Showing papers on "Boundary value problem published in 1988"

Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media.

Abstract: The transfer of monochromatic radiation in a scattering, absorbing, and emitting plane-parallel medium with a specified bidirectional reflectivity at the lower boundary is considered. The equations and boundary conditions are summarized. The numerical implementation of the theory is discussed with attention given to the reliable and efficient computation of eigenvalues and eigenvectors. Ways of avoiding fatal overflows and ill-conditioning in the matrix inversion needed to determine the integration constants are also presented.

The Boltzmann equation and its applications

Abstract: I. Basic Principles of The Kinetic Theory of Gases.- 1. Introduction.- 2. Probability.- 3. Phase space and Liouville's theorem.- 4. Hard spheres and rigid walls. Mean free path.- 5. Scattering of a volume element in phase space.- 6. Time averages, ergodic hypothesis and equilibrium states.- References.- II. The Boltzmann Equation.- 1. The problem of nonequilibrium states.- 2. Equations for the many particle distribution functions for a gas of rigid spheres.- 3. The Boltzmann equation for rigid spheres.- 4. Generalizations.- 5. Details of the collision term.- 6. Elementary properties of the collision operator. Collision invariants.- 7. Solution of the equation Q(f,f) = 0.- 8. Connection between the microscopic description and the macroscopic description of gas dynamics.- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation.- 10. Model equations.- References.- III. Gas-Surface Interaction and the H-Theorem.- 1. Boundary conditions and the gas-surface interaction.- 2. Computation of scattering kernels.- 3. Reciprocity.- 4. A remarkable inequality.- 5. Maxwell's boundary conditions. Accommodation coefficients.- 6. Mathematical models for gas-surface interaction.- 7. Physical models for gas-surface interaction.- 8. Scattering of molecular beams.- 9. The H-theorem. Irreversibility.- 10. Equilibrium states and Maxwellian distributions.- References.- IV, Linear Transport.- 1. The linearized collision operator.- 2. The linearized Boltzmann equation.- 3. The linear Boltzmann equation. Neutron transport and radiative transfer.- 4. Uniqueness of the solution for initial and boundary value problems.- 5. Further investigation of the linearized collision term.- 6. The decay to equilibrium and the spectrum of the collision operator.- 7. Steady one-dimensional problems. Transport coefficients.- 8. The general case.- 9. Linearized kinetic models.- 10. The variational principle.- 11. Green's function.- 12. The integral equation approach.- References.- V. Small and Large Mean Free Paths.- 1. The Knudsen number.- 2. The Hilbert expansion.- 3. The Chapman-Enskog expansion.- 4. Criticism of the Chapman-Enskog method.- 5. Initial, boundary and shock layers.- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients.- 7. Free molecule flow past a convex body.- 8. Free molecule flow in presence of nonconvex boundaries.- 9. Nearly free-molecule flows.- References.- VI. Analytical Solutions of Models.- 1. The method of elementary solutions.- 2. Splitting of a one-dimensional model equation.- 3. Elementary solutions of the simplest transport equation.- 4. Application of the general method to the Kramers and Milne problems.- 5. Application to the flow between parallel plates and the critical problem of a slab.- 6. Unsteady solutions of kinetic models with constant collision frequency.- 7. Analytical solutions of specific problems.- 8. More general models.- 9. Some special cases.- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency.- 11. Analytic continuation.- 12. Sound propagation in monatomic gases.- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies.- 14. Fluctuations and light scattering.- References.- VII. The Transition Regime.- 1. Introduction.- 2. Moment and discrete ordinate methods.- 3. The variational method.- 4. Monte Carlo methods.- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders.- 6. Shock-wave structure.- 7. External flows.- 8. Expansion of a gas into a vacuum.- References.- VIII. Theorems on the Solutions of the Boltzmann Equation.- 1. Introduction.- 2. The space homogeneous case.- 3. Mollified and other modified versions of the Boltzmann equation.- 4. Nonstandard analysis approach to the Boltzmann equation.- 5. Local existence and validity of the Boltzmann equation.- 6. Global existence near equilibrium.- 7. Perturbations of vacuum.- 8. Homoenergetic solutions.- 9. Boundary value problems. The linearized and weakly nonlinear cases.- 10. Nonlinear boundary value problems.- 11. Concluding remarks.- References.- References.- Author Index.

Reassessment of the scale-determining equation for advanced turbulence models

Abstract: A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: 1) conventional A:-e and A>w formulations generally are inaccurate for boundary layers in adverse pressure gradient; 2) using "wall functions'' tends to mask the shortcomings of such models; and 3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows. With no viscous damping of the model's closure coefficients and without the aid of wall functions, the model equations can be integrated through the viscous sublayer. Surface boundary conditions are presented that permit accurate predictions for flow over rough surfaces and for flows with surface mass addition.

Boundary conditions for integrable quantum systems

Abstract: A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

Fourth-order finite-difference P-SV seismograms

Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

Boundary regularity for solutions of degenerate elliptic equations

Abstract: On etudie la regularite limite des solutions faibles de l'equation divA(x,u,Du)+B(x,u,Du)=0 avec des conditions aux limites conormales et de Dirichlet

Numerical solution of boundary value problems for ordinary differential equations

Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.

Boundary Integral Operators on Lipschitz Domains: Elementary Results

Abstract: The simple and double layer potentials for second order linear strongly elliptic differential operators on Lipschitz domains are studied and it is shown that in a certain range of Sobolev spaces, r...

Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions

Abstract: Preliminaries.- Fredholm and semi-Fredholm results.- Proofs.- Two-dimensional domains.- Singularities along the edges.- Laplace operator.- Variational boundary value problems on smooth domains.- Variational boundary value problems on polyhedral domains.

Nonlocal Continuum Damage, Localization Instability and Convergence

Abstract: A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.

Particlelike solutions of the Einstein-Yang-Mills equations.

Abstract: We study the static spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group and find numerical solutions which are nonsingular and asymptotically flat. These solutions have a high-density interior region with sharp boundary, a near-field region where the metric is approximately Reissner-N\o{}rdstrom with Dirac monopole curvature source, and a far-field region where the metric is approximately Schwarzschild.

Three-dimensional vortex dynamics in superfluid He 4 : Homogeneous superfluid turbulence

Abstract: The behavior of a tangle of quantized vortex lines subject to uniform superfluid and normal-fluid driving velocities is investigated. The dynamical equation of the quantized vortices in the local approximation is supplemented by the assumption that when two such singularities cross, they undergo a reconnection. The properties of the dynamical equation, when combined with the assumption of homogeneity, imply numerous scaling relations, which are in fact observed experimentally. The primitive dynamical rules are utilized to perform extensive numerical simulations of the vortex tangle, using not only periodic, but also smooth-wall and rough-wall boundary conditions. All lead to the same homogeneous vortex-tangle state, although the case of periodic boundary conditions requires an additional trick to eliminate artificial features. The quantitative results obtained from these simulations are found to be in excellent absolute agreement with a large variety of experiments, including recent studies of the vortex-tangle anisotropy.

Three-dimensional elasticity

Abstract: Part A Description of Three-Dimensional Elasticity 1 Geometrical and other preliminaries 2 The equations of equilibrium and the principle of virtual work 3 Elastic materials and their constitutive equations 4 Hyperelasticity 5 The boundary value problems of three-dimensional elasticity Part B Mathematical Methods in Three-Dimensional Elasticity 6 Existence theory based on the implicit function theorem 7 Existence theory based on the minimization of the Energy Bibliography Index

Creeping flow of dilute polymer solutions past cylinders and spheres

Abstract: A dilute polymer solution is modelled as a suspension of dumbbells with finite extensibility. Time-dependent numerical calculations are performed of flow part cylindrical and spherical surfaces at low Reynolds number. A finite-difference scheme is employed in which the evolution in time of the dumbbells is followed from an initially unstretched equilibrium. Results are calculated with (i) a no-slip, and (ii) a zero-tangential-stress boundary condition at the body surface. At large Deborah number, D , the polymer is most highly stretched in thin regions of fluid close to and downstream of stagnation points of the flow. The most important region dynamically is found to be at the rear of the obstacle. Numerical refinements in space and time are included in order properly to resolve this fine-scale structure. Numerically stable results are obtained for values of D up to 16, and show that the flow field and drag force on the obstacle tend toward finite values at large D . Experimental measurements of the drag on a falling rigid sphere, and the velocity distribution around it, are compared with the numerical results for the no-slip boundary. Observations of bubble behaviour are discussed in the light of the results for the slip boundary.

Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks

Abstract: We present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls. We show that in a general case, when the controls are distributed, stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks. We also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems.

Soil-structure-interaction analysis in time domain

Abstract: The procedures to perform nonlinear soil-structure-interaction analysis in the time domain are summarized. The nonlinearity is restricted to the structure and possibly an adjacent irregular soil region. The unbounded soil (far field) must remain linear in this formulation. Besides the direct method where local frequency-independent boundary conditions are enforced on the artificial boundary, various formulations based on the substructure method are addressed, ranging from a discrete model with springs, dashpots and masses to boundary-element methods with convolution integrals involving either the dynamic-stiffness coefficients or the Green's functions in the time domain via the iterative hybrid-frequency-time-domain analysis procedure with the nonlinearities affecting only the right-hand side of the equations of motion.

Molecular dynamics of Poiseuille flow and moving contact lines.

Abstract: We report on molecular-dynamics simulations of the low--Reynolds-number flow of Lennard-Jones fluids through a channel. Application of a pressure gradient to a single fluid produces Poiseuille flow with a no-slip boundary condition and Taylor-Aris hydrodynamic dispersion. For an immiscible two-fluid system we find a (predictable) static contact angle and, when accelerated, velocity-dependent advancing and receding contact angles. The approximate local velocity field is obtained, in which the no-slip condition appears to break down near the contact line.

Symmetric vortex merger in two dimensions - Causes and conditions

Abstract: Two like-signed vorticity regions can pair or merge into one vortex. This phenomenon occurs if the original two vortices are sufficiently close together, that is, if the distance between the vorticity centroids is smaller than a certain critical merger distance, which depends on the initial shape of the vortex distributions. Our conclusions are based on an analytical/numerical study, which presents the first quantitative description of the cause and mechanism behind the restricted process of symmetric vortex merger. We use two complementary models to investigate the merger of identical vorticity regions. The first, based on a recently introduced low-order physical-space moment model of the two-dimensional Euler equations, is a Hamiltonian system of ordinary differential equations for the evolution of the centroid position, aspect ratio and orientation of each region. By imposing symmetry this system is made integrable and we obtain a necessary and sufficient condition for merger. This condition involves only the initial conditions and the conserved quantities. The second model is a high-resolution pseudospectral algorithm governing weakly dissipative flow in a box with periodic boundary conditions. When the results obtained by both methods are juxtaposed, we obtain a detailed kinematic insight into the merger process. When the moment model is generalized to include a weak Newtonian viscosity, we find a ‘metastable’ state with a lifetime depending on the dissipation timescale. This state attracts all initial configurations that do not merge on a convective timescale. Eventually, convective merger occurs and the state disappears. Furthermore, spectral simulations show that initial conditions with a centroid separation slightly larger than the critical merger distance initially cause a rapid approach towards this metastable state.

Boundary Element Methods in Elastodynamics

Abstract: Governing equations of motions boundary integral formulation in elastodynamics numerical treatment of boundary equations other boundary methods wave propagation analysis soil structure interaction vibrations of structures other linear material models computer implementation aspects.

Boundary conditions for nonlinear hyperbolic systems of conservation laws

Abstract: We propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws. A first approach is based on the vanishing viscosity method and a second one is related to the Riemann problem. The equivalence between these two conditions is studied. The latter formulation is extended to treat numerically physically relevant boundary conditions. Monodimensional experiments are presented.

Existence and uniqueness theorems for the bending of an elastic beam equation

Abstract: This paper concerns the existence and uniqueness theorems for nonlinear analogue of the boundary value problems describing the deformations of an elastic beam under an external load.

Parabolic evolution equations and nonlinear boundary conditions

Abstract: Nonlinear oscillations and other motions of large axially symmetric liquid drops in zero gravity are studied numerically by a boundary-integral method. The effect of small viscosity is included in the computations by retaining first-order viscous terms in the normal stress boundary condition. This is accomplished by making use of a partial solution of the boundary-layer equations which describe the weak vortical surface layer. Small viscosity is found to have a relatively large effect on resonant mode coupling phenomena.

Oscillations of drops in zero gravity with weak viscous effects

Abstract: Nonlinear oscillations and other motions of large axially symmetric liquid drops in zero gravity are studied numerically by a boundary-integral method. The effect of small viscosity is included in the computations by retaining first-order viscous terms in the normal stress boundary condition. This is accomplished by making use of a partial solution of the boundary-layer equations which describe the weak vortical surface layer. Small viscosity is found to have a relatively large effect on resonant mode coupling phenomena.

An n -dimensional Borg-Levinson theorem

Abstract: We show that the potentialq is uniquely determined by the spectrum, and boundary values of the normal derivatives of the eigenfunctions of the Schrodinger operator −Δ+q with Dirichlet boundary conditions on a bounded domain Ω in ℝ n . This and related results can be viewed as a direct generalization of the theorem in the title, which states that the spectrum and the norming constants determine the potential in the one dimensional case.