A variant of Nitsche's method

We present a method that aims to reconcile Nitsche's method with the traditional finite element method ('weak' versus 'strong implementation' of essential boundary conditions). We retain the original idea of a variational formulation based on an extended energy, but replace the original boundary terms by domain terms involving weak derivatives. The solution of the proposed method coincides, for the Poisson problem, with the one of the traditional method, which in particular shows monotonicity under the standard angle condition for the Courant element. For more general second-order problems, it allows for the weighting of boundary terms inherent to Nitsche's method. This is of particular interest for singularly perturbed problems.     

On the Average Distribution of Inversive Pseudorandom Numbers

The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. The authors have recently introduced a new method for obtaining nontrivial upper bounds on the multidimensional discrepancy of inversive congruential pseudorandom numbers in parts of the period. This method has also been used to study the multidimensional distribution of several other similar families of pseudorandom numbers. Here we apply this method to show that, “on average” over all initial values, much stronger results than those known for “individual” sequences can be obtained.     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Practical enhancements to the Magnanti–Wong method

The Magnanti–Wong method–accelerating Benders decomposition–is shown to exhibit difficulties due to its dependence on the subproblem; an independent version is therefore introduced. The method additionally requires a–sometimes intractable–master problem core point; for several applications it is proved and experimentally verified that alternative points may be used.     

Ultrasonic method of determining the elastic characteristics of anisotropic materials

An ultrasonic method of determining the elastic characteristics, developed for isotropic materials and based on the measurement of the angles of total internal reflection of ultrasound at a liquid-solid interface, is applied to an anisotropic Compreg of the DSPA type. The results obtained for the shear modulus using the ultrasonic method at 2.5 MHz and an independent method based on measuring the frequency of the damped torsional vibrations of a prismatic bar at 10–30 Hz are compared. It is shown that the shear modulus varies only slightly (10–12%) on the frequency interval investigated.Leningrad Kirov Forest Engineering Academy. Translated from Mekhanika Polimerov, No. 4, pp. 728–731, July–August, 1969.     

Successive Proportional Additive Numeration Using Fuzzy Linguistic Labels (Fuzzy Linguistic SPAN)

The SPAN (Successive Proportional Additive Numeration or Social Participatory Allocation Network) is a procedure that converts individual judgments into a group decision. The procedure is based on a voting design by which individual experts allocate their votes iteratively between their preferred options and other experts. The process ends when all the votes are allocated to options, and the one with the highest number of votes is selected. The method requires the experts to specify an exact allocation of votes to both options and other experts. The Fuzzy Linguistic SPAN allows experts to allocate their votes using linguistic labels such as “most of” or “a few”, and determine the preferred option. This method is demonstrated using the Max–Min aggregation function used to develop a proportional representation of the option and member voting schemes. The method is also demonstrated using the LOWA aggregation function. The Fuzzy Linguistic SPAN method is beneficial since the linguistic voting process is easier for the experts and significantly reduces the computational process compared to the traditional SPAN. The paper presents the method and two examples with comparisons to the numerical SPAN method.     

Adaptivity and computational complexity in the numerical solution of ODEs

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τ_j} to a non-equidistant grid in the original independent variable, {t_j}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.     

Circle measurements in ancient China

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π. A translation of Liu's method is given in the Appendix. Also examined are the values for π given by Zu Chongzhi (429–500) and unsurpassed for a millenium. Although the method used by Zu is not extant, it is almost certain that the applied Liu's method. With the help of an electronic computer, a table of computations adhering to Liu's method is given to show the derivation of Zu's results. The paper concludes with a survey of circle measurements in China.     

Bifurcation-free design method of pulse energy converter controllers

In this paper, a design method of pulse energy converter (PEC) controllers is proposed. This method develops a classical frequency domain design, based on the small signal modeling, by means of an addition of a nonlinear dynamics analysis stage. The main idea of the proposed method consists in fact that the PEC controller, designed with an application of the small signal modeling, is tuned after with taking into the consideration an essentially nonlinear nature of the PEC that makes it possible to avoid bifurcation phenomena in the PEC dynamics at the design stage (bifurcation-free design). Also application of the proposed method allows an improvement of the designed controller performance. The application of this bifurcation-free design method is demonstrated on an example of the controller design of direct current–direct current (DC–DC) buck converter with an input electromagnetic interference filter.     

On the optimization of constrained functions: comparison of sequential gradient-restoration algorithm and gradient-projection algorithm

The problem of minimizing a function fnof(x) subject to the nonlinear constraint ?(x) = 0 is considered, where fnof is a scalar, x is an n-vector, and ? is a q-vector, with q < n. The sequential gradient-restoration algorithm (SGRA: Miele, [1, 2]) and the gradient-projection algorithm (GPA: Rosen, [3, 4]) are considered. These algorithms have one common characteristic: they are all composed of the alternate succession of gradient phases and restoration phases. However, they are different in several aspects, namely, (a) problem formulation, (b) structure of the gradient phase, and (c) structure of the restoration phase. First, a critical summary of SGRA and GPA is presented. Then, a comparison is undertaken by considering the speed of convergence and, above all, robustness (that is, the capacity of an algorithm to converge to a solution). The comparison is done through 16 numerical examples. In order to understand the separate effects of characteristics (a), (b), (c), six new experimental algorithms are generated by combining parts of Miele's algorithm with parts of Rosen's algorithm. Thus, the total number of algorithms investigated is eight. The numerical results show that Miele's method is on the average faster than Rosen's method. More importantly, regarding robustness, Miele's method compares favorably with Rosen's method. Through the examples, it is shown that Miele's advantage in robustness is more prominent as the curvature of the constraint increases. While this advantage is due to the combined effect of characteristics (a), (b), (c), it is characteristic (c) that plays the dominant role. Indeed, Miele's restoration provides a better search direction as well as better step-size control than Rosen's restoration.     

Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations. II

A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an nonlinear potentials method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step.Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations.In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in R^d with variable coefficients.     

A conjugate gradient method with descent direction for unconstrained optimization

A modified conjugate gradient method is presented for solving unconstrained optimization problems, which possesses the following properties: (i) The sufficient descent property is satisfied without any line search; (ii) The search direction will be in a trust region automatically; (iii) The Zoutendijk condition holds for the Wolfe–Powell line search technique; (iv) This method inherits an important property of the well-known Polak–Ribière–Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. The global convergence and the linearly convergent rate of the given method are established. Numerical results show that this method is interesting.     

Extinction and permanence in delayed stage-structure predator–prey system with impulsive effects

Based on the classical stage-structured model and Lotka–Volterra predator–prey model, an impulsive delayed differential equation to model the process of periodically releasing natural enemies at fixed times for pest control is proposed and investigated. We show that the conditions for global attractivity of the ‘pest-extinction’ (‘prey-eradication’) periodic solution and permanence of the population of the model depend on time delay. We also show that constant maturation time delay and impulsive releasing for the predator can bring great effects on the dynamics of system by numerical analysis. As a result, the pest maturation time delay is considered to establish a procedure to maintain the pests at an acceptably low level in the long term. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy-pest) model with age structure, exhibit a new modelling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

On the approximate solution to an initial boundary valued problem for the Cahn–Hilliard equation

The particular approximate solution of the initial boundary valued problem to the Cahn–Hilliard equation is provided. The Fourier Method is combined with the Adomian’s decomposition method in order to provide an approximate solution that satisfy the initial and the boundary conditions. The approximate solution also satisfies the mass conservation principle.     

Function evaluation using expansions in Bessel functions of the first kind

An iterative method is proposed for evaluation of fonctions that are expandable in series of Bessel functions of the first kind. The Bessel functions are evaluated by Miller's method, avoiding the need to determine their exact values. As an example, we describe algorithms for evaluation of the integral sine and the normal probability integral with an accuracy of to 10–12 significant digits.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 37–42, 1987.     

Worst case behavior of the steepest edge simplex method

At each nondegenerate iteration of the steepest-edge simplex method one moves from a vertex of the polyhedron, P, of feasible points to an adjacent vertex along an edge that is steepest with respect to the linear objective function ψ. In this paper we show how to construct a sequence of linear programs (P_n,ψ_n) in n variables for which the number of iterations required by the steepest edge simplex method is 2ⁿ−1.     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

The selection of optimum parameters for a cylindrical fiberglass plastic casing under axial compression

A modern mathematical optimization method — the random search method — is applied to nonlinear programming with the aim to select the parameters of an axially compressed fiberglass plastic cylindrical casing which minimize its weight and, at the same time, satisfy the strength and stability requirements.Dnepropetrovsk Engineering Constructions Institute. Translated from Mekhanika Polimerov, No. 5, pp. 945–946, September–October, 1972.     

Procedure for measuring the thermophysical properties of polymers at high hydrostatic pressures

A procedure is suggested for determining the thermophysical properties of solids under hydrostatic pressure conditions by the instantaneous cylindrical heat source method without allowance for heat exchange at the sample-liquid boundary. A calculation performed showed that the temperature field of a cylinder of finite radius is identical to the temperature field of an infinite body up to Fo=0.05–0.07 with an accuracy far better than 0.5%. It is shown that the sample dimensions can be small thereupon at measurement times sufficient for performing the measurements on a standard apparatus of the order of 40–70 sec.Moscow Chemical Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 568–570, May–June, 1973.