Dual Reciprocity Boundary Element Method



Key words: dual reciprocity BEM (DR-BEM), multiple reciprocity BEM (MR-BEM), boundary-only, meshfree, radial basis function (RBF), differential quadrature method (DQM), time integrator, impact, numerical damping, precise integration method, dynamic programming filter, regularization, measurement error, inverse elastodynamic problems.

 

Transformation of the domain integrals has been a central task in the BEM to preserve its boundary-only merit. There are several different approaches available now for this purpose. Among them, the dual reciprocity BEM (DRBEM), introduced in the 1980s by Nordini and Brebbia, stands out the method of choice in engineering computations due to its ease of implementation, meshfree feature and strong flexibility of applying fundamental solutions. The radial basis function (RBF) is the very basis of the prowess of this method. The RBF approach also leads to the intrinsic meshree merit. The DR-BEM can compete over the FEM for general linear and nonlinear problems. In our study, the DR-BEM was applied to the transient diffusion, elastodynamic and inverse dynamic problems. Below outlines our works:

#1: The DRBEM and differential quadrature method (DQM) were combined to approximate the partial derivatives in space and time respectively for the solution of the transient diffusion problems. Our work shows that the inefficiency of the DR-BEM solution of the Dirichlet problems is due to the time integrators used in the literature rather than the DR-BEM itself.

#2: A comprehensive comparative study was carried out to investigate various standard time integrators in the context of the DR-BEM formulation of elastodynamic problems. Although the Houbolt method seems predominant currently in the solution of the DR-BEM formulation of elastodynamics problems, our numerical experiments and analysis show that the Newmark method should be in general preferred. Also, for the cases where response is primarily dominated by low and intermediate frequency modes, the DQM in time approximation exhibits an impressive advantage in the solution accuracy. For the systems on which high modes have important affect such as plate-impact problems, the damped Newmark method is preferred. The precise integration method, which was recently introduced by Zhong and Williams and in fact equivalent to the exponential matrix approach, was also investigated. We found that the method produced nearly the same accuracy as the DQM but required far more computational effort. In addition, the method lacks the numerical damping and is therefore inefficient for impact problems.

#3: The coupling application of the DR-BEM and dynamic programming with regularization was first presented to solve inverse elastodynamic problems. The affect of noise level, regularization parameter, and measurement and input force types on the estimation was investigated. The study showed that the combined method is accurate, robust and generally applicable. It was found that the strategy is insensitive to measurement errors and can give good estimation even using heavily noisy data.

#4: The meshfree boundary knot method and boundary particle method were developed. In addition, a recusive multiple reciprocity scheme was presented which dramatically reduces computing cost of the multiple reciprocity technique. For details see Kernel distance functions (radial basis functions) and wavelets .

#5: We also discovered the high-order fundamental and general solutions of convection-diffusion, Bergers, Winkler, vibration thin plate equations.


Related Publications

 

International Journals

  1. W. Chen and M. Tanaka, New Insights into Boundary-only and Domain-type RBF Methods, Int. J. Nonlinear Sci. & Numer. Simulation, 1(3), 145-151, 2000.
  2. W. Chen and J. He, A study on radial basis function and quasi-Monte Carlo methods, Int. J. Nonlinear Sci. & Numer. Simulation, 1(4), 343-342, 2000.
  3. M. Tanaka and W. Chen, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods in Applied Mechanics and Engineering, 190, 2331-2347, 2001.

 

  1. M. Tanaka and W. Chen, Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems, Applied Mathematical Modelling, 25(3), 257-268, 2001.

 

  1. W. Chen and M. Tanaka, A meshless, exponential convergence, integration-free, and boundary-only RBF technique, Computers and Mathematics with Applications, 43, 379-391, 2002.
  2. W. Chen and M. Tanaka, A study on time schemes for DRBEM analysis of scalar impact wave, Comput. Mech. 28, 331-338, 2002.
  3. W. Chen, Symmetric boundary knot method, Engng. Anal. Bound. Elem., 26(6), 489-494, 2002.
  4. W. Chen, High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method, Engng. Anal. Bound. Elem., 26(7), 571-575, 2002.
  5. W. Chen, Meshfree boundary particle method applied to Helmholtz problems, Engng. Anal. Bound. Elem., 26(7), 577-581, 2002.
  6. Y.C. Hon and W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems with complicated geometry, Int. J. Numer. Meth. Engng. 1931-1948, 56(13), 2003.

Japanese Journals

  1. M. Tanaka and W. Chen, Dual reciprocity BEM and dynamic programming filter for inverse elastodynamic problems, Transactions of the Japan Society for Computational Engineering and Science, 2, 2000.



Proceedings

  1. M. Tanaka and W. Chen, "Transient diffusion problems by a combined use of dual reciprocity and differential quadrature method", Proceedings of Computational Engineering Conference, Vol.4, 970-974,1999.

 

  1. M. Tanaka and W. Chen, "A combined use of BEM and differential quadrature method for transient elastodynamic problems", BETC-99, 13-18, 1999.

 

  1. M. Tanaka and W. Chen, "Combination of dual reciprocity BEM in space and differential quadrature method in time for elastodynamic problems", Proc.of the 12th JSME Comput. Mech. Conference, 415-416, Nov. 1999, Japan.

 

  1. M. Tanaka and W. Chen, "On highly accurate time integration scheme for the DR-BEM analysis plate-impact responses", The 16th Japan National Symposium on BEM (JASCOME), 45-50, Dec. 1999, Japan.

 

  1. M. Tanaka and W. Chen, "Identification of elastodynamic load using DRBEM and dynamic programming filter", International Symposium on Inverse Problems in Engineering Mechanics 2000, Mar. 2000, Nagano, Japan.

 

  1. M. Tanaka and W. Chen, "Dual reciprocity BEM solution of an inverse heat conduction boundary problem", Proceedings of JSME conference on thermoelastic dynamic problems, Aug. 2000, Nagoya, Japan.

 

  1. M. Tanaka and W. Chen, "A Combined Method of Solution by DRBEM and DQM for the Forced Vibration Problem of Damped Elastic Plates", IABEM 2000, 233-236, July, 2000, Brescia, Italy.

 

  1. W. Chen and M. Tanaka, Relationship between boundary integral equation and radial basis function, Invited talk to the 52th Symposium of Japan Society for Computational Methods in Engineering (JASCOME) on BEM, Tokyo, 2000.
  2. W. Chen and Y.C. Hon, "Relationship between radial basis function and Monte Carlo methods", The fourth International Conference of Monte Carlo and quasi-Monte Carlo Methods, Hong Kong, Dec. 2000.

 

  1. W. Chen and M. Tanaka, Solution of some inverse heat conduction problems by the dynamic programming filter and BEM, International Symposium on Inverse Problems in Engineering Mechanics 2001, 23-28, Feb. 2001, Nagano, Japan.

 

  1. W. Chen, New RBF collocation schemes and kernel RBFs with applications, Lecture Notes in Computational Science and Engineering, Vol 26, 73-84, 2002.
  2. W. Chen, "Boundary knot method for Laplace and biharmonic problems". The 14th Nordic Seminar on Computational Mechanics, Lund, Sweden, Oct. 2001.
  3. W. Chen, "RBF-based meshless boundary knot method and boundary particle method". The China Congress on Computational Mechanics's 2001, Guangzhou, China, Dec. 2001.
  4. W. Chen, Some recent advances on the RBF. Proceedings of BEM 24, pp. 125-134, Portugal, June, 2002.
  5. W. Chen, "Kernel and wavelets RBFs based on fundamental and general solutions of partial differential equations". the Fifth International Conference of Curves & Surfaces, (abstract pp. 12), France, June 27 - July 3, 2002.


Projects

  1. JSPS (Japan): "A combined use of the differential quadrature and boundary element methods" 1998, 10-2000, 9 (Postdoctoral research fellow).




    [ DQ-type methods || Modeling || BEM || Inverse analysis || RBF || Wavelet || Patent ]


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Last updated 08/07/2002