This paper provides examples of the numerical solution of boundary-value problems in nonlinear magnetoelasticity involving finite geometry based on the theoretical framework developed by Dorfmann and co-workers. Speci...
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This paper provides examples of the numerical solution of boundary-value problems in nonlinear magnetoelasticity involving finite geometry based on the theoretical framework developed by Dorfmann and co-workers. Specifically, using a prototype constitutive model for isotropic magnetoelasticity, we consider two two-dimensional problems for a block with rectangular cross-section and of infinite extent in the third direction. In the first problem the deformation induced in the block by the application of a uniform magnetic field far from the block and normal to its larger faces without mechanical load is examined, while in the second problem the same magnetic field is applied in conjunction with a shearing deformation produced by in-plane shear stresses on its larger faces. For each problem the distribution of the magnetic field throughout the block and the surrounding space is illustrated graphically, along with the corresponding deformation of the block. The rapidly (in space) changing magnitude of the magnetic field in the neighbourhood of the faces of the block is highlighted. (C) 2010 Elsevier Ltd. All rights reserved.
boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral...
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boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered.
A Galerkin's finite element approach based on weighted-residual formulation is presented to find approximate solutions to obstacle, unilateral and contact second-order boundary-value problems. The approach utilize...
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A Galerkin's finite element approach based on weighted-residual formulation is presented to find approximate solutions to obstacle, unilateral and contact second-order boundary-value problems. The approach utilizes a piece-wise linear approximations utilizing linear Langrange polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of this scheme in comparison to collocation, finite-difference and spline methods.
In this article, with the help of fractal quintic spline, numerical solutions of singularly perturbed boundaryvalue problem of ordinary differential equations are obtained. To check the accuracy of the proposed metho...
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In this article, with the help of fractal quintic spline, numerical solutions of singularly perturbed boundaryvalue problem of ordinary differential equations are obtained. To check the accuracy of the proposed method, convergence analysis is derived. The developed method has fourth order convergence. To check the correctness of the proposed method, numerical examples are provided.
This paper describes an improvement of England and Mattheij's code MUTSSYM for solving linear boundary-value problems for Ordinary Differential Equations, which may or not give rise to sharp boundary Layers. The m...
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This paper describes an improvement of England and Mattheij's code MUTSSYM for solving linear boundary-value problems for Ordinary Differential Equations, which may or not give rise to sharp boundary Layers. The method is based on Multiple Shooting with a Decoupling strategy, allowing the calculation of stable solutions according to the increasing or decreasing fundamental modes. The integration of the associated Initial-valueproblems is performed using a 4(th)-order symmetric implicit Runge-Kutta method with the Dichotomic Stability property. If the problem is well conditioned, the method calculates discrete decaying (growing) modes controlled by initial (terminal) conditions corresponding to similar continuous modes. A special step-size control strategy permits efficient calculation of the numerical solution throughout the interval. (C) 1999 Elsevier Science Ltd. All rights reserved.
The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirro...
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The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirrors the natural decomposition of the original constraints. Second, it details a library of realizations of the zero functions, monitor functions, and contributions to the adjoint equations associated with each stage of problem construction. Third, it illustrates the application of this framework to multisegment trajectory problems, for example, in the analysis of spacecraft optimal control problems. The proposed construction paradigm generalizes a formulation for the construction of integro-differential-algebraic continuation problems in the software package COCO to also include automatically generated adjoint terms. The implementation enables arbitrarily many levels of nested construction with larger problems assembled from smaller ones, and multiple instances of a problem class combined inside a composite problem with the help of suitably de fined coupling conditions. The application of this paradigm to problems of constrained optimization relies on the successive continuation paradigm introduced by Kernevez and Doedel [Optimization in bifurcation problems using a continuation method, in Bifurcation: Analysis, Algorithms, Applications, Birkhauser Verlag, Basel, 1987, pp. 153-160], in which solutions to the necessary conditions for locally optimal solutions are found at the end of a sequence of easily initialized separate stages of continuation. The numerical examples illustrate the paradigm of construction and the efficacy of the continuation approach in locating stationary solutions to high-dimensional problems.
A review of various techniques for the approximate solution of boundary-value problems is presented. The analysis is based on a simple one-dimensional example. A differential equation describing steady-state voltage d...
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A review of various techniques for the approximate solution of boundary-value problems is presented. The analysis is based on a simple one-dimensional example. A differential equation describing steady-state voltage distribution along a transmission line excited by a dc source was taken as the example. The problem was solved by the commonly applied methods of moments and by the variational Ritz method using polynomial and trigonometric approximations. The specific features of the methods applied are discussed. Although the example does not deal with wave propagation, it is valuable for teaching numerical methods in electromagnetics.
In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-val...
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In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed. (C) 2007 Elsevier B.V. All rights reserved.
When applying the multiple-shooting code BNDSCO [7] to boundary-value problems (BVPs) on a vector computer, a vectorized version of a numerical integration method for ordinary differential equations (ODEs) has to be d...
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When applying the multiple-shooting code BNDSCO [7] to boundary-value problems (BVPs) on a vector computer, a vectorized version of a numerical integration method for ordinary differential equations (ODEs) has to be developed. Since the dominating part of the computing time is spent in solving a series of initial valueproblems (IVPs), the computing time can be considerably reduced by the use of a vectorized Runge-Kutta method. The method presented is applicable for BVPs with nearly all kinds of discontinuities in the solution as well as in the right-hand side at a finite number of interior points and is therefore applicable for many types of optimal-control problems (OCPs) with constraints.
This study puts forward construction of an efficient nonpolynomial twin parameter cubic spline-based numerical scheme for approximations to the solution of heat transfer and defection in cables problems represented as...
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This study puts forward construction of an efficient nonpolynomial twin parameter cubic spline-based numerical scheme for approximations to the solution of heat transfer and defection in cables problems represented as system of second-order boundary-value problems. The introduction of an additional parameter in trigonometric part of nonpolynomial cubic spline makes this scheme a better one as compared to other existing numerical methods. The Icing on the cake is the applicability of the proposed scheme for unequal step size. The present algorithm gives better approximations in comparison to other spline, collocation, and finite-difference methods. The convergence analysis of the proposed algorithm is talked about to make a strong foundation to the proposed algorithm. Practical usefulness of the proposed method is illustrated through numerical examples.
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