An algorithm and a method to search bifurcation points in non‐linear problems

This paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non‐linear time‐independent continuum mechanic problems. First, some theoretical results about uniqueness are recalled. In the framework of the large‐strain assumption, the differences between the classical finite‐step problem and the rate problem are presented. An iterative algorithm able to solve the rate problem is given. Using different initializations, it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property. The constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems. Finally, a method is given which is able in some case to give an upper bound of the bifurcation threshold. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.     

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.     

Mixed-hybrid scheme of the finite element method for solving the problems on bending,free vibration,and stability of plates

To solve a problem on bending, vibration, and stability of plates, a hybrid finite element has been constructed on the basis of Zienkiewicz’s triangle. A mixed approximation is used for the plate deflection and turning angles. It is shown that with a decrease in the triangle dimensions the mixed approach ensures convergence both for the plate deflection and the bending moments, which is practically independent of the way the plate is split into triangular elements. In the problems on free vibrations and stability of plates, the mixed approach yields more exact values of the eigenfrequencies and critical loads as compared to a classical Zienkiewicz’s triangle. The results of the numerical analysis of the convergence and accuracy of the solutions to a number of test problems on bending, free vibration, and stability of a square plate are presented. __________ Translated from Problemy Prochnosti, No. 4, pp. 108–122, July–August, 2008.     

The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media

In this paper, the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady‐state heat conduction in an anisotropic medium. Since the resulting matrix equation is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, while the optimal regularization parameter is chosen according to the L‐curve criterion. Numerical results are presented for both two‐ and three‐dimensional problems, as well as exact and noisy data. The convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries, and decreasing the amount of noise added into the input data, respectively, are analysed. A sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed. The numerical results obtained show that the proposed numerical method is accurate, convergent, stable and computationally efficient, and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady‐state heat conduction. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for computation of discontinuous wave propagation in heterogeneous solids: basic algorithm description and application to one‐dimensional problems

An explicit integration algorithm for computations of discontinuous wave propagation in heterogeneous solids is presented, which is aimed at minimizing spurious oscillations when the wave fronts pass through several zones of different wave speeds. The essence of the present method is a combination of two wave capturing characteristics: a new integration formula that is obtained by pushforward–pullback operations in time designed to filter post‐shock oscillations, and the central difference method that intrinsically filters front‐shock oscillations. It is shown that a judicious combination of these two characteristics substantially reduces both spurious front‐shock and post‐shock oscillations. The performance of the new method is demonstrated as applied to wave propagation through a uniform bar with varying courant numbers, then to heterogeneous bars. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of the crack compliance method to long axial cracks in pipes with allowance for geometrical nonlinearity and shape imperfections (dents)

Application of the crack compliance method to the analysis of thin-walled rings with a radial crack has two features: a crack is considered as a concentrated angular compliance and the deformation of all other sections of the rings is calculated as for a curvilinear beam. The latter can be most conveniently found by the method of initial parameters where the values of generalized forces and displacements at the end of some zone are determined as a linear combination of their values at the beginning of the zone. The goal of the study is to derive and apply the method of initial parameters equations taking into account the influence of circumferential stresses on the ring curvature. As far as the authors know, this is the first time that the stress intensity factor has been derived for an elastic thin-walled pipe with a radial crack in a geometrically nonlinear formulation. Here, an increase in pressure leads to a somewhat slowed increase in the stress intensity factor. In addition, a number of problems for dents are considered. The effect of the dent shape on the stress-strain state is analyzed. An expression for the stress intensity factor for a complex defect, a crack emanating from the dent apex, is presented.     

Development of a genetic algorithm‐based lookup table approach for efficient numerical integration in the method of finite spheres with application to the solution of thin beam and plate problems

It is observed that for the solution of thin beam and plate problems using the meshfree method of finite spheres, Gaussian and adaptive quadrature schemes are computationally inefficient. In this paper, we develop a novel technique in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline computational step. During online computations, this lookup table is used much like a table of Gaussian integration points and weights in the finite element computations. This technique offers significant reduction of computational time without sacrificing accuracy. Example problems are solved which demonstrate the effectiveness of the procedure. Copyright © 2006 John Wiley & Sons, Ltd.