solutions,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 383-394, Mar. 1991.
J. P. Webb, “Efficient generation of divergence-free fields for the finite element analysis of 3D cavity resonances,” IEEE Trans. Magn, vol. 24, pp. 162-165, 1981.
A. Konrad, “A method for rendering 3D finite element vector field solutions non-divergent,” IEEE Trans. Magn, vol. 25, pp. 2822-2834, 1989.
J. C. Nedelec, “Mixed finite elements in R
,” Numerische Mathematik, vol. 35, pp. 315-341, 1980.
M. Hano, “Finite-element analysis of dielectric-loaded waveguides,” IEEE Trans. Microw. Theory Tech., vol. 32, pp. 1275-1279, 1984.
M. L. Barton and Z. J. Cendes, “New vector finite elements for three-dimensional magnetic field computation,” J. Appl. Physics, vol. 61, pp. 3919-3921, 1987.
C. W. Crowley, P. P. Silvester and H. Hurwitz, “Covariant projection elements for 3D vector field problems,” IEEE Trans. Magn., vol. 24, pp. 397-400, 1988.
J. F. Lee, “Analysis of passive microwave devices by using three-dimensional tangential vector finite elements,” Int. J. Numer. Model., vol. 3, pp. 235-246, 1990.
H. Whitney, Geometric Integration Theory. Princeton University Press, Princeton, NJ, 1957.
D. Sun, J. Manges, X. Yuan and Z. Cendes, “Spurious modes in finite-element methods,” IEEE Antennas Propagat. Mag., vol. 37, pp. 12-24,1995.
R. D. Graglia, A. F. Peterson and F. P. Andriulli, “Curl-conforming hierarchical vector bases for triangles and tetrahedra,” IEEE Trans. Antennas Propagat., vol. 59, pp. 950-959, 2011.
R. D. Graglia and G. Lombardi, “Singular higher order complete vector bases for finite methods,” IEEE Trans. Antennas Propagat., vol. 52, pp. 1672-1685, 2004.
A. F. Peterson and R. D. Graglia, “Basis functions for vertex, edge, and corner singularities: a review,” IEEE J. Multiscale Multiphys. Comput. Tech., vol. 1, pp. 161-175, 2016.
K. K. Mei, “Unimoment method of solving antenna and scattering problems,” IEEE Trans. Antennas Propagat., vol. 22, pp. 760-766, 1974.
S. K. Jeng and C. H. Chen, “On variational electromagnetics; theory and application,” IEEE Trans. Antennas Propagat., vol. 32, pp. 902-907, 1984.
A. C. Cangellaris and R. lee, “The bymoment method for two-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propagat., vol. 38, pp. 1429-1437, 1990.
P. P. Silvester and M. S. Hsieh, “Finite-element solution of 2-dimensional exterior-field problems,” IEE Proc. H, vol. 118, pp. 1743-1747, 1971.
J.-M. Jin and J. L. Volakis, “A hybrid finite element method for scattering and radiation by microstrip patch antennas and arrays radiating in a cavity,” IEEE Trans. Antennas Propagat., vol. 39, pp. 1598-1604, 1991.
M. A. Morgan and B. E. Welch, “Field feedback formulation for electromagnetic scattering computations,” IEEE Trans. Antennas Propagat., vol. 34, pp. 1377-1382, 1986.
A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: Numerical consequences and a survey of remedies,” Electromagnetics, vol. 10, pp. 293-312, 1990.
B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp., vol. 31, pp. 629-651, 1977.
A. F. Peterson, “Absorbing boundary conditions for the vector wave equation,” Microw. Opt. Techn. Lett., vol. 1, pp. 62-64, 1988.
J.-M. Jin, L. Volakis and V. V. Liepa, “Ficticious absorber for truncating finite element meshes in scattering,” IEEE Proc. H, vol. 139, pp. 472-476, 1992.
J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Physics, vol. 114, pp. 185-200, 1994.
R. Coccioli, T. Itoh, G. Pelosi and P. P. Silvester, “Finite-element methods in microwaves: a selected bibliography,” IEEE Antennas Propagat. Mag., vol. 38, pp. 34-48, 1996.
R. D. Graglia, G. Pelosi and S. Selleri [Eds.], “International Workshop on Finite Elements for Microwave Engineering - From 1992 to present & Proceedings of the 13th Workshop,” Firenze University Press, Florence, I, 2016
S. Selleri, “13th International workshop on finite elements for microwave engineering [Meeting Reports],” IEEE Antennas Propagat. Mag., vol. 58, pp. 13-14, 2016.
J. T. Oden, “Historical Comments on Finite Elements,” in Stephen G. Nash (ed.), A history of scientific computing, ACM Press, New York, NY, 1990.
S. Selleri, “Finite Elements for Engineers,” Ph.D. course notes; course held at the Politecnico di Milano, Milan, Italy, 2021.
P. P. Silvester and G. Pelosi [Eds.], Finite Elements for Wave Electromagnetics: Methods and Techniques, IEEE Press, New York, NY,1994.
