Showing papers on "Boundary value problem published in 1984"

Free and moving boundary problems

Abstract: 1. Moving boundary problems: formulation 2. Free boundary problems: formulation 3. Analytical solutions 4. Front-tracking methods 5. Front-fixing methods 6. Fixed-domain methods 7. Analytical solution of seepage problems 8. Numerical solution of free boundary problems References Author index Subject index

Boundary Element Techniques: Theory and Applications in Engineering

Abstract: 1 Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definitions.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.4.1. The Collocation Method.- 1.4.2. Method of Collocation by Subregions.- 1.5. Method of Galerkin.- 1.6. Weak Formulations.- 1.7. Inverse Problem and Boundary Solutions.- 1.8. Classification of Approximate Methods.- References.- 2 Potential Problems.- 2.1. Introduction.- 2.2. Elements of Potential Theory.- 2.3. Indirect Formulation.- 2.4. Direct Formulation.- 2.5. Boundary Element Method.- 2.6. Two-Dimensional Problems.- 2.6.1. Source Formulation.- 2.7. Poisson Equation.- 2.8. Subregions.- 2.9. Orthotropy and Anisotropy.- 2.10. Infinite Regions.- 2.11. Special Fundamental Solutions.- 2.12. Three-Dimensional Problems.- 2.13. Axisymmetric Problems.- 2.14. Axisymmetric Problems with Arbitrary Boundary Conditions.- 2.15. Nonlinear Materials and Boundary Conditions.- 2.15.1. Nonlinear Boundary Conditions.- References.- 3 Interpolation Functions.- 3.1. Introduction.- 3.2. Linear Elements for Two-Dimensional Problems.- 3.3. Quadratic and Higher-Order Elements.- 3.4. Boundary Elements for Three-Dimensional Problems.- 3.4.1. Quadrilateral Elements.- 3.4.2. Higher-Order Quadrilateral Elements.- 3.4.3. Lagrangian Quadrilateral Elements.- 3.4.4. Triangular Elements.- 3.4.5. Higher-Order Triangular Elements.- 3.5. Three-Dimensional Cell Elements.- 3.5.1. Tetrahedron.- 3.5.2. Cube.- 3.6. Discontinuous Boundary Elements.- 3.7. Order of Interpolation Functions.- References.- 4 Diffusion Problems.- 4.1. Introduction.- 4.2. Laplace Transforms.- 4.3. Coupled Boundary Element - Finite Difference Methods.- 4.4. Time-Dependent Fundamental Solutions.- 4.5. Two-Dimensional Problems.- 4.5.1. Constant Time Interpolation.- 4.5.2. Linear Time Interpolation.- 4.5.3. Quadratic Time Interpolation.- 4.5.4. Space Integration.- 4.6. Time-Marching Schemes.- 4.7. Three-Dimensional Problems.- 4.8. Axisymmetric Problems.- 4.9. Nonlinear Diffusion.- References.- 5 Elastostatics.- 5.1. Introduction to the Theory of Elasticity.- 5.1.1. Initial Stresses or Initial Strains.- 5.2. Fundamental Integral Statement.- 5.2.1. Somigliana Identity.- 5.3. Fundamental Solutions.- 5.4. Stresses at Internal Points.- 5.5. Boundary Integral Equation.- 5.6. Infinite and Semi-Infinite Regions.- 5.7. Numerical Implementation.- 5.8. Boundary Elements.- 5.9. System of Equations.- 5.10. Stresses and Displacements Inside the Body.- 5.11. Stresses on the Boundary.- 5.12. Surface Traction Discontinuities.- 5.13. Two-Dimensional Elasticity.- 5.14. Body Forces.- 5.14.1. Gravitational Loads.- 5.14.2. Centrifugal Load.- 5.14.3. Thermal Loading.- 5.15. Axisymmetric Problems.- 5.15.1. Extension to Nonaxisymmetric Boundary Values.- 5.16. Anisotropy.- References.- 6 Boundary Integral Formulation for Inelastic Problems.- 6.1. Introduction.- 6.2. Inelastic Behavior of Materials.- 6.3. Governing Equations.- 6.4. Boundary Integral Formulation.- 6.5. Internal Stresses.- 6.6. Alternative Boundary Element Formulations.- 6.6.1. Initial Strain.- 6.6.2. Initial Stress.- 6.6.3. Fictitious Tractions and Body Forces.- 6.7. Half-Plane Formulations.- 6.8. Spatial Discretization.- 6.9. Internal Cells.- 6.10. Axisymmetric Case.- References.- 7 Elastoplasticity.- 7.1. Introduction.- 7.2. Some Simple Elastoplastic Relations.- 7.3. Initial Strain: Numerical Solution Technique.- 7.3.1. Examples - Initial Strain Formulation.- 7.4. General Elastoplastic Stress-Strain Relations.- 7.5. Initial Stress: Outline of Solution Techniques.- 7.5.1. Examples: Kelvin Implementation.- 7.5.2. Examples: Half-Plane Implementation.- 7.6. Comparison with Finite Elements.- References.- 8 Other Nonlinear Material Problems.- 8.1. Introduction.- 8.2. Rate-Dependent Constitutive Equations.- 8.3. Solution Technique: Viscoplasticity.- 8.4. Examples: Time-Dependent Problems.- 8.5. No-Tension Materials.- References.- 9 Plate Bending.- 9.1. Introduction.- 9.2. Governing Equations.- 9.3. Integral Equations.- 9.3.1. Other Fundamental Solutions.- 9.4. Applications.- References.- 10 Wave Propagation Problems.- 10.1. Introduction.- 10.2. Three-Dimensional Water Wave Propagation Problems.- 10.3. Vertical Axisymmetric Bodies.- 10.4. Horizontal Cylinders of Arbitrary Section.- 10.5. Vertical Cylinders of Arbitrary Section.- 10.6. Transient Scalar Wave Equation.- 10.7. Three-Dimensional Problems: The Retarded Potential.- 10.8. Two-Dimensional Problems.- References.- 11 Vibrations.- 11.1. Introduction.- 11.2. Governing Equations.- 11.3. Time-Dependent Integral Formulation.- 11.4. Laplace Transform Formulation.- 11.5. Steady-State Elastodynamics.- 11.6. Free Vibrations.- References.- 12 Further Applications in Fluid Mechanics.- 12.1. Introduction.- 12.2. Transient Groundwater Flow.- 12.3. Moving Interface Problems.- 12.4. Axisymmetric Bodies in Cross Flow.- 12.5. Slow Viscous Flow (Stokes Flow).- 12.6. General Viscous Flow.- 12.6.1. Steady Problems.- 12.6.2. Transient Problems.- References.- 13 Coupling of Boundary Elements with Other Methods.- 13.1. Introduction.- 13.2. Coupling of Finite Element and Boundary Element Solutions.- 13.2.1. The Energy Approach.- 13.3. Alternative Approach.- 13.4. Internal Fluid Problems.- 13.4.1. Free-Surface Boundary Condition.- 13.4.2. Extension to Compressible Fluid.- 13.5. Approximate Boundary Elements.- 13.6. Approximate Finite Elements.- References.- 14 Computer Program for Two-Dimensional Elastostatics.- 14.1. Introduction.- 14.2. Main Program and Data Structure.- 14.3. Subroutine INPUT.- 14.4. Subroutine MATRX.- 14.5. Subroutine FUNC.- 14.6. Subroutine SLNPD.- 14.7. Subroutine OUTPT.- 14.8. Subroutine FENC.- 14.9. Examples.- 14.9.1. Square Plate.- 14.9.2. Cylindrical Cavity Problem.- References.- Appendix A Numerical Integration Formulas.- A.1. Introduction.- A.2. Standard Gaussian Quadrature.- A.2.1. One-Dimensional Quadrature.- A.2.2. Two- and Three-Dimensional Quadrature for Rectangles and Rectangular Hexahedra.- A.2.3. Triangular Domain.- A.3. Computation of Singular Integrals.- A.3.1. One-Dimensional Logarithmic Gaussian Quadrature Formulas.- A.3.3. Numerical Evaluation of Cauchy Principal Values.- References.- Appendix B Semi-Infinite Fundamental Solutions.- B.1. Half-Space.- B.2. Half-Plane.- References.- Appendix C Some Particular Expressions for Two-Dimensional Inelastic Problems.

Physical and Computational Aspects of Convective Heat Transfer

Abstract: 1 Introduction- 11 Momentum Transfer- 12 Heat and Mass Transfer- 13 Relations between Heat and Momentum Transfer- l4 Coupled and Uncoupled Flows- 15 Units and Dimensions- l6 Outline of the Book- Problems- References- 2 Conservation Equations for Mass, Momentum, and Energy- 21 Continuity Equation- 22 Momentum Equations- 23 Internal Energy and Enthalpy Equations- 24 Conservation Equations for Turbulent Flow- 25 Equations of Motion: Summary- Problems- References- 3 Boundary-Layer Equations- 3l Uncoupled Flows- 32 Estimates of Density Fluctuations in Coupled Turbulent Flows- 33 Equations for Coupled Turbulent Flows- 34 Integral Equations- 35 Boundary Conditions- 36 Thin-Shear-Layer Equations: Summary- Problems- References- 4 Uncoupled Laminar Boundary Layers- 41 Similarity Analysis- 42 Two-Dimensional Similar Flows- 43 Two-Dimensional Nonsimilar Flows- 44 Axisymmetric Flows- 45 Wall Jets and Film Cooling- Problems- References- 5 Uncoupled Laminar Duct Flows- 51 Fully Developed Duct Flow- 52 Thermal Entry Length for a Fully Developed Velocity Field- 53 Hydrodynamic and Thermal Entry Lengths- Problems- References- 6 Uncoupled Turbulent Boundary Layers- 61 Composite Nature of a Turbulent Boundary Layer- 62 The Inner Layer- 63 The Outer Layer- 64 The Whole Layer- 65 Two-Dimensional Boundary Layers with Zero Pressure Gradient- 66 Two-Dimensional Flows with Pressure Gradient- 67 Wall Jets and Film Cooling- Problems- References- 7 Uncoupled Turbulent Duct Flows- 71 Fully Developed Duct Flow- 72 Thermal Entry Length for a Fully Developed Velocity Field- 73 Hydrodynamic and Thermal Entry Lengths- Problems- References- 8 Free Shear Flows- 81 Two-Dimensional Laminar Jet- 82 Laminar Mixing Layer between Two Uniform Streams at Different Temperatures- 83 Two-Dimensional Turbulent Jet- 84 Turbulent Mixing Layer between Two Uniform Streams at Different Temperatures- 85 Coupled Flows- Problems- References- 9 Buoyant Flows- 91 Natural-Convection Boundary Layers- 92 Combined Natural- and Forced-Convection Boundary Layers- 93 Wall Jets and Film Heating or Cooling- 94 Natural and Forced Convection in Duct Flows- 95 Natural Convection in Free Shear Flows- Problems- References- 10 Coupled Laminar Boundary Layers- 101 Similar Flows- 102 Nonsimilar Flows- 103 Shock-Wave/Shear-Layer Interaction- 104 A Prescription for Computing Interactive Flows with Shocks- Problems- References- 11 Coupled Turbulent Boundary Layers- 111 Inner-Layer Similarity Analysis for Velocity and Temperature Profiles- 112 Transformations for Coupled Turbulent Flows- 113 Two-Dimensional Boundary Layers with Zero Pressure Gradient- 114 Two-Dimensional Flows with Pressure Gradient- 115 Shock-Wave/Boundary-Layer Interaction- References- 12 Coupled Duct Flows- 121 Laminar Flow in a Tube with Uniform Heat Flux- 122 Laminar, Transitional and Turbulent Flow in a Cooled Tube- References- 13 Finite-Difference Solution of Boundary-Layer Equations- 131 Review of Numerical Methods for Boundary-Layer Equations- 132 Solution of the Energy Equation for Internal Flows with Fully Developed Velocity Profile- 133 Fortran Program for Internal Laminar and Turbulent Flows with Fully Developed Velocity Profile- 134 Solution of Mass, Momentum, and Energy Equations for Boundary-Layer Flows- 135 Fortran Program for Coupled Boundary-Layer Flows- References- 14 Applications of a Computer Program to Heat-Transfer Problems- 141 Forced and Free Convection between Two Vertical Parallel Plates- 142 Wall Jet and Film Heating- 143 Turbulent Free Jet- 144 Mixing Layer between Two Uniform Streams at Different Temperatures- References- Appendix A Conversion Factors- Appendix B Physical Properties of Gases, Liquids, Liquid Metals, and Metals- Appendix C Gamma, Beta and Incomplete Beta Functions- Appendix D Fortran Program for Head's Method

Quick iterative scheme for the calculation of transfer matrices: application to Mo (100)

Abstract: The transfer matrix of a solid described by the stacking of principal layers is obtained by an iterative procedure which takes into account 2 layers after n iterations, in contrast to usual schemes where each iteration includes just one more layer. The Green function and density of states at the surface of the corresponding semi-infinite crystal are then given by well known formulae in terms of the transfer matrix. This method, especially convenient near singularities, is applied to the calculation of the spectral as well as the total densities of states for the (100) face of molybdenum. The Slater-Koster algorithm for the calculation of tight-binding parameters is used with a basis of nine orbitals per atom (4d, 5s, 5p). Surface states and resonances are first identified and then analysed into orbital components to find their dominant symmetry. Their evolution along the main symmetry lines of the two-dimensional Brillouin zone is given explicitly. The surface-state peak just below the Fermi level (Swanson hump) is not obtained. This is traced to the difficulty in placing an appropriate boundary condition at the surface with the tight-binding parameterisation scheme.

Numerical Simulation of Turbulent Flows

Abstract: Computational models of turbulence in incompressible Newtonian fluids governed by the Navier-Stokes equations are reviewed. The governing equations are presented, and both direct and large-eddy-simulations are examined. Resolution requirements and numerical techniques of spatial representation, definition of initial and boundary conditions, and time advancement are considered. Results of simulations of homogeneous turbulence in uniform shear, the evolution of a turbulent mixing layer, and turbulent channel flow are presented graphically and discussed.

Stochastic differential equations with reflecting boundary conditions

Abstract: On resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de Skorokhod

A global compactness result for elliptic boundary value problems involving limiting nonlinearities

Abstract: On demontre l'existence de solutions nontriviales de −Δu−λu=u|u| 2 * −2 dans Ω∈R n , u/∂Ω=0 arbitrairement proches des valeurs propres λ k de −Δ:H 0 1,2 (Ω)→H −1 (Ω)

A transmitting boundary for transient wave analyses

Abstract: An artificial boundary condition at the edge of finite computational grids is devised. Itcan simulate the transmitting process of clastic surface waves and body waves incident atarbitrary angles under any accuracy required. It may be used for two- or three-dimensionaltransient wave analyses in laterally heterogeneous media and easily incorporated into exist-ing finite element or finite difference computational codes.

The three‐dimensional flow due to a stretching flat surface

Abstract: An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

Boundedly nonhomogeneous elliptic and parabolic equations in a domain

Abstract: In this paper the Dirichlet problem is studied for equations of the form and also the first boundary value problem for equations of the form , where and are positive homogeneous functions of the first degree in , convex upwards in , that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on and when the second derivatives of with respect to are bounded above, the solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in on the boundary are constructed, and convexity and restrictions on the second derivatives of are not used in the derivation.Bibliography: 13 titles.

Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates

Abstract: While component dissipation patterns and system operating modes vary widely, many electronic packaging configurations can be modeled by symmetrically or asymmetrically isothermal or isoflux plates. The idealized configurations are amenable to analytic optimization based on maximizing total heat transfer per unit volume or unit primary area. To achieve this anlaytic optimization, however, it is necessary to develop composite relations for the variation of the heat transfer coefficient along the plate surfaces. The mathematical development and verification of such composite relations as well as the formulation and solution of the optimizing equations for the various boundary conditions of interest constitute the core of this presentation.

Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique

Abstract: We present here a brief description of a numerical technique suitable for solving axisymmetric (or two-dimensional) free-boundary problems of fluid mechanics. The technique is based on a finite-difference solution of the equations of motion on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The overall solution is achieved via a global iterative process, with the condition of balance between total normal stress and the capillary pressure at the free boundary being used to drive the boundary shape to its ultimate equilibrium position.

Boundary Value Problems and Sharp Inequalities for Martingale Transforms

Abstract: Let $p^\ast$ be the maximum of $p$ and $q$ where $1 0$. This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if $\varepsilon$ is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case $p = 2$. The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of $L^p(0, 1)$ is $p^\ast - 1$. The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.

On some properties of force-free magnetic fields in infinite regions of space

Abstract: Techniques for solving boundary value problems (BVP) for a force free magnetic field (FFF) in infinite space are presented. A priori inequalities are defined which must be satisfied by the force-free equations. It is shown that upper bounds may be calculated for the magnetic energy of the region provided the value of the magnetic normal component at the boundary of the region can be shown to decay sufficiently fast at infinity. The results are employed to prove a nonexistence theorem for the BVP for the FFF in the spatial region. The implications of the theory for modeling the origins of solar flares are discussed.

Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment

Abstract: We present a numerical method that allows us to solve the Navicr-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties ; the bestmeasured speeds have fractional uncertainties of less than 0.2 o/o.

Coating flow theory by finite element and asymptotic analysis of the navier-stokes system

Abstract: Coating flows are laminar free surface flows, preferably steady and two-dimensional, by which a liquid film is deposited on a substrate. Their theory rests on mass and momentum accounting for which Galerkin's weighted residual method, finite element basis functions, isoparametric mappings, and a new free surface parametrization prove particularly well-suited, especially in coping with the highly deformed free boundaries, irregular flow domains, and the singular nature of static and dynamic contact lines where fluid interfaces intersect solid surfaces. Typically, short forming zones of rapidly rearranging two-dimensional flow merge with simpler asymptotic regimes of developing or developed flow upstream and downstream. The two-dimensional computational domain can be shrunk in size by imposing boundary conditions from asymptotic analysis of those regimes or by matching to one-dimensional finite element solutions of asymptotic equations. The theory is laid out with special attention to conditions at free surfaces, contact lines, and open inflow and outflow boundaries. Efficient computation of predictions is described with emphasis on a grand Newton iteration that converges rapidly and brings other benefits. Sample results for curtain coating and roll coating flows of Newtonian liquids illustrate the power and effectiveness of the theory.

Application of a fractional-step method to incompressible Navier-Stokes equation

Abstract: A numerical method for computing three dimensional, time dependent incompressible flows is presented. The method is based on a fractional step, or time-splitting, scheme in conjunction with the approximate-factorization technique. The use of velocity boundary conditions for the intermediate velocity field leads to inconsistent numerical solutions. Appropriate boundary conditions for the intermediate velocity field are derived and tested. Numerical solutions for flow inside a driven cavity and over a backward-facing step are presented and compared with experimenal data and other numerical results.

Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions

Abstract: Approximate expressions are given for the characteristic impedance and propagation wavenumber for linear acoustic transmission through a gas enclosed in a rigid cylindrical duct. These expressions are most complicated in the transition zone where the thermoviscous boundary layers are on the order of the tube radius. The approximations are accurate to within 1% for all frequencies and tube diameters except within the transition zone where the approximations are accurate to within 10%. A simple modification of the transmission line parameters is presented for the case where the tube walls are nonisothermal.

Iterative computational techniques in scattering based upon the integrated square error criterion

Abstract: An iterative technique is developed to rigorously compute the electromagnetic time- and frequency-domain scattering problems. The method is based upon a wave-function expansion technique (this also includes the integral-representation techniques), in which the electromagnetic field equations and causality conditions are satisfied analytically, while the boundary conditions or the constitutive relations have to be satisfied in a computational manner. The latter is accomplished by an iterative minimization of the integrated square error. For the solution of an integral equation, it is shown how to obtain optimum convergence. Some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method.

Feedback and adaptive finite element solution of one-dimensional boundary value problems

Abstract: This paper examines the concepts of feedback and adaptivity for the Finite Element Method. The model problem concernsC 0 elements of arbitrary, fixed degree for a one-dimensional two-point boundary value problem. Three different feedback methods are introduced and a detailed analysis of their adaptivity is given.

Diffusion of electromagnetic fields into a two-dimensional earth; a finite-difference approach

Abstract: We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space...