Abstract: Introduction to Numerical Methods for Problems with Boundary Layers Numerical Methods on Uniform Meshes Layer Resolving Methods for Convection-Diffusion Problems in One Dimension The Limitations of Non-Monotone Numerical Methods Convection-Diffusion Problems in a Moving Medium Convection-Diffusion Problems with Frictionless Walls Convection-Diffusion Problems with No Slip Boundary Conditions Experimental Estimation of Errors Non-Monotone Methods in Two Dimensions Linear and Nonlinear Reaction-Diffusion Problems Prandtl Flow past a Flat Plate-Blasius' Method Prandtl Flow past a Flat Plate-Direct Method References.

Abstract: A full numerical solution for the mixed elastohydrodynamic lubrication (EHL) in point contacts is presented in this paper, using a new numerical approach that is simple and robust, capable of handling three-dimensional measured engineering rough surfaces moving at different rolling and sliding velocities. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed and boundary lubrication, In the hydrodynamically lubricated areas the Reynolds equation is used. In the asperity contact areas, where the film thickness is zero, the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. In order to save computing time, a multi-level integration method is used to calculate surface deformation. Sample cases under severe condition show that this approach is capable of analyzing different cases in a full range of λ ratio, from infinitely large down to nearly zero (less than 0.03).

Abstract: A number of physical phenomena are described by nonlinear hyperbolic equations Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems Construction of such methods for systems more complicated than the Euler gas dynamic equations requires the investigation of existence and uniqueness of the self-similar solutions to be used in the development of discontinuity-capturing high-resolution numerical methods This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion We discuss these problems in the application to the magnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic wave propagation in magnetics

Abstract: The paper deals with topology optimization of structures undergoing large deformations. The geometrically nonlinear behaviour of the structures are modelled using a total Lagrangian finite element formulation and the equilibrium is found using a Newton-Raphson iterative scheme. The sensitivities of the objective functions are found with the adjoint method and the optimization problem is solved using the Method of Moving Asymptotes. A filtering scheme is used to obtain checkerboard-free and mesh-independent designs and a continuation approach improves convergence to efficient designs. Different objective functions are tested. Minimizing compliance for a fixed load results in degenerated topologies which are very inefficient for smaller or larger loads. The problem of obtaining degenerated "optimal" topologies which only can support the design load is even more pronounced than for structures with linear response. The problem is circumvented by optimizing the structures for multiple loading conditions or by minimizing the complementary elastic work. Examples show that differences in stiffnesses of structures optimized using linear and nonlinear modelling are generally small but they can be large in certain cases involving buckling or snap-through effects.

Abstract: This work reports results of numerical simulations of viscous incompressible flow past a sphere. The primary objective is to identify transitions that occur with increasing Reynolds number, as well as their underlying physical mechanisms. The numerical method used is a mixed spectral element/Fourier spectral method developed for applications involving both Cartesian and cylindrical coordinates. In cylindrical coordinates, a formulation, based on special Jacobi-type polynomials, is used close to the axis of symmetry for the efficient treatment of the ‘pole’ problem. Spectral convergence and accuracy of the numerical formulation are verified. Many of the computations reported here were performed on parallel computers. It was found that the first transition of the flow past a sphere is a linear one and leads to a three-dimensional steady flow field with planar symmetry, i.e. it is of the ‘exchange of stability’ type, consistent with experimental observations on falling spheres and linear stability analysis results. The second transition leads to a single-frequency periodic flow with vortex shedding, which maintains the planar symmetry observed at lower Reynolds number. As the Reynolds number increases further, the planar symmetry is lost and the flow reaches a chaotic state. Small scales are first introduced in the flow by Kelvin–Helmholtz instability of the separating cylindrical shear layer; this shear layer instability is present even after the wake is rendered turbulent.

Abstract: Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question "for what choices of stepsize does the numerical method reproduce the characteristics of the test equation?" We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the theta method. We extend some mean-square stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33 (1996), pp. 2254--2267]. In particular, we show that an extension of the deterministic A-stability property holds. We also plot mean-square stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992] has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain some numerical results reported in [Milstein, Platen, and Schurz, SIAM J. Numer. Anal., 35 (1998), pp. 1010--1019.]

Abstract: Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations.

Abstract: We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.

Abstract: We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to the total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals.

Abstract: We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations. Assuming access to the element stiffness matrices, we have that AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.

Abstract: We consider H (curl ;Ω)-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

Abstract: Nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3, et sa discretisation par elements finis. Le but de cet article est de demontrer des estimations d'erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients.

Abstract: A finite element with a spatially varying material property field is formulated and compared to a conventional, homogeneous element for solving boundary value problems involving continuously nonhomogeneous materials. The particular element studied is a two-dimensional plane stress element with linear interpolation and an exponential material property gradient. However, the main results are applicable to other types of elements and property gradients. Exact solutions for a finite rectangular plate subjected to either uniform displacement or traction either perpendicular or parallel to the property gradient are used as the basis for comparison. The results show that for identical meshes with equal number of degrees-of-freedom, the graded elements give more accurate local stress values than conventional elements in some boundary value problems, while in other problems the reverse is true.

Abstract: The meshless method based on the local boundary integral equation (LBIE) is a promising method for solving boundary value problems, using an local unsymmetric weak form and shape functions from the moving least squares approximation. In the present paper, the meshless method based on the LBIE for solving problems in linear elasticity is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation can be easily imposed even when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples show that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

Abstract: DERIVATION OF THE EQUATIONS WELL-POSEDNESS OF THE LINEARIZED SYSTEM AND GENERAL ASYMPTOTICS Linear Well-Posedness First Results on the Time-Asymptotic Behavior ASYMPTOTIC BEHAVIOR FOR LINEARIZED ONE-DIMENSIONAL MODELS Large Time Behavior Bounded Domains The Cauchy Problem The Semi-Axis Propagation of Singularities ASYMPTOTIC BEHAVIOR FOR LINEARIZED MULTI-DIMENSIONAL MODELS Large-Time Behavior Bounded Domains The Cauchy Problem Isotropic Media Cubic media Propagation of singularities LOCAL EXISTENCE Initial Boundary Value Problems The Cauchy Problem NONLINEAR ONE-DIMENSIONAL THERMOELASTICITY Bounded Domains The Cauchy Problem The Semi-Axis Stationary Forces Blow-up of Smooth Solutions for Large Data Weak Solutions NONLINEAR MULTI-DIMENSIONAL THERMOELASTICITY Bounded Domains The Cauchy Problem Blow-Up CONTACT PROBLEMS Fully Dynamical Contact Problems Quasi-Static Contact Problems Linear Quasi-Static Problems Quasi-Static Contact Smoothing Property RELATED RESULTS Exponential Decay in the Case of Damping Asymptotic Behavior of Solutions as 1/2x1/2 (R)* Numerical Analysis APPENDIX Existence Theory for Linear Equations Existence for Linear Evolution Systems Linear Hyperbolic Systems Linear Parabolic Equations Regularity for Linear Elliptic Systems and Inequalities

Abstract: The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.