Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

Boundary element approach for identification of boundary conditions of distributed-parameter systems

An identification problem is discussed for the boundary conditions of a deterministic distributed-parameter system governed by a partial differential equation of parabolic type. A method based on the fundamental ideas of the boundary element method is proposed for identification of unknown boundary conditions. The identification problem is formulated by using the ideas of boundary partition and weighted residual expression corresponding to the given partial differential equation. The boundary conditions are estimated by the method of least squares using the state observations taken at the interior points. A numerical example illustrates the applicability of the proposed method for boundary identification.     

Finite Pointset Method for biharmonic equations

Finite pointset method is one of the grid free methods that is used to solve differential equations arising from physical problems. It is a local iterative procedure based on weighted least square approximation technique. In this paper, biharmonic equation with simply supported, clamped and Cahn–Hilliard type boundary conditions, is solved using the finite pointset method. Numerical examples illustrate the efficiency of the method.     

Adaptive simulation of two dimensional hyperbolic problems by Collocated Discrete Least Squares Meshless method

An adaptive refinement technique is presented in this paper and used in conjunction with the Collocated Discrete Least Squares Meshless (CDLSM) method for the effective simulation of two-dimensional shocked hyperbolic problems. The CDLSM method is based on minimizing the least squares functional calculated at collocation points chosen on the problem domain and its boundaries. The functional is defined as the weighted sum of the squared residuals of the differential equation and its boundary conditions. A Moving Least Squares (MLS) method is used here to construct the meshless shape functions. An error estimator based on the value of functional at nodal points used to discretize the problem domain and its boundaries is developed and used to predict the areas of poor solutions. A node moving strategy is then used to refine the predicted zones of poor solutions before the problem is resolved on the refined distribution of nodes. The proposed methodology is applied to some two dimensional hyperbolic benchmark problems and the results are presented and compared to the exact solutions. The results clearly show the capabilities of the proposed method for the effective and efficient solution of hyperbolic problems of shocked and high gradient solutions.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

基于无网格自然邻接点Petrov-Galerkin法求解带源参数瞬态热传导问题

基于无网格自然邻接点Petrov-Galerkin法,本文建立了一种求解带源参数瞬态热传导问题的新方法.为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻接点插值构造试函数.在局部多边形子域上采用局部Petrov-Galerkin方法建立瞬态热传导问题的积分弱形式.这些多边形子域可由Delaunay三角形创建.时间域则通过传统的两点差分法进行离散.最后通过算例验证了该数值算法的有效性和正确性.     

Parallel implementation of the Kronecker product technique for numerical solution of parabolic partial differential equations

Using the alternating directional Galerkin technique we show that the approximate solution of the initial boundary value problem of parabolic partial differential equations is equivalent to the least squares solution of the linear system A B = b. In the full rank case, an efficient method for obtaining the solution of the least squares problem suitable for distributive memory computers was presented in (Fausett et al., 1994). This method is extended to solve the rank deficient case using the RRQR factorization of matrices A and B together with the commutatively property of the Kronecker product. Solution algorithm and parallel implementation are discussed. Timing results are presented and compared with previous work.     

Numerical solution of laplace equation in three dimensions for a pierce type electron gun

In this paper, as part of a three-dimensional ribbon beam gun design, Laplace equation is solved in 3D for the case of boundaries which can be represented as simple surfaces which are subjected to specified voltages. Curve fitting by the least squares method is used to characterize the accelerator electrode shape as an extension of the 1D laminar electron flow in a Pierce type electron gun. The method involves solution of governing equations by an iterative finite difference technique. Modification of the standard Leibmann procedure is used to greatly increase the rapidity of convergence. The technique of handling irregular boundary points is considered in detail.     

Modified integral equation solution of viscous flows near sharp corners

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's Theorem is used to reformulate the differential equation as a pair of coupled integral equations. The classical BBIE gives poor convergence in the presence of singularities arising in the solution domain. The rate of convergence is improved dramatically by including the analytic behaviour of the flow in the neighbourhood of the singularities. The modified BBIE (MBBIE) effectively ‘subtracts out’ this analytic behaviour in terms of a series representation whose coefficients are initially unknown. In this way the modified flow variables are regular throughout the entire solution domain. Also presented is a method for including the asymptotic nature of the flow when the solution domain is unbounded.     

基于移动最小二乘法的气动力数据建模方法

针对一体化飞行器高度耦合的非线性气动问题,提出了一种基于移动最小二乘法的气动力数据建模方法;首先,对影响模型精度的因素进行了分析;接着,在构建移动最小二乘模型时采用遗传算法获取最佳支撑域半径以及最佳影响因子β,提高近似精度从而达到减少样本点的目的;得到泛化能力较强的气动力模型,并与偏最小二乘方法的建模结果进行对比;实验结果表明:移动最小二乘法的建模效果优于偏最小二乘方法,预测误差较小,证明了将该方法应用于气动数据建模是可行的。     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.     

The element-free Galerkin method for the nonlinear p-Laplacian equation

The element-free Galerkin (EFG) method is developed in this paper for solving the nonlinear p-Laplacian equation. The moving least squares approximation is used to generate meshless shape functions, the penalty approach is adopted to enforce the Dirichlet boundary condition, the Galerkin weak form is employed to obtain the system of discrete equations, and two iterative procedures are developed to deal with the strong nonlinearity. Then, the computational formulas of the EFG method for the p-Laplacian equation are established. Numerical results are finally given to verify the convergence and high computational precision of the method.     

Local integration of population dynamics via moving least squares approximation

This paper applies an approach based on the Galerkin and collocation methods so-called meshless local Petrov–Galerkin (MLPG) method to treat a nonlinear partial integro-differential equation arising in population dynamics. In the proposed method, the MLPG method is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The moving least squares approximation is proposed to construct shape functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. Convergence in both time and spatial discretizations is shown and more, stability of the method is illustrated.     

Dirichlet feedback control for the stabilization of the wave equation: a numerical approach

In (J. Differential Equations 66 (1987) 340) a uniform stabilization method of the wave equation by boundary control à la Dirichlet has been discussed. In this article, we investigate the numerical implementation of the above stabilization process by a numerical scheme which mimics the energy decay properties of its continuous counterpart. The practical implementation of that scheme leads to a biharmonic problem of a new type which is solved by a method directly inspired by some related work of Glowinski and Pironneau on the solution of the Dirichlet problem for the biharmonic operator (SIAM Rev. 21(2) (1979) 167). Numerical experiments show that the decay properties of the energy are well-preserved by our numerical methodology.     

Boundary element method applied to the analysis of thin plates

The classic Kirchhoff plate theory is formulated through integral representation decomposing the biharmonic field equation in two coupled harmonic equations. Two domain integrals appear, one inherent to the problem and the other due to the previous decomposition. Both of them are evaluated defining equivalent boundary integrations, using collocation method in several points on the boundary and the domain. These equivalent integrations are the same as those involved in the integral representation of the Poisson equation, therefore no extra integrations are needed.     

Identification of hazards with impulsive sources

Given some observations downstream can one determine the location and intensities of point sources of a hazard (pollutant chemical or biological)? The unknown concentrations are governed by the diffusion-advection partial differential equation. The corresponding algebraic system is studied. The fixed location problem is considered using reordering, the Schur complement and nonnegative least squares. A nonlinear problem is proposed, and an iterative method is formulated based on nonnegative least squares and Newton's method. The variable location problem is tackled with simulated annealing. The complexities of controlling aquatic populations, which are nonlinear, time-dependent and have multiple sources, will be illustrated.     

Unsteady flow computations for flow past multiple moving boundaries using LSKUM

We present the latest developments in the least squares kinetic upwind method (LSKUM), a kinetic theory based grid free approach for the solution of Euler equations. A single step higher order scheme through modified CIR splitting is presented. A new weighted least squares method has been used in the present work which simplifies the 2-D formulae to an equivalent 1-D form. This is achieved through diagonalisation of the least squares matrix through suitable choices of the weights. All these developments have been extended to problems with moving nodes and boundaries. A 2-D unsteady Euler code has been developed incorporating all the above ideas along with the well known dual time stepping procedure. The code has been verified and validated for the standard test case AGARD CT(5) which corresponds to unsteady flow past oscillating NACA0012 airfoil pitching about quarter chord. Good comparisons with the experimental values have been obtained. In order to demonstrate the ability of the method to handle multiple moving bodies we have computed unsteady flow past two oscillating NACA0012 airfoils one behind the other. Some interesting results are presented for this case.     

Finite difference methods for solving

In this report, finite difference methods of orders 2 and 4 are developed and analysed for the solution of a two-point boundary value problem associated with a fourth- order linear differential equation. A sufficient condition guaranteeing a unique solution of the boundary value problem is also given. Numerical results for a typical boundary value problem are tabulated and compared with the shooting technique using the fourth-order Runge-Kutta method.     

The structured total least squares algorithm research for passive location based on angle information

Based on the constrained total least squares (CTLS) passive location algorithm with bearing-only measurements, in this paper, the same passive location problem is transformed into the structured total least squares (STLS) problem. The solution of the STLS problem for passive location can be obtained using the inverse iteration method. It also expatiates that both the STLS algorithm and the CTLS algorithm have the same location mean squares error under certain condition. Finally, the article presents a kind of location and tracking algorithm for moving target by combining STLS location algorithm with Kalman filter (KF). The efficiency and superiority of the proposed algorithms can be confirmed by computer simulation results.     

Integral Equation Formulation of the Biharmonic Dirichlet Problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman–Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.