Basis Function-based Differential Quadrature Method Research Articles

Efficient solution for transient laminar mixed convection problems using semi-explicit DQ/RBF-DQM

This article proposes a meshless, semi-explicit solution technique for time-dependent transient laminar mixed convection problems. The approach approximates both spatial and temporal derivatives using high-order differential quadrature rules, with spatial derivatives approximated using the radial basis function-based differential quadrature method (RBF-DQM) and temporal derivatives using the conventional global differential quadrature method (DQM). The method offers the unique ability to utilize DQM for both spatial and temporal derivatives of partial differential equations simultaneously. The RBF-DQM's meshless nature makes it suitable for irregular spatial domains, allowing for the analysis of problems with irregular geometries. The proposed algorithm was evaluated using transient laminar mixed convection simulations within both rectangular and irregular cavities, with results validated against benchmark solutions. The study concludes that the proposed algorithm is a reliable and accurate tool for the analysis of transient laminar mixed convection problems, with the added advantages of being easy to program and mesh-free, albeit with a need for suitable shape parameter selection.

A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation

Numerical simulation technique of two-dimensional variable-order time fractional advection-diffusion equation is developed in this paper using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors' knowledge, this is the first application of this method to variable-order time fractional advection-diffusion equations. For the general case of irregular geometries, the meshless local form of RBF-DQ is used and the multiquadric type of radial basis functions is selected for the computations. This approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the Dirichlet and Neumann boundary conditions on the irregular boundaries. The method is validated by the well documented test examples involving variable-order fractional modeling of air pollution. The numerical results demonstrate that the proposed method provides accurate solutions for two-dimensional variable-order time fractional advection-diffusion equations.

Hermite type radial basis function-based differential quadrature method for higher order equations

In the paper, the radial basis function-based differential quadrature method (RBF-DQM) that uses Hermite type interpolation is developed. The method is an extension of the known RBF-DQM, which is a meshless numerical technique for solving differential equations. According to this technique, derivatives in a governing equation are approximated by a linear weighted sum of the sought function values defined at scattered nodes. To allow the method to be applied in higher order equations, where more than one boundary condition is imposed at an edge, Hermite type interpolation for the radial basis functions is used and appropriate weighting coefficients for differential quadrature method are determined in the paper. As a numerical test the method is used to discretize the governing equation for the free vibration of thin plates with various boundary conditions. Different shaped plates with various boundary conditions are analyzed. The convergence tests carried out in the work confirm usefulness of the method as a truly meshless technique.

Unsteady seepage analysis using local radial basis function-based differential quadrature method

Numerical simulation of two-dimensional transient seepage is developed using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to seepage analysis. For the general case of irregular geometry and unstructured node distribution, the local form of RBF-DQ was used. The multiquadric type of radial basis functions was selected for the computations, and the results were compared with analytical, finite element method, and existing numerical solutions from the literature. Results of this study show that localized RBF-DQ can produce accurate results for the analysis of seepage. The method is meshfree and easy to program, but as with previous applications of RBFs, requires careful selection of suitable shape parameters. A practical method for estimating suitable shape parameters is discussed. For time integration, Crank–Nicolson, Galerkin and finite difference methods were applied, leading to stable results.

Meshless local RBF-DG for 2-D heat conduction: A comparative study

Meshless local radial basis function-based differential quadrature method is applied to 2-D conduction problem. Numerical results are compared with those gained by homotopy perturbation method. Outcomes are presented through graphs which prove the accuracy of homotopy perturbation method and its applicability in heat transfer problems.

Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method

In this study, a recently developed local radial basis function-based differential quadrature (LRBFDQ) method is applied for the vibration analysis of arbitrarily shaped membranes. LRBFDQ method combines the good features of differential quadrature (DQ) approximation of derivatives and mesh-free nature of the radial basis functions (RBFs) in a local region. The derivative at a reference point is approximated as a linear weighted sum of functional values at a set of scattered points in the local supporting region of the reference point. The Helmholtz equation governing membrane vibration is directly discretized into algebraic equations, from which the wavenumbers (natural frequencies) and mode shapes of freely vibrating membranes are easily calculated. Owing to the properties of mesh-free and local approximation of the LRBFDQ method, the problems with arbitrarily shaped domains can be solved readily and accurately. In particular, for highly concave-shaped membranes and multi-connected membranes with a hole, very accurate numerical results can be easily obtained without the use of any domain decomposition technique. It is also shown that the LRBFDQ method can produce more accurate solutions than FEM when the two methods use nearly the same number of points in a domain.

Solution of partial differential equations by a global radial basis function-based differential quadrature method

The conventional differential quadrature (DQ) method is limited in its application to regular regions by using functional values along a mesh line to approximate derivatives. In this work, we extend the idea of DQ method to a general case. In other words, any spatial derivative is approximated by a linear weighted sum of all the functional values in the whole physical domain. The weighting coefficients in the new approach are determined by the radial basis functions (RBFs). The proposed method combines the advantages of the conventional DQ method and the RBFs. Since the method directly approximates the derivative, it can be consistently well applied to linear and nonlinear problems. It also remains mesh free feature of RBFs. Numerical examples of linear and nonlinear cases showed that RBF-based DQ method has a potential to become an efficient approach for solving partial differential equations.

Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations

Local radial basis function-based differential quadrature method is presented in detail in this paper. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with “truly” mesh-free property. In this paper, we mainly concentrate on the multiquadric RBFs since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to the simulation of natural convection in a square cavity. Excellent numerical results are obtained on an irregular knot distribution.