Showing papers on "Boundary value problem published in 1990"

Finite element interpolation of nonsmooth functions satisfying boundary conditions

Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

Introduction to modeling of transport phenomena in porous media.

Abstract: A General Theory.- 1 The Porous Medium.- 1.1 Definition and Classification of Porous Media.- 1.1.1 Definition of a porous medium.- 1.1.2 Classification of porous media.- 1.1.3 Some geometrical characteristics of porous media.- 1.1.4 Homogeneity and isotropy of a porous medium.- 1.2 The Continuum Model of a Porous Medium.- 1.2.1 The need for a continuum approach.- 1.2.2 Representative Elementary Volume (REV).- 1.2.3 Selection of REV.- 1.2.4 Representative Elementary Area (REA).- 1.3 Macroscopic Values.- 1.3.1 Volume and mass averages.- 1.3.2 Areal averages.- 1.3.3 Relationship between volume and areal averages.- 1.4 Higher-Order Averaging.- 1.4.1 Smoothing out macroscopic heterogeneity.- 1.4.2 The hydraulic approach.- 1.4.3 Compartmental models.- 1.5 Multicontinuum Models.- 1.5.1 Fractured porous media.- 1.5.2 Multilayer systems.- 2 Macroscopic Description of Transport Phenomena in Porous Media.- 2.1 Elements of Kinematics of Continua.- 2.1.1 Points and particles.- 2.1.2 Coordinates.- 2.1.3 Displacement and strain.- 2.1.4 Processes.- 2.1.5 Material derivative.- 2.1.6 Velocities.- 2.1.7 Flux and discharge.- 2.1.8 Gauss' theorem.- 2.1.9 Reynolds' transport theorem.- 2.1.10 Green's vector theorem.- 2.1.11 Pathlines, transport lines and transport functions.- 2.1.12 Velocity potential and complex potential.- 2.1.13 Movement of a front.- 2.2 Microscopic Balance and Constitutive Equations.- 2.2.1 Derivation of balance equations.- 2.2.2 Particular cases of balance equations.- 2.2.3 Constitutive equations.- 2.2.4 Coupled transport phenomena.- 2.2.5 Phase equilibrium.- 2.3 Averaging Rules.- 2.3.1 Average of a sum.- 2.3.2 Average of a product.- 2.3.3 Average of a time derivative.- 2.3.4 Average of a spatial derivative.- 2.3.5 Average of a spatial derivative of a scalar satisfying ?2G = 0.- 2.3.6 The coefficient T?*.- 2.3.7 Average of a material derivative.- 2.4 Macroscopic Balance Equations.- 2.4.1 General balance equation.- 2.4.2 Mass balance of a phase.- 2.4.3 Volume balance of a phase.- 2.4.4 Mass balance equation for a component of a phase.- 2.4.5 Balance equation for the linear momentum of a phase.- 2.4.6 Heat balance for a phase and for a saturated porous medium.- 2.4.7 Mass balance in a fractured porous medium.- 2.4.8 Megascopic balance equation.- 2.5 Stress and Strain in a Porous Medium.- 2.5.1 Total stress.- 2.5.2 Effective stress.- 2.5.3 Forces acting on the solid matrix.- 2.6 Macroscopic Fluxes.- 2.6.1 Advective flux of a single Newtonian fluid.- 2.6.2 Advective fluxes in a multiphase system.- 2.6.3 Diffusive flux.- 2.6.4 Dispersive flux.- 2.6.5 Transport coefficients.- 2.6.6 Coupled fluxes.- 2.6.7 Macrodispersive flux.- 2.7 Macroscopic Boundary Conditions.- 2.7.1 Macroscopic boundary.- 2.7.2 The general boundary condition.- 2.7.3 Boundary conditions between two porous media in single phase flow.- 2.7.4 Boundary conditions between two porous media in multiphase flow.- 2.7.5 Boundary between two fluids.- 2.7.6 Boundary with a 'well mixed's vector theorem.- 2.1.11 Pathlines, transport lines and transport functions.- 2.1.12 Velocity potential and complex potential.- 2.1.13 Movement of a front.- 2.2 Microscopic Balance and Constitutive Equations.- 2.2.1 Derivation of balance equations.- 2.2.2 Particular cases of balance equations.- 2.2.3 Constitutive equations.- 2.2.4 Coupled transport phenomena.- 2.2.5 Phase equilibrium.- 2.3 Averaging Rules.- 2.3.1 Average of a sum.- 2.3.2 Average of a product.- 2.3.3 Average of a time derivative.- 2.3.4 Average of a spatial derivative.- 2.3.5 Average of a spatial derivative of a scalar satisfying ?2G = 0.- 2.3.6 The coefficient T?*.- 2.3.7 Average of a material derivative.- 2.4 Macroscopic Balance Equations.- 2.4.1 General balance equation.- 2.4.2 Mass balance of a phase.- 2.4.3 Volume balance of a phase.- 2.4.4 Mass balance equation for a component of a phase.- 2.4.5 Balance equation for the linear momentum of a phase.- 2.4.6 Heat balance for a phase and for a saturated porous medium.- 2.4.7 Mass balance in a fractured porous medium.- 2.4.8 Megascopic balance equation.- 2.5 Stress and Strain in a Porous Medium.- 2.5.1 Total stress.- 2.5.2 Effective stress.- 2.5.3 Forces acting on the solid matrix.- 2.6 Macroscopic Fluxes.- 2.6.1 Advective flux of a single Newtonian fluid.- 2.6.2 Advective fluxes in a multiphase system.- 2.6.3 Diffusive flux.- 2.6.4 Dispersive flux.- 2.6.5 Transport coefficients.- 2.6.6 Coupled fluxes.- 2.6.7 Macrodispersive flux.- 2.7 Macroscopic Boundary Conditions.- 2.7.1 Macroscopic boundary.- 2.7.2 The general boundary condition.- 2.7.3 Boundary conditions between two porous media in single phase flow.- 2.7.4 Boundary conditions between two porous media in multiphase flow.- 2.7.5 Boundary between two fluids.- 2.7.6 Boundary with a 'well mixed' domain.- 2.7.7 Boundary with fluid phase change.- 2.7.8 Boundary between a porous medium and an overlying body of flowing fluid.- 3 Mathematical Statement of a Transport Problem.- 3.1 Standard Content of a Problem Statement.- 3.1.1 Conceptual model.- 3.1.2 Mathematical model.- 3.2 Multicontinuum Models.- 3.3 Deletion of Nondominant Effects.- 3.3.1 Methodology.- 3.3.2 Examples.- 3.3.3 Concluding remarks.- B Application.- 4 Mass Transport of a Single Fluid Phase Under Isothermal Conditions.- 4.1 Mass Balance Equations.- 4.1.1 The basic equation.- 4.1.2 Stationary rigid porous medium.- 4.1.3 Deformable porous medium.- 4.2 Boundary Conditions.- 4.2.1 Boundary of prescribed pressure or head.- 4.2.2 Boundary of prescribed mass flux.- 4.2.3 Semipervious boundary.- 4.2.4 Discontinuity in solid matrix properties.- 4.2.5 Sharp interface between two fluids.- 4.2.6 Phreatic surface.- 4.2.7 Seepage face.- 4.3 Complete Mathematical Model.- 4.4 Inertial Effects.- 5 Mass Transport of Multiple Fluid Phases Under Isothermal Conditions.- 5.1 Hydrostatics of a Multiphase System.- 5.1.1 Interfacial tension and capillary pressure.- 5.1.2 Capillary pressure curves.- 5.1.3 Three fluid phases.- 5.1.4 Saturation at medium discontinuity.- 5.2 Advective Fluxes.- 5.2.1 Two fluids.- 5.2.2 Two-phase effective permeability.- 5.2.3 Three-phase effective permeability.- 5.3 Mass Balance Equations.- 5.3.1 Basic equations.- 5.3.2 Nondeformable porous medium.- 5.3.3 Deformable porous medium.- 5.3.4 Buckley-Leverett approximation.- 5.3.5 Flow with interphase mass transfer.- 5.3.6 Immobile fluid phase.- 5.4 Complete Model of Multiphase Flow.- 5.4.1 Boundary and initial conditions.- 5.4.2 Complete model.- 5.4.3 Saturated-unsaturated flow domain.- 6 Transport of a Component in a Fluid Phase Under Isothermal Conditions.- 6.1 Balance Equation for a Component of a Phase.- 6.1.1 The dispersive flux.- 6.1.2 Diffusive flux.- 6.1.3 Sources and sinks at the solid-fluid interface.- 6.1.4 Sources and sinks within the liquid phase.- 6.1.5 Mass balance equation for a single component.- 6.1.6 Variable fluid density and deformable porous medium.- 6.1.7 Balance equations with immobile liquid.- 6.1.8 Fractured porous media.- 6.2 Boundary Conditions.- 6.2.1 Boundary of prescribed concentration.- 6.2.2 Boundary of prescribed flux.- 6.2.3 Boundary between two porous media.- 6.2.4 Boundary with a body of fluid.- 6.2.5 Boundary between two fluids.- 6.2.6 Phreatic surface.- 6.2.7 Seepage face.- 6.3 Complete Mathematical Model.- 6.4 Multicomponent systems.- 6.4.1 Radionuclide and other decay chains.- 6.4.2 Two multicomponent phases.- 6.4.3 Three multicomponent phases.- 7 Heat and Mass Transport.- 7.1 Fluxes.- 7.1.1 Advective flux.- 7.1.2 Dispersive flux.- 7.1.3 Diffusive flux.- 7.2 Balance Equations.- 7.2.1 Single fluid phase.- 7.2.2 Multiple fluid phases.- 7.2.3 Deformable porous medium.- 7.3 Initial and Boundary Conditions.- 7.3.1 Boundary of prescribed temperature.- 7.3.2 Boundary of prescribed flux.- 7.3.3 Boundary between two porous media.- 7.3.4 Boundary with a 'well mixed' domain.- 7.3.5 Boundary with phase change.- 7.4 Complete Mathematical Model.- 7.5 Natural Convection.- 8 Hydraulic Approach to Transport in Aquifers.- 8.1 Essentially Horizontal Flow Approximation.- 8.2 Integration Along Thickness.- 8.3 Conditions on the Top and Bottom Surfaces.- 8.3.1 General flux condition on a boundary.- 8.3.2 Conditions for mass transport of a single fluid phase.- 8.3.3 Conditions for a component of a fluid phase.- 8.3.4 Heat.- 8.3.5 Conditions for stress.- 8.4 Particular Balance Equations for an Aquifer.- 8.4.1 Single fluid phase.- 8.4.2 Component of a phase.- 8.4.3 Fluids separated by an abrupt interface.- 8.5 Aquifer Compaction.- 8.5.1 Integrated flow equation.- 8.5.2 Integrated equilibrium equation.- 8.6 Complete Statement of a Problem of Transport in an Aquifer.- 8.6.1 Mass of a single fluid phase.- 8.6.2 Mass of a component of a fluid phase.- 8.6.3 Saturated-unsaturated mass and component transport.- References.- Problems.

Nonreflecting boundary conditions for Euler equation calculations

Abstract: We present a unified theory for the construction of steady-state and unsteady nonreflecting boundary conditions for the Euler equations. These allow calculatios to be performed on truncated domains without the generation of spurious nonphysical reflections at the far-field boundaries.

Crack Extension Force in a Piezoelectric Material

Abstract: A conservation law that leads to a path-independent integral of fracture mechanics is derived along with the governing equations and boundary conditions for linear piezoelectric materials. A closed-form solution to the antiplane fracture problem is obtained for an unbounded piezoelectric medium

Artificially soft and hard surfaces in electromagnetics

Abstract: A transversely corrugated surface as used in corrugated horn antennas represents a soft boundary. A hard boundary is made by using longitudinal corrugations filled with dielectric material. The concept of soft and hard surfaces is treated in detail, considering different geometries. It is shown that both the hard and soft boundaries have the advantage of a polarization-independent reflection coefficient for geometrical optics ray fields, so that a circularly polarized wave is circularly polarized in the same sense after reflection. The hard boundary can be used to obtain strong radiation fields along a surface for any polarization, whereas the soft boundary makes the fields radiated along the surface zero. >

Boundary conditions for open quantum systems driven far from equilibrium

Abstract: This is a study of simple kinetic models of open systems, in the sense of systems that can exchange conserved particles with their environment. The system is assumed to be one dimensional and situated between two particle reservoirs. Such a system is readily driven far from equilibrium if the chemical potentials of the reservoirs differ appreciably. The openness of the system modifies the spatial boundary conditions on the single-particle Liouville---von Neumann equation, leading to a non-Hermitian Liouville operator. If the open-system boundary conditions are time reversible, exponentially growing (unphysical) solutions are introduced into the time dependence of the density matrix. This problem is avoided by applying time-irreversible boundary conditions to the Wigner distribution function. These boundary conditions model the external environment as ideal particle reservoirs with properties analogous to those of a blackbody. This time-irreversible model may be numerically evaluated in a discrete approximation and has been applied to the study of a resonant-tunneling semiconductor diode. The physical and mathematical properties of the irreversible kinetic model, in both its discrete and its continuum formulations, are examined in detail. The model demonstrates the distinction in kinetic theory between commutator superoperators, which may become non-Hermitian to describe irreversible behavior, and anticommutator superoperators, which remain Hermitian and are used to evaluate physical observables.

Shear flow near solids: Epitaxial order and flow boundary conditions.

Abstract: We report on molecular-dynamics simulations of Lennard-Jones liquids sheared between two solid walls. The velocity fields, flow boundary conditions, and fluid structure were studied for a variety of wall and fluid properties. A broad spectrum of boundary conditions was observed including slip, no-slip, and locking. We show that the degree of slip is directly related to the amount of structure induced in the fluid by the periodic potential from the solid walls. For weak wall-fluid interactions there is little ordering and slip was observed. At large interactions, substantial epitaxial ordering was induced and the first one or two fluid layers became locked to the wall. This epitaxial ordering was enhanced when the wall and fluid densities were equal. For unequal densities, high-order commensurate structures formed in the first fluid layer creating slip within the fluid.

Boundary Value Problems for Operator Differential Equations

Abstract: The aim of this lecture is to give a brief exposition of the main points of the test and generalized vectors theory for a closed linear operator on a Banach space and to apply this theory for investigating the solutions of differential equations with operator coefficients smooth inside an interval. Some spectral properties of boundary value problems for such equations are studied as well.

Parallel multilevel preconditioners

Abstract: In this paper, we shall report on some techniques for the development of preconditioners for the discrete systems which arise in the approximation of solutions to elliptic boundary value problems. Here we shall only state the resulting theorems. It has been demonstrated that preconditioned iteration techniques often lead to the most computationally effective algorithms for the solution of the large algebraic systems corresponding to boundary value problems in two and three dimensional Euclidean space. The use of preconditioned iteration will become even more important on computers with parallel architecture. This paper discusses an approach for developing completely parallel multilevel preconditioners. In order to illustrate the resulting algorithms, we shall describe the simplest application of the technique to a model elliptic problem.

An analysis of tensile decohesion along an interface

Abstract: A COHESIVE zone type interface model, taking full account of finite geometry changes, is used to study the decohesion of a viscoplastic block from a rigid substrate. Dimensional considerations introduce a characteristic length into the formulation. The specific boundary value problem analysed is one of plane strain tension with a superposed hydrostatic stress. For a perfect interface, if the maximum traction that the viscoplastic block can support is greater than the interfacial strength, decohesion takes place in a primarily tensile mode. If this maximum traction is lower than the interfacial strength, a shear dominated decohesion initiates at the block edge. Imperfections in the form of a non-bonded portion of the interface are considered. The effects of imposed stress triaxiality, size scale, loading rate and interfacial properties on the course of defect dominated decohesion are illustrated. The characterization of decohesion initiation and propagation in terms of rice's (J. appl. Mech. 35, 379, 1968) J-integral is investigated for a variety of interface descriptions and values of the superposed hydrostatic stress.

An analysis of decohesion along an imperfect interface

Abstract: A cohesive zone type interface model, taking full account of finite geometry changes, is used to study the decohesion of a viscoplastic block from a rigid substrate. The specific boundary value problem analyzed is a plane strain one with the imposed loading corresponding to overall uniaxial straining. The imperfection takes the form of a non-bonded portion of the interface. Dimensional considerations introduce a characteristic length into the formulation and the decohesion mode shifts from more or less uniform separation along the bond line to crack-like propagation as the ratio of block size to characteristic length increases. Field distributions prior to and accompanying propagation are displayed.

Boundary Value Problems in Mechanics of Nonhomogeneous Fluids

Abstract: Models of the Dynamics of Heterogeneous Media and the Body of Mathematics Correctness ``In the Whole'' of the Boundary Problems for Equations of One-Dimensional Non-Stationary Motion of a Viscous Gas Initial-Boundary Value Problems for the Navier-Stokes Equations of Nonhomogeneous Viscous Incompressible Fluid Correctness of the Problem of Flow through an Ideal Incompressible Liquid Filtration of Immiscible Liquids References

Height and gradient from shading

Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented. The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell. The new scheme can find an exact solution of a given shape-from-shading problem even though a regularizing term is included. This is a reflection of the fact that shape-from-shading problems are {\it not} ill-posed when boundary conditions are available or when the image contains singular points.

Origin of stick-slip motion in boundary lubrication.

Abstract: Molecular dynamics simulations of atomically thin, fluid films confined between two solid plates are described. For a broad range of parameters, a generic stick-slip motion is observed, consistent with the results of recent boundary lubrication experiments. Static plates induce crystalline order in the film. Stick-slip motion involves periodic shear-melting transitions and recrystllization of the film. Uniform motion occurs at high velocities where the film no longer has time to order. These results indicate that the origin of stick-slip motion is thermodynamic instability of the sliding state, rather than a dynamic instability as usually assumed.

Theory of semiconductor superlattice electronic structure

Abstract: The authors review the theory of semiconductor superlattice electronic structure. First a survey of theoretical methods is presented. These methods can be divided into two general classes: the supercell approach in which the superlattice is viewed as a material with a large unit cell, and the boundary-condition approach in which bulk wave functions in the constituent semiconductors are matched at the superlattice interfaces. Supercell approaches are essentially the same as conventional band-structure methods. They can only be applied to thin-layer superlattices because of numerical cost. The authors discuss problems of interface matching that occur in various boundary-condition methods and relate these methods to each other. A particular boundary-condition method is used to discuss the electronic structure of various III-V semiconductor superlattices. Emphasis is placed on discussing the qualitatively different behavior that can arise because of different energy-band lineups, strain conditions, and growth orientations. The authors compare the results of three commonly used boundary-condition methods and find generally good agreement.

Tidal dynamics of the water table in beaches

Abstract: Tidal motions of the water table height inside a sloping beach are investigated via field measurements and theoretical considerations. Only the movements forced by the tide are considered, so a beach with negligible wave activity was chosen for the field measurements. The data show that even in the absence of precipitation the time averaged inland water table stands considerably above the mean sea level. Also the water table at a fixed point inside the beach is far from sinusoidal even though its variation is forced by an essentially sinusoidal tide. This latter effect is due to the boundary condition along the sloping beach face which acts as a highly nonlinear filter. The observed behavior of the water table is explained in terms of perturbation extensions to the classical “deep aquifer solution.” One extension deals with the nonlinearity in the interior, the other with the boundary condition at the sloping beach face.

The quantum transmitting boundary method

Abstract: A numerical algorithm for the solution of the two‐dimensional effective mass Schrodinger equation for current‐carrying states is developed. Boundary conditions appropriate for such states are developed and a solution algorithm constructed that is based on the finite element method. The utility of the technique is illustrated by solving problems relevant to submicron semiconductor quantum device structures.

A test problem for outflow boundary conditions—flow over a backward-facing step

Abstract: A numerical solution for steady incompressible flow over a two-dimensional backward-facing step is developed using a Galerkin-based finite element method. The Reynolds number for the simulations is 800. Computations are performed on an extended channel length to minimize the effect of the outflow boundary on the upstream recirculation zones. A thorough mesh refinement study is performed to validate the results. Extensive profile data at several channel locations are provided to allow future testing and evaluation of outflow boundary conditions.

On some control problems in fluid mechanics

Abstract: The issue of minimizing turbulence in an evolutionary Navier-Stokes flow is addressed from the point of view of optimal control. We derive theoretical results for various physical situations: distributed control, Benard-type problems with boundary control, and flow in a channel. For each case that we consider, our results include the formulation of the problem as an optimal control problem and proof of the existence of an optimal control (which is not expected to be unique). Finally, we describe a numerical algorithm based on the gradient method for the corresponding cost function. For readers who are not interested in the mathematical details and the mathematical justifications, a nontechnical description of our results is included in Section 5.

Rapid, stable fluid dynamics for computer graphics

Abstract: We present a new method for animating water based on a simple, rapid and stable solution of a set of partial differential equations resulting from an approximation to the shallow water equations. The approximation gives rise to a version of the wave equation on a height-field where the wave velocity is proportional to the square root of the depth of the water. The resulting wave equation is then solved with an alternating-direction implicit method on a uniform finite-difference grid. The computational work required for an iteration consists mainly of solving a simple tridiagonal linear system for each row and column of the height field. A single iteration per frame suffices in most cases for convincing animation.Like previous computer-graphics models of wave motion, the new method can generate the effects of wave refraction with depth. Unlike previous models, it also handles wave reflections, net transport of water and boundary conditions with changing topology. As a consequence, the model is suitable for animating phenomena such as flowing rivers, raindrops hitting surfaces and waves in a fish tank as well as the classic phenomenon of waves lapping on a beach. The height-field representation prevents it from easily simulating phenomena such as breaking waves, except perhaps in combination with particle-based fluid models. The water is rendered using a form of caustic shading which simulates the refraction of illuminating rays at the water surface. A wetness map is also used to compute the wetting and drying of sand as the water passes over it.

Twisted boundary conditions and effective mass in Heisenberg-Ising and Hubbard rings.

Abstract: We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e., the Mott-Hubbard transition or Density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.

An algorithm for evolutionary surfaces

Abstract: An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem forH-surfaces.