Skip to main content
SpringerLink
Log in

Nonlinear thermal analysis with a boundary element zone condensation technique

  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

This paper presents a substantially more economical technique for the boundary element analysis (BEA) of a large class of nonlinear heat transfer problems including those with temperature dependent conductivity, temperature dependent convection coefficients, and radiation boundary conditions. The technique involves an exact static condensation of boundary element zones in a multi-zone boundary element model. The condensed boundary element zone contributions to be overall sparse blocked boundary element system matrices are formed once in the first step of the iterative nonlinear solution process and subsequently reused as the nonlinear parts of the overall problem are evolved to a convergent solution. Through a series of example problems it is demonstrated that the zone condensation technique facilitates the use of highly convergent iterative strategies for the solution of the nonlinear heat transfer problem involving modification and subsequent factorization of the overall boundary element system left had side matrix. For heat transfer problems with localized nonlinear effects, the condensation technique is shown to allow for the solution of nonlinear problems in less than half the CPU time required by methods that do not employ condensation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akkuratov, Y. N.; Mikhailov, V. N. (1980): The method of boundary integral equations for the solution of nonlinear heat transmission problems. USSR Comput. Maths. Math. Phys. 20, 117–125

    Google Scholar 

  • Anza, J. J.; Ahedo, E.; DaRiva, I.; Alarcon, E. (1982): A new boundary condition solved with BEM In: Brebbia, C. (ed): Boundary Element Methods in Engineering, pp. 607–618. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Azevendo, J. P. S.; Wrobel, L. C. (1982): Nonlinear heat conduction in composite bodies: A boundary element formulation. International Journal for Numerical Methods in Engineering, 26, 19–38

    Google Scholar 

  • Bialecki, R.; Nowak, A. J. (1981): Boundary value problems for nonlinear material and nonlinear boundary conditions. Appl. Math. Modell. 4, 417–421

    Google Scholar 

  • Bailecki, R.; Nahlik, R. (1987): Linear equations solver for large block matrices arising in boundary element methods. Boundary Elements IX, Vol. 1, Computational Mechanics Publ. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Bailecki, R. (1987): Nonlinear equations solver for large equation sets arizing when using BEM in homogeneous regions of nonlinear material. Boundary Elements IX, Vol. 1, Computational Mechanics Publ. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Banerjee, P. K. (1979): Nonlinear problems of potential flow. In: Developments in Boundary Element Methods—1, pp. 21–30. Banerjee, P. K; Butterfield, R. (eds.). London: Appl. Sci.

    Google Scholar 

  • Banerjee, P. K.; Butterfield, R. (1981): Boundary element methods in enger. sci. London: McGraw Hill

    Google Scholar 

  • Beer, G. (1986): Implementation of combined boundary element-finite element analysis with applications in geomechanics, Chapter 7. In: Banerjee, P. K.; Watson, J. P. (eds). Developments in boundary elements methods. London: Elsevier

    Google Scholar 

  • Brebbia, C. A.; Walker, S. (1980): Boundary element techniques in engineering. London: Butterworths

    Google Scholar 

  • Brebbia, C. A.; Telles, J. C. F.; Wrobel, L. C. (1984): Boundary element techniques. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Butterfield, R.; Tomlin, G. R. (1971): Integral techniques for solving zoned anistropic continuum problmes, pp. 9/31–51: Proc. International Conference on Variational Methods in Engineering, Southampton Univ.

  • Butterfield, R. (1978): An application of the boundary element method to potential flow problems in generally inhomogeneous bodies. In: Brebbia, C. (ed). Recent advances in boundary element methods, pp. 123–135. London: Pentech Press

    Google Scholar 

  • Crotty, J. M. (1981): A block equation solver for large unsymmetric matrices arizing in the boundary element method. Numer. Meth. in Eng. 18, 997–1017

    Google Scholar 

  • Das, P. C. (1978): A disc based block elimination technique used for the solution of non-symmetrical fully populated matrix systems encountered in the boundary element method, pp. 391–404. Proc. Int. Symp. On Recent. Dev. In Boundary Element Meth., Southampton Univ.

  • Davies, T. G. (1979): Linear and Nonlinear Analysis of Pile Groups. Ph.D. Thesis, University of Wales, University College, Cardiff

  • Jin, H.; Runesson, K.; Simuelsson, A. (1987); Application of the boundary element method to contact problems in elasticity with a nonclassical friction law, Boundary Elements IX, Computational Mechanics Publications, Vol. 2 pp. 397–415. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Kane, J. H. (1986); Shape optimization utilizing a boundary element formulation. BETECH 86, Proceedings, 1986 Boundary Element Technology Conference, MIT, Computational Mechanics Publications, Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Kane, J. H.; Saigal, S. (1988): Design sensitivity analysis of solids using BEM. J. Engg. Mech. ASCE, 114, 10, 1703–1722

    Google Scholar 

  • Kane, J. H.; Saigal, S.; Gallagher, R. H. (1988): Design sensitivity analysis of boundary element substructures. Second NASA/Airforce Symposium on Recent Experiences in Multi Disciplinary Analysis and Optimization, September, NASA CP-3031, Part 2, p. 777

  • Kane, J. H.; Saigal, S. (1989): An arbitrary condensing, noncondensing solution strategy for large scale, multi-zone boundary element analysis. Comput. Meth. Appl. Mech. & Engg. 29/2, 219

    Google Scholar 

  • Kane, J. H.; Kumar, B. L. K.; Gallagher, R. H. (1990): Boundary element iterative reanalysis techniques for continuum structures. J. Engg. Mech. ASCE accepted for pub. (in press)

  • Khader, M. S.; Ingham, D. B. (19 ): An iterative boundary integral numerical solution for general steady heat conduction problems. J. Heat Transf. Trans, ASME 103 2, 26–31

  • Lachat, J. C.; Watson, J. O. (1975): A second generation boundary integral program for three dimensional elastic analysis. Chapter In: Cruse, T. A.; Rizzo, F. J. (eds): Boundary Integral Equation Method: Computational Applications in Applied Mechanics. Appl. Mech. Div. ASME 11, 000-000

  • Lachat, J. C. (1975): Further developments of boundary integral technique for elasto-statics. Ph.D. Thesis, Southampton Univ.

  • Lachat, J. C.; Watson, J. O. (19 ): Progress in the use of boundary integral equations, illustrated by examples. Comput. Meth. in Appl. Mech. Engg. 10, 273–289

  • Margenov, S.; Georgiev, K.; Hadjikov, L.; Novakova, M. (1987): An effective approach for boundary element method application to friction contact problems. Boundary Elements IX, Vol. 1, pp. 439–445. Computational Mechanics Publications. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Mustoe, G. G. W. (19 ): A combination of the finite method and boundary solution procedures for continuum problems. Ph.D. Thesis, University of Wales, University College, Sawansea

  • Onishi, K.; Kuroki, T. (1982): Boundary element methods in singular and nonlinear heat transfer. In: Brebbia, C. (ed): Boundary Element Methods in Engineering, pp. 141–155. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Onishi, K.; Kuroki, T. (1986): On nonlinear heat transfer problems. In: Banerjee, P. K.; Watson, J. O. (eds): Developments in Boundary Element Methods—4. pp. 191–226. London: Elsevier

    Google Scholar 

  • Saigal, S.; Kane, J. H. (1989a): A boundary element shape optimization system for aircraft components. AIAA J. (28/7, 1203–1204

    Google Scholar 

  • Saigal, S.; Aithal R.; Kane, J. H. (1989a): Conforming boundary elements in plane elasticity for shape design sensitivity. Intern. J. for Numer. Meth. in Engg. (to appear)

  • Saigal, S.; Kane, J. H.; Aithal, R. (1989b): Semi-analytical structural sensitivity formulation using a boundary elements. AIAA J. 27/11, 1615–1621

    Google Scholar 

  • Saigal, S.; Borggaard, J. T.; Kane, J. H. (1989c): Boundary element implicit differentiation equations for design sensitivities of axisymmetric structures. Intern. J. of Solids & Struct. 25/5, 527–538

    Google Scholar 

  • Saigal, S.; Kane, J. H. (1989b): An arbitrary multi-zone condensation technique for boundary element design sensitivity analysis. AIAA J. 28/7, 1277–1284

    Google Scholar 

  • Skerget, P.; Brebbia, C. (1983): Nonlinear potential problems. Progress in Boundary Element methods, Vol. 2, London, Pentech, New York: Springer

    Google Scholar 

  • Tomlin, G. R. (1982): Numerical analysis of continuum problems in zoned anistropic media. Ph.D. Thesis, Southampton Univ. Wrobel, L. C.: Azevedo, J. P. S. (1985: A boundary element analysis of nonlinear heat conduction. In: Lewis, R.; Morgan, K. (eds): Proc Fourth International Conference on Numerical Methods in Thermal Problems Swansea: Pineridge

  • Wrobel, L. C.; Brebbia, C. A. (19 ): The dual reciprocity boundary formulation for nonlinear diffusion problems. Comp. Methods Appl. Mech. Eng. (to appear)

Download references

Author information

Authors and Affiliations

  1. Clarkson University, 13676, Potsdam, NY, USA

    J. H. Kane, H. Wang & B. L. K. Kumar

Authors
  1. J. H. Kane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. L. K. Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S. N. Atluri, November 6, 1989

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kane, J.H., Wang, H. & Kumar, B.L.K. Nonlinear thermal analysis with a boundary element zone condensation technique. Computational Mechanics 7, 107–122 (1990). https://doi.org/10.1007/BF00375925

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00375925

Keywords

Search

Navigation