An alternative approach for numerical solutions of the Navier–Stokes equations

Conventional approaches for solving the Navier–Stokes equations of incompressible fluid dynamics are the primitive‐variable approach and the vorticity–velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher‐order partial differential equation with one unknown for two‐dimensional problems or two higher‐order partial differential equations with two unknowns for three‐dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier–Stokes equations using this approach is presented. The proposed method is used to solve a two‐ and a three‐dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.     

A Meshless Numerical Method for Kirchhoff Plates under Arbitrary Loadings

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of Kirchhoff plates under arbitrary loadings. In the solution procedure, a arbitrary distributed loading is first approximated by either the multiquadrics (MQ) or the augmented polyharmonic splines (APS), which are constructed by splines and monomials. The particular solutions of multiquadrics, splines and monomials are all derived analytically and explicitly. Then, the complementary solutions are solved formally by the MFS. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of multiquadrics, splines and polynomials as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas. In these numerical experiments, homogeneous problems are first considered to find the best location of the MFS sources by the way proposed by Tsai, Lin, Young and Atluri (2006). Then the corresponding nonhomogeneous problems are solved by the DRM based on both the MQ and APS. The numerical results demonstrate that the MQ is in general more accurate than the thin plate spline, or the first order APS, but less accurate than the high order APSs. Overall, this paper derives a meshless numerical method for solving problems of Kirchhoff plates under arbitrary loadings with all kinds of boundary conditions by both the MQ and APS.     

H-Adaptive Local Radial Basis Function Collocation Meshless Method

This paper introduces an effective H-adaptive upgrade to solution of the transport phenomena by the novel Local Radial Basis Function Collocation Method (LRBFCM). The transport variable is represented on overlapping 5-noded influence-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the variable are calculated from the respective derivatives of the RBFs. The transport equation is solved through explicit time stepping. The H-adaptive upgrade includes refinement/derefinement of one to four nodes to/from the vicinity of the reference node. The number of the nodes added or removed depends on the topology of the reference node vicinity. The refinement/derefinement is triggered by an error indicator, which very simply depends on the ratio between the norm of the collocation coefficients and collocation matrix. The refinement/derefinement is proportional with the growth/decay of this indicator. Such adaptivity much increases the accuracy/performance ratio of the method. The performance of the method is numerically tested on two-dimensional Burger's equation. The results are compared with different numerical solutions, published in literature. Outstanding CPU efficiency and accuracy are clearly demonstrated from the results. The paper probably for the first time shows such a simple and effective H-adaptive meshless method, designed on five noded influence domain. The advantages of the represented meshless approach are its simplicity, accuracy, similar coding in 2D and 3D, straightforward applicability in non-uniform node arrangements, and native parallel implementation.     

Low and Intermediate Re Solution of Lid Driven Cavity Problem by Local Radial Basis Function Collocation Method

This paper explores the application of Local Radial Basis Function Collocation Method (LRBFCM) [Šarler and Vertnik (2006)] for solution of Newtonian incompressible 2D fluid flow for a lid driven cavity problem [Ghia, Ghia, and Shin (1982)] in primitive variables. The involved velocity and pressure fields are represented on overlapping five-noded sub-domains through collocation by using Radial Basis Functions (RBF). The required first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The momentum equation is solved through explicit time stepping. The method is alternatively structured with multiquadrics and inverse multiquadrics RBF’s. In addition, two different approaches are used for pressure velocity coupling (Fractional Step Method (FSM) [Chorin (1968)] and Artificial Compressibility Method (ACM) [Chorin (1967)] with Characteristic Based Split (CBS) [Zienkiewicz and Codina (1995); Zienkiewicz, Morgan, Sai, Codina and Vasquez (1995)]). The method is tested for several low and intermediate Reynolds numbers (100, 400, 1000 and 3200) and node arrangements (41x41, 81x81, 101x101, 129x129). The original contribution of the paper represents extension of the LRBFCM to Reynolds number beyond 400 and assessment of the method for two different types of RBFs and two different types of pressure-velocity couplings. The obtained numerical results, in terms of mid-plane velocities, are in a good agreement with the data calculated in several reference publications and by commercial code. Both RBF’s used give approximately the same results. Both pressure-velocity coupling methods give approximately the same results, however the FSM turns out to be slightly more efficient. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements.     

Some comments on the use of radial basis functions in the dual reciprocity method

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.     

基于遗传算法和复合二次径向基函数的复合材料层合板自由振动分析

为了提高复合材料层合板自由振动分析的精度,采用无网格径向基配点法分析复合材料材料层合板的自由振动问题,径向基函数的形状参数对计算精度有很大影响。利用遗传算法对复合二次径向基函数的形状参数进行优化,用优化后的形状参数的复合二次径向基函数计算复合材料层合板的固有频率,计算结果与文献中的结果具有较好的一致性。遗传算法在形状参数优化方面具有很大的潜力,所提出的方法具有较高的计算精度。