×

Heritage and early history of the boundary element method. (English) Zbl 1182.65005

Summary: This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green’s functions, Green’s identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s.

MSC:

65-03 History of numerical analysis
65N38 Boundary element methods for boundary value problems involving PDEs
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
74S15 Boundary element methods applied to problems in solid mechanics
76M15 Boundary element methods applied to problems in fluid mechanics
01A60 History of mathematics in the 20th century
01A55 History of mathematics in the 19th century
01A50 History of mathematics in the 18th century
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Altiero, N. J.; Sikarskie, D. L., An integral equation method applied to penetration problems in rock mechanics, (Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, AMD-vol. 11 (1975)), 119-141
[2] Atkinson, K. E., A survey of numerical methods for the solution of fredholm integral equations of the second kind, SIAM (1976) · Zbl 0353.65069
[3] Ayres, D. J., A numerical procedure for calculating stress and deformation near a slit in a three dimensional elastic plastic solid, Eng Fract Mech, 2, 87-106 (1970)
[4] Baker, C. T.H., The numerical treatment of integral equations (1977), Oxford University Press: Oxford University Press Oxford · Zbl 0373.65060
[5] Banaugh, R. P.; Goldsmith, W., Diffraction of steady acoustic waves by surfaces of arbitrary shape, J Acoust Soc Am, 35, 1590-1601 (1963) · Zbl 0134.44704
[6] Banerjee PK. A contribution to the study of axially loaded pile foundations. PhD Thesis. University of Southampton; 1970.; Banerjee PK. A contribution to the study of axially loaded pile foundations. PhD Thesis. University of Southampton; 1970.
[7] Banerjee, P. K.; Butterfield, R., Boundary element method in geomechanics, (Gudehus, G., Finite elements in geomechanics (1977), Wiley: Wiley New York), 529-570, Chapter 16
[8] Belhoste, B., Augustin-Louis Cauchy. A biography (1991), Springer: Springer Berlin · Zbl 0726.01015
[9] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int J Numer Meth Engng, 37, 229-256 (1994) · Zbl 0796.73077
[10] Betti, E., Teoria della elasticità, Nuovo Cimento, Ser, 2, 6-10 (1872) · JFM 04.0504.01
[11] Betti, E., Sopra l’equazioni di equilibrio dei corpi solidi elastici, Annali delle Università Toscane, 10, 143-158 (1874)
[12] Bilby, B. A., Continuous distributions of dislocations, (Sneddon, I. N.; Hill, R., Progress in solid mechanics (1960), North-Holland: North-Holland Amsterdam), 329-398
[13] Birkhoff, G. D., Oliver Dimon Kellogg—in memoriam, Bull Am Math Soc, 39, 171-177 (1933) · JFM 59.0042.07
[14] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J Numer Anal, 22, 644-669 (1985) · Zbl 0579.65121
[15] Boyer, C. B., A history of mathematics (1991), Wiley: Wiley New York · Zbl 0182.30401
[16] Brady, B. H.G.; Bray, J. W., Boundary element method for determining stresses and displacements around long opening in a triaxial stress-field, Int J Rock Mech Min Sci, 15, 21-28 (1978)
[17] Brady, B. H.G.; Bray, J. W., Boundary element method for elastic analysis of tabular orebody extraction, assuming complete plane strain, Int J Rock Mech Min Sci, 15, 29-37 (1978)
[18] Brebbia CA. On hyperbolic paraboloidal shells. PhD Thesis, University of Southampton; 1968.; Brebbia CA. On hyperbolic paraboloidal shells. PhD Thesis, University of Southampton; 1968.
[19] Brebbia, C. A., The boundary element method for engineers (1978), Pentech Press/Halstead Press: Pentech Press/Halstead Press London/New York · Zbl 0414.65060
[20] (Brebbia, C. A., Recent advances in boundary element methods. Recent advances in boundary element methods, Proceedings of first internatonal conference on boundary element methods, University of Southampton (1978)) · Zbl 0384.00014
[21] Brebbia, C. A., Introduction to boundary element methods, (Brebbia, C. A., Recent advances in boundary element methods, Proceedings of first international confeerence on boundary element methods (1978), University of Southampton: University of Southampton Pentech), 1-42
[22] Brebbia, C. A., Tribute to Tom Cruse, Tom Cruse Commemorative Issue, (Brebbia, C. A.; Cheng, A. H.-D., Eng Anal Bound Elem, 24 (2000)), 701 · Zbl 0968.00074
[23] Brebbia CA. personal communication; 2004.; Brebbia CA. personal communication; 2004.
[24] Brebbia, C. A.; Connor, J. J., Fundamentals of finite element techniques for structural engineers (1973), Butterworth: Butterworth London
[25] Brebbia, C. A.; Dominguez, J., Boundary element methods for potential problems, Appl Math Modell, 1, 372-378 (1977) · Zbl 0373.31007
[26] Brebbia, C. A., Boundary elements XXVI, Proceedings of 26th international conference on boundary element methods (2004), WIT Press: WIT Press Bologna, Italy · Zbl 1051.74001
[27] Brebbia, C. A.; Tottenham, H., The first international conference on variational methods in engineering, University of Southampton, UK (1972)
[28] Brunner, H., 1896-1996: One hundred years of Volterra integral equations of the first kind, Appl Numer Math, 24, 83-93 (1997) · Zbl 0879.45001
[29] Bückner, H. F., Die praktische Behandlung von Integralgleichungen (1952), Springer: Springer Berlin · Zbl 0048.35701
[30] Bühler, W. K.; Gauss, A., Gauss, a biographical study (1981), Springer: Springer Berlin · Zbl 0455.01001
[31] Burton, D. M., The history of mathematics—an introduction (1985), Allyn and Bacon: Allyn and Bacon Newton, MA · Zbl 0657.01003
[32] (Cahan, D., Hermann von Helmholtz and the foundation of nineteenth century science (1993), U. California Press: U. California Press Berkeley, CA) · Zbl 0868.01019
[33] Cannell, D. M., George Green, mathematician and physicist 1793-1841, the background to his life and work (2001), SIAM · Zbl 0973.01097
[34] Cauchy A-L. Mémoire sur les intégrales définies, Mémoires des divers savants, ser 2 1;1827:601-799 [written 1814].; Cauchy A-L. Mémoire sur les intégrales définies, Mémoires des divers savants, ser 2 1;1827:601-799 [written 1814].
[35] Cauchy A-L. Leçons sur les applications de calcul infinitésimal à la géométries; 1826-1828.; Cauchy A-L. Leçons sur les applications de calcul infinitésimal à la géométries; 1826-1828.
[36] Chen, L. H.; Schweikert, D. G., Sound radiation from an arbitrary body, J Acoust Soc Am, 35, 1626-1632 (1963)
[37] Chertock, G., Sound radiation from vibrating surfaces, J Acoust Soc Am, 36, 1305-1313 (1964)
[38] Clough RW. The finite element method in plane stress analysis. Proceedings of second ASCE Conference on electronic computation; 1960.; Clough RW. The finite element method in plane stress analysis. Proceedings of second ASCE Conference on electronic computation; 1960.
[39] Connor, J. J.; Brebbia, C. A., Finite element techniques for fluid flow (1976), Newnes-Butterworths: Newnes-Butterworths London · Zbl 0431.76001
[40] Copley, L. G., Integral equation method for radiation from vibrating bodies, J Acoust Soc Am, 41, 807-816 (1967)
[41] Copley, L. G., Fundamental results concerning integral representation in acoustic radiation, J Acoust Soc Am, 44, 28-32 (1968) · Zbl 0162.57204
[42] Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J Math Anal, 19, 613-626 (1988) · Zbl 0644.35037
[43] Cruse TA. The transient problem in classical elastodynamics solved by integral equations. Doctoral dissertation. University of Washington; 1967, 117 pp.; Cruse TA. The transient problem in classical elastodynamics solved by integral equations. Doctoral dissertation. University of Washington; 1967, 117 pp.
[44] Cruse, T. A., A direct formulation and numerical solution of the general transient elastodynamic problem—II, J Math Anal Appl, 22, 341-355 (1968) · Zbl 0167.16302
[45] Cruse, T. A., Numerical solutions in three dimensional elastostatics, Int J Solids Struct, 5, 1259-1274 (1969) · Zbl 0181.52404
[46] Cruse, T. A., Lateral constraint in a cracked, three-dimensional elastic body, Int J Fract Mech, 6, 326-328 (1970)
[47] Cruse, T. A.; VanBuren, W., Three-dimensional elastic stress analysis of a fracture specimen with an edge crack, Int J Fract Mech, 7, 1-15 (1971)
[48] Cruse, T. A., (Brebbia, C. A.; Tottenham, H., Application of the boundary-integral equation solution method in solid mechanics. Application of the boundary-integral equation solution method in solid mechanics, Variational method in engineering, vol. II, Proceedings of an international conference (1972), Southampton University Press: Southampton University Press Southampton), 9-929
[49] Cruse, T. A., Boundary-integral equation fracture mechanics analysis, (Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, AMD-vol. 11 (1975)), 31-46
[50] Cruse, T. A., BIE fracture mechanics: 25 years of developments, Comp Mech, 18, 1-11 (1996) · Zbl 0946.74073
[51] Cruse, T. A., Boundary integral equations—a personal view, (Brebbia, C. A.; Cheng, A. H.-D., Tom Cruse commemorative issue: III. Tom Cruse commemorative issue: III, Eng Anal Bound Elem, 25 (2001)), 709-712 · Zbl 1102.74300
[52] Cruse TA, Lachat J-C. Proceedings of the international symposium on innovative numerical analysis in applied engineering sciences, Versailles, France; 1977.; Cruse TA, Lachat J-C. Proceedings of the international symposium on innovative numerical analysis in applied engineering sciences, Versailles, France; 1977.
[53] Cruse, T. A.; Rizzo, F. J., A direct formulation and numerical solution of the general transient elastodynamic problem—I, J Math Anal Appl, 22, 244-259 (1968) · Zbl 0167.16301
[54] Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, Applied mechanics conference, ASME, Rensselaer Polytechnic Institute, June 23-25,, AMD-vol. 11 (1975)
[55] Delves, L. M.; Mohamed, J. L., Computational methods for integral equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
[56] Dominguez J. Stress analysis around anchor plates: a boundary element method application. PhD Thesis. Universidad de Sevilla; 1977.; Dominguez J. Stress analysis around anchor plates: a boundary element method application. PhD Thesis. Universidad de Sevilla; 1977.
[57] Dominguez J. personal communication; 2003.; Dominguez J. personal communication; 2003.
[58] Dunham, W., Euler, the master of us all, Math Assoc Am (1999) · Zbl 0951.01012
[59] Ehrlich P. Kellogg, Bliss, Hedrick—Mizzou math pioneers. Critical Points, 4. Department of Mathematics, University of Missouri; 1999 [http://www.math.missouri.edu/∼;news/issue4/pioneers.html].; Ehrlich P. Kellogg, Bliss, Hedrick—Mizzou math pioneers. Critical Points, 4. Department of Mathematics, University of Missouri; 1999 [http://www.math.missouri.edu/∼;news/issue4/pioneers.html].
[60] Erdogan, F.; Gupta, G. D., Numerical solution of singular integral-equations, Q J Appl Math, 29, 525-534 (1972) · Zbl 0236.65083
[61] Erdogan, F.; Gupta, G. D.; Cook, T. S., Numerical solution of singular integral equations, (Sih, G. C., Mechanics of fracture 1, methods of analysis and solutions of crack problems (1973)), 368-425
[62] Eshelby, J. D., The continuum theory of lattice defects, Solid State Phys, 3, 79-114 (1956)
[63] Euler, L., Principes géneraux du mouvement des fluides, Mém Acad Sci Berlin, 11, 274-315 (1755)
[64] Euler L. Opera Omnia, Series prima: Opera mathematica (29 volumes), Series secunda: Opera mechanica et astronomica (31 volumes), Series tertia: Opera physica, Miscellanea (12 volumes), Series quarta: A. Commercium epistolicum (10 volumes), edited by the Euler Committee of the Swiss Academy of Science. Basel: Birkhäuser; 2003.; Euler L. Opera Omnia, Series prima: Opera mathematica (29 volumes), Series secunda: Opera mechanica et astronomica (31 volumes), Series tertia: Opera physica, Miscellanea (12 volumes), Series quarta: A. Commercium epistolicum (10 volumes), edited by the Euler Committee of the Swiss Academy of Science. Basel: Birkhäuser; 2003.
[65] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv Comput Math, 9, 69-95 (1998) · Zbl 0922.65074
[66] Fourier JBJ. Théorie Analytique de la Chaleur (The analytical theory of heat); 1822.; Fourier JBJ. Théorie Analytique de la Chaleur (The analytical theory of heat); 1822.
[67] Fredholm, I., Sur une classe d’équations fonctionelles, Acta Math, 27, 365-390 (1903) · JFM 34.0422.02
[68] Friedman, M. B.; Shaw, R., Diffraction of pulse by cylindrical obstacles of arbitrary cross section, J Appl Mech, Trans ASME, 29, 40-46 (1962) · Zbl 0108.40204
[69] Fritzius, R. S., (Hsu, J. P.; Zhang, Y. Z., Abbreviated biographical sketch of Walter Ritz, Appendix. Abbreviated biographical sketch of Walter Ritz, Appendix, Lorentz and Poincaré invariance—100 years of realtivity (2003), World Scientific Publishers: World Scientific Publishers Singapore)
[70] Gauss, C. F., Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractata, Commentationes societatis regiae scientiarum Gottingensis recentiores, II, 2-5 (1813)
[71] Gavelya, S. P., Periodical problems for shallow shells of arbitrary curvature with apertures, Rep Ukrain Acad Sci, 8, 703-708 (1969) · Zbl 0175.22903
[72] Gegelia, T.; Jentsch, L., Potential methods in continuum mechanics, Georgian Math J, 1, 6, 599-640 (1994) · Zbl 0815.35120
[73] (Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; Küstner, H., The VNR concise encyclopedia of mathematics (1989), Van Nostrand Reinhold: Van Nostrand Reinhold New York) · Zbl 0744.00007
[74] Gerasoulis, A., The use of piecewise quadratic polynomials for the solution of singular integral equations of the Cauchy type, Comp Math Appl, 8, 15-22 (1982) · Zbl 0467.65071
[75] Geroski, P. A., Models of technology diffusion, Res Policy, 29, 603-625 (2000)
[76] Gillispie, C. C., Pierre-Simon Laplace, 1749-1827, a life in exact science (1997), Princeton University Press: Princeton University Press Princeton
[77] Golberg, M. A., A survey of numerical methods for integral equations, (Golberg, M. A., Solution methods for integral equations (1978), Plenum Press: Plenum Press New York), 1-58, Chapter 1
[78] Gorgidze, A. Y.; Rukhadze, A. K., On the numerical solution of the integral equations of the plane theory of elasticity, Soobshchenia Gruz. Filiala A.N.S.S.S.R., 1, 255-258 (1940), [in Russian]
[79] Grabiner, J. V., The origin of Cauchy’s rigorous calculus (1981), MIT Press: MIT Press Cambridge, MA · Zbl 0517.01002
[80] Grattan-Guinness, I., Joseph Fourier, 1768-1830 (1972), MIT Press: MIT Press Cambridge, MA · Zbl 0245.01008
[81] Green G. An essay on the application of mathematical analysis to the theories of electricity and magnetism. Printed for the Author by Wheelhouse T. Nottigham; 1828. 72 p. Also, Mathematical papers of George Green. Chelsea Publishing Co.; 1970. p. 1-115.; Green G. An essay on the application of mathematical analysis to the theories of electricity and magnetism. Printed for the Author by Wheelhouse T. Nottigham; 1828. 72 p. Also, Mathematical papers of George Green. Chelsea Publishing Co.; 1970. p. 1-115.
[82] Green, G., (Ferrers, N. M., Mathematical papers of George Green (1871)), [reprinted by Chelsea Publ. Co., New York, 1970]
[83] Greenberg, M. D., Application of Green’s functions in science and engineering (1971), Prentice Hall: Prentice Hall Englewood Cliffs, NJ
[84] Gupta, K. K.; Meek, J. L., A brief history of the beginning of the finite element method, Int J Numer Meth Eng, 39, 3761-3774 (1996) · Zbl 0884.73067
[85] Hadamard, J., Theorie des équations aux dérivées partielles linéaires hyperboliques et du problème de Cauchy, Acta Math, 31, 333-380 (1908) · JFM 39.0425.02
[86] Hahn R. Laplace as a Newtonian Scientist, a paper delivered at a Seminar on the Newtonian influence. Clark Library, UCLA; 1967.; Hahn R. Laplace as a Newtonian Scientist, a paper delivered at a Seminar on the Newtonian influence. Clark Library, UCLA; 1967.
[87] Helmholtz, H., Theorie der Luftschwingungen in Röhren mit offenen Enden, Journal für die reine und angewandte Mathematik, 57, 1-72 (1860) · ERAM 057.1499cj
[88] Helmholtz H. Die Lehre von den Tonempfindungen, (On the Sensations of Tone); 1863.; Helmholtz H. Die Lehre von den Tonempfindungen, (On the Sensations of Tone); 1863.
[89] Hess, J. L., Calculations of potential flow about bodies of revolution having axes perpendicular to the free stream direction, J Aero Sci, 29, 726-742 (1962)
[90] Hess, J. L., Review of integral-equation techniques for solving potential-flow problems with emphasis on the surface source method, Comp Meth Appl Mech Eng, 5, 145-196 (1975) · Zbl 0299.76011
[91] Hess, J. L.; Smith, A. M.O., Calculations of nonlifting potential flow about arbitrary three-dimensional bodies, J Ship Res, 8, 22-44 (1964)
[92] Ivanov, V. V., The theory of approximate methods and their application to the numerical solution of singular integral equations (1976), Noordhoff, [Russian edition 1968] · Zbl 0346.65065
[93] Jaswon, M. A., Integral equation methods in potential theory I, Proc R Soc, A, 275, 23-32 (1963) · Zbl 0112.33103
[94] Jaswon, M. A., An introduction to mathematical crystallography (1965), American Elsevier: American Elsevier New York · Zbl 0131.46504
[95] Jaswon, M. A.; Ponter, A. R., An integral equation solution of the torsion problem, Proc R Soc, A, 273, 237-246 (1963) · Zbl 0112.16703
[96] Jaswon, M. A.; Rose, M. A., Crystal symmetry: theory of colour crystallography (1983), Halsted Press: Halsted Press New York · Zbl 0507.20001
[97] Jaswon, M. A.; Symm, G. T., Integral equation methods in potential theory and elastostatics (1977), Academic Press: Academic Press London · Zbl 0414.45001
[98] Kagiwada, H.; Kalaba, R. E., Integral equations via imbedding methods (1974), Addison-Wesley: Addison-Wesley New York · Zbl 0331.45001
[99] Katz, V. J., The history of Stokes theorem, Math Mag, 52, 146-156 (1979) · Zbl 0439.26001
[100] Kellogg, O. D., Foundations of potential theory (1953), Dover · Zbl 0053.07301
[101] Kelvin, W. T., Note on the integrations of the equations of equilibrium of an elastic solid, Cambridge Dublin Math J, 3 (1848)
[102] Kolosov, G. V., On an application of complex function theory to a plane problem of the mathematical theory of elasticity (1909), Yuriev, [in Russian]
[103] Kulakov, V. M.; Tolkachev, V. M., Bending of plates of an arbitrary shape, Rep Russ Acad Sci, 230 (1976)
[104] Kupradze, V. D., Potential methods in the theory of elasticity, (Sneddon, I. N.; Hills, R., Israeli program for scientific translation, 1965 (1965)), [earlier edition: Kupradze VD, Dynamical problems in elasticity, vol. III in Progress in solid mechanics, eds. Sneddon IN, Hills R. North-Holland; 1963]
[105] Kupradze, V. D.; Aleksidze, M. A., The method of functional equations for the approximate solution of some boundary value problems, Zh vichisl mat i mat fiz, 4, 683-715 (1964), [in Russian]
[106] Kupradze, V. D.; Gegelia, T. G.; Basheleishvili, M. O.; Burchuladze, T. V., Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity (1979), North-Holland, [Russian edition 1976]
[107] Lachat JC. A further development of the boundary integral technique for elastostatics. Doctoral dissertation. University of Southampton; 1975.; Lachat JC. A further development of the boundary integral technique for elastostatics. Doctoral dissertation. University of Southampton; 1975.
[108] Lachat, J. C.; Watson, J. O., Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics, Int J Numer Meth Eng, 10, 991-1005 (1976) · Zbl 0332.73022
[109] Laplace PS. Histoire de l’Académie des Sciences de Paris, p. 135, 1782/85; p. 252, 1787/89.; Laplace PS. Histoire de l’Académie des Sciences de Paris, p. 135, 1782/85; p. 252, 1787/89.
[110] Lagrange, J., Miscellanea Taurinasia, 2, 273 (1760)
[111] Lagrange, J.-L., Mémoires de l’Académie Royale des Sciences de Paris, Savants étrangèrs, VII (1773)
[112] Lebesgue, H., Sur des cas d’impossibilité du problème de Dirichlet, Comptes Rendus de la Société Mathématique de France, 41, 17 (1913) · JFM 44.0456.01
[113] Lotz, I., Calculation of potential flow past airship body in a yaw, NACA TM 675 (1932)
[114] Love, A. E.H., A treatise on the mathematical theory of elasticity (1944), Dover: Dover New York · Zbl 0063.03651
[115] Massonet, C. E., (Zienkiewicz, O. C.; Hollister, G. S., Numerical use of integral procedures. Numerical use of integral procedures, Stress analysis (1965), Wiley: Wiley New York), 198-235, Chapter 10
[116] Mathon, R.; Johnston, R. L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J Numer Anal, 14, 638-650 (1977) · Zbl 0368.65058
[117] Maxwell, J. C., On the calculation of the equilibrium and stiffness of frames, Phil Mag, 27, 294-299 (1864)
[118] Maz’ya, V.; Shaposhnikova, T., Jacques Hadamard, a universal mathematician, Am Math Soc (1998) · Zbl 0906.01031
[119] Mendelson, A.; Albers, L. U., Application of boundary integral equation to elastoplastic problems, (Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, AMD-vol. 11 (1975)), 47-84
[120] Mikhlin, S. G., Integral equations and their applications to certain problems in mechanics, mathematical physics and technology (1964), Pergamon Press: Pergamon Press New York, [first Russian edition 1949] · Zbl 0117.31902
[121] Mikhlin, S. G.; Smolitsky, J. L., Approximation methods for the solution of differential and integral equations (1967), Elsevier: Elsevier Amsterdam
[122] Mitzner, K. M., Numerical solution for transient scattering from a hard surface of arbitrary shape—retarded potential technique, J Acoust Soc Am, 42, 391-397 (1967) · Zbl 0149.45903
[123] Morse, P. M.; Feshbach, H., Methods of theoretical physics (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.40603
[124] Muskhelishvili, N. I., On the numerical solution of the plane problems of the theory of elasticity, Trudy Tbilisskogo Matematicheskogo Instituta, 1, 83-87 (1937), [in Georgian]
[125] Muskhelishvili, N. I., Singular integral equations (1953), Noordhoff, [Russian edition 1946] · Zbl 0051.33203
[126] Muskhelishvili, N. I., Some basic problems of the mathematical theory of elasticity (1959), Noordhoof, [first Russian edition 1933]
[127] Oden, J. T., Historical comments on finite elements, (Nash, S. G., A history of scientific computing (1990), ACM Press), 152-166
[128] O’Connor JJ, Robertson EF. MacTutor history of mathematics archive, [http://www-groups.dcs.st-and.ac.uk/history/; 2003.; O’Connor JJ, Robertson EF. MacTutor history of mathematics archive, [http://www-groups.dcs.st-and.ac.uk/history/; 2003.
[129] O’Donnell, S., William Rowan Hamilton—portrait of a prodigy (1983), Boole Press: Boole Press Dublin · Zbl 0536.01022
[130] Oliveira, E. R.A., Plane stress analysis by a general integral method, J Eng Mech Div, ASCE, 94, 79-101 (1968)
[131] Peaceman, D. W., A personal retrospection of reservoir simulation, (Nash, S. G., A history of scientific computing (1990), ACM Press), 106-129
[132] Poisson, P.-S., Bulletin de in soclété philomatique, 3, 388 (1813)
[133] Ponter, A. R.S., An integral equation solution of the inhomogeneous torsion problem, J SIAM Appl Math, 14, 819-830 (1966) · Zbl 0146.21705
[134] Prager, W., Die Druckverteilung an Körpern in ebener Potentialströmung, Physik Zeitschr., 29, 865 (1928) · JFM 54.0900.04
[135] Rankine, W. J.M., On plane water-lines in two dimensions, Phil Trans (1864)
[136] Rayleigh, J. W.S., Proc London Math Soc, 4, 357-368 (1873)
[137] Rayleigh, J. W.S., The theory of sound, vols 1 and 2 (1945), Dover: Dover New York, [originally published in 1877]
[138] Rayleigh, R. J.S., Life of John William Strutt, Third Baron Rayleigh (1924), London
[139] Reissner, E., On a variational theorem in elasticity, J Math Phys, 29, 90-95 (1950) · Zbl 0039.40502
[140] Ritz, W., Über eine neue methode zur Lösung gewissen variations—Problems der mathematischen physik, J Reine Angew Math, 135, 1-61 (1908) · JFM 39.0449.01
[141] Rizzo FJ. Some integral equation methods for plane problems of classical elastostatics. Doctoral dissertation. University of Illinois, Urbana-Champaign; 1964, p. 34.; Rizzo FJ. Some integral equation methods for plane problems of classical elastostatics. Doctoral dissertation. University of Illinois, Urbana-Champaign; 1964, p. 34.
[142] Rizzo, F. J., An integral equation approach to boundary value problems of classical elastostatics, Q Appl Math, 25, 83-95 (1967) · Zbl 0158.43406
[143] Rizzo, F. J., The boundary element method, some early history—a personal view, (Beskos, D. E., Boundary element methods in structural analysis (1989), ASCE), 1-16
[144] Rizzo, F. J., In honor of T.A. Cruse, (Brebbia, C. A.; Cheng, A. H.-D., Tom Cruse Commemorative Issue. Tom Cruse Commemorative Issue, Eng Anal Bound Anal, 24 (2000)), 703-705 · Zbl 0968.01016
[145] Rizzo, F. J., Springs, formulas and flatland: a path to boundary integral methods in elasticity, Electron J Bound Elem, 1, 1-7 (2003), [http://tabula.rutgers.edu/EJBE/]
[146] Rizzo, F. J.; Shippy, D. J., Formulation and solution procedure for the general non-homogeneous elastic inclusion problem, Int J Solids Struct, 4, 1161-1179 (1968) · Zbl 0253.73010
[147] Rizzo, F. J.; Shippy, D. J., A method for stress determination in plane anisotropic bodies, J Compos Mater, 4, 36-61 (1970)
[148] Rizzo, F. J.; Shippy, D. J., A method of solution for certain problems of transient heat conduction, AIAA J, 8, 2004-2009 (1970) · Zbl 0237.65074
[149] Rizzo, F. J.; Shippy, D. J., An application of the correspondence principle of linear viscoelasticity theory, SIAM J Appl Math, 21, 321-330 (1971) · Zbl 0211.28303
[150] Schenck, H. A., Improved integral formulation for acoustic radiation problems, J Acoust Soc Am, 44, 41-58 (1968) · Zbl 0187.50302
[151] Sharlin, H. I., Kelvin, William Thomson, Baron,, Encyclopædia Britannica (2003)
[152] Shaw, R. P., Diffraction of acoustic pulses by obstacles of arbitrary shape with a Robin boundary condition, J Acoust Soc Am, 41, 855-859 (1967)
[153] Shaw, R. P., Boundary integral equation methods applied to water waves, (Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, AMD-vol. 11 (1975)), 7-14
[154] Shaw, R. P., A history of boundary elements, in Boundary Elements XV, (Brebbia, C. A.; Rencis, J. J., Fluid flow and computational aspects, Worcester, Massachusetts, vol. 1 (1993), CMP/Elsevier), 265-280
[155] Shippy, D. J., Application of the boundary-integral equation method to transient phenomena in solids, (Cruse, T. A.; Rizzo, F. J., Boundary-integral equation method: computational applications in applied mechanics, AMD-vol. 11 (1975)), 15-30
[156] Shippy, D. J., Early development of the BEM at the University of Kentucky, Electron J Bound Elem, 1, 26-33 (2003), [http://tabula.rutgers.edu/EJBE/]
[157] Somigliana, C., Sopra l’equilibrio di un corpo elastico isotropo, Nuovo Cimento, ser, 3, 17-20 (1885)
[158] Somigliana, C., Sulla teoria delle distorsioni elastiche, Note I e II Atti Accad Naz Lincei Classe Sci Fis Mat e Nat, 23, 463-472 (1914) · JFM 45.1030.02
[159] Somigliana, C., Sulla teoria delle distorsioni elastiche, Note I e II Atti Accad Naz Lincei Classe Sci Fis Mat e Nat, 24, 655-666 (1915) · JFM 45.1031.01
[160] Stein, E., An appreciation of Erich Trefftz, Comput Assist Mech Eng Sci, 4, 301-304 (1997) · Zbl 0947.01015
[161] Sternberg, W. J.; Smith, T. L., The theory of potential and spherical harmonics (1944), University of Toronto Press · Zbl 0063.07182
[162] Stokes GG. A Smith’s prize paper. Cambridge University Calendar; 1854.; Stokes GG. A Smith’s prize paper. Cambridge University Calendar; 1854.
[163] Swedlow, J. L.; Cruse, T. A., Formulation of boundary integral equations for three-dimensional elasto-plastic flow, Int J Solids Struct, 7, 1673-1683 (1971) · Zbl 0228.73040
[164] Symm, G. T., Integral equation methods in potential theory, II, Proc R Soc, A, 275, 33-46 (1963) · Zbl 0112.33201
[165] Symm GT. Integral equation methods in elasticity and potential theory. Doctoral dissertation. Imperial College; 1964.; Symm GT. Integral equation methods in elasticity and potential theory. Doctoral dissertation. Imperial College; 1964.
[166] Tazzioli, R., Green’s function in some contributions of 19th century mathematicians, Historia Mathematica, 28, 232-252 (2001) · Zbl 0995.01003
[167] Thompson, S. P., The life of William Thomson, Baron Kelvin of Largs, vols. 1 and 2 (1910), Macmillan: Macmillan New York · JFM 41.0019.01
[168] Timoshenko, S. P., History of strength of materials (1983), Dover
[169] Tomlin GR. Numerical analysis of continuum problems in zoned anisotropic media. PhD Thesis. University of Southampton; 1973.; Tomlin GR. Numerical analysis of continuum problems in zoned anisotropic media. PhD Thesis. University of Southampton; 1973.
[170] Tracey, D. M., Finite elements for determination of crack tip elastic stress intensity factors, Eng Fract Mech, 3, 255-265 (1971)
[171] Trefftz, E., Ein Gegenstück zum Ritz’schen verfahren, Verh d.2. Intern Kongr f Techn Mech (Proc second international congress applied mechanics), Zurich, 131-137 (1926) · JFM 52.0483.02
[172] Trefftz, E., Konvergenz und fehlerabschatzung beim Ritz’schen verfahren, Math Ann, 100, 503-521 (1928) · JFM 54.0537.01
[173] Tricomi GF. Matematici italiani del primo secolo dello stato unitario, Memorie dell’Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, ser IV I;1962.; Tricomi GF. Matematici italiani del primo secolo dello stato unitario, Memorie dell’Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, ser IV I;1962.
[174] (Twaites, B., Incompressible aerodynamics (1960), Oxford University Press: Oxford University Press Oxford)
[175] Twomey, S., On numerical solution of Fredholm integral equations of first kind by inversion of linear system produced by quadrature, J ACM, 10, 97-101 (1963) · Zbl 0125.36102
[176] Vainberg, D. V.; Sinyavskii, A. L., Application of the potential method for numerical analysis of deformation in cylindrical shells, Rep Ukrain Acad Sci, 7, 907-912 (1960)
[177] Vandrey, F., On the calculation of the transverse potential flow past a body of revolution with the aid of the method of Mrs Flügge-Lotz, Astia AD40 089 (1951)
[178] Vandrey, F., A method for calculating the pressure distribution of a body of revolution moving in a circular path through a prefect incompressible fluid, Aero Res Council R&M No. 3139 (1960)
[179] Vekua, I. N., New methods for solving elliptic equations (1967), North-Holland: North-Holland Amsterdam · Zbl 0146.34301
[180] Vekua, N. P., Systems of singular integral equations and some boundary value problems (1967), Noordhoof
[181] Veriuzhskii, Y. V., Numerical potential methods in some problems of applied mechanics (1978), Kiev State University Publishers
[182] Volterra, V., Sulla inversione degli integrali multipli, Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur, 5, 289-300 (1896) · JFM 27.0309.02
[183] Volterra, V., Sur l’équilibre des carps élastiques multiplement connexes, Anna Sci de l’École Norm Super, Paris, 24, 401-517 (1907) · JFM 38.0814.01
[184] von Kármán, T., Berechnung der druckverteilung an luftschiffkörpern, Abhandl Aerodynamischen Inst Tech Hoch Aachen, 6, 3-13 (1927) · JFM 53.0810.16
[185] von Kármán, T., Calculation of pressure distribution on airship hulls, NACA TM 574 (1930)
[186] von Kármán T. (with Edson L.). The wind and beyond. Little Brown and Co.; 1967.; von Kármán T. (with Edson L.). The wind and beyond. Little Brown and Co.; 1967.
[187] Washizu, K., Variational methods in elasticity and plasticity (1968), Pergamon Press · Zbl 0164.26001
[188] Waterman, P. C., Matrix formulation of electromagnetic scattering, Proc IEEE, 53, 805-812 (1965) · Zbl 1400.78014
[189] Waterman, P. C., New formulation of acoustic scattering, J Acoust Soc Am, 45, 1417-1429 (1969) · Zbl 0187.24504
[190] Watson JO. The analysis of stress in thick shells with holes, by integral representation with displacement. PhD Thesis. University of Southampton; 1973.; Watson JO. The analysis of stress in thick shells with holes, by integral representation with displacement. PhD Thesis. University of Southampton; 1973.
[191] Watson, J. O., Boundary elements from 1960 to the present day, Electron J Bound Elem, 1, 34-36 (1960), [http://tabula.rutgers.edu/EJBE/]
[192] Williams LP. Helmholtz, Hermann von. Encyclopædia Britannica; 2003.; Williams LP. Helmholtz, Hermann von. Encyclopædia Britannica; 2003.
[193] Young, D. M., A historical review of iterative methods, (Nash, S. G., A history of scientific computing (1990), ACM Press), 180-194
[194] Zienkiewicz, O. C.; Taylor, R. L., Finite element method (2000), Butterworth-Heinemann · Zbl 0991.74003
[195] Science Citation Index Expanded, the Web of Science, Institute for Scientific Information, http://www.isinet.com/isi/products/citation/scie/index.html; 2003.; Science Citation Index Expanded, the Web of Science, Institute for Scientific Information, http://www.isinet.com/isi/products/citation/scie/index.html; 2003.
[196] Niko Muskhelishvili, Curriculum Vitae, http://www.rmi.acnet.ge/person/muskhel/; 2003.; Niko Muskhelishvili, Curriculum Vitae, http://www.rmi.acnet.ge/person/muskhel/; 2003.
[197] A survey of Muskhelishvili’s scientific heritage, http://kr.cs.ait.ac.th/∼;radok/mus/mus10.htm; 2003.; A survey of Muskhelishvili’s scientific heritage, http://kr.cs.ait.ac.th/∼;radok/mus/mus10.htm; 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.