Abstract: In this paper, the algorithms of the automatic mesh generator NETGEN are described. The domain is provided by a Constructive Solid Geometry (CSG). The whole task of 3D mesh generation splits into four subproblems of special point calculation, edge following, surface meshing and finally volume mesh generation. Surface and volume mesh generation are based on the advancing front method. Emphasis is given to the abstract structure of the element generation rules. Several techniques of mesh optimization are tested and quality plots are presented.

Abstract: Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds. The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

Abstract: This is a text in numerical analysis which is taken to mean the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis, approximations theory, the theory of equations, and ordinary differential equations. The topics in this book are presented with a view towards stressing basic principles and maintaining simplicity and teachability as far as possible. Topics that require a level of technicality that goes beyond the standard of simplicity imposed are referenced in bibliographic notes at the end of each chapter. This book does not cover numerical linear algebra, nor the numerical solution of partial differential equations, as the author takes the view that these are now separate disciplines. It is intended that the student has a good background in calculus and advanced calculus and some knowledge of linear algebra, complex analysis, and differential equations.

Abstract: We propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution. They are obtained by considering a maximum likelihood (ML) solution corresponding to some given distributions of the sources and relaxing this assumption afterward. The first method is specially adapted to temporally independent non-Gaussian sources and is based on the use of nonlinear separating functions. The second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters. A theoretical analysis of the performance of the methods has been made. A simple procedure for optimally choosing the separating functions is proposed. Further, in the second method, a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided. Finally, some numerical and simulation results are given, illustrating the performance of the method and the good agreement between the experiments and the theory.

Abstract: T hat numerical analysis is an extremely useful tool for solving problems and exploring fundamental concepts comes as no surprise to any student of dynamical systems. However , preoccupation with learning the mathematical theory of nonlinear dynamics often precludes a deeper understanding of the variety of numerical methods available and prevents a proper appreciation of the possibilities and limitations of these methods. On the other hand, many numerical analysts spend a large part of their careers not realizing that they are actually solving problems of nonlinear dynamics and struggling with issues of dynamical-systems and chaos theory. Clearly, their work would benefit greatly from a closer familiarity with the main ideas and results of that theory. These are precisely the two scientific communities that Stuart and Hum-phries' book aims to address. Its purpose is evidently to help researchers in these communities get better acquainted with each other. It is a timely publication which, in my opinion, will appeal to many members of the two groups and succeed in achieving its purpose to a considerable extent. Of differential equations The book is exclusively concerned with the solution of the initial-value problem of ordinary differential equations, (1) with u(t) ∈ ‫ޒ‬ p , t > 0; and f : ‫ޒ‬ p → ‫ޒ‬ P where f is (at least) continuous and Lipschitz, so that a solution of Equation 1 exists and is unique. Since an analytical expression of this solution is hardly ever available, one attempts to solve Equation 1 numerically, by writing it as a dynamical system (2) with G : D → ‫ޒ‬ p. The solution of Equation 2 exists, if the U n 's remain bounded within D ⊆ ‫ޒ‬ p for all n, and is unique as long as G is single-valued. U n is, of course, the approximation of u(t n) at the nth time step, t n = n∆t, n = 0, 1, 2, …, and is expected to be increasingly accurate as ∆t → 0. Clearly, the two most crucial issues facing a researcher who attempts this approximation are convergence and stability of the numerical strategy adopted. Convergence here means the ability to determine bounds for the norm . of the error: (3) where c 1 , c 2 , and r are appropriate constants. Stability refers to one's capacity to control the effect of small perturbations on the particular method chosen. More precisely, let us …

Abstract: In this paper, which is part II in a series of two, the investigation of the Galerkin finite element solution to the Helmholtz equation is continued. While part I contained results on the $h$ version with piecewise linear approximation, the present part deals with approximation spaces of order $p \ge 1$. As in part I, the results are presented on a one-dimensional model problem with Dirichlet--Robin boundary conditions. In particular, there are proven stability estimates, both with respect to data of higher regularity and data that is bounded in lower norms. The estimates are shown both for the continuous and the discrete spaces under consideration. Further, there is proven a result on the phase difference between the exact and the Galerkin finite element solutions for arbitrary $p$ that had been previously conjectured from numerical experiments. These results and further preparatory statements are then employed to show error estimates for the Galerkin finite element method (FEM). It becomes evident that the error estimate for higher approximation can---with certain assumptions on the data---be written in the same form as the piecewise linear case, namely, as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference. The paper is concluded with a numerical evaluation.

Abstract: A new numerical method is presented for propagating elastic waves in heterogeneous earth media, based on spectral approximations of the wavefield combined with domain decomposition techniques. The flexibility of finite element techniques in dealing with irregular geologic structures is preserved, together with the high accuracy of spectral methods. High computational efficiency can be achieved especially in 3D calculations, where the commonly used finite-difference approaches are limited both in the frequency range and in handling strongly irregular geometries. The treatment of the seismic source, introduced via a moment tensor distribution, is thoroughly discussed together with the aspects associated with its numerical implementation. The numerical results of the present method are successfully compared with analytical and numerical solutions, both in 2D and 3D.

Abstract: Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements.

Abstract: A computationally efficient numerical method for the solution of nonlinear sea ice dynamics models employing viscous-plastic rheologies is presented. The method is based on a semi-implicit decoupling of the x and y ice momentum equations into a form having better convergence properties than the coupled equations. While this decoupled form also speeds up solutions employing point relaxation methods, a line successive overrelaxation technique combined with a tridiagonal matrix solver procedure was found to converge particularly rapidly. The procedure is also applicable to the ice dynamics equations in orthogonal curvilinear coordinates which are given in explicit form for the special case of spherical coordinates.

Abstract: Recent Developments in Dense Numerical Linear Algebra Sparse Numerical Linear Algebra: Direct Methods and Preconditioning Closer to the Solution: Iterative Linear Solvers 150 Years Old and Still Alive: Eigenproblems Geometric Integration Convergence and Stability in the Numerical Approximation of Dynamical Systems Beyond the Classical Theory of Computational Ordinary Differential Equations Numerical Analysis of Volterra Functional and Integral Equations The Numerical Solution of Boundary Integral Equations Aspects of Approximation with Emphasis on the Univariate Case A Review of Methods for Multivariable Interpolation at Scattered Data Points Large Scale Unconstrained Optimization Interior Point Methods for Linear and Nonlinear Programming Methods for Nonlinear Constraints in Optimization Calculations Stabilization Techniques and Subgrid Scales Capturing Approximation of Curvature Dependent Interface Motion Finite Element Methods for Hyperbolic Problems: a Posteriori Error Analysis and Adaptivity Approximation of Multidimensional Hyperbolic Partial Differential Equations Algorithms in Tomography Partial Differential Equations and Image Iterative Filtering

Abstract: An asymptotic approximation is developed for evaluating the probability integrals that arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on second-order reliability methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte Carlo simulation. Simple structures subjected to white noise stochastic excitation are used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method.

Abstract: Field time integrators with second-order-accurate numerical schemes for both the fluid and the structure are considered for unsteady Euler aeroelastic computations. We show that if these schemes are simply coupled and used straightforwardly with subcycling, then accuracy and stability properties may be lost. We present new coupling staggered procedures where momentum conservation is enforced at the interface. This is done by using a structural predictor. Continuity of structural and fluid grid displacements is not satisfied at the fluid/structure interface. However, we show on a two-degree-of-freedom aerofoil that this new type of method has many advantages, e.g. accuracy of conservation at the interface and extended stability. The supersonic flutter of a flat panel is simulated in order to numerically prove that the algorithm gives accurate results with arbitrary subcycling for the fluid in the satisfying limit of 30 time steps per period of coupled oscillation. © 1997 John Wiley & Sons, Ltd.

Abstract: We describe some numerical techniques for the total variation image restoration method, namely a primal-dual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Abstract: A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals ofmotion can be constructed in nonresonant case. In a resonant case with a resonant relations, an (n +α)-dimensional averaged FPK equation governing the transition probability density of n action variables and a combinations ofphase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

Abstract: A finite-volume integration method is proposed for computing the pressure gradient force in general vertical coordinates. It is based on fundamental physical principles in the discrete physical space, rather than on the common approach of transforming analytically the pressure gradient terms in differential form from the vertical physical (i.e., height or pressure) coordinate to one following the bottom topography. The finite-volume discretization is compact, involving only the four vertices of the finite volume. The accuracy of the method is evaluated statically in a two-dimensional environment and dynamically in three-dimensional dynamical cores for general circulation models. The errors generated by the proposed method are demonstrated to be very low in these tests.

Abstract: A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.

Abstract: Free damped vibrations of an oscillator, whose viscoelastic properties are described in terms of the fractional calculus Kelvin-Voight model, Maxwell model, and standard linear solid model are determined. The problem is solved by the Laplace transform method. When passing from image to pre-image one is led to find the roots of an algebraic equation with fractional exponents. The method for solving such equations is proposed which allows one to investigate the roots behaviour in a wide range of single-mass system parameters. A comparison between the results obtained on the basis of the three models has been carried out. It has been shown that for all models the characteristic equations do not possess real roots, but have one pair of complex conjugates, i.e. the test single-mass systems subjected to the impulse excitation do not pass into an aperiodic regime in none of magnitudes of the relaxation and creep times. Main characteristics of vibratory motions of the single-mass system as functions of the relaxation time or creep time, which are equivalent to the temperature dependencies, are constructed and analyzed for all three models.

Abstract: The loose coupling of computational fluid dynamics and computational structural dynamics solvers introduces some problems related to the information transfer between the codes. Some techniques developed to solve the problems of the load transfer and interface surface tracking are presented. The main criterion is to achieve conservation of total loads and total energy. The load projection scheme is based on Gaussian integration and fast interpolation algorithms for unstructured grids. The surface tracking algorithm, also based on interpolation, is important for many applications, including aeroelastic deformation of wings due to aerodynamic loads. The methodologies not only improve present fluid-structure interaction simulations, but also increase the range of their applicability. These techniques are of general character and can be used in other multidisciplinary applications as well.

Abstract: We develop high-resolution shock-capturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order-1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one cannot decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a second-order scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a first-order scheme and numerical convergence proof for the second-order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.

Abstract: Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem.

Abstract: We present a new, naturally parallelizable, accurate numerical method for the solution of transport-dominated diffusion processes in heterogeneous porous media For the discretization in time of one of the governing partial differential equations, we introduce a new characteristics-based procedure which is mass conservative, the modified method of characteristics with adjusted advection (MMOCAA) Hybridized mixed finite elements are used for the spatial discretization of the equations and a new strip-based domain decomposition procedure is applied towards the solution of the resulting algebraic problems We consider as a model problem the two-phase immiscible displacement in petroleum reservoirs A very detailed description of the numerical method is presented Following that, numerical experiments are presented illustrating the important features of the new method and comparing computed results with ones derived from previous, related techniques

Abstract: Approximation of Functions. Direct Methods for Linear Systems. Solution of Nonlinear Equations. Iterative Methods for Linear Systems. Eigenvalue Problems. Numerical Integration. Ordinary Differential Equations. Difference Methods for PDEs. Introduction to Finite Elements. Appendices. Index.