Skip to main content

Abstract

In recent years, by a boundary method, it is usually understood a numerical procedure in which a subregion or the entire region, is left out of the numerical treatment, by use of available analytical solutions (or, more generally, previously computed solutions). Boundary methods reduce the dimensions involved in the problem leading to considerable economy in the numerical work and constitute a very convenient manner of treating adequately unbounded regions by numerical means. Generally, the dimensionality of the problem is reduced by one, but even when part of the region is treated by finite elements, the size of the discretized domain is reduced [1–2].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Zienkiewicz, O. C., The Finite Element Method in Engineering Science. McGraw-Hill, New York 1977

    Google Scholar 

  2. Zienkiewicz, O. C., Kelly, D. W., Bettess, P., The coupling of the finte element method and boundary solution procedures. Int. J. Num. Meth. Eng., 11, pp. 355–375, 1977

    Article  MathSciNet  MATH  Google Scholar 

  3. Brebbia, C. A., The Boundary Element Method for Engineers. Pentech Press, London 1978

    Google Scholar 

  4. Brebbia, C. A., Boundary element methods. Proc. Third Int. Seminar, Irvine, Ca., 1981; Proc. Fourth Int. Seminar, Southampton 1982, Springer-Verlag, Berlin-Heidelberg, New York

    Google Scholar 

  5. Sânchez-Sesma, F. J., Herrera, I., Avilés, J., Boundary methods for elastic wave diffraction-application to scattering of SH waves by surface irregularities. Bull. Seism. Soc. Am., 72 (2), pp. 473–490, 1982

    Google Scholar 

  6. Rektorys, K, Survey of Applicable Mathematics. ILIFFE Books Ltd, London 1969

    MATH  Google Scholar 

  7. Trefftz, E., Ein Gegenstück zum Ritz’schen Verfahren, Proc. Second Int. Congress Appl. Mech., Zürich 1926

    Google Scholar 

  8. Herrera, I., Boundary Methods for Fluids. Finite Elements for Fluids. IV, Gallagher, R. H. (Ed.), John Wiley & Sons Limited, Chapter 19, pp. 403–432, 1982

    Google Scholar 

  9. Mikhlin, S. G., Variational methods in mathematical physics. Pergamon Press, 1964

    Google Scholar 

  10. Rektorys, K., Variational methods in mathematics, science and engineering. D. Reidel Pub., Co., 1977

    Google Scholar 

  11. Kupradze, V. D. et al., Three dimensional problems of the mathematical theory of elasticity and thermoelasticity. North Holland, 1979

    Google Scholar 

  12. Amerio, L., Sul calculo delle autosoluzioni dei problemi al contorno per le equazioni lineari del secondo ordine di tipo ellitico. Rend. Acc. Lincei, 1, pp. 352–359 and 505–509, 1946

    Google Scholar 

  13. Fichera, G., Teoremi di completezza sulla frontiera di un dominio per taluni sistema di funzioni. Ann. Mat. Pura e Appl, 27, pp. 1–28, 1948

    Article  MathSciNet  MATH  Google Scholar 

  14. Picone, M., Nouvi metodi risolutivi per i problemi d’integrazione delle equazioni lineari a derivati parziali e nuova applicazione delle transformate multipla di Laplace nel caso delle equazioni a coefficienti constanti. Atti Acc. Sc. Torino, 76, pp. 413–426, 1940

    MathSciNet  Google Scholar 

  15. Kupradze, V. D., On the approximate solution of problems in mathematical physics. Russian Math. Surveys, 22, (2), pp. 58–108, 1967, (Uspehi Mat. Nauk, 22 (2), pp. 59107, 1967 )

    MathSciNet  Google Scholar 

  16. Vekua, I. N., New methods for solving elliptic equations. North Holland Pub. Co., 1967

    Google Scholar 

  17. Colton, D., Watzlawek, W., Complete families of solutions to the heat equation and generalized heat equation in R n. Jour. Differential Equations, 25 (1), pp. 96–107, 1977

    Article  MathSciNet  Google Scholar 

  18. Colton, D., The approximation of solutions to initial boundary value problems for parabolic equations in one space variable. Quart. Appl. Math., 34 (4), pp. 377–386, 1976

    MathSciNet  Google Scholar 

  19. Herrera, I., Boundary Methods. An Algebraic Theory, Pitman Publishing Co., 1984

    MATH  Google Scholar 

  20. Sabina, F. J., Herrera, I., England, R., Theory of connectivity: Applications to scattering of seismic waves. SH-wave motion, Proc. Second Int. Conference on Microzonation, San Francisco, Ca., 1979

    Google Scholar 

  21. England, R., Sabina, F. J., Herrera, I., Scattering of SH-waves by surface cavities of arbitrary shape using boundary methods. Physics of the Earth and Planetary Interiors, 21, pp. 148–157, 1980

    Article  Google Scholar 

  22. Herrera, I., Boundary methods in flow problems. Proc. Third International Conference on Finite Elements in Flow Problems, Banff, Canada, 1, pp. 30–42, 1980 (Invitéd General Lecture)

    Google Scholar 

  23. Herrera, I., General variational principles applicable to the hybrid element method. Proc. Nat. Acad. Sci. USA, 74, pp. 2595–2597, 1977

    Article  MathSciNet  MATH  Google Scholar 

  24. Herrera, I., Theory of connectivity for formally symmetric operators. Proc. Nat. Acad. Sci. USA, 74, pp. 4722–4725, 1977

    Article  MathSciNet  MATH  Google Scholar 

  25. Herrera, I., On the variational principles of mechanics. Trends in Applications of Pure Mathematics to Mechanics, II, Zorsky, H. (Ed.), Pitman Publishing Limited, pp. 115–128, 1979

    Google Scholar 

  26. Herrera, I., Theory of connectivity: A systematic formulation of boundary element methods. Applied Mathematical Modelling, 3, pp. 151–156, 1979

    Article  MATH  Google Scholar 

  27. Herrera, I., Theory of connectivity: A unified approach to boundary methods. Variational Methods in the Mechanics of Solids. Nemat-Nasser, S. (Ed.), Pergamon Press, Oxford and New York, pp. 77–82, 1980

    Google Scholar 

  28. Herrera, I., Variational principles for problems with linear constraints. Prescribed jumps and continuation type restrictions. Jour. Inst. Maths. Applics., 25, pp. 67–96, 1980

    Article  MathSciNet  MATH  Google Scholar 

  29. Herrera, I., Sabina, F. J., Connectivity as an alternative to boundary integral equations. Construction of bases. Proc. Nat. Acad. Sci. USA, 75 (5), pp. 2059–2063, 1978

    Article  MathSciNet  MATH  Google Scholar 

  30. Herrera, I., Boundary methods. A criterion for completeness. Proc. Nat. Acad. Sci. USA, 77 (8), pp. 4395–4398, ( 1980

    Article  MathSciNet  MATH  Google Scholar 

  31. Gourgeon, H., Herrera, I., Boundary methods. C-complete systems for the biharmonic equation. Boundary Element Methods. Brebbia, C. A. (Ed.), Springer-Verlag, Berlin, pp. 431–441, 1981

    Google Scholar 

  32. Herrera, I., Gourgeon, H., Boundary methods. C-complete systems for Stokes problems. Computer Methods in Applied Mechanics and Engineering, 30, pp. 225–241, 1982

    Article  MathSciNet  MATH  Google Scholar 

  33. Herrera, I., Boundary methods: Development of complete systems of solutions. Finite Element Flow Analysis. Kawai, T. (Ed.), University of Tokyo Press, pp. 897–906, 1982

    Google Scholar 

  34. Herrera, K., An algebraic theory of boundary value problems. Kinam, 3 (2), pp. 161–230, 1981

    MathSciNet  MATH  Google Scholar 

  35. Herrera, I., Spence, D. A., Framework for biorthogonal Fourier series. Proc. Nat. Acad. Sci. USA, 78 (12), pp. 7240–7244, 1981

    Article  MathSciNet  MATH  Google Scholar 

  36. Alduncin, G., Herrera, I., Solution of free boundary problems using C-complete systems. Boundary Element Methods in Engineering. Brebbia, C. A. (Ed.), Springer-Verlag, Berlin, Heidelberg, New York, pp. 34–42, 1982

    Chapter  Google Scholar 

  37. Alduncin, G., Herrera, I., Contribution to free boundary problems using boundary elements. Trefftz Approach, Communicaciones Técnicas, IIMAS-UNAM, 1983 ( Computers Methods in Applied Mechanics and Engineering, submitted)

    Google Scholar 

  38. Liggett, J. A., Liu, P. L.-F., The boundary integral equation method applied to flow in porous media. Allen and Unwin, 1982

    Google Scholar 

  39. Kikuchi, N., Oden, J. T., Contact problems in elasticity. TICOM Report 79–8, 1979

    Google Scholar 

  40. Oden, J. T., Kim, S. J., Interior penalty methods for finite element approximations of the Signorini problem in elastostatics. Comp. & Math. with Applics., 8, pp. 35–56, 1982

    Google Scholar 

  41. Lions, J. L., Magenes, E., Non-Homogeneous Boundary Value Problems and Applications. 3 Volumes, Springer-Verlag, New York-Heidelberg-Berlin, 1972

    Google Scholar 

  42. Oden, J. T., Reddy, J. N., An introduction to the mathematical theory of finite elements. Pure & Applied Mathematics Series, J. Wiley & Sons, New York, London, Sydney, Toronto, 1976

    Google Scholar 

  43. Bates, R. H. T., Analytic constraints on electro-magnetic field computations. IEEE Trans. on Microwave Theory and Techniques, 23, pp. 605–623, 1975

    Google Scholar 

  44. Abraham, R., Marsden, J. E., Foundations of mechanics. The Benjamin Cummins Publishing Co. Inc., pp. 159–187, 1978

    Google Scholar 

  45. Herrera, I., A general formulation of variational principles. Instituto de Ingenieria, UNAM, E-10, 1974

    Google Scholar 

  46. Gurtin, M. E., Variational principles for linear initial value problems. Quart. Appl. Math. 22, pp. 252–256, 1964

    MathSciNet  MATH  Google Scholar 

  47. Herrera, I., Bielak, J., A simplified version of Gurtin’s variational principles. Arch. Rat. Mechanics and Analysis, 53 (2), pp. 131–149, 1974

    Article  MathSciNet  MATH  Google Scholar 

  48. Gurtin, M., The linear theory of elasticity. In: Encyclopedia of Physics. VI a/2, Springer-Verlag, Berlin, pp. 1–295, 1972

    Google Scholar 

  49. Prager, W., Variational principles of linear elastodynamics for discontinuous displacements, strains and stresses, in recent progress in applied mechanics. In: The Folke-Adqvist Volume. Broberg, B. Hult, J. & Niordson, F. (Ed.), Almgvist and Wiksell, Stockholm, pp. 463–474, 1967

    Google Scholar 

  50. Nemat-Nasser, S., General variational principles in non-linear and linear elasticity with applications, in mechanics today. Nemat-Nasser, S. (Ed.), Pergamon, 1, pp. 214–261, 1972

    Google Scholar 

  51. Nemat-Nasser, S., On variational methods in finite and incremental elastic deformation problems with discontinuous fields. Quart. Appl. Math., 30 (2), pp. 143–156, 1972

    MathSciNet  MATH  Google Scholar 

  52. Jackson, J. D., Classical Electrodynamics. Wiley, New York, pp. 65, 69, 86, 541, 1962

    Google Scholar 

  53. Morse, P., Feshbach, H., Methods of Theoretical Physics. McGraw-Hill, New York, 1953

    MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Herrera, I. (1984). Trefftz Method. In: Brebbia, C.A. (eds) Topics in Boundary Element Research. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2877-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4899-2877-1_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-387-13097-2

  • Online ISBN: 978-1-4899-2877-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics