Abstract: We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49] In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction

Abstract: A cognitive journey towards the reliable simulation of scattering problems using finite element methods, with the pre-asymptotic analysis of Galerkin FEM for the Helmholtz equation with moderate and large wave number forming the core of this book. Starting from the basic physical assumptions, the author methodically develops both the strong and weak forms of the governing equations, while the main chapter on finite element analysis is preceded by a systematic treatment of Galerkin methods for indefinite sesquilinear forms. In the final chapter, three dimensional computational simulations are presented and compared with experimental data. The author also includes broad reference material on numerical methods for the Helmholtz equation in unbounded domains, including Dirichlet-to-Neumann methods, absorbing boundary conditions, infinite elements and the perfectly matched layer. A self-contained and easily readable work.

Abstract: SUMMARY The numerical method used in this study is the moving particle semi-implicit (MPS) method, which is based on particles and their interactions. The particle number density is implicitly required to be constant to satisfy incompressibility. A semi-implicit algorithm is used for two-dimensional incompressible non-viscous flow analysis. The particles whose particle number densities are below a set point are considered as on the free surface. Grids are not necessary in any calculation steps. It is estimated that most of computation time is used in generation of the list of neighboring particles in a large problem. An algorithm to enhance the computation speed is proposed. The MPS method is applied to numerical simulation of breaking waves on slopes. Two types of breaking waves, plunging and spilling breakers, are observed in the calculation results. The breaker types are classified by using the minimum angular momentum at the wave front. The surf similarity parameter which separates the types agrees well with references. Breaking waves are also calculated with a passively moving float which is modelled by particles. Artificial friction due to the disturbed motion of particles causes errors in the flow velocity distribution which is shown in comparison with the theoretical solution of a cnoidal wave. © 1998 John Wiley & Sons, Ltd.

Abstract: Numerical methods for time stepping the Cahn-Hilliard equation are given and discussed. The methods are unconditionally gradient stable, and are uniquely solvable for all time steps. The schemes require the solution of ill-conditioned linear equations, and numerical methods to accurately solve these equations are also discussed.

Abstract: The paper describes a method for modelling progressive mixed-mode delamination in fibre composites. The procedure, which is incorporated within the non-linear finite element method, is based on the use of interface elements in conjunction with softening relationships between the stresses and the relative displacements. Fracture mechanics is indirectly introduced by relating the areas under the stress/displacement curves to the critical fracture energies.

Abstract: In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal $L^2$ -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical.

Abstract: We analyze the optimal velocity model (OVM) with explicit delay. The properties of congestion and the delay time of car motion are investigated by analytical and numerical methods. It is shown that the small explicit delay time has almost no effects. In the case of the large explicit delay time, a new phase of congestion pattern of OVM seems to appear.

Abstract: A practical method is presented for numerical analysis of problems in solid (in particular soil) mechanics which involve large strains or deformations. The method is similar to what is referred to as ‘arbitrary Lagrangian–Eulerian’, with simple infinitesimal strain incremental analysis combined with regular updating of co-ordinates, remeshing of the domain and interpolation of material and stress parameters. The technique thus differs from the Lagrangian or Eulerian methods more commonly used. Remeshing is accomplished using a fully automatic remeshing technique based on normal offsetting, Delaunay triangulation and Laplacian smoothing. This technique is efficient and robust. It ensures good quality shape and distribution of elements for boundary regions of irregular shape, and is very quick computationally. With remeshing and interpolation, small fluctuations appeared initially in the load-deformation results. In order to minimize these, different increment sizes and remeshing frequencies were explored. Also, various planar linear interpolation techniques were compared, and the unique element method found to work best. Application of the technique is focused on the widespread problem of penetration of surface foundations into soft soil, including deep penetration of foundations where soil flows back over the upper surface of the foundation. Numerical results are presented for a plane strain footing and an axisymmetric jack-up (spudcan) foundation, penetrating deeply into soil which has been modelled as a simple Tresca or Von Mises material, but allowing for increase of the soil strength with depth. The computed results are compared with plasticity solutions for bearing capacity. The numerical method is shown to work extremely well, with potential application to a wide range of soil–structure interaction problems. © 1998 John Wiley & Sons, Ltd.

Abstract: A numerical technique (FGVT) for solving the time-dependent incompressible Navier-Stokes equations in fluid flows with large density variations is presented for staggered grids. Mass conservation is based on a volume tracking method and incorporates a piecewise-linear interface reconstruction on a grid twice as fine as the velocity pressure grid. It also uses a special flux-corrected transport algorithm for momentum advection, a multigrid algorithm for solving a pressure-correction equation and a surface tension algorithm that is robust and stable. In principle, the method conserves both mass and momentum exactly, and maintains extremely sharp fluid interfaces. Applications of the numerical method to prediction of two-dimensional bubble rise in an inclined channel and a bubble bursting through an interface are presented

Abstract: 1. Differential Equations 2. Linear Difference Equations with Constant Coefficients 3. Polynomials and Toeplitz Matrices 4. Numerical Methods for Initial Value Problems 5. Generalized Backward Differentiation Formulae 6. Symmetric Schemes 7. Generalized Adams Methods 8. Hamiltonian Problems 9. Boundary Value Problems 10. Mesh Selection Strategies 11. Block BVMs 12. Parallel Implementation of B2VMs 13. Extensions and Applications to Special Problems

Abstract: This paper describes an Arbitrary Lagrangian- Eulerian (ALE) finite element method for the simulation of fluid domains with moving structures. The fluid is viscous, incompressible and unsteady and the fluid motion is solved by a fractional step discretization of the Navier-Stokes equations. The emphasis is on convection dominated flows, and a three-step method is used for the convection term. The moving structure causes the mesh of the fluid domain to move, and a new algorithm is proposed to solve the important and crucial problem of the calculation of the mesh velocities. Numerical calculations of the added mass and added damping of a vibrating two-dimensional circular cylinder in the frequency Reynolds number range Re w =20−2000 are performed to evaluate the proposed ALE finite element method. The numerically calculated added mass and added damping are compared to both analytical and numerical results. To further demonstrate the generality of the method, a numerical simulation of flow past an oscillating schematic sports car is presented.

Abstract: The classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope (e.g., specialized recursions for centered and staggered approximations, for Adams--Bashforth-, Adams--Moulton-, and BDF-formulas for ODEs, etc.). Two recent algorithms overcome these problems. For equispaced grids, such weights can be found very conveniently with a two-line algorithm when using a symbolic language such as Mathematica (reducing to one line in the case of explicit approximations). For arbitrarily spaced grids, we describe a computationally very inexpensive numerical algorithm.

Abstract: We introduce a well-developed Newton iterative (truncated Newton) algorithm for solving large-scale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and features. The structure offers the user great flexibility in addressing problem specificity through preconditioning and other means and allows easy adaptation to parallel environments. Features and capabilities are illustrated in numerical experiments.

Abstract: This paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Numerical experiments compare the stability properties of these schemes with explicit ones.

Abstract: We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.

Abstract: This paper provides a review of numerical methods for the solution of smooth semi-infinite programming problems. Fundamental and partly new results on level sets, discretization, and local reduction are presented in a primary section. References to algorithms for real and complex continuous Chebyshev approximation are given for historical reasons and in order to point out connections.

Abstract: We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty(L2) and L2(H1) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.

Abstract: A fast and accurate numerical method for free-space beam propagation between arbitrarily oriented planes is developed. The only approximation made in the development of the method was that the vector nature of light was ignored. The method is based on evaluating the Rayleigh–Sommerfeld diffraction integral by use of the fast Fourier transform with a special transformation to handle tilts and offsets of planes. The fundamental aspects of a software package based on the developed method are presented. A numerical example realized with the software package is presented to establish the validity of the method.

Abstract: For a general Stewart-Gough platform, two rigid bodies connected by six rods attached via spherical joints, it is known that the maximum number of assembly modes can be at most 40 (counting complex modes as well). However, it was not known yet if there exist examples of Stewart-Gough platforms which actually possess 40 real (the only realizable) assembly modes or postures. This article presents a numerical method which systematically changes the parameters of a given Stewart-Gough platform with the goal to increase the number of real postures and ultimately to obtain an example which possesses 40 real postures. The proposed method is exemplified by way of one particular example of a Stewart-Gough platform for which we obtained 40 real postures.