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The generalized geometric programming algorithm GOP has been found to be one of the better algorithms for optimizing algebraic functions subject to algebraic constraints. The paper discusses two problems with the GOP algorithm presented by Avriel et al. (1980). The first problem is that, because of the way that constraint tolerances are checked, the checks may pass even when a constraint is violated beyond the acceptable tolerance. This problem can affect an acceleration technique suggested for the algorithm. This paper presents two possible solutions to this problem. The second problem is the algorithm's weakness in handling equality constraints. This problem seems to especially affect the `Phase I' part of the algorithm, which finds a feasible solution from a random starting point. While extensions have been made to get around this limitation, no discussion on why the original algorithm has problems with equality constraints has been made. This paper discusses why the problems exist. [ABSTRACT FROM AUTHOR]

Wave-based numerical methods often require to integrate products of polynomials and exponentials. With quadrature methods, this task can be particularly expensive at high frequencies as large numbers of integration points are required. This paper presents a set of closed-form solutions for the integrals of polynomial–exponential products in two and three dimensions. These results apply to arbitrary polygons in two dimensions, and for arbitrary polygonal surfaces or polyhedral volumes in three dimensions. Quadrature methods are therefore not required for this class of integrals that can be evaluated quickly and exactly. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The adaptive variational multiscale method is an extension of the variational multiscale method where the fine-scale part of the solution is approximated by a sum of numerically computed solutions to localized subgrid problems. Furthermore, the crucial discretization parameters are chosen automatically by an adaptive algorithm based on a posteriori error estimates. This method has been developed for diffusion-dominated problems and applied to multiscale problems that arise in oil reservoir simulation. In this paper, we extend the method to convection–diffusion problems. We present a duality based a posteriori error representation formula and an adaptive algorithm that tunes the fine-scale mesh size and the patch sizes of the local problems. Numerical results show rapid convergence of the adaptive algorithm. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Weakly compressible and advection approximations of incompressible isothermal flows were developed and tested in (Commun. Numer. Methods Eng. 2006; 22:831–847). In this paper, we extend the method to solve equations governing incompressible thermal flows. The emphasis is again on the reconstruction of unconditionally stable numerical scheme such that, restriction on time steps, projection procedures, solution of linear system of algebraic equations and staggered grids are completely avoided in its implementation. These features are achieved by combining a low-Mach asymptotic in compressible flow equations with a semi-Lagrangian method for the weakly compressible approach. The time integration is carried out using an explicit Runge–Kutta with variable stages. The method is applied to the natural convection flows in a squared cavity for both steady and transient computations. The numerical results demonstrate high resolution of the proposed method and confirm its capability to provide accurate and efficient simulations for thermal flow problems. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, an eigenvalue problem that involves uncertain-but-non-random parameters is discussed. A new method is developed to evaluate the reliable upper and lower bounds on frequencies of structures for these problems. In this method the matrix in the deviation amplitude interval is considered to be a perturbation around the nominal value of the interval matrix, and the upper and lower bounds to the maximum and minimum eigenvalues of this perturbation matrix are computed, respectively. Then based on the matrix perturbation theory, the eigenvalue bounds of the original interval eigenvalue problem can be obtained. Finally, two numerical examples are provided and the results show that the proposed method is reliable and efficient. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Symbolic computer algebra systems relieve one from the tedious task of different mathematical operations which are essential to obtain solutions. Due to their highly advanced features they have come to be used widely in computational mechanics. This paper describes an application of the modern computer algebra system Mathematica to the derivation of fundamental solutions necessary for the application of the boundary integral equation method. The problem treated is an asymmetric cross-section Timoshenko beam in free vibration. For this problem, the derivation of fundamental solutions involves lengthy mathematical operations which are very tedious if performed explicitly by hand. Therefore, using Mathematica we derive the fundamental solutions and generate the influence matrices from which the natural frequencies can be found. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The integrals required in the computation of influence coefficient matrices of the boundary element method (BEM) depend on the distance r(x,x′) from the collocation point or field point x to the source or load point x′. As a consequence, a distinction must be made between the case where the collocation point does not belong to the integration domain (proper integrals) and the case where the collocation point does belong to the integration domain (improper integrals). Moreover, situations arise where x comes close to x′ and the integrals, albeit of a regular character, behave almost as improper, this case being referred to as nearly singular integration. Analytical integration captures best the singular or nearly singular kernel behavior, but this technique can only be carried out in very simple situations as, for instance, boundary integrals over straight elements. In the present paper a set of useful analytical integration formulas for the 2D BEM with curved elements is deduced, employing a symbolic computational algebra system. © 1997 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In elasticity theory involving anisotropic materials, the rotational transformation of stress, strain and constitutive relations is of major importance. Also, when these problems are represented in matrix form there are several different contracted notations used to express the stress and strain tensors as column matrices. In general, the contracted notations used lead to unnecessary complications and may lead to errors. In the paper, contracted forms which yield orthonormal rotational transformations are advocated. These transformations are illustrated for stress, strain, modulus and compliance and are also applicable for strength transformation. The gains to be made are procedural and mathematically simple, leading to easier understanding. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of two- or three-dimensional contact problems with Coulomb friction and large deformation. The classical approach is based on two minimum principles or two variational inequalities: the first for unilateral contact and the second for friction. A coupled approach using only one principle or one inequality is presented. This new approach allows us to extend the notion of normality law to dissipative behaviors with a non-associated flow rule, such as surface friction. Non-differentiable contact potentials are regularized by means of the augmented Lagrangian method. Using the C++ language, an object-oriented finite element database is created, which allows us to implement the contact and friction in an existing code in a very simple and neat way. Numerical examples are carried out in many difficult cases such as shock absorber and three-dimensional contact. The numerical results prove that this approach is robust and efficient concerning numerical stability. [ABSTRACT FROM AUTHOR]

In this paper I re-examine the mechanics of the residual bending flexibility correction and show that it is an extravariational trick. [ABSTRACT FROM AUTHOR]

In this paper, I discuss the convergence and divergence rates for a class of generalized matrix multisplitting relaxation methods in a detailed manner. In particular, when the coefficient matrix is an L-matrix, I obtain several sufficient and necessary conditions ensuring the convergence of these methods. [ABSTRACT FROM AUTHOR]

Many finite elements based on Reissner-Mindlin plate bending theory with discrete constraints have been developed in the past few years. This approach avoids the C continuity required by the classical Kirchhoff plate theory. However, in those elements the shape functions for deflections (or their derivatives) are not integrated. Particularly in DKT (discrete Kirchhoff triangle) and in DST (discrete shear triangle) elements, the lateral deflections are defined only on the boundaries. In the paper we briefly discuss the definition of the transverse displacement for these elements, and present a simple approach to define a pseudoconsistent load vector and mass matrix. [ABSTRACT FROM AUTHOR]

Most commonly used second-order-accurate, dissipative time integration algorithms for structural dynamics possess a spurious root. For an algorithm to be accurate, it has been suggested that the spurious root must be small and ideally be zero in the low-frequency limit. In the paper we show that good accuracy can be achieved even lithe spurious root does not tend towards zero in the low-frequency limit. This permits more flexibility in the design of time integration algorithms. As an example, we present an algorithm that has greater accuracy than several other dissipative algorithms even though for all frequencies its spurious root is non-zero. We also show that the degraded performance of the Bazzi-ρ algorithm is not due to its non-zero spurious root. [ABSTRACT FROM AUTHOR]

In the paper we present a superconvergent patch recovery technique for obtaining higher-order-accurate finite-element solutions and thus a postprocessed type of L2 norm error estimate. Two modifications make our procedure different from the one proposed by Zienkiewicz and Zhu (1992), in which higher- order-accurate derivatives of the finite-element solution at nodes are determined. Firstly, the recovery process is made for elements, not for nodes. An 'element patch', which represents the union of an element under consideration and the surrounding elements, is introduced. Secondly, the local error estimate is calculated directly from the improved solution for this element. Numerical tests on both 1D and 2D model problems show that this method can provide an asymptotically exact a posteriori L2 norm error estimate if the used element possesses superconvergent points for the solutions. [ABSTRACT FROM AUTHOR]

The paper deals with the solution of the quasi-harmonic equation by the finite element method in a domain with corners on its boundary. A mapping technique, which can be used for locating the nodes of standard isoparametric elements in a subgrid (patch) of finite elements in the vicinity of a corner singularity point, is presented. The equations of the mapping are based on the analytical singular solution of the problem. Some analytical and numerical results to show the efficiency of the technique are presented. [ABSTRACT FROM AUTHOR]

A method for analyzing eigenvalues of the Helmholtz equation using a standard existing subroutine for eigenvalue determination is developed in this paper. It is based on the boundary integral equation formulation known as the multiple reciprocity method (MRM). The above-mentioned formulation, having polynomial matrices in terms of the eigenvalue as the coefficient matrices, is transformed into the standard-type eigenvalue problem. The resulting formulation makes it possible to determine the required eigenvalues only by boundary discretisation without any initial rough estimation. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for obtaining the inverse of a tridiagonal matrix numerically. The algorithm does not require diagonal dominance in the matrix and is also computationally efficient. [ABSTRACT FROM AUTHOR]

The static bifurcation criterion, i.e. singularity of the tangential stiffness matrix, is discussed in the sense of non-linear dynamics since it is sometimes used in the engineering literature. In this paper the connection between static stability, i.e. that the tangential stiffness matrix is positive-definite, and insensitivity, i.e. local damping of small perturbations, is proved. Since, in general, there is no connection between insensitivity and asymptotic stability, the concept of sensitivity cannot replace the classical stability theory of motion [ABSTRACT FROM AUTHOR]

This paper compares two formulations of Lax-Wendroff TVD methods. The basis of comparison is to the results of several test problems providing both qualitative and quantitative results. Results show that using an upwind biased limiter provides higher resolution of both smooth and discontinuous solutions from a lower amount of induced numerical viscosity. The conclusion is that a limiter should have as small a support as possible in order to limit its effects if high resolution is the object. [ABSTRACT FROM AUTHOR]