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To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a new solution strategy for the minimal compliance problem in topology optimization. This problem contains the system of linear elasticity partial differential equations (PDEs) as a constraint resulting in a large scaled optimization problem after the finite element discretization. Due to the repeated solution of the direct field problem given by the PDE constraints, efficient solution techniques are required. In this paper we present a new solution method involving adaptive multilevel techniques combined with a multigrid approach for the direct problem. Topology optimization problems are ill-posed, so regularization is needed. In our algorithm we combine two regularization techniques, in fact filter methods, such that their disadvantages are eliminated and only their positive properties remain. Numerical experiments are performed with several benchmark problems, where our multilevel approach turns out to be quite efficient. For solving the optimization problems arising in each iteration step, the method of moving asymptotes is used. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

By using the natural boundary reduction an overlapping domain decomposition method is designed to solve some exterior two-dimensional time-dependent parabolic problems. The governing equation is first discretized in time, leading to a sequence of boundary value problems with respect to time step in an unbounded domain. Then artificial boundaries are introduced. For each time level, an overlapping domain decomposition method, which is based on the natural boundary reduction, is constructed to solve the exterior elliptic boundary value problem on a two-dimensional domain. It is shown that the algorithm is equivalent to Schwarz alternating method. The convergence of this algorithm is given. The contraction factor for the exterior circular domain is also discussed. In the end of this paper, some numerical examples are presented, which illustrate the feasibility and the effectiveness of the proposed methods in this paper. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The idea of the paper is to present theoretical aspects and finite element method implementation of the sensitivity analysis in homogenization of linear elastic fibre composites by the use of effective modules homogenization approach. The deterministic sensitivity approach to the homogenization problem is presented in a general form for n-component composite and is illustrated in the example of two-dimensional fibre-reinforced periodic composite structure. The results of sensitivity analysis shown in the paper confirm the usefulness of the homogenization method to computational analysis of composite materials and may be successfully applied to numerical optimization of engineering composites and to their shape sensitivity studies. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper demonstrates a simple finite element implementation of Lagrange multipliers to model the mechanical behaviour of an orthotropic composite material. The research shows the proper set of kinematic boundary conditions that must be applied in 2D plane stress elasticity to achieve the correct unit strain vectors that, upon interrogation of the associated Lagrange multipliers, give the stresses induced by these strain vectors. From these stresses the terms in the elasticity matrix can be evaluated. As well as demonstrating the correct kinematic conditions required, the paper presents the consequences of applying intuitive but incorrect conditions. © 1997 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper describes a preconditioner for the iterative solution to the indefinite linear equation system that is obtained when the vector wave equation is discretized by higher-order H(curl)-conforming finite elements. The preconditioner is based on a decomposition of the H(curl)-conforming functions into blocks comprising lower-order functions and higher-order functions. Fast convergence is obtained when the grid spacing is sufficiently small. A method for selecting suitable grid spacings is proposed and a series of numerical examples are included to demonstrate the effectiveness of the approach. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a new fictitious-domain technique for numerically solving elliptic second-order partial differential equations (PDEs) in complex geometries. The proposed technique is based on the use of integral-collocation schemes and Chebyshev polynomials. The boundary conditions on the actual boundary are implemented by means of integration constants. The method works for both Dirichlet and Neumann boundary conditions. Several test problems are considered to verify the technique. Numerical results show that the present method yields spectral accuracy for smooth (analytic) problems. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper illustrates the use of simplified PN approximations as a tools of achieving verification of codes and simulations of radiative transfer in combustion systems. The main advantage of considering these models is the fact that the integro-differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable, compatible to the partial differential equations of flow and combustion, and easy to solve using standard numerical discretizations. Validation of these models is then performed by comparing predictions to measurements for a three-dimensional diffusion flame. The good agreement between measurements and predictions indicates that the simplified PN models can be used to incorporate radiation transfer in combustion systems at very low computational cost without relying on discrete ordinates or Monte Carlo methods. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Free-surface boundary is not only unknown in a priori but also the boundary conditions are overspecified. In this paper, the Laplace problem with overspecified boundary conditions on the free surface is solved by using the hypersingular equation instead of singular equation used conventionally in boundary element method. The free surface can be determined using an iterative procedure starting from an initial guess. By introducing the hypersingular equation, the convergence rate of free surface can be accelerated. Finally, numerical examples including rectangular dams and canals were demonstrated and were compared with others to show the validity of the present method. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a meshfree interpolating moving least squares (IMLS) method based on singular weights for the solution of partial differential equations. Due to the specific singular choice of weight functions, which is needed to guarantee the interpolation, there arises a problem of finding the inverse of the occurring singular matrix. The inverse is carried out using a regularization of weight functions. It turns out that a stable inverse is obtained by considering the vanishing regularization parameter. Moreover, the use of this perturbation technique allows the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost. Unlike standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to really interpolating functions which satisfy the Kronecker-delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems. Solution to a model BVP as well as an experimental convergence study of the method to analytical solutions, confirming the mathematical derivations, are given. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyse some local and parallel covolume algorithms for second-order elliptic problems. We first prove some local a priori error estimates for covolume method. With these results, we shall apply the two-grid strategy to covolume approximation to give two novel covolume algorithms, and show that the convergence rates of these algorithms are optimal in piecewise H1-norm. Numerical results confirming the theory are also provided. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Unstructured mesh based discretization techniques can offer certain advantages relative to standard finite difference approaches which are commonly used in reservoir simulation, due to their flexibility to model complex geological features and to their ability to easily incorporate mesh adaptation. Within such class of methods the most frequently used are the finite element method (FEM) and the finite volume method (FVM). The latter is particularly attractive in reservoir problems due to its local and global conservation properties. The main goal of the present paper is to describe an unstructured edge-based FVM formulation which is used to solve the partial differential equations resulting from the modelling of two-phase fluid flow of oil and water in homogeneous porous media, when an IMPES (IMplicit Pressure Explicit Saturation) approach is used. This finite volume formulation is similar to the edge-based FEM formulation when linear triangular or tetrahedral elements are employed. Some model examples are presented in order to validate the presented formulation. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a unified method is developed for calculating the eigenvalues of the weighted adjacency and Laplacian matrices of three different graph products. These products have many applications in computational mechanics, such as ordering, graph partitioning, and subdomaining of finite element models. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The redistribution and embedding are two main adaptive-grid methods. With use of initial structured grid, it is possible to combine these adaptive-grid techniques. It means that before the grid becomes unstructured, we employ redistribution technique and then use embedding method. It is used to solve three model problems, transonic and supersonic inviscid flows in channels with circular arc bump and with convergent part. The Euler equations are integrated to steady state by an explicit, finite volume, Ni's Lax–Wendroff type. The combination of adaptive-grid redistribution and embedding is complex, but it is shown in this paper that it is very efficient and worth to use it in large CPU time consuming problems. The results also show that the combination requires less grid points and therefore less memory space for prescribed accuracy. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper numerical methods for solving first-order hyperbolic partial differential equations are developed. These methods are developed by approximating the first-order spatial derivative by third-order finite-difference approximations and a matrix exponential function by a third-order rational approximation having distinct real poles. Then parallel algorithms are developed and tested on a sequential computer for an advection equation with constant coefficient and a non-linear problem. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper describes a strategy to control errors in finite element approximations of the time-dependent incompressible Navier–Stokes equations. The approach involves estimating the errors due to the discretization in space, using information from the residuals in the momentum and continuity equations. Following a numerical stability analysis of channel flows past a cylinder, it is concluded that the errors due to the residual in the continuity equation should be carefully controlled since it appears to be the source of unphysical perturbations artificially created by the spatial discretization. The performance of the adaptive strategy is then tested for lid-driven oblique cavity flows. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper the Tikhonov regularization method is analysed for the ill-conditioned system of linear equations which arises from the discretization of an ill-posed boundary value problem in linear elasticity. The choice of the regularization parameter is achieved according to the L-curve method. Then the regularization method is numerically implemented using the boundary element method (BEM). Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we provide an example to illustrate the checkerboard mode Ker(B*h)={qh∈Qh: (qh,div vh)=0, ∀vh∈Vh} for velocity–pressure finite element space Vh-Qh for the Stokes problem, which varies with three different types of triangular meshes of the domain.We prove that the inf–sup stable condition is valid, independent of mesh size h for the Union Jack mesh but depends on mesh size h with O(h) for the uniform triangular mesh and fish-bone mesh. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, by using classical tensor calculus, we derive the compressible Navier–Stokes equation on a so-called stream surface which is a two-dimensional (2-D) manifold that gives a definition of a stream function with the equation satisfied by it. Based on this, a new algorithm is proposed which is called dimension split algorithm. This new method is different from the domain decomposition method. In the domain decomposition method, we have to solve a three-dimensional problem in each subdomain but we solve a 2-D problem in each subdomain. A numerical experiment for the turbomachine is provided. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper compares finite element and finite volume schemes for some Poisson-type problems. Special attention is devoted to the conditioning properties of the linear systems to be solved. For the benchmark which is considered, we show that the finite element scheme leads to the lowest condition numbers. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Domain decomposition methods for Laplace's equation are investigated in the present paper. An optimization approach is adopted to satisfy the Dirichlet and/or Neumann interface conditions. The adjoint variable method is used to find the sensitivity of the objective function of optimization. The boundary element method is utilized for the discretization of the primary and adjoint problems. Two example problems are studied with the proposed solution method. Comparisons are also provided by using two other techniques, namely the Uzawa and Schwarz methods, already available in the literature. It is also suggested that a parallel implementation of the method could be performed leading to computational efficiency. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The numerical dispersion relation is an important measure of the discretization error inherent in grid-based numerical methods for solving Maxwell's equations. In this paper we calculate the numerical dispersion of a vector finite element method for the case of a distorted three-dimensional hexahedral grid. The main result is that the numerical dispersion relation remains second-order accurate as the grid is distorted, although the dispersion becomes quite anisotropic. Published in 2000 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper describes an analytical model for a beam system, based on a modified Timoshenko theory, where the beam is pinned to a hub driven by an actuator at one end and is subject to a heavy load at the other end. A new efficient computational algorithm is then proposed for solving the higher-order non-canonical partial differential equation model, which is developed based on the generalized difference method. This allows a suitable selection of different trial and test spaces, so as to improve the computational efficiency while preserving the high convergence rate of the standard finite element method. With the trial space of cubic Hermite finite elements and the test space of piecewise linear functions, the computational scheme reduces to a semi-discretized or even fully discretized computational algorithm. A numerical simulation result is included to visualize the theoretical modelling and computational results. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper a numerical solution for incompressible Stokes equations using moving least-squares interpolators is developed. This approach does not require an element discretization; just a cloud of points is necessary. This is very attractive for 3D problems and deformable domains. First, taking into consideration that Dirichlet boundary conditions are not applicable a posteriori as in finite elements, a variational weak formulation that includes all kinematic restrictions (Dirichlet and incompressibility) is derived. Then the discretized resultant equations using a moving least-squares (MLS) interpolant for velocity and pressure fields are presented. Finally, the performance of the MLS interpolation is analysed by comparing numerical and analytical solutions, paying attention to the selection of the polynomial degree for the basis function and its orthogonalization. Different aspects of numerical implementation are discussed. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Together with the finite element method (FEM), the control volume method (CVM) is of particular interest for the numerical solution of partial differential equations. The accuracy of computation of the CVM almost matches that of FEM in contour-adapted co-ordinates or block-structured meshes of a single element type. The CVM is superior to the FEM in terms of physical interpretation and ease of handling. The paper presents an interpretation of the FEM that allows it to be treated like a CVM. This formal CVM, which is based on FEM techniques, combines the advantages of both methods. The method is applied to the discretization of the Navier-Stokes equation. [ABSTRACT FROM AUTHOR]

The paper deals with the discretization of any given multi-connected curved surface into triangular elements with straight sides. The method evolves from an initial rough triangular mesh generated from a set of input points which describe the geometry of the problem domain. Interior nodes are distributed according to user-established node-spacing functions of pre-specified spacing control parameters, and they are linked using the advancing front technique. Particular attention is paid to obtaining good distribution of interior nodes in the vicinity of the domain limits. Surface geometry representation is established using triangular B&etilde;zier patches with G1 continuity. This approach ensures a geometrically well-defined working platform for the subsequent discretization of the problem domain. The proposed method requires minimum input from the user and allows mesh gradation and remeshing to be carried out in a straightforward manner. Furthermore, problems associated with variations in the domain geometry as a result of local remeshing are eliminated with the aid of the geometrically pie-defined discretization platform. Results are presented for a range of both curved and planar surfaces, typical of those which might be encountered in hydrodynamics modelling involving flows with a free surface. The presented results demonstrate the flexibility and power of the technique. [ABSTRACT FROM AUTHOR]

We present a new procedure for the numerical calculation of the transient response of systems characterized by partial differential equations in several space variables and lime. The procedure is based on: (i) spatially discretizing the system, (ii) deriving an equivalent circuit for each volume element, (iii) applying a time Laplace transformation and obtaining an admittance matrix, (iv) writing the expressions for the Laplace transforms of the desired variables according to Cramer's determinant rule, (v) interpreting the determinants as generalized eigenvalue problems, (vi) obtaining the proper eigenvalues of the system (poles and zeros), and (vii) finding the time-dependent solutions by 'trivial' inverse Laplace transformations. Although the procedure proposed applies to many partial differential equations (fluid flow in porous media, conductive heat transfer with convection, pressure tests in oil wells, electrical conduction, etc.), in the paper we illustrate the procedure in detail for a conduction heat transfer system (nine nodes). The procedure presented is much more efficient numerically than the direct numerical integration (in space and time) of the ruling PDE, or than solutions which require 'standard' numerical inverse Laplace transformation after matrix inversion. [ABSTRACT FROM AUTHOR]

The paper presents a new finite difference solution for the 3D non-linear heat equation. This solution is based on the approach of making finite difference substitutions into the solution of the partial differential equation, rather than into the partial differential equation itself, which is the classical approach. Numerical results show that the new FD equation is approximately ten times more accurate than the classical solutions. [ABSTRACT FROM AUTHOR]

Recent advances in planar mesh generation of arbitrary domains incorporate methods for placing points as well as connecting them into a triangulation. Postprocessing techniques such as Laplacian smoothing and mesh relaxation enhance the shape of triangles in these meshes but do not address the construction of meshes near boundaries or provide a criterion to determine the number of interior points of the initial triangulation. The paper addresses these issues by investigating the construction of degree-6 triangulations, the primary goal of the mesh relaxation method. [ABSTRACT FROM AUTHOR]

Based on the concept of the Laplacian matrix of a graph, this paper presents the SGPD (spectral graph pseudoperipheral and pseudodiameter) algorithm for finding a pseudoperipheral vertex or the end-points of a pseudodiameter in a graph. This algorithm is compared with the ones by Grimes eta!. (1990), George and Liu (1979), and Gibbs et a!. (1976). Numerical results from a collection of benchmark test problems show the effectiveness of the proposed algorithm. Moreover, it is shown that this algorithm can be efficiently used in conjunction with heuristic algorithms for ordering sparse matrix equations. Such heuristic algorithms, of course, must be the ones which use the pseudoperipheral vertex or pseudodiameter concepts [ABSTRACT FROM AUTHOR]

A technique is presented whereby numerical calculations of vibration modes can be improved. The paper looks at the classical two-dimensional wave equation using finite difference approximations. Analysis of the numerical dispersion of the approximations is used to develop a correction method. In general the numerical dispersion is dependent upon both the frequency and the direction of a wave, but if a 9-point formula is used the directional dependence is much reduced. This enables correction factors to be obtained using only the frequency of a vibration mode. The method was tested on the vibration of a square membrane and of an L-shaped region; in both cases a marked improvement in accuracy was obtained, at very little computational cost. [ABSTRACT FROM AUTHOR]