An effective spectral collocation method for the direct solution of high-order ODEs

This paper reports a new Chebyshev spectral collocation method for directly solving high-order ordinary differential equations (ODEs). The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows the multiple boundary conditions to be incorporated more efficiently. Numerical results show that the proposed formulation significantly improves the conditioning of the system and yields more accurate results and faster convergence rates than conventional formulations

Numerical analysis of corrugated tube flow using RBFNs

This paper reports the application of neural networks for the numerical analysis of steady-state axisymmetric flow through an indefinitely long corrugated tube. Meshless global radial basis function networks (RBFNs) are employed to represent all dependent variables in the governing differential equations. For a better quality of approximation, the networks used here are constructed based on the integration process rather than the usual differentiation process. Multiple spaces of network weights for each variable are converted into the single space of nodal variable values, resulting in the square system of equations with usual size. The governing equations are discretized in the strong form by point collocation and the resultant nonlinear system is solved with trust-region methods. The corrugated tube flow of a Newtonian fluid, power-law fluid and Oldroyd-B fluid are considered. With relatively low numbers of data points, flow resistance predictions obtained are in good agreement with the benchmark solutions

A second-order continuity domain-decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C2 (C squared) function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials

Computation of transient viscous flows using indirect radial basis function networks

In this paper, an indirect/integrated radial-basis-function network (IRBFN) method is further developed to solve transient partial differential equations (PDEs) governing fluid flow problems. Spatial derivatives are discretized using one- and two-dimensional IRBFN interpolation schemes, whereas temporal derivatives are approximated using a method of lines and a finite-difference technique. In the case of moving interface problems, the IRBFN method is combined with the level set method to capture the evolution of the interface. The accuracy of the method is investigated by considering several benchmark test problems, including the classical lid-driven cavity flow. Very accurate results are achieved using relatively low numbers of data points

A spectral collocation technique based on integrated Chebyshev polynomials for biharmonic problems in irregular domains

In this paper, an integral collocation approach based on Chebyshev polynomials for numerically solving biharmonic equations [N. Mai-Duy, R.I. Tanner, A spectral collocation method based on integrated Chebyshev polynomials for biharmonic boundary-value problems, J. Comput. Appl. Math. 201 (1) (2007) 30–47] is further developed for the case of irregularly shaped domains. The problem domain is embedded in a domain of regular shape, which facilitates the use of tensor product grids. Two relevant important issues, namely the description of the boundary of the domain on a tensor product grid and the imposition of double boundary conditions, are handled effectively by means of integration constants. Several schemes of the integral collocation formulation are proposed, and their performances are numerically investigated through the interpolation of a function and the solution of 1D and 2D biharmonic problems. Results obtained show that they yield spectral accuracy

An improved quadrilateral flat element with drilling degrees of freedom for shell structural analysis

This paper reports the development of a simple and efficient 4-node flat shell element with six degrees of freedom per node for the analysis of arbitrary shell structures. The element is developed by incorporating a strain smoothing technique into a flat shell finite element approach. The membrane part is formulated by applying the smoothing operation on a quadrilateral membrane element using Allman-type interpolation functions with drilling DOFs. The plate-bending component is established by a combination of the smoothed curvature and the substitute shear strain fields. As a result, the bending and a part of membrane stiffness matrices are computed on the boundaries of smoothing cells which leads to very accurate solutions, even with distorted meshes, and possible reduction in computational cost. The performance of the proposed element is validated and demonstrated through several numerical benchmark problems. Convergence studies and comparison with other existing solutions in the literature suggest that the present element is efficient, accurate and free of lockings

A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures

This paper reports a study of linear elastic analysis of two-dimensional piezoelectric structures using a smoothed four-node piezoelectric element. The element is built by incorporating the strain smoothing method of mesh-free conforming nodal integration into the standard four-node quadrilateral piezoelectric finite element. The approximations of mechanical strains and electric potential fields are normalized using a constant smoothing function. This allows the field gradients to be directly computed from shape functions. No mapping or coordinate transformation is necessary so that the element can be used in arbitrary shapes. Through several examples, the simplicity, efficiency and reliability of the element are demonstrated. Numerical results and comparative studies with other existing solutions in the literature suggest that the present element is robust, computationally inexpensive and easy to implement

RBF interpolation of boundary values in the BEM for heat transfer problems

[Abstract]: This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10-5) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10-2) with the same boundary points employed. The IRBFN-BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h1.38) and O(h4.78) for the quadratic BEM and the IRBFN-based BEM, respectively, where h is the nodal spacing

A control volume scheme using compact integrated radial basis function stencils for solving the Richards equation

A new control volume approach is developed based on compact integrated radial basis function (CIRBF) stencils for solution of the highly nonlinear Richards equation describing transient water flow in variably saturated soils. Unlike the conventional control volume method, which is regarded as second-order accurate, the proposed approach has high-order accuracy owing to the use of a compact integrated radial basis function approximation that enables improved flux predictions. The method is used to solve the Richards equation for transient flow in 1D homogeneous and heterogeneous soil profiles. Numerical results for different boundary conditions, initial conditions and soil types are shown to be in good agreement with Warrick's semi-analytical solution and simulations using the HYDRUS-1D software package. Results obtained with the proposed method were far less dependent upon the grid spacing than the HYDRUS-1D finite element solutions

A comprehensive evaluation of polygenic score and genotype imputation performances of human SNP arrays in diverse populations

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