B-spline finite-element method in polar coordinates

A numerical solution method for two-dimensional electromagnetic field problems is presented using the B-spline finite-element expression based on polar coordinates. The technique has two main advantages: (1) to avoid the truncation errors at some curved boundaries and (2) to improve the accuracy of singular boundary-value problems with a sharp corner. The B-spline finite-element formulation in polar coordinates is derived and its numerical applications are illustrated by an eddy current problem and several waveguide eigenvalue problems.     

Free-Edge Stresses around Holes in Laminates by the Boundary Finite-Element Method

In the present paper the boundary finite-element method is presented as a highly efficient technique for the numerical investigation of the free-edge stresses around a circular hole in laminates. In this method, as in the boundary element method, only the boundary needs to be discretized, whereas the element formulation in essence is finite-element based. The surface discretization provides a high numerical efficiency and requires less computation time compared to finite-element analyses. Numerical results for the concentration of interlaminar stresses at holes in composite laminates show a very good agreement with comparative finite-element calculations.     

二维非线性对流扩散方程特征有限元的两重网络算法

针对二维非线性对流扩散方程,构造了特征有限元两重网格算法．该算法只需要在粗网格上进行非线性迭代运算,而在所需要求解的细网格上进行一次线性运算即可．对于非线性对流占优扩散方程,不仅可以消除因对流占优项引起的数值振荡现象,还可以加快收敛速度、提高计算效率．误差估计表明只要选取粗细网格步长满足一定的关系式,就可以使两重网格解与有限元解保持同样的计算精度．算例显示:两重网格算法比特征有限元算法的收敛速度明显加快．     

Analysis of quadruple corner-cut ridged square waveguide using a scaled boundary finite element method

This paper describes the development of an efficient semi-analytical method, namely scaled boundary finite-element method (SBFEM) for a quadruple corner-cut ridged square waveguide. Thinking about its symmetry, only a quarter of its cross-section needs to be considered and divided into a few sub-domains. Only the boundaries of the sub-domains are discretized with line elements leading to great flexibility in mesh generation. The singularities in the re-entrant corners are represented analytically by locating the scaling center in those points. Variational principle approach is presented to formulate the basis SBFE equations for the sub-domains. Then, an equation of the ‘stiffness matrix’ on the discretized boundary is established. Finally, by using the continued-fraction solution and introducing auxiliary variables, a generalized eigenvalue equation with respect to the cutoff wave number is obtained without introducing an internal mesh. Numerical results are presented to verify the accuracy and efficiency of the present technique. Variations of the cutoff wave numbers of the dominant and higher-order modes for both TE and TM cases with the corner-cut ridge dimensions are investigated in details. Simple approximate equations are found to accurately predict the cutoff wave number of TE_20U, TE₂₂, TM₁₁ and TM_13L modes. The single mode bandwidth of the waveguide is also calculated.     

Mesh shape-quality optimization using the inverse mean-ratio metric

Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finite-element method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program that relocates the vertex coordinates of a given mesh to optimize the average element shape quality as measured by the inverse mean-ratio metric. To solve the resulting large-scale optimization problems, we apply an efficient implementation of an inexact Newton algorithm that uses the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional functions, applying this result to components of the inverse mean-ratio metric, and showing that each block in the preconditioner is invertible. Numerical results obtained with this special-purpose code on several test meshes are presented and used to quantify the impact on solution time and memory requirements of using a modeling language and general-purpose algorithm to solve these problems.     

Numerical solution method for the electric impedance tomography problem in the case of piecewise constant conductivity and several unknown boundaries

We study the electrical impedance tomography problem with piecewise constant electric conductivity coefficient, whose values are assumed to be known. The problem is to find the unknown boundaries of domains with distinct conductivities. The input information for the solution of this problem includes several pairs of Dirichlet and Neumann data on the known external boundary of the domain, i.e., several cases of specification of the potential and its normal derivative. We suggest a numerical solution method for this problem on the basis of the derivation of a nonlinear operator equation for the functions that define the unknown boundaries and an iterative solution method for this equation with the use of the Tikhonov regularization method. The results of numerical experiments are presented.     

The Numerical Solution of Sets of Linear Equations Arising from Ritz-Galerkin Methods

The Ritz-Galerkin solution of a linear integral or differentialequation or set of equations leads to a set of linear algebraicequations, the structure of which depends on the type of expansionset used. For a finite-element expansion, the matrix involvedis sparse, and reasonably efficient solution techniques areknown. We study here the alternative case when a "global" expansionis chosen. Then the matrix involved is in general full, buthas nonetheless a characteristic structure; we discuss the waysin which this structure can be used to yield efficient solutionmethods. Our main result is that a block iterative method canyield an arbitrarily high convergence rate; however, we alsoconsider the stability of a direct solution of the equations.     

Galerkin-petrov limit schemes for the convection-diffusion equation

In the present paper, we suggest a method for constructing grid schemes for the multidimensional convection-diffusion equation. The method is based on the approximation of the integral identity that is used in the definition of a weak solution of the differential problem. The use of spaces of smooth trial functions and spaces of functions with possible discontinuities in which the solution of the original problem is sought naturally leads to Galerkin-Petrov methods. The suggested method for the construction of grid schemes is based on a finite-element semidiscretization of the original space with respect to space variables, which constructs the space of trial functions on the basis of the direction of the convective transport near the boundaries of finite elements, the limit passage from a scheme with smooth trial functions to schemes with discontinuous trial functions, and the further discretization of the resulting equations with respect to the time variable. We prove the stability of the constructed difference schemes and present the results of computations for model problems.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

On the ritz penalty method for solving the control of a diffusion equation

The study of the finite-element method for the solution of a variety of complicated scientific problems has enjoyed a period of intense activity and stimulation, because of its simplicity in concept and elegance in development. These qualities have led eventually to its growing acceptance as a promising technique equipped with a powerful mathematical basis. The finite-element method operates on the subdomain principle; this means that the domain of the equation to be solved is usually divided into a number of separate regions or subdomains. The unknown solution function is then approximated in each subdomain by some functions, generally known as pyramid functions or basis functions. In the present article, the Ritz penalty method, which is based on finite-element processes, is the programming method used for establishing our numerical results. Our intent is to demonstrate extensively the effect of large penalty constants on the profile of the inputU(x,t) that minimizes the diffusion control problem, Problem (P1). We show that, as the penalty constant tends to infinity in an attempt to attain close constraint satisfaction, the rate of convergence of our method deteriorates sharply.     

Applications of RBF collocation method to elliptic PDEs in an arbitrary domain by fictitious domain extensions

A fictitious domain extension approach is introduced to study elliptic PDE's defined in arbitrary domains by the radial basis function (RBF) collocation method. In this approach, arbitrary physical geometries are extended to domains which are topologically rectangular. The solution domain is also extended to the fictitious area and assumed to satisfy the same governing equation in it and on its extended boundaries. The boundary conditions are still specified on the boundaries of the original physical domain. The problem in the extended domain becomes ill-posed. However, it can be easily circumvented by the collocation method. We demonstrate that the solution can be directly obtained without domain decompositions and iterations. The new approach is simple, efficient and accurate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-element calculation of a plate compliant in transverse shear

A finite-element calculation of a plate with a low transverse shear stiffness is presented. As the basic kinematic parameters, the angles of transverse shear at each of four nodes of the finite element of the plate are selected. The results found for the stress-strain state of an isotropic homogeneous composite and a three-layer plates confirm that the finite-element model elaborated is also efficient in the cases of nonclassical boundary conditions for plates, including conditions for the angles of transverse shear.     

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.     

On the shear boundary layer of tension plates

The present paper deals with a generic class of problems for plates subjected to loadings combining a high in-plane tension and a small lateral pressure. It develops the governing differential equations in the singular pertubation form, through the postulation of retaining only one of the Kirchhoff's assumptions, that the plate thickness in the boundary layer region is invariant. The solution by using the standard perturbation method is discussed. The postulation is justified when it is demonstrated that in the shear boundary layer the plate thickness is of higher-order smallness. The general method of solution by the standard perturbation technique is applied to an annular plate problem. Problems of different combinations of supports at the inner and the outer boundaries are solved. The case in which both edges are simply supported is presented as an illustration of the solution technique. In other cases results only are presented. The effect of support on the boundaries is also discussed. The shear effect is found to be most significant at a clamped edge. In the special geometry, it is possible to demonstrate that, when the condition on membrane force is not met as required in the general theory, thagnitude of the boundary layer changes. Specifically, the paper presents a case in which the membrhich the membrane force is zero at the inner edge.     

Finite element solution of the Kolmogorov-Feller equation

For the Kolmogorov-Feller integro-differential equation, a solution method is proposed based on its finite-element approximation.     

Numerical studies of optimal grid construction

This article is concerned with the construction of optimal grids for convection dominated problems. We consider the redistribution approach to generate the optimal grids. With an optimal grid, the number of unknowns can be dramatically reduced and this generally gives a computationally efficient model. Both Galerkin finite-element and least-squares finite-element approximations are considered with mesh redistribution. Numerical results of models problems illustrating the efficiency of the minimum mesh size approach are presented. Discussions of the capabilities and limitations of the schemes are also provided. © 1996 John Wiley & Sons, Inc.     

New Optimization Approach to Multiphase Flow

A new optimization formulation for simulating multiphase flow in porous media is introduced. A locally mass-conservative, mixed finite-element method is employed for the spatial discretization. An unconditionally stable, fully-implicit time discretization is used and leads to a coupled system of nonlinear equations that must be solved at each time step. We reformulate this system as a least squares problem with simple bounds involving only one of the phase saturations. Both a Gauss–Newton method and a quasi-Newton secant method are considered as potential solvers for the optimization problem. Each evaluation of the least squares objective function and gradient requires solving two single-phase self-adjoint, linear, uniformly-elliptic partial differential equations for which very efficient solution techniques have been developed.     

On computational proofs of the existence of solutions to nonlinear parabolic problems

This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C⁰ finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.     

Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems

We consider the iterative solution of linear systems of equations arising from the discretization of singularly perturbed reaction-diffusion differential equations by finite-element methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be arbitrarily small, and correspondingly, their solutions feature layers: regions where the solution changes rapidly. Therefore, numerical solutions are computed on specially designed, highly anisotropic layer-adapted meshes. Usually, the resulting linear systems are ill-conditioned, and so, careful design of suitable preconditioners is necessary in order to solve them in a way that is robust, with respect to the perturbation parameter, and efficient. We propose a boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method (MacLachlan and Madden, SIAM J. Sci. Comput. 35(5), A2225–A2254 2013). We prove the optimality of this preconditioner and establish a suitable stopping criterion for one-dimensional problems. Numerical results are presented which demonstrate that the ideas extend to problems in two dimensions.     

On one approach to realization of the branch-and-bound method on a distributed system

The branch-and-bound method is one of most efficient search algorithms. However, parallel decomposition of this method is difficult even on architectures with common memory, not to mention distributed systems. The paper describes an algorithm for solution of the problems that can be solved with the branch-and-bound method on a distributed computer system. The results of testing of the proposed algorithm via solution of the chess-game problem are presented.