Showing papers on "Numerical analysis published in 1990"

An all‐electron numerical method for solving the local density functional for polyatomic molecules

Abstract: A method for accurate and efficient local density functional calculations (LDF) on molecules is described and presented with results The method, Dmol for short, uses fast convergent three‐dimensional numerical integrations to calculate the matrix elements occurring in the Ritz variation method The flexibility of the integration technique opens the way to use the most efficient variational basis sets A practical choice of numerical basis sets is shown with a built‐in capability to reach the LDF dissociation limit exactly Dmol includes also an efficient, exact approach for calculating the electrostatic potential Results on small molecules illustrate present accuracy and error properties of the method Computational effort for this method grows to leading order with the cube of the molecule size Except for the solution of an algebraic eigenvalue problem the method can be refined to quadratic growth for large molecules

A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures With Participating Media

Abstract: A new finite-volume method is proposed to predict radiant heat transfer in enclosures with participating media. The method can conceptually be applied with the same nonorthogonal computational grids used to compute fluid flow and convective heat transfer. A fairly general version of the method is derived, and details are illustrated by applying it to several simple benchmark problems. Test results indicate that good accuracy is obtained on coarse computational grids, and that solution errors diminish rapidly as the grid is refined.

A self-consistent solution of Schrödinger-Poisson equations using a nonuniform mesh

Abstract: A self‐consistent, one‐dimensional solution of the Schrodinger and Poisson equations is obtained using the finite‐difference method with a nonuniform mesh size. The use of the proper matrix transformation allows preservation of the symmetry of the discretized Schrodinger equation, even with the use of a nonuniform mesh size, therefore reducing the computation time. This method is very efficient in finding eigenstates extending over relatively large spatial areas without loss of accuracy. For confirmation of the accuracy of this method, a comparison is made with the exactly calculated eigenstates of GaAs/AlGaAs rectangular wells. An example of the solution of the conduction band and the electron density distribution of a single‐heterostructure GaAs/AlGaAs is also presented.

Advanced probabilistic structural analysis method for implicit performance functions

Abstract: In probabilistic structural analysis, the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite-elemen t methods. In such cases, the commonly used probabilistic analysis tool is the mean-based second-moment method, which provides only the first two statistical moments. This paper presents an advanced mean-based method, which is capable of establishing the full probability distributions to provide additional information for reliability design. The method requires slightly more computations than the mean-based second-moment method but is highly efficient relative to the other alternative methods. Several examples are presented to demonstrate the method. In particular, the examples show that the new mean-based method can be used to solve problems involving nonmonotonic functions that result in truncated distributions.

Proximity control in bundle methods for convex

Abstract: Proximal bundle methods for minimizing a convex functionf generate a sequence {x k } by takingx k+1 to be the minimizer of $$\hat f^k (x) + u^k |x - x^k |^2 /2$$ , where $$\hat f^k $$ is a sufficiently accurate polyhedral approximation tof andu k > 0. The usual choice ofu k = 1 may yield very slow convergence. A technique is given for choosing {u k } adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.

On a stress resultant geometrically exact shell model. Part IV: variable thickness shells with through-the-thickness stretching

Abstract: This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.

Numerical Fracture Mechanics

Abstract: This text bridges the gap between existing specialist books on theoretical fracture mechanics, and texts on numerical methods. It concentrates on the application of the boundary element method to fracture mechanics, although an account of recent advances in other numerical methods is presented.

Hybrid method for computing demagnetizing fields

Abstract: A hybrid finite-element-boundary integral method is proposed for computing demagnetizing field. For a three-dimensional (3-D) mesh with N nodes in the magnetic region, the method requires storage O(N/sup 4/3/), and no nodes outside the magnetic region need to be considered. The method is suitable for nonlinear calculations, can be adapted to handle curved boundaries, and is especially convenient when the magnetic region consists of several parts whose relative positions are variable. >

Matrix methods applied to acoustic waves in multilayers

Abstract: Matrix methods for analyzing the electroacoustic characteristics of anisotropic piezoelectric multilayers are described. The conceptual usefulness of the methods is demonstrated in a tutorial fashion by examples showing how formal statements of propagation, transduction, and boundary-value problems in complicated acoustic layered geometries such as those which occur in surface acoustic wave (SAW) devices, in multicomponent laminates, and in bulk-wave composite transducers are simplified. The formulation given reduces the electroacoustic equations to a set of first-order matrix differential equations, one for each layer, in the variables that must be continuous across interfaces. The solution to these equations is a transfer matrix that maps the variables from one layer face to the other. Interface boundary conditions for a planar multilayer are automatically satisfied by multiplying the individual transfer matrices in the appropriate order, thus reducing the problem to just having to impose boundary conditions appropriate to the remaining two surfaces. The computational advantages of the matrix method result from the fact that the problem rank is independent of the number of layers, and from the availability of personal computer software that makes interactive numerical experimentation with complex layered structures practical. >

Numerical methods in rock mechanics

Abstract: Keywords: Mecanique des roches ; Analyse numerique ; Methode des elements finis Reference Record created on 2004-09-07, modified on 2016-08-08

Non-parallel stability of a flat-plate boundary layer using the complete Navier-Stokes equations

Abstract: Non-parallel effects which are due to the growing boundary layer are investigated by direct numerical integration of the complete Navier-Stokes equations for incompressible flows The problem formulation is spatial, ie disturbances may grow or decay in the downstream direction as in the physical experiments In the past various non-parallel theories were published that differ considerably from each other in both approach and interpretation of the results In this paper a detailed comparison of the Navier-Stokes calculation with the various non-parallel theories is provided It is shown, that the good agreement of some of the theories with experiments is fortuitous and that the difference between experiments and theories concerning the branch I neutral location cannot be explained by non-parallel effects

MSLiP: a computer code for the multistage stochastic linear programming problem

Abstract: This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.

A Nyström method for boundary integral equations in domains with corners

Abstract: We give a convergence and error analysis for a Nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners.

A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods

Abstract: In this paper we discuss the numerical implementation of a systematic method for the exact boundary controllability of the wave equation, concentrating on the particular case of Dirichlet controls. The numerical methods described here consist in a combination of: finite element approximations for the space discretization; explicit finite difference schemes for the time discretization; a preconditioned conjugate gradient algorithm for the solution of the discrete problems; a pre/post processing technique based on a biharmonic Tychonoff regularization. The efficiency of the computational methodology is illustrated by the results of numerical experiments.

New numerical method to study phase transitions.

Abstract: A powerful method of detecting first order transitions by numerical simulations of finite systems is presented. The method relies on simulations and the finite size scaling properties of free energy barriers between coexisting states. It is demonstrated that the first order transitions in d = 2, q = 5 and d = q = 3 Potts models are easily seen with modest computing time. The method can also be used to obtain quite accurate estimates of critical exponents by studying the barriers in the vicinity of a critical point. Some new results on exponents and conformal charge in frustrated XY models and a related coupled XY-Ising model in d = 2 are presented. These show that the transitions in these models are in new universality classes and that the conformal charge varies with a parameter.

Algorithm 681: INTBIS, a portable interval Newton/bisection package

Abstract: We present a portable software package for finding all real roots of a system of nonlinear equations within a region defined by bounds on the variables. Where practical, the package should find all roots with mathematical certainty. Though based on interval Newton methods, it is self-contained. It allows various control and output options and does not require programming if the equations are polynomials; it is structured for further algorithmic research. Its practicality does not depend in a simple way on the dimension of the system or on the degree of nonlinearity.

A numerical analysis of phase-change problems including natural convection

Abstract: Fixed grid solutions for phase-change problems remove the need to satisfy conditions at the phase-change front and can be easily extended to multidimensional problems. The two most important and widely used methods are enthalpy methods and temperature-based equivalent heat capacity methods. Both methods in this group have advantages and disadvantages. Enthalpy methods (Shamsundar and Sparrow, 1975; Voller and Prakash, 1987; Cao et al., 1989) are flexible and can handle phase-change problems occurring both at a single temperature and over a temperature range. The drawback of this method is that although the predicted temperature distributions and melting fronts are reasonable, the predicted time history of the temperature at a typical grid point may have some oscillations. The temperature-based fixed grid methods (Morgan, 1981; Hsiao and Chung, 1984) have no such time history problems and are more convenient with conjugate problems involving an adjacent wall, but have to deal with the severe nonlinearity of the governing equations when the phase-change temperature range is small. In this paper, a new temperature-based fixed-grid formulation is proposed, and the reason that the original equivalent heat capacity model is subject to such restrictions on the time step, mesh size, and the phase-change temperature range will alsomore » be discussed.« less

Convergence of the point vortex method for the 2-D euler equations

Abstract: We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l p norm. Consequently the method converge in l p norm for all time. The convergence is also illustrated by a numerical experiment

Magnetostatic field computations in terms of two‐component vector potentials

Abstract: In this paper the magnetostatic problem is stated in terms of two-component electric and magnetic vector potentials. An associated numerical method, based on the adoption of edge elements, is proposed. This procedure overcomes the cancellation problems and the complexity of the interface conditions encountered by similar approaches in the presence of magnetic inhomogeneities and discontinuities of currents and magnetic fields.

Projected Implicit Runge-Kutta Methods for Differential-Algebraic Equations

Abstract: In this paper we introduce a new class of numerical methods, Projected Implicit Runge-Kutta methods, for the solution of index-two Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs). These types of systems arise in a variety of applications, including the modelling of singular optimal control problems and parameter estimation for differential-algebraic equations such as multibody systems. The new methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation and loss of accuracy; the new methods overcome these difficulties. For linear problems we define an essential underlying boundary value ODE and prove well-conditioning of the differential (or state-space) solution components. This is then used to prove stability and superconvergence for the corresponding numerical approximations for linear and nonlinear problems.

Convergence properties of the Runge-Kutta-Chebyshev method

Abstract: The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length β proportional tos 2. The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of β withs 2. The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations.

Abstract: In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.

2‐D Depth‐Averaged Flow Computation near Groyne

Abstract: The depth-averaged velocity and bottom shear stress distributions in a rectangular channel near a groyne are computed by using a 2-D depth averaged model. The model uses a hybrid finite difference scheme and an iterative method to solve the governing equations of flow and turbulence transport. Due to streamline curvature effects in the region near the groyne tip, a correction factor is incorporated into the \Ik\N=ϵ\N turbulence model that significantly improves the agreement between the computed and experimental data of the velocities and of the streamline pattern compared to previous numerical methods. In this region the bottom shear stress is found to be largely influenced by the 3-D effects. A 3-D correction factor is introduced which considerably improves the computed bottom shear stresses. Sensitivity analysis is made on the \Ik\N=ϵ\N model coefficients and on the correction factors of the streamline curvature and the 3-D effects. The experimental errors in the velocity and bottom shear stress measurements are analyzed. The average errors between the computed and previous experimental results are presented with confidence intervals.