Showing papers on "Boundary value problem published in 1989"

Optimal shape design as a material distribution problem

Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers

Abstract: Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of spanwise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, ‘vortex dislocations’ are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (S_θ) can be collapsed onto the parallel-shedding Strouhal curve (S_0) by the transformation, S_0 = S_θ/cosθ, where θ is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.

KIVA-II: A computer program for chemically reactive flows with sprays

Abstract: This report documents the KIVA-II computer program for the numerical calculation of transient, two- and three-dimensional, chemically reactive fluid flows with sprays. KIVA-II extends and enhances the earlier KIVA code, improving its computational accuracy and efficiency and its ease-of-use. The KIVA-II equations and numerical solution procedures are very general and can be applied to laminar or turbulent flows, subsonic or supersonic flows, and single-phase or dispersed two-phase flows. Arbitrary numbers of species and chemical reactions are allowed. A stochastic particle method is used to calculate evaporating liquid sprays, including the effects of droplet collisions and aerodynamic breakups. Although the initial and boundary conditions and mesh generation have been written for internal combustion engine calculations, the logic for these specifications can be easily modified for a variety of other applications. Following an overview of the principal features of the KIVA-II program, we describe in detail the equations solved, the numerical solution procedure, and the structure of the computer program. Sixteen appendices provide additional details concerning the numerical solution procedure. 67 refs., 29 figs.

Numerical shape from shading and occluding boundaries

Abstract: An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly .used gradient space. Use of the stereographic plane makes it possible to incorporate occluding boundary information, but forces us to employ a smoothness constraint different from the one previously proposed. The new constraint follows directly from a particular definition of surface smoothness. We solve the set of equations arising from the smoothness constraints and the image-irradiance equation iteratively, using occluding boundary information to supply boundary conditions. Good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution. Numerical experiments show that the method is effective and robust. Finally, we analyze scanning electron microscope (SEM) pictures using this method. Other applications are also proposed.

Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials

Abstract: 1 Formulation of Elementary Boundary Value Problems- 1 The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients- 2 The Concept of Generalized Solution- 3 Generalized Formulations of Problems for the Basic Equations of Mathematical Physics- 2 The Concept of Asymptotic Expansion A Model Example to Illustrate the Averaging Method- 1 Asymptotic expansion A Formal Asymptotic Solution- 2 Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3- 3 Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method- 4 Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient- 5 Averaging the System of Differential Equations- 3 Averaging Processes in Layered Media- 1 Problem of Small Longitudinal Vibrations of a Rod- 2 Nonstationary Problem of Heat Conduction- 3 Averaging Maxwell Equations- 4 Averaging Equations of a Viscoelastic Medium- 5 Media with Slowly Changing Geometric Characteristics- 6 Heat Transfer Through a System of Screens- 7 Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod- 8 The System of Equations of Elasticity Theory in a Layered Medium- 9 Considerations Permitting Reduction of Calculations in Constructing Averaged Equations- 10 Nonstationary Nonlinear Problems- 11 Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients- 12 Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables- 4 Averaging Basic Equations of Mathematical Physics- 1 Averaging Stationary Thermal Fields in a Composite- 2 Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem- 3 Stationary Thermal Field in a Porous Medium- 4 Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials- 5 Nonstationary Systems of Equations of Elasticity and Diffusion Theory- 6 Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory- 7 Averaging Stokes and Navier-Stokes Equations The Derivation of the Percolation Law for a Porous Medium (Darcy's Law)- 8 Averaging in case of Short-Wave Propagation- 9 Averaging the Transition Equation for a Periodic Medium- 10 Eigenvalue Problems- 5 General Formal Averaging Procedure- 1 Averaging Nonlinear Equations- 2 Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory- 3 A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables- 6 Properties of Effective Coefficients Relationship Among Local and Averaged Characteristics of a Solution- 1 Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging- 2 On the Principle of Equivalent Homogeneity- 3 The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems- 4 Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment- 7 Composite Materials Containing High-Modulus Reinforcement- 1 The Stationary Field in a Layered Material- 2 Composite Materials with Grains for Reinforcement- 3 Dissipation of Waves in Layered Media- 4 High-Modulus 3D Composite Materials- 5 The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites- 8 Averaging of Processes in Skeletal Structures- 1 An Example of Averaging a Problem on the Simplest Framework- 2 A Geometric Model of a Framework- 3 The Splitting Principle for the Averaged Operator for a Periodic Framework- 4 The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures- 5 On Refining the Splitting Principle for the Averaged Operator- 2 The Concept of Generalized Solution- 3 Generalized Formulations of Problems for the Basic Equations of Mathematical Physics- 2 The Concept of Asymptotic Expansion A Model Example to Illustrate the Averaging Method- 1 Asymptotic expansion A Formal Asymptotic Solution- 2 Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3- 3 Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method- 4 Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient- 5 Averaging the System of Differential Equations- 3 Averaging Processes in Layered Media- 1 Problem of Small Longitudinal Vibrations of a Rod- 2 Nonstationary Problem of Heat Conduction- 3 Averaging Maxwell Equations- 4 Averaging Equations of a Viscoelastic Medium- 5 Media with Slowly Changing Geometric Characteristics- 6 Heat Transfer Through a System of Screens- 7 Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod- 8 The System of Equations of Elasticity Theory in a Layered Medium- 9 Considerations Permitting Reduction of Calculations in Constructing Averaged Equations- 10 Nonstationary Nonlinear Problems- 11 Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients- 12 Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables- 4 Averaging Basic Equations of Mathematical Physics- 1 Averaging Stationary Thermal Fields in a Composite- 2 Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem- 3 Stationary Thermal Field in a Porous Medium- 4 Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials- 5 Nonstationary Systems of Equations of Elasticity and Diffusion Theory- 6 Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory- 7 Averaging Stokes and Navier-Stokes Equations The Derivation of the Percolation Law for a Porous Medium (Darcy's Law)- 8 Averaging in case of Short-Wave Propagation- 9 Averaging the Transition Equation for a Periodic Medium- 10 Eigenvalue Problems- 5 General Formal Averaging Procedure- 1 Averaging Nonlinear Equations- 2 Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory- 3 A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables- 6 Properties of Effective Coefficients Relationship Among Local and Averaged Characteristics of a Solution- 1 Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging- 2 On the Principle of Equivalent Homogeneity- 3 The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems- 4 Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment- 7 Composite Materials Containing High-Modulus Reinforcement- 1 The Stationary Field in a Layered Material- 2 Composite Materials with Grains for Reinforcement- 3 Dissipation of Waves in Layered Media- 4 High-Modulus 3D Composite Materials- 5 The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites- 8 Averaging of Processes in Skeletal Structures- 1 An Example of Averaging a Problem on the Simplest Framework- 2 A Geometric Model of a Framework- 3 The Splitting Principle for the Averaged Operator for a Periodic Framework- 4 The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures- 5 On Refining the Splitting Principle for the Averaged Operator- 6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework- 7 Skeletal Structures with Random Properties- 9 Mathematics of Boundary-Layer Theory in Composite Materials- 1 Problem on the Contact of Two Layered Media- 2 The Boundary Layer for an Elliptic Equation Defined on a Half-Plane- 3 The Boundary Layer Near the Interface of Two Periodic Structures- 4 Problem on the Contact of Two Media Divided by a Thin Interlayer- 5 The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory- 6 On the Ultimate Strength of a Composite- 7 Boundary Conditions of Other Types- 8 On the Averaging of Fields in Layer Media with Layers of Composite Materials- 9 The Time Boundary Layer for the Cauchy Parabolic Problem- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell

Coherent structure of the convective boundary layer derived from large-eddy simulations

Abstract: Turbulence in the convective boundary layer (CBL) uniformly heated from below and topped by a layer of uniformly stratified fluid is investigated for zero mean horizontal flow using large-eddy simulations (LES). The Rayleigh number is effectively infinite, the Froude number of the stable layer is 0.09 and the surface roughness height relative to the height of the convective layer is varied between 10−6 and 10−2. The LES uses a finite-difference method to integrate the three-dimensional grid-volume-averaged Navier–Stokes equations for a Boussinesq fluid. Subgrid-scale (SGS) fluxes are determined from algebraically approximated second-order closure (SOC) transport equations for which all essential coefficients are determined from the inertial-range theory. The surface boundary condition uses the Monin–Obukhov relationships. A radiation boundary condition at the top of the computational domain prevents spurious reflections of gravity waves. The simulation uses 160 × 160 × 48 grid cells. In the asymptotic state, the results in terms of vertical mean profiles of turbulence statistics generally agree very well with results available from laboratory and atmospheric field experiments. We found less agreement with respect to horizontal velocity fluctuations, pressure fluctuations and dissipation rates, which previous investigations tend to overestimate. Horizontal spectra exhibit an inertial subrange. The entrainment heat flux at the top of the CBL is carried by cold updraughts and warm downdraughts in the form of wisps at scales comparable with the height of the boundary layer. Plots of instantaneous flow fields show a spoke pattern in the lower quarter of the CBL which feeds large-scale updraughts penetrating into the stable layer aloft. The spoke pattern has also been found in a few previous investigations. Small-scale plumes near the surface and remote from strong updraughts do not merge together but decay while rising through large-scale downdraughts. The structure of updraughts and downdraughts is identified by three-dimensional correlation functions and conditionally averaged fields. The mean circulation extends vertically over the whole boundary layer. We find that updraughts are composed of quasi-steady large-scale plumes together with transient rising thermals which grow in size by lateral entrainment. The skewness of the vertical velocity fluctuations is generally positive but becomes negative in the lowest mesh cells when the dissipation rate exceeds the production rate due to buoyancy near the surface, as is the case for very rough surfaces. The LES results are used to determine the root-mean-square value of the surface friction velocity and the mean temperature difference between the surface and the mixed layer as a function of the roughness height. The results corroborate a simple model of the heat transfer in the surface layer.

Triply-differential cross sections for ionisation of hydrogen atoms by electrons and positrons

Abstract: A derivation is given of the exact form of the three-body Coulomb wavefunction in the asymptotic region where the separation of all particles tends to infinity. Using a modification of the method of Pluvinage (1951), an approximate three-body scattering wavefunction is derived that satisfies this boundary condition. Triply-differential cross sections (TDCS) for electron impact ionisation of atomic hydrogen calculated with this scattering wavefunction, which contains no free parameters, show excellent agreement with measurements at impact energies greater than 150 eV. The corresponding TDCS for positron impact ionisation are also presented.

The time‐dependent Schrödinger equation: Application of absorbing boundary conditions

Abstract: In this work the time‐dependent Schrodinger equation, which is solved by employing absorbing boundary conditions, is considered. To achieve this, a negative imaginary short‐range potential was added to the potential at the asymptotic regions. The study concentrated on deriving the necessary conditions for the absorbing potential to be almost perfect. The theoretical findings are supported by a numerical study carried out for a one‐dimensional (reactive) model.

Quaternionic analysis and elliptic boundary value problems

Abstract: 1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU's Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of the T-Operator.- 2.4. VEKUA's Theorems.- 2.5. Some Integral Operators on the Manifold.- 3. Orthogonal Decomposition of the Space L2,H(G).- 4. Some Boundary Value Problems of DIRICHLET's Type.- 4.1. LAPLACE Equation.- 4.2. HELMHOLTZ Equation.- 4.3. Equations of Linear Elasticity.- 4.4. Time-independent MAXWELL Equations.- 4.5. STOKES Equations.- 4.6. NAVIER-STOKES Equations.- 4.7. Stream Problems with Free Convection.- 4.8. Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity.- 5. H-regular Boundary Collocation Methods.- 5.1. Complete Systems of H-regular Functions.- 5.2. Numerical Properties of H-complete Systems of H-regular Functions.- 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems.- 5.4. Numerical Examples.- 6. Discrete Quaternionic Function Theory.- 6.1. Fundamental Solutions of the Discrete Laplacian.- 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator.- 6.3. Elements of a Discrete Quaternionic Function Theory.- 6.4. Main Properties of Discrete Operators.- 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations.- 6.6. Concluding Remarks.- References.- Notations.

Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms

Abstract: Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.

Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface

Abstract: Paper 1 [1988 g][1] of this series began an investigation whose goal is a thermomechanics of two-phase continua based on Gibbs’s notion of a sharp phase-interface endowed with thermomechanical structure. In that paper a balance law, balance of capillary forces, was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics; the chief results are thermodynamic restrictions on constitutive equations, exact and approximate free-boundary conditions at the interface, and a hierarchy of free-boundary problems. The simplest versions of these problems (the Mullins-Sekerka problems) are essentially the classical Stefan problem with the free-boundary condition u = 0 for the temperature replaced by the condition u = h K, where K is the curvature of the free-boundary and h > 0 is a material constant. This dependence on curvature renders the problem difficult, and apart from numerical studies involving linearization stability, there are almost no supporting theoretical results.

Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard‐sphere molecules

Abstract: The Poiseuille and thermal transpiration flows of a rarefied gas between two parallel plates are investigated on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. The velocity distribution functions of the gas molecules as well as the gas velocity and heat flow profiles and mass fluxes are obtained for the whole range of the Knudsen number with good accuracy by the numerical method recently developed by the authors.

New approximate boundary conditions for large eddy simulations of wall-bounded flows

Abstract: Two new approximate boundary conditions have been applied to the large eddy simulation of channel flow with and without transpiration. These new boundary conditions give more accurate results than those previously in use, and allow significant reduction of the required CPU time over simulations in which no‐slip conditions are applied. Mean velocity profiles and turbulence intensities compare well both with experimental data and with the results of resolved simulations. The influence of the approximate boundary conditions remains confined near the point of application and does not affect the turbulence statistics in the core of the flow.

Finite Element Approximation for Optimal Shape Design: Theory and Applications

Abstract: Preliminaries - Green's formula abstract setting of optimal shape design problem and its approximation shape optimization of systems governed by unilateral boundary value state problem - scalar case approximation of the optimal shape design problems by finite elements - scalar case numerical realization shape optimization in unilateral boundary value problems with "flux" cost functional optimal shape design in contact problems - elastic case shape optimization of elastic/perfectly plastic bodies in contact on the design of the optimal covering supported by an obstacle state constrained optimal control problems and their approximations FE-grid optimization concluding remarks on references on optimal shape design and related topics. Appendices: algorithms for FEM on the differentation of stiffness and mass matrices and force vectors subgradient method for convex linearly constrained optimization description of the sequential quadratic programming (SQP) algorithm on the differentiability of a projection on a convex set in Hilbert space.

Numerical Methods for Viscous Flows With Moving Boundaries

Abstract: A review of numerical algorithms for the analysis of viscous flows with moving interfaces is presented. The review is supplemented with a discussion of methods that have been introduced in the context of other classes of free boundary problems, but which can be generalized to viscous flows with moving interfaces. The available algorithms can be classified as Eulerian, Langrangian, and mixed, ie, Eulerian-Lagrangian. Eulerian algorithms consist of fixed grid methods, adaptive grid methods, mapping methods, and special methods. Langrangian algorithms consist of strictly Langrangian methods, Langrangian methods with rezoning, free Lagrangian methods and particle methods. Mixed methods rely on both Lagrangian and Eulerian concepts. The review consists of a description of the present state-of-the-art of each group of algorithms and their applications to a variety of problems. The existing methods are effective in dealing with small to medium interface deformations. For problems with medium to large deformations the methods produce results that are reasonable from a physical viewpoint; however, their accuracy is difficult to ascertain.

Boundary integral formulations for homogeneous material bodies

Abstract: There are many boundary integral formulations for the problem of electromagnetic scattering from and transmission into a homogeneous material body. The only formulations which give a unique solution at all frequencies are those which involve both electric and magnetic equivalent currents, and satisfy boundary conditions on both tangential E and tangential H. Formulations which involve only electric (or magnetic) equivalent currents, and those which involve boundary conditions on only tangential E (or tangential H) are singular at frequencies corresponding to the resonant frequencies of a resonator formed by a perfect conductor covering the surface of the body and filled with the material exterior to the body in the original problem.

Buckling and vibration of laminated composite plates using various plate theories

Abstract: Analytical and finite-element solutions of the classical, first-order, and third-order laminate theories are developed to study the buckling and free-vibration behavior of cross-ply rectangular composite laminates under various boundary conditions

Nonparabolicity effects in a quantum well: Sublevel shift, parallel mass, and Landau levels

Abstract: The influence of nonparabolicity on the subband structure in a quantum well is analyzed. Starting from an accurate expression for the bulk conduction-band structure expanded up to fourth order in k, we determine both the shift of the confinement energies and the energy dispersion parallel to the layers E${(\mathrm{k}}_{?}$). The resulting eigenvalue equations are of the same form as in the parabolic case, but somewhat more complicated. The anisotropy of the bulk conduction band is included, and it is found to have a larger effect in quantum wells than in the bulk. The results can be expressed in terms of the perpendicular mass, which is relevant for the determination of confinement energies, and the parallel mass, which gives the curvature of E${(\mathrm{k}}_{?}$) at the bottom of a subband. We derive approximate expressions for these masses in the form of explicit functions of the confinement energy, which is experimentally accessible. The enhancement of the parallel mass relative to the bulk mass is found to be 2--3 times stronger than that of the perpendicular mass. It is shown that the boundary conditions need to be modified in the nonparabolic case. The nonintuitive result is that the confinement energy for the ground state usually is increased relative to a similar calculation in the parabolic approximation. We include the effect of a perpendicular magnetic field and derive an analytic expression for the Landau levels. The cyclotron mass is found to increase with magnetic field and approach the parallel mass in the limit of small magnetic fields. The parallel mass is also relevant for transport parallel to the layers, density of states, and exciton properties. The agreement with experiment is encouraging. Previous theoretical approaches are critically reviewed and the differences and similarities with this work are pointed out.

Optimal finite-element interpolation on curved domains

Abstract: This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.

Finite element analysis of the compressible Euler and Navier-Stokes equations

Abstract: A space-time finite element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms, boundary conditions, and solution strategies. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. A multi-element group, domain decomposition algorithm is presented for solving the nonsymmetric linear systems. This algorithm employs an iterative strategy based on the generalized minimal residual (GMRES) procedure. A two-level preconditioning technique is presented, which significantly accelerates the convergence of the GMRES procedure. Numerical results are presented to demonstrate the performance of the method.

The collimation of magnetized winds

Abstract: It is established that any stationary axisymmetric magnetized wind will collimate along the symmetry axis at large distances from the source. This result is proved by consideration of the asymoptotic properties of the transfield equation, keeping the exact conserved quantities along field lines. The only consistent nonsingular solution with a nonvanishing poloidal current approaches a cylindrical structure. For singular solutions or those with a vanishing poloidal current, the asymptotic solutions can be paraboloidal. This result only applies to pure wind boundary conditions on the surface of the source. It is shown how the boundary conditions and the critical point analysis are related in our asymptotic analysis. This result demonstrates that axisymmetric magnetized flows tend generally to collimate, and it is hypothesized that this is the natural reason why there are so many collimated flows and jets. 15 refs.

Flow channeling in a single fracture as a two-dimensional strongly heterogeneous permeable medium

Abstract: The void space of a rock fracture is conceptualized as a two-dimensional heterogeneous system with variable apertures as a function of position in the fracture plane. The apertures are generated using geostatistical methods. Fluid flow is simulated with constant head boundary conditions on two opposite sides of the two-dimensional flow region, with closed boundaries on the remaining two sides. The results show that the majority of flow tends to coalesce into certain preferred flow paths (channels) which offer the least resistance. Tracer transport is then simulated using a particle tracking method. The apertures along the paths taken by the tracer particles are found to obey a distribution different from that of all the apertures in the fracture. They obey a distribution with a larger mean and a smaller standard deviation. The shift in the distribution parameters increases with increasing values of variance for the apertures in the two-dimensional fracture. Provided that the correlation length is no greater than one fifth of the scale of measurement, the aperture density distributions of tracer particle paths remain similar for flow in two orthogonal directions, even with anisotropy ratio of spatial correlation up to 5. These results may be applicable in general to flow and transport through a two-dimensional strongly heterogeneous porous medium with a broad permeability distribution, where the dispersion of the system may be related to the parameters of the permeability distribution along preferred flow channels.

Boundary Conditions at the Cartilage-Synovial Fluid Interface for Joint Lubrication and Theoretical Verifications

Abstract: The objective of this study is to establish and verify the set of boundary conditions at the interface between a biphasic mixture (articular cartilage) and a Newtonian or non-Newtonian fluid (synovial fluid) such that a set of well-posed mathematical problems may be formulated to investigate joint lubrication problems. A "pseudo-no-slip" kinematic boundary condition is proposed based upon the principle that the conditions at the interface between mixtures or mixtures and fluids must reduce to those boundary conditions in single phase continuum mechanics. From this proposed kinematic boundary condition, and balances of mass, momentum and energy, the boundary conditions at the interface between a biphasic mixture and a Newtonian or non-Newtonian fluid are mathematically derived. Based upon these general results, the appropriate boundary conditions needed in modeling the cartilage-synovial fluid-cartilage lubrication problem are deduced. For two simple cases where a Newtonian viscous fluid is forced to flow (with imposed Couette or Poiseuille flow conditions) over a porous-permeable biphasic material of relatively low permeability, the well known empirical Taylor slip condition may be derived using matched asymptotic analysis of the boundary layer at the interface.

Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules

Abstract: Shear flow and thermal creep flow (flow induced by the temperature gradient along the boundary wall) of a rarefied gas over a plane wall are considered on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. These fundamental rarefied gas dynamic problems, typical half‐space boundary‐value problems of the linearized Boltzmann equation, are analyzed numerically by the finite‐difference method developed recently by the authors, and the velocity distribution functions, as well as the macroscopic variables, are obtained with good accuracy. From the results, the shear and thermal creep slip coefficients and their associated Knudsen layers of a slightly rarefied gas flow past a body are derived. The results for the slip coefficients and Knudsen layers are compared with experimental data and various results by the Boltzmann–Krook–Welander (BKW) equation, the modified BKW equation, and a direct simulation method.

Coherent structures in multidimensions.

Abstract: We solve an initial-boundary value problem for the Davey-Stewartson equation, a multidimensional analog of the nonlinear Schr\"odinger equation. It is shown that for large time, an arbitrary initial disturbance will, in general, decompose into a number of two-dimensional coherent structures. These structures exhibit interesting novel features not found in one-dimensional solitons.