Efficient numerical solution of constrained multibody dynamics systems

A numerical algorithm for conducting coupled system dynamical simulation is presented. The interconnected system, comprising numerous modules, is treated as a constrained multibody dynamics system. Of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules. The proposed algorithm, Maggi’s equations with perturbed iteration (MEPI) emanates from numerical methods for differential-algebraic equations. Separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks MEPI’s main characteristic.     

Efficient parallel solution of structural eigenvalue problems

An efficient parallelisation of an existing sequential method for obtaining the eigenvalues of a structure by an exact analytical procedure is presented. Results are given which illustrate finding the undamped natural frequencies of a rigidly jointed plane frame, but the method is also applicable to buckling problems and to other types of structure. The parallel method is suited to both distributed and shared-memory parallel machines. It seeks to equate the workload of each processor (node) by initially sharing out the work and by subsequently passing work from working nodes to idle nodes. Experimental runs on an nCUBE2 computer show that reasonably high levels of efficiency are possible.     

On the numerical solution of nonlinear eigenvalue problems

We consider the numerical solution of the nonlinear eigenvalue problemA(λ)x=0, where the matrixA(λ) is dependent on the eigenvalue parameter λ nonlinearly. Some new methods (the BDS methods) are presented, together with the analysis of the condition of the methods. Numerical examples comparing the methods are given.     

Efficient multistep procedures for nonlinear parabolic problems with nonlinear Neumann boundary conditions

Efficient multistep procedures for time-stepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a precoditioned iterative method for approximately solving the differet linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained. Sponsored by the United States Army under Contract Nos. DAAG29-75-C-0024 and DAAG29-78-G-0161. This material is based on work supported by National Science Foundation under Grant No. MCS78-09525.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

On numerical solution of some problems of minimax control

Consideration was given to two classes of problems of optimal control with functionals of the discrete maximum type. The problem of improving the permissible controls was solved on the basis of a system of integral equations in the functional variation parameters. The necessary optimality conditions were due to the inconsistency of the introduced system of relations. The improvement iteration required solution of special optimal control problems minimizing the final state norm.     

Efficient numerical solution of Fisher's equation by using B-spline method

This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction–diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.     

Finite difference procedure for solution of poisson equation over complex domains with Neumann boundary conditions

A generalized finite difference scheme for solving Poisson equation over multiply connected domain bounded by irregular boundaries at which Neumann boundary conditions are specified, is presented in this paper. The method used to treat the Neumann condition is a six-point gradient approximation method given by Greenspan[6]. The method is generalized to treat all types of grid intersections with the boundary. An efficient computational procedure is devised by eliminating the calculations at the boundary during the interations.The scheme is applied to the problem of forced convection heat transfer in a fully developed laminar flow through seven and nineteen rod-cluster assemblies. Fluid properties are assumed to be uniform. In arriving at the fast converging and efficient method from computational point of view, different iterative techniques, overrelaxation methods and boundary treatments were tried. The results of computations and the computer times are reported in the present paper.     

On the numerical solution of double-periodic elliptic eigenvalue problems

In the present paper, numerical solving of the double-periodic elliptic eigenvalue problems $$M(u,\lambda ): = \Delta u + \lambda (u + f(u)) = 0, 0 \leqslant x< 2\pi ,0 \leqslant y< 2\pi /\sqrt {3,} $$ is considered regarding special symmetry properties. At first, subspacesV with the desired symmetry are constructed then a classical Ritz method is applied for the discretization inV and the resulting finite-dimensional bifurcation problem is solved by an algorithm proposed by Keller and Langford representing anumerical implementation of the Ljapunov-Schmidt procedure. Iff(u) is an entire function or a polynomial andV is an algebra then the computed solutions reveal to be stable with respect to perturbations of less symmetry. Some examples demonstrate the efficiency of the procedure.     

A finite element solution of diffraction problems in unbounded domains

In this paper, the method introduced in [3] is extended and applied to diffraction problems of acoustics and hydrodynamics. The problems dealt with are linear elliptic and may involve non-constant coefficients; they are set in unbounded domains. The method uses an integral representation formula with a regular kernel which allows an equivalent problem to be set in an interior annular closed region; it is shown how irregular frequencies are avoided. A variational formulation and its finite element discretization are detailed. Some numerical results are shown which support the validity of the technique and corroborate the theoretical analysis.     

Efficient difference scheme for numerical solution of a convective diffusion problem

Problems of modeling of atmospheric circulation are investigated. A new method for solution of a one-dimensional nonstationary inhomogeneous initial-boundary-value problem of convective diffusion is considered. The problem is solved using a new unconditionally stable and efficient difference scheme. The results of a theoretical analysis of the scheme are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 64–74, May–June 2007. An erratum to this article is available at .     

Analytic and numerical solution of coupled implicit semi-infinite diffusion problems

This paper deals with singular coupled implicit semi-infinite mixed diffusion problems. By application of the sine Fourier transform, existence conditions and an analytic closed form solution is first obtained. Given an admissible error and a rectangular bounded closed domain, analytic-numerical approximations whose error with respect to the exact solution is less than the admissible error in the bounded domain are constructed. An algorithm and an illustrative example are included.     

A stabilizing transformation for numerical solution of maximum principle problems

To aid in solving numerically a two-point boundary problem, a transformation of variables is proposed. This transformation is useful when an analog computer is being used to solve the differential equations, for it keeps the computations on scale and reduces the sensitivity of terminal conditions to initial conditions. Its performance in the presence of computational error appears satisfactory.     

General formulation for the numerical solution of optimal control problems

A new improved computational method for a class of optimal control problems is presented. The state and the costate (adjoint) variables are approximated using a set of basis functions. A method, similar to a variational virtual work approach with weighing coefficients, is used to transform the canonical equations into a set of algebraic equations. The method allows approximating functions that need not satisfy the initial conditions a priori. A Lagrange multiplier technique is used to enforce the terminal conditions. This enlarges the space from which the approximating functions can be chosen. Orthogonal polynomials are used to obtain a set of simultaneous equations with fewer non-zero entries. Such a sparse system results in substantial computational economy. Two examples, a time-invariant system and a time-varying system with quadratic performance index, are solved using three different sets of orthogonal polynomials and the power series to demonstrate the feasibility and efficiency of this method.     

An improved numerical process for solution of solid mechanics problems

This paper gives an overview of the development and status of an improved numerical process for the solution of solid mechanics problems. The proposed process uses a mixed formulation with the fundamental unknowns consisting of both stress and displacement parameters. The problem is formulated either by means of first-order partial differential equations or in a variational form by using a Hellinger-Reissner-type mixed variational principle.

For presentation purposes, the components of a numerical process are characterized and the criteria for an ideal process are outlined. Commonly used finite-difference and finite-element procedures arc examined in the light of these criteria and it is shown that they fall short in a number of ways. The proposed numerical process, on the other hand, satisfies most of the optimality criteria and appears to be particularly suited for use with the forthcoming generation computers (e.g. STAR-100 computer).

The paper includes a number of examples showing application of the proposed process to a broad spectrum of solid mechanics problems. These examples demonstrate the versatility and high accuracy of the numerical process obtained by using mixed formulations in conjunction with improved discretization techniques.     

Nonlinearity problem in the numerical solution of superstiff Cauchy problems

For the numerical solution of Cauchy stiff initial problems, many schemes have been proposed for ordinary differential equation systems. They work well on linear and weakly nonlinear problems. The article presents a study of a number of well-known schemes on nonlinear problems (which include, for example, the problem of chemical kinetics). It is shown that on these problems, the known numerical methods are unreliable. They require a sufficient step reducing at some critical moments, and to determine these moments, sufficiently reliable algorithms have not been developed. It is shown that in the choice of time as an argument, the difficulty is associated with the boundary layer. If the length of the integral curve arc is taken as an argument, difficulties are caused by the transition zone between the boundary layer and regular solution.     

Efficient solution of some problems in free partially commutative monoids

Linear-time algorithms are presented for several problems concerning words in a partially commutative monoid, including whether one word is a factor of another and whether two words are conjugate in the monoid.