Generic domain decomposition and iterative solvers for 3D BEM problems.

*(English)*Zbl 1191.74052Authors’ abstract: In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi-singular boundary element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones, also including non-Hermitian matrices. The former fact has implied in CPU-time reduction during the assembling of the system of equations, and the latter fact in its faster solution.

In this paper, a triangle-polar-co-ordinate transformation and the Telles co-ordinate transformation, applied in previous works independently for evaluating singular and quasi-singular integrals [J. C. F. Telles, ibid. 24, 959–973 (1987; Zbl 0622.54014)], are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrix-assembly routines. In addition, the Jacobi-preconditioned biconjugate gradient solver is used to develop a generic substructuring boundary element algorithm. In this way, not only the system solution is accelerated, but also the computer memory is optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted.

In this paper, a triangle-polar-co-ordinate transformation and the Telles co-ordinate transformation, applied in previous works independently for evaluating singular and quasi-singular integrals [J. C. F. Telles, ibid. 24, 959–973 (1987; Zbl 0622.54014)], are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrix-assembly routines. In addition, the Jacobi-preconditioned biconjugate gradient solver is used to develop a generic substructuring boundary element algorithm. In this way, not only the system solution is accelerated, but also the computer memory is optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted.

Reviewer: Ján Sládek (Bratislava)

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

80M15 | Boundary element methods applied to problems in thermodynamics and heat transfer |

74L10 | Soil and rock mechanics |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

##### Keywords:

quasi-singular integration algorithms; Krylov’s solvers; Jacobi-preconditioned biconjugate gradient solver##### Software:

CGS
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\textit{F. C. Araújo} et al., Int. J. Numer. Methods Eng. 68, No. 4, 448--472 (2006; Zbl 1191.74052)

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##### References:

[1] | (eds), In Boundary Element Advances in Solid Mechanics Springer: Wien, 2003. |

[2] | Boundary Integral Equation Methods for Fluids and Solids. Wiley: New York, 1999. |

[3] | . The Boundary Element Method, vols. I & II. Wiley: New York, 2002. |

[4] | Cruse, Computational Mechanics 18 pp 1– (1996) · Zbl 0946.74073 |

[5] | Gray, International Journal for Numerical Methods in Engineering 29 pp 1135– (1990) |

[6] | Boundary Elements in Dynamics. Computational Mechanics Publication: Southampton, 1993. |

[7] | Araújo, Engineering Analysis with Boundary Elements 25 pp 795– (2001) |

[8] | , , . Analysis of 3D time-harmonic soil–foundation interaction problems by using efficient BE substructuring algorithms. Proceedings of the European Conference on Dynamics, vol. 2. A.A. Balkema Publishers: Rotterdam, Munich, 2002; 1267–1272. |

[9] | , . Dynamic cross-interaction between footings–3D frequency-domain parallelized analysis via the BEM. Proceedings of the International Conference on Boundary Element Techniques IV, vol. I, Queen Mary, University of London: London, Granada, 2003; 253–258. |

[10] | . Boundary Element Methods in Elastodynamics. Unwin Hyman, 1988. |

[11] | , . Boundary Element Techniques: Theory and Applications in Engineering. Springer: Berlin, 1984. |

[12] | , . Time-domain three dimensional analysis. In Boundary Element Acoustics–Fundamentals and Computer Codes, (ed.). Computational Mechanics Publications: London, 2000; 159–216. |

[13] | Guiggiani, International Journal for Numerical Methods in Engineering 24 pp 1711– (1987) |

[14] | Lu, International Journal for Numerical Methods in Engineering 32 pp 295– (1991) |

[15] | Liao, International Journal for Numerical Methods in Engineering 35 pp 1473– (1992) |

[16] | Lachat, International Journal for Numerical Methods in Engineering 10 pp 991– (1976) |

[17] | Jun, Engineering Analysis with Boundary Elements 2 pp 118– (1985) |

[18] | Voutsinas, Applied Mathematical Modeling 14 pp 618– (1990) |

[19] | Li, International Journal for Numerical Methods in Engineering 21 pp 2071– (1985) |

[20] | Zhang, International Journal for Numerical Methods in Engineering 28 pp 2059– (1989) |

[21] | Telles, International Journal for Numerical Methods in Engineering 24 pp 959– (1987) |

[22] | Telles, Engineering Analysis with Boundary Elements 13 pp 135– (1994) |

[23] | Luo, Computational Mechanics 22 pp 404– (1998) |

[24] | Cruse, International Journal for Numerical Methods in Engineering 39 pp 3273– (1996) |

[25] | Krishnasamy, International Journal for Numerical Methods in Engineering 37 pp 107– (1994) |

[26] | Liu, International Journal for Numerical Methods in Engineering 41 pp 541– (1998) |

[27] | Iterative Solution Methods. Cambridge University Press: New York, 1994. |

[28] | Iterative Methods for Sparse Linear Systems. PWS Publishing: New York, 1996. |

[29] | van der Vorst, Journal of Computational and Applied Mathematics 149 pp 251– (2002) |

[30] | Iterative Krylov Methods for Large Linear Systems. Cambridge University Press: Cambridge, MA, 2003. · Zbl 1023.65027 |

[31] | Iterative techniques for solving linear systems of equations originated from the boundary element method. M.Sc. Thesis, COPPE–Federal University of Rio de Janeiro, Brazil, 1989 (in Portuguese). |

[32] | Mansur, International Journal for Numerical Methods in Engineering 33 pp 1823– (1992) |

[33] | Prasad, International Journal for Numerical Methods in Engineering 37 pp 1651– (1994) |

[34] | Leung, International Journal for Numerical Methods in Engineering 40 pp 2227– (1997) |

[35] | Valente, Engineering Analysis with Boundary Elements 25 pp 423– (2001) |

[36] | Davey, Computers and Structures 80 pp 643– (2002) |

[37] | Fang, Engineering Analysis with Boundary Elements 26 pp 789– (2002) |

[38] | Bucher, Communications in Numerical Methods in Engineering 19 pp 387– (2003) |

[39] | Briceño, Engineering Analysis with Boundary Elements 28 pp 333– (2004) |

[40] | Sonneveld, SIAM Journal on Scientific and Statistical Computing 10 pp 36– (1989) |

[41] | van der Vorst, SIAM Journal on Scientific and Statistical Computing 13 pp 631– (1992) |

[42] | Araújo, Journal of the Brazilian Society of Mechanical Sciences and Engineering 26 pp 231– (2004) |

[43] | Kane, Computer Methods in Applied Mechanics and Engineering 79 pp 219– (1990) |

[44] | Rigby, International Journal for Numerical Methods in Engineering 38 pp 1507– (1995) |

[45] | Bialecki, International Journal for Numerical Methods in Engineering 39 pp 4215– (1996) |

[46] | Ganguly, International Journal for Numerical Methods in Engineering 44 pp 991– (1999) |

[47] | Tadeu, Engineering Analysis with Boundary Elements 23 pp 671– (1999) |

[48] | Tadeu, Engineering Analysis with Boundary Elements 23 pp 757– (1999) |

[49] | Johan, Computer Methods in Applied Mechanics and Engineering 87 pp 281– (1991) |

[50] | Baum, AIAA Journal 32 pp 1183– (1994) |

[51] | Dutto, Computer Methods in Applied Mechanics and Engineering 188 pp 441– (2000) |

[52] | Catabriga, Computer Methods in Applied Mechanics and Engineering 191 pp 3477– (2002) |

[53] | Whitman, Journal of the Soil Mechanics and Foundations Division 93 pp 169– (1967) |

[54] | . Handbook of Impedance Functions (Quest edn). Surface Foundations, 1991. |

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