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.

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.

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.

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.

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

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.

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.

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

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.

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.

Abstract: The Newton form is a convenient representation for interpolation polynomials. Its sensitivity to perturbations depends on the distribution and ordering of the interpolation points. The present paper bounds the growth of the condition number of the Newton form when the interpolation points are Leja points for fairly general compact sets K in the complex plane. Because the Leja points are defined recursively, they are attractive to use with the Newton form. If K is an interval, then the Leja points are distributed roughly like Chebyshev points. Our investigation of the Newton form defined by interpolation at Leja points suggests an ordering scheme for arbitrary interpolation points.

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.

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.

Abstract: With the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.

Abstract: The Burgers equation with a small viscosity term, initial and periodic boundary conditions is resolved using a spatial approximation constructed from an orthonormal basis of wavelets. The algorithm is directly derived from the notions of multiresolution analysis and tree algorithms. Before the numerical algorithm is described these notions are first recalled. The method uses extensively the localization properties of the wavelets in the physical and Fourier spaces. Moreover, the authors take advantage of the fact that the involved linear operators have constant coefficients. Finally, the algorithm can be considered as a time marching version of the tree algorithm. The most important point is that an adaptive version of the algorithm exists: it allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution. Numerical results and description of the different elements of the algorithm are provided in combination with different mathematical comments on the method and some comparison with more classical numerical algorithms.

Abstract: This paper describes a damped-Newton method for solving the nonlinear complementarity problem when it is formulated as a system of B-differentiable equations through the use of the Minty-map. This general Newton algorithm contains a one-dimensional line search and possesses a global convergence property under certain conditions; modifications and heuristic implementations of the algorithm for the case when these conditions do not hold are also discussed. The numerical experiments show that, in general, this new scheme is more efficient and robust than the traditional Josephy-Newton algorithm.

Abstract: An effort to unify three major numerical methods in electromagnetics is presented. Harrington's direct method of moments, the iterative methods, and the reaction integral equation method are shown to be generally equivalent and are unified as the generalised method of moments. It is shown that the reaction integral equation method is in general a moment method, and that the moment method, when defined in a symmetric space, generally satisfies the reaction theorem, and therefore reciprocity. A broad, though limited, equivalence between the moment and the iterative methods is also demonstrated. A numerical example is discussed to illustrate these and other points.

Abstract: Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.

Abstract: We give a numerical method for the computation of heteroclinic orbits connecting two saddle points in R 2 . These can be computed to very high period due to an integral phase condition and an adaptive discretization. We can also compute entire branches of such orbits. The method can be extended to compute an invariant manifold that connects two fixed points in R n . As an example we compute branches of traveling wave front solutions to the Huxley equation. Using weighted Sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period

Abstract: This paper deals with th stability analysis of numerical methods for the solution of delay differential equations. We focus on the behaviour of the one-leg θ-method and the linear θ-method in the solution of the linear test equation U'(t)=λU(t)+μU(t−τ), with τ>0 and complex λ,μ. The stability regions for both of these methods are determined. The regions turn out to be equal to each other only if θ=0 or θ=1

Abstract: The present book is an edition of the manuscripts to the courses "Numerical Methods I" and "Numerical Mathematics I and II" which Professor H. Rutishauser held at the E.T.H. in Zurich. The first-named course was newly conceived in the spring semester of 1970, and intended for beginners, while the two others were given repeatedly as elective courses in the sixties. For an understanding of most chapters the funda mentals of linear algebra and calculus suffice. In some places a little complex variable theory is used in addition. However, the reader can get by without any knowledge of functional analysis. The first seven chapters discuss the direct solution of systems of linear equations, the solution of nonlinear systems, least squares prob lems, interpolation by polynomials, numerical quadrature, and approxima tion by Chebyshev series and by Remez' algorithm. The remaining chapters include the treatment of ordinary and partial differential equa tions, the iterative solution of linear equations, and a discussion of eigen value problems. In addition, there is an appendix dealing with the qd algorithm and with an axiomatic treatment of computer arithmetic."

Abstract: We consider conjugate gradient type methods for the solution of large linear systemsA x=b with complex coefficient matrices of the typeA=T+i?I whereT is Hermitian and ? a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure ofA can be preserved by using polynomial preconditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.

Abstract: A method is described to optimally determine the elastic constants of anisotropic solids from wave‐speeds measurements in arbitrary nonprincipal planes. For such a problem, the characteristic equation is a degree‐three polynomial which generally does not factorize. By developing and rearranging this polynomial, a nonlinear system of equations is obtained. The elastic constants are then recovered by minimizing a functional derived from this overdetermined system of equations. Calculations of the functional are given for two specific cases, i.e., the orthorhombic and the hexagonal symmetries. Some numerical results showing the efficiency of the algorithm are presented. A numerical method is also described for the recovery of the orientation of the principal acoustical axes. This problem is solved through a double‐iterative numerical scheme. Numerical as well as experimental results are presented for a unidirectional composite material.