A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems

The asymptotic accuracies of structural shell theories are reviewed. Several finite element models to solve arch problems are formulated by utilizing the shell theories. The asymptotic rate of energy convergence is determined by the ability of the approximate strains to assume arbitray polynomial states. The optimal choice of interpolation functions for tangential and normal displacement is treated. Stress resultants and stress couples are evaluated by using nodal forces (strain integration method) or internal strain patterns (strain method). The accuracy of these methods are investigated. In particular the distribution of errors in the strains calculated by the strain method is determined and utilized for accuracte stress evaluation. The theoretical considerations are supported by a vaste number of numerical experiments, which confirm the theoretical results.     

A note on complete problems for complexity classes

We show that every class of recursive sets which is closed under p-Turing equivalence possesses a ⩽_pm-complete set if and only if it possesses a ⩽_pT-complete set. One of the consequences hereof is that the classes Δ_n^p of the polynomial hierarchy possess ⩽_pm-complete problems.     

A note on practical bubbles for advection-diffusion problems

In this note, we show that, in the one-dimensional case, as an approximation to residual-free bubbles (RFB), certain practical bubbles can be applied to obtain a scheme which is uniformly convergent with respect to small viscosity in the energy norm for advection-diffusion problems. Received: January 2002 / Accepted: August 2002     

A note on the condition for convergence of nonlinear ABS method

This paper is concerned with the assumptions in the convergence theorem of the nonlinear ABS algorithm. In the published papers, it has been assumed that the condition number of the scaling matrices is uniformly bounded above. In this paper it is shown that the above condition can be replaced by the following simpler condition: the orthogonality measure of scaling vector sets is uniformly bounded from below. It is also shown that the latter condition is weaker than the original one.     

Concept convergence process: A framework for improving product concepts

Concept selection is one of the most important decisions in product development, since success of the final product depends on the selected concept. The exploration and evaluation of alternatives early in the product development (PD) process reduces the amount and magnitude of changes in later stages and increases the likelihood of success of new product development (NPD) projects. Though, currently available methods attempt to select the best concept from the available set of initial concepts, they do not help create an improved concept based on the learning and knowledge generated through the evaluation of initial concepts. The paper proposes a framework for selecting and/or evolving improved concepts through a rigorous concept evaluation and convergence process. The concept convergence process allows bringing together the best (desirable) traits from the initial set of concepts and creates a new set of hybrid concepts. The framework uses a fuzzy inference process for evaluating each initial concept against identified decision criteria, thus generating hybrid concepts to select the best feasible concept under given cost and technological constraints. The approach is demonstrated using a steering wheel concept generation example.     

A note on the convergence of the mean shift

Mean shift is an effective iterative algorithm widely used in computer vision community. However, to our knowledge, its convergence, a key aspect of any iterative algorithm, has not been rigorously proved up to now. In this paper, by further imposing some commonly acceptable conditions, its convergence is proved.     

A note on robustness tolerances for combinatorial optimization problems

We consider so-called generic combinatorial optimization problem, where the set of feasible solutions is some family of nonempty subsets of a finite ground set with specified positive initial weights of elements, and the objective function represents the total weight of elements of the feasible solution. We assume that the set of feasible solutions is fixed, but the weights of elements may be perturbed or are given with errors. All possible realizations of weights form the set of scenarios.A feasible solution, which for a given set of scenarios guarantees the minimum value of the worst-case relative regret among all the feasible solutions, is called a robust solution. The maximum percentage perturbation of a single weight, which does not destroy the robustness of a given solution, is called the robustness tolerance of this weight with respect to the solution considered.In this paper we give formulae for computing the robustness tolerances with respect to an optimal solution obtained for some initial weights and we show that this can be done in polynomial time whenever the optimization problem is polynomially solvable itself.     

A note on convergence of semi-implicit Euler methods for stochastic pantograph equations

In the literature [1] [Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equation, J. Math. Anal. Appl. 325 (2007) 1142–1159], Fan and Liu investigated the existence and uniqueness of the solution for stochastic pantograph equation and proved the convergence of the semi-implicit Euler methods under the Lipschitz condition and the linear growth condition. Unfortunately, the main result of convergence derived by the conditions is somewhat restrictive for the purpose of practical application, because there are many stochastic pantograph equations that only satisfy the local Lipschitz condition. In this note we improve the corresponding results in the above-mentioned reference.     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

High order A-stable numerical methods for stiff problems

Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge-Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order.     

Curvature-based grid step selection for stiff Cauchy problems

A new method of automatic step selection is proposed for the numerical integration of the Cauchy problem for ordinary differential equations. The method is based on using the geometrical characteristics (cuvature and slope) of the integral curve. Formulas have been constructed for the curvature of the integral curve for different choices of multidimensional space. In the two-dimensional case, they turn into well-known formulas, but their general multidimensional form is nontrivial. These formulas have a simple form, are convenient for practical use, and are of independent interest for the differential geometry of multidimensional spaces. For the grids constructed by our method, a procedure of step splitting is proposed that allows one to apply Richardson’s method and to calculate posterior asymptotically precise error estimation for the obtained solution (no such estimates have been found for traditional algorithms of automatic step selection). Therefore, the proposed methods demonstrate significantly superior reliability and validity of the results as compared to calculations by conventional algorithms. In the existing automatic procedures for step selection, steps can be unexpectedly reduced by 2–4 orders of magnitude for no apparent reason. This undermines the reliability of the algorithms. The cause of this phenomenon is explained. The proposed methods are especially effective for highly stiff problems, which is illustrated by examples of calculations.     

A note on two-sided stochastic control problems

We consider a class of two-sided stochastic control problems. For each continuous process π_t = π_t⁺ − π_t⁻ with bounded variation, the state process (x_t) is defined by x_t = B_t + f₀^t I(x_s - a)^d_πs+ − f₀^t I(x_s a)^dπ_s−, where a is a positive constant and (B_t) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function J(π) = E [f₀^∞ e^−αsX_s² ds], with discount rate α > 0, associated with π.     

Uniform and optimal schemes for stiff initial-value problems

We formulate a class of difference schemes for stiff initial-value problems, with a small parameter ε multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ε, that is the solution of the difference scheme u_i^h satisfies |u_i^h−u(x_i)| Ch, where C is independent of h and ε. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |u_i^h−u(x_i)| C min (h, ε). Finally, we show that this class contains higher-order schemes.     

A note on continuity in scalarization for multicriteria optimization problems

We establish sufficient conditions for solutions of scalarization replacements of multicriteria optimization problems to exhibit continuous dependence on scaling factors and other parameters.     

A note on the value function for constrained control problems

We consider the Mayer optimal control problem with dynamics given by a nonconvex differential inclusion, whose trajectories are constrained to a given set and we obtain a relation between the costate function that appears in the maximum principle and the value function. This relation extends the known conditions existing in the literature for unconstrained problems to those for problems under state constraints.     

A note on convergence requirements for nonlinear maximum-likelihood estimation of parameters from mixture models

Modifications are offered for some algorithms previously established by the authors for the maximum-likelihood estimation of the parameters of mixed exponential and Weibull distributions. The prior work assumed that the likelihood function was “well behaved” and that good starting points were available. We now provide enhancements of the methods to permit the handling of any alternative possibility.     

A note on the theoretical convergence properties of the SIMP method

The solid isotropic material with penalization (SIMP) method is used in topology optimization to solve problems where the variables are 0 or 1. The theoretical convergence properties have not been exhaustively studied. In this paper a convergence theorem with weaker assumptions than earlier conditions is given.