Differential Quadrature Method



Key words: Differential quadrature method (DQM), pseudo-spectral method, collocation method, sampling points, centrosymmetric matrix, structural mechanics, truncation error, orthogonal polynomials, initial value problem, nonlinear matrix computation, Hadamard product, SJT product, Jacobian matrix.

 

The pseudo-spectral (collocation) method is so far the only method in the direct numerical simulation (DNS) of turbulent flows and widely used in computational physics and fluid mechanics, for instance, weather prediction, due to so-called spectral accuracy (exponential convergence). The differential quadrature method (DQM) can be regarded as the "direct approach" of the normal collocation (pseudo-spectral) methods in that the governing equations are analogized in terms of practical physical variables instead of usually fictitious expansion (spectral) coefficients. The advantages of the DQM over the normal pseudo-spectral method are its ease in implementation and more flexibility in choosing the sampling points. The direct use of the physical variables manifests the DQM in easy-to-choose starting solutions of nonlinear iterations, while, in contrast, the fictitious expansion variables in the collocation methods usually have not physical meanings and are therefore difficult to do so.

The essential difficulty in applying the DQM (pseudo-spectral, collocation) and even finite difference method to practical engineering is complex domain problem. Recent some studies have launched the geometry flexibility by means of the coordinate mappings (grid generation) and element (multidomain) techniques. Although some preliminary successes were achieved, the flexibility of irregular geometry is still a major deterrence in the broad application of this type methods compared with the current dominant FEM. Now we are planning to develop a very promising meshless technique of the DQM and pseudo-spectral method by using some new approach for arbitrary geometry.

My contributions in differential quadrature method and its applications to solid, structure and fluid mechanics problems (applicable for the pseudo-spectral and collocation methods due to the actual equivalence between them) include:

#1: We first noticed the fact that the rank of the DQM weighting coefficient matrix is M-i for the i-th order derivative when the number of grid points is M. In fact, the DQM coefficient matrix is power zero matrix. Based on this fact, we proposed a new approach to accurately implement the multiple boundary conditions in the DQM solution of high-order boundary value problems. The presented methodology solves the difficulty applying the boundary conditions at corner points especially for solid and structure mechanics problems. The numerical experiments of linear and nonlinear plate and shell problems show its easy use, good stability, wide applicability and high accuracy in comparison with the other approaches.

#2: We validated the centrosymmetric or skew-centrosymmetric structures of the DQM coefficient matrix if symmetric gird spacing is employed. Using these matrix structure features, the computational effort can be reduced by 75% or so for some problems. For example, plate deflection and vibration problems.

#3: The conventional formulas of the truncation error estimate in the DQM do not involve the grid interval and are too imprecise for the practical applications. We presented new formulas to more accurately estimate the truncation error at any discrete grid points. It is noted that the formulas are also applicable for the collocation (pseudo-spectral) method. The given formulas may validate the exponential convergence of the DQM, while, in contrast, the traditional proof for the collocation and pseudo-spectral methods used the norm approach and is mathematically far more complex than mine.

#4: We gave the simplified formulas for accurately and rapidly computing the DQM weighting coefficients for equally spaced grid points and the zeros of the Chebyshev or Legendre polynomials, and presented a simple transformation approach to overcome the difficulty that the conventional application of the zeros of the orthogonal polynomials can not encompass the boundary points. It is noted that the formulas are different from and simpler than those for the collocation method.

#5: Through the numerical experiments, it was found that the sampling points using the zeros of the Chebyshev or Legendre polynomials are often not optimal for many cases, especially in structural and solid mechanics. A new efficient approach was presented to choose sampling points.

#6: We presented that the DQM approximate formulas in matrix form for multi-dimension problems. By using these formulas, the fast algorithms in the solution of the Lyapunov matrix equation were introduced to the DQM solution of initial and boundary value problems with three orders of magnitude less computing effort, which was demonstrated in the solution of the Possion equation.

#7: We presented the DQM approximate formulas in matrix form for initial value problem and first applied this method to approximate temporal derivative successfully. Two new approaches were presented to apply the multiple initial conditions for initial value problems of two order or above by analogy with the techniques applying multiple boundary conditions. It was noted that the formulations of initial value problems can be expressed as the Lyapunov algebraic matrix equation. Several fast algorithms in the solution of the Lyapunov matrix equations are applied to reduce the computing effort and storage requirements by an order of N^3 and N^2, respectively, where N is the number of interior grid points. Consequently, the DQM requires comparable computational effort in the solution of linear dynamic problems as the existing multistep and single step methods such as the Newmark and Gear methods, etc. while its high order of computational accuracy is maintained. Numerical experiments were done in structural dynamics and stiff dynamic systems. It should be pointed out that the method is unconditional stability, namely, A-stable.

#8: We first used the Hadamard product, a special matrix product, to express the DQM nonlinear numerical discretization. By using Hadamard product power concept, we can easily construct some very simple and efficient iteration formulas of the simple iteration method for the solution of the Hadamard product representation of nonlinear formulations without the use of linearization procedure. We also presented a new special matrix product, SJT product, for easy, efficient and accurate calculation of the Jacobian matrix of the nonlinear analogous equations of the Hadamard product representation. The work also leads to a general approach of uncoupling the numerical formulation of the coupled nonlinear partial differential systems, which causes a considerable saving of computer resources. For example, the computational effort and storage requirements in the DQM solution of geometrically nonlinear bending of orthotropic plates were reduced by about one twenty-seventh and one-ninth, respectively.


Related Publications

 

Ph.D. dissertation:

Differential Quadrature Method and Its Applications to Engineering Problems - Application special matrix product to nonlinear computations  (in English, Dec. 1996).

 

International Journals

  1. W. Chen, Y. Yu and X. Wang, Reducing the computational requirement of differential quadrature method, Numerical Methods for Partial Differential Equations, 12, 565-577, 1996.
  2. W. Chen, X. Wang and T. Zhong, The structure of weighting coefficient matrices of harmonic differential quadrature and its application, Communications in Numerical Methods in Engineering, 12, 455-460, 1996.
  3. W. Chen and T. Zhong, The study on the nonlinear computations of the DQ and DC methods, Numerical Methods for Partial Differential Equations, 13, 57-75, 1997.
  4. W. Chen, T. Zhong and W. He, A note on the DQ analysis of anisotropic plates, Journal of Sound and Vibration, 204(1), 180-182, 1997.
  5. W. Chen, S. Liang and T. Zhong, On the DQ analysis of geometrically nonlinear vibration of immovably simply supported beams, Journal of Sound and Vibration, 206(5), 745-748, 1997.
  6. W. Chen and T. Zhong, A Lyapunov formulation for efficient solution of the Poisson and convection-diffusion equations by the differential quadrature method, J. of Computational Physics, 139, 1-7, 1998.
  7. Q. Xu, Z. Li and W. Chen, Simulation of Nonuniform Interconnects by Harmonic Differential Quadrature Method, IEE Electronics Letters, 34(22), 2137-2138, 1998.
  8. C. Shu and W. Chen, On Optimal Selection of Interior Points for Applying Discretized Boundary Conditions in DQ Vibration Analysis of Beams And Plates, J. of Sound & Vibration, 222(2), 239-257, 1999.
  9. W. Chen, C. Shu, W. He and T. Zhong, The DQ solution of geometrically nonlinear bending of orthotropic rectangular plates by using Hadamard and SJT product, Computers & Structures, 74(1), 65-74 2000.
  10. W. Chen, Relationship Theorem between Nonlinear Polynomial Equation and the Corresponding Jacobian Matrix, Int. J. of Nonlinear Sciences and Numerical Simulation, 1, 5-14, 2000.
  11. C. Shu, W. Chen and H. Du, Free vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method, J. Computational Physics, 163(2), 452-466, 2000.
  12. 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.
  13. 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.
  14. C. Shu, W. Chen, H. Xue and H. Du, Numerical Study of Grid Distribution Effect on Accuracy of DQ Analysis of Beams and Plates by Error Estimation of Derivative Approximations, Int. J. Numer. Meth. Engng., 51(2), 159-179, 2001.
  15. W. Chen and M. Tanaka, A study on time schemes for DRBEM analysis of scalar impact wave, Comput. Mech. 28, 331-338, 2002.


Chinese Journals

  1. X. Wang, B. He and W. Chen, "On the reducibility of centrosymmetric and skew centrosymmetric linear algebraic equations", J. Nanjing Univ. of Aeronautics & Astronautics, 28, 599-607, 1996 (English Abstracts).
  2. S. Liang, H. Li and W. Chen, "The DQ solution of buckling of circular arches under water loading", J. Huazhong Univ. of Sci. & Tech., 25, 81-82, 1997 (English Abstracts).
  3. S. Liang, H. Li and W. Chen, "The quadrature solution of geometrically nonlinear vibration of beams with variable cross-sections", J. Huazhong Univ. of Sci. & Tech., 25, 87-89, 1997 (English Abstracts).
  4. S. Liang and W. Chen, The DQ solution of nonlinear static problem of simply-supported beam in consideration of axial force", J. Huazhong Univ. of Sci. & Tech., 26, 65-67, 1998 (English Abstracts).
  5. W. Chen, T. Zhong and Y. Yu, "Applying special matrix product to nonlinear numerical computations", J. Appl. Comp. Math. Vol. 12, No. 1, 51-58, 1998. (English Abstracts).
  6. Q. Xu, Z. Li and W. Chen, "Application of harmonic differential quadrature method to solving the transient responses of lossy interconnects in the high speed VLSI", J. Shanghai Jiao Tong Univ. No. 8, 1-4, 1998.
  7. Q. Xu, Z. Li and W. Chen, "Application of differential quadrature method to the transient simulation of interconnects in the high speed VLSI", Journal of circuits and systems, 3, No. 2, 76-81, 1998.
  8. Q. Xu, Z. Li and W. Chen, "An Efficient Numerical Method for Transient Simulation of high speed interconnects", Acta Electronica Sinica, 27(11), 114-116, 1999.

 

Proceedings

  1. W. Chen and Y. Yu, "Calculation and analysis of weighting coefficient matrices in differential quadrature method", Proc. of the 1st Pan-Pacific Conference on Computational Engineering, B. M. Kwak and M. Tanaka Eds., Elsevier Science Publ. B. V. , Netherlands, pp. 157-162, 1993.
  2. W. Chen and Y. Yu, "Differential quadrature method for high order boundary value problems", Proc. of the 1st Pan-Pacific Conference on Computational Engineering, B. M. Kwak and M. Tanaka Eds., Elsevier Science Publ. B. V. , Netherlands, pp. 163-168, 1993.
  3. 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.
  4. M. Tanaka and W. Chen, "A combined use of BEM and differential quadrature method for transient elastodynamic problems", BETC-99, 13-18, 1999.
  5. 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.
  6. M. Tanaka and W. Chen, "On highly accurate time integration scheme for the DRBEM analysis plate-impact responses", The 16th Japan National Symposium on BEM (JASCOME), 45-50, Dec. 1999, Japan.



Projects

  1. JSPS (Japan): "A combined use of the differential quadrature and boundary element methods" 1998, 10-2000, 9 (Postdoctoral research fellow).
  2. ONR (Singapore): "The extension of the GDQ method to the discontinuous and complex problems" 1997, 11-1998, 10 (Research engineer).




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Last updated 28/09/2003