An overdetermined B-spline collocation method for Poisson problems on complex domains

The aim of this work is to show how the B-spline collocation method may be used for the approximate solution of Poisson problems considered on complex shaped planar domains in a simple and stable way. The most important aspect of this work consists in the use of approximate Fekete points recently developed by Sommariva and Vianello. Numerical experiments concerning the collocation solution of Poisson problems defined on an amoeba-like domain, star shaped domain and a square with eight holes subject to Dirichlet boundary conditions are presented.     

Numerical solution of Cauchy type singular integral equations with logarithmic weight,based on arbitrary collocation points

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points x _k. Until now these x _k have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of x _k without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.     

A numerical solution of singular integral equations without using special collocation points

A new technique for the solution of singular integral equations is proposed, where the unknown function may have a particular singular behaviour, different from the one defined by the dominant part of the singular integral equation. In this case the integral equation may be discretized by two different quadratures defined in such a way that the collocation points of the one correspond to the integration points of the other. In this manner the system is reduced to a n × n system of discrete equations and the method preserves, for the same number of equations, the same polynomial accuracy. The main advantage of the method is that it can proceed without using special collocation points. This new technique was tested in a series of typical examples and yielded results which are in good agreement with already existing solutions.     

The quasi-linear method of fundamental solution applied to transient non-linear Poisson problems

This paper proposes the use of a quasi-linear method of fundamental solution(QMFS) and explicit Euler method to treat the transient non-linear Poisson-type equations. The MFS, which is a fully meshless method, often deals with the linear and non-linear poisson equations by approximating a particular solution via employing radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the nonhomogeneous term in terms of the dependent variable and finite differencing in time resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the particular solution is no longer needed and the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method.     

On an approximate method of solution of nonlinear heat-conduction problems

An approximate method is presented for the solution of problems of transient heat conduction in solids with thermal conductivity and specific heat linearly dependent on temperature.     

An approximate solution of machine scheduling problems by decomposition method

To obtain an approximate solution for a large-scale job-shop scheduling problem the decomposition method was investigated. This means that an original problem is decomposed into subproblems, which are solved separately, and then the solution of the original problem is composed from the subproblems' solutions. Different methods to decompose the problem were tested by computational experiments and evaluated from the viewpoint of the goodness of schedule and computation time.     

A global collocation technique for the solution of opening mode crack problems

A review of various experimental and numerical techniques for determination of fracture mechanics calibration functions (i.e., the variation of K ₁ and CMOD with crack length) revealed that neither technique, employed independently, can determine K ₁, CMOD, and full field stresses in closed form over a wide range of crack lengths. To fill this void, a combined experimental/numerical collocation technique based on a series expansion of the modified Westergaard functions was developed. This technique uses both boundary conditions, known a priori, and interior stress field conditions, determined using a suitable experimental technique, for analysis of two dimensional, finite body, opening mode crack problems. This paper reports on an investigation of the accuracy of this technique and its sensitivity to errors in experimental data for a sample problem of practical interest.     

Numerical solution of first passage problems using an approximate Chapman-Kolmogorov relation

A new deterministic numerical method for solving first passage time problem is described, analyzed and computationally tested. The method is based on recursively solving an integral equation for the reliability function. The integral equation is derived from the Chapman-Kolmogorov relation and involves an approximation to the Green's function for the forward Kolmogorov equation. An error analysis yields estimates of convergence rates. Numerical experiments indicate that the method is stable and can accurately approximate the reliability function and first passage times.     

Numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points

An integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points (ACP). As a first step a Gaussian quadrature of order n (hence of polynomial accuracy 2n? 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x?_k, determined as roots of appropriate transcedental functions, in order to retain the 2n ? 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that x_k may be arbitrary and yet the high (2n ? 1) accuracy of the Gaussian quadrature is retained.     

Comparison of different methods for choosing the collocation points in the boundary collocation method for 2D-harmonic problems with special purpose Trefftz functions

Based on special purpose Trefftz functions, this paper presents a comparison of different methods for choosing collocation points when a boundary collocation method is used to fulfill the boundary conditions. First, it is shown that for some geometries when applying a boundary collocation method with special purpose Trefftz functions, equidistant collocation points give unacceptable results. Next, for four 2-D harmonic boundary value problems, four cases of different placements of the collocation points are described and compared. Based on the results of the comparison of the test methods for the location of the collocation points it is shown that one of the proposed methods for placing the collocation points is simple to implement and accurate. The essence of this method is an adaptive determination of the consequent collocation points.     

Perturbation technique and method of fundamental solution to solve nonlinear Poisson problems

We show in this work that the Asymptotic Numerical Method (ANM) combined with the Method of Fundamental Solution (MFS) can be a robust algorithm to solve the nonlinear Poisson problem. The ANM transforms the nonlinear problem into a sequence of linear ones which can be solved by MFS. This last method consists of approximating the solution of the linear Poisson problem by a linear combination of fundamental solutions. Some examples are presented to show the efficiency of the proposed method.     

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.     

A method of approximate solution of boundary value problems for rigid,viscoplastic structures

Summary It is shown that under the condition of simple (proportional) loading the Hohenemser-Prager non-linear constitutive equation describing the rigid, viscoplastic material can be reduced to the linear relation first suggested byH. J. Plass. Corresponding flow rules in generalized quantities are then formulated. These relations are used next to solve three boundary value problems for thin plates loaded by a transverse continuous load slowly variable in time. Under the above conditions simple loading is realized and the proposed approach is, therefore, a close approximation to the exact solutions. In the case of simply supported and built-in circular plates closed form solutions are obtained. For square plates an elastic analogy is used to find the velocity fields. Comparison with the solutions already existing in the literature indicates the usefulness of the proposed approximate method.

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Zusammenfassung Es wird gezeigt, daß die nichtlineare Zustandsgleichung von Hohenemser-Prager für ein starr-viskoplastisches Material unter der Voraussetzung einfacher (proportionaler) Belastung auf eine lineare Beziehung zurückgeführt werden kann, die zuerst vonH. J. Plass vorgeschlagen wurde. Entsprechende Fließregeln in verallgemeinerten Größen werden formuliert und benützt, um drei Randwertprobleme für dünne Platten unter kontinuierlicher, langsam veränderlicher Querbelastung zu lösen. Einfache Belastung ist unter diesen Bedingungen realisiert, und das vorgeschlagene Verfahren führt nahe an die exakten Lösungen. Im Falle der drebhar gelagerten und der eingespannten Kreisplatte werden geschlossene Lösungen erhalten; für die quadratische Platte wird eine elastische Analogie benützt, um zu den Fließfeldern zu gelangen. Ein Vergleich mit den in der Literatur bekannten Lösungen zeigt die Brauchbarkeit der vorgeschlagenen Näherungsmethode.

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With 5 Figures     

Integration‐free Coons macroelements for the solution of 2D Poisson problems

Large isoparametric macroelements with closed‐form cardinal global shape functions under the label ‘Coons‐patch macroelements’ (CPM) have been previously proposed and used in conjunction with the finite element method and the boundary element method. This paper continues the research on the performance of CPM in conjunction with the collocation method. In contrast to the previous CPM that was based on a Galerkin/Ritz formulation, no domain integration is now required, a fact that justifies the name ‘integration‐free Coons macroelements’. Therefore, in addition to avoiding mesh generation, and saving human effort, the proposed technique has the additional advantage of further reducing the computer effort. The theory is supported by five test cases concerning Poisson and Laplace problems within 2D smooth quadrilateral domains. Copyright © 2008 John Wiley & Sons, Ltd.     

Radial basis function Hermite collocation approach for the solution of time dependent convection–diffusion problems

This work presents a meshless numerical approach for the solution of time dependent convection–diffusion problems in terms of a Hermite radial basis function interpolation numerical scheme. To test the proposed scheme several numerical examples are analysed including problems with variable convective velocity and reaction coefficient. Comparisons are made with available analytical solutions.     

Direct solution of ill‐posed boundary value problems by radial basis function collocation method

Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.     

A simple solution procedure to 3D-piezoelectric problems: Isotropic BEM coupled with a point collocation method

This paper presents a simple strategy allowing to adapt well established isotropic BEM approach for the solution of coupled problems with anisotropic material parameters. The method which is illustrated on the case of a piezoelectric material is based on the partition of the primary fields into complementary and particular parts. The complementary fields solve the isotropic form of the partial differential equation while particular fields are obtained by a point collocation of the strong form equation. Using the local radial point interpolation method, the effectiveness and accuracy of approach is demonstrated on some examples allowing a comparison with literature results.     

Understanding knee points in bicriteria problems and their implications as preferred solution principles

A knee point is almost always a preferred trade-off solution, if it exists in a bicriteria optimization problem. In this article, an attempt is made to improve understanding of a knee point and investigate the properties of a bicriteria problem that may exhibit a knee on its Pareto-optimal front. Past studies are reviewed and a couple of new definitions are suggested. Additionally, a knee region is defined for problems in which, instead of one, a set of knee-like solutions exists. Edge-knee solutions, which behave like knee solutions but lie near one of the extremes on the Pareto-optimal front, are also introduced. It is interesting that in many problem-solving tasks, despite the existence of a number of solution methodologies, only one or a few of them are commonly used. Here, it is argued that often such common solution principles are knee solutions to a bicriteria problem formed with two conflicting goals of the underlying problem-solving task. The argument is illustrated on a number of tasks, such as regression, sorting, clustering and a number of engineering designs.