Showing papers on "Boundary value problem published in 1993"

Computing Discrete Minimal Surfaces and Their Conjugates

Abstract: We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations. Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

Abstract: It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations

Abstract: LSODE, the Livermore Solver for Ordinary Differential Equations, is a package of FORTRAN subroutines designed for the numerical solution of the initial value problem for a system of ordinary differential equations. It is particularly well suited for 'stiff' differential systems, for which the backward differentiation formula method of orders 1 to 5 is provided. The code includes the Adams-Moulton method of orders 1 to 12, so it can be used for nonstiff problems as well. In addition, the user can easily switch methods to increase computational efficiency for problems that change character. For both methods a variety of corrector iteration techniques is included in the code. Also, to minimize computational work, both the step size and method order are varied dynamically. This report presents complete descriptions of the code and integration methods, including their implementation. It also provides a detailed guide to the use of the code, as well as an illustrative example problem.

In-plane response of laminates with spatially varying fiber orientations - Variable stiffness concept

Abstract: A solution to the plane elasticity problem for a symmetrically laminated composite panel with spatially varying fiber orientations has been obtained. The fiber angles vary along the length of the composite laminate, resulting in stiffness properties that change as a function of location. This work presents an analysis of the stiffness variation and its effects on the elastic response of the panel. The in-plane response of a variable stiff ness panel is governed by a system of coupled elliptic partial differential equations/Solving these equations yields the displacement fields, from which the strains, stresses, and stress resultants can be subsequently calculated. A numerical solution has been obtained using an iterative collocation technique. Corresponding closed-form solutions are presented for three sets of boundary conditions, two of which have exact solutions, and therefore serve to validate the numerical model. The effects of the variable fiber orientation on the displacement fields, stress resultants, and global stiffness are analyzed.

A Reynolds stress model for near-wall turbulence

Abstract: A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied. The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data. The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery. A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.

A unified approach to a posteriori error estimation using element residual methods

Abstract: This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283---301 (1985)].

A simple approach to solve boundary-value problems in gradient elasticity

Abstract: We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.

Introduction to Regularity Theory for Nonlinear Elliptic Systems

Abstract: These lectures are an introduction to the regularity theory of minimizers of variational integrals, in the calculus of variations and of solutions of nonlinear elliptic systems. Topics covered include the Hilbert-space approach to boundary value problems and energy estimates.

Integral Transforms in Computational Heat and Fluid Flow

Abstract: INTRODUCTION THE CLASSICAL INTEGRAL TRANSFORM TECHNIQUE General Solution for a Base Problem - Parabolic System Steady Multidimensional Problem - Elliptic System Eigenvalue Problem SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS Linear Systems Numerical Methods and Stiff Systems Infinite Systems and Approximate Solutions PROBLEMS WITH VARIABLE EQUATION COEFFICIENTS Transient Problem (Parabolic System) Steady-State Problem (Elliptic System) Applications PROBLEMS WITH VARIABLE BOUNDARY CONDITION COEFFICIENTS Time-dependent Boundary Condition Coefficients An Alternative Approach Space-dependent Boundary Condition Coefficients Applications PROBLEMS WITH VARIABLE BOUNDARIES Moving Boundary Problems Diffusion within Irregular Domains - Elliptic Problem Diffusion within Irregular Domains - Parabolic Problem Applications PROBLEMS THAT INVOLVE DIFFICULT AUXILIARY PROBLEMS Sturm-Liouville Problem with a Laplace Transform Variable Sturm-Liouville Problem with Complex Variables Non-separable Sturm-Liouville Systems Non-classical Eigenvalue Problems AN INTRODUCTION TO THE SOLUTION OF NONLINEAR DIFFUSION-CONVECTION PROBLEMS Nonlinear Source Terms Nonlinear Convective Terms Nonlinear Diffusion Coefficients Computational Procedure Applications REFERENCES APPENDICES A - Notes B - Alternative Solution of Nonhomogeneous Problems C - Integral Transform Method for Eigenvalue Problems

Crustal thickening versus lateral expulsion in the Indian‐Asian continental collision

Abstract: Since the beginning of the continental collision between India and Asia there has been about 2500 km of convergence, and the northward movement of India has been accommodated by major internal deformation of the Asian lithosphere. The crustal thickening in and around the Tibetan Plateau is clearly a direct consequence of this collision, but there is considerable debate as to whether a large fraction of the indentation has been accommodated by eastward motion of the lithospheric blocks of southeastern Asia and southern China. Numerical experiments described here test this hypothesis for a range of indentation geometries and rheological models of the lithosphere. We employ a thin viscous sheet model of the lithosphere with a depth-averaged nonlinear viscous rheology described by a stress-strain rate exponent n and including gravitational buoyancy forces scaled by the dimensionless Argand number Ar. The eastern boundary for the collision region is described as a lithostatic boundary; the precollision normal stress is determined by static balance, and that constant stress is applied throughout the collision. The experiments show that during collision the eastern boundary is smoothly displaced to the east at a rate about 1/4 of the indentation rate, with only minor variation due to geometry or rheology. Crustal thickening rates in the region of the plateau are reduced by between 10% and 25% of the corresponding rates determined for similar experiments with a rigid eastern boundary. To generalize from the results of these experiments, the total north-south shortening strain produced by the collision is partitioned between crustal thickening and eastward displacement in the ratio of at least 3:1 and more probably 4:1.

Boundary conditions for direct computation of aerodynamic sound generation

Abstract: A numerical scheme suitable for the computation of both the near field acoustic sources and the far field sound produced by turbulent free shear flows utilizing the Navier-Stokes equations is presented. To produce stable numerical schemes in the presence of shear, damping terms must be added to the boundary conditions. The numerical technique and boundary conditions are found to give stable results for computations of spatially evolving mixing layers.

Maximally smooth image recovery in transform coding

Abstract: The authors consider the reconstruction of images from partial coefficients in block transform coders and its application to packet loss recovery in image transmission over asynchronous transfer mode (ATM) networks. The proposed algorithm uses the smoothness property of common image signals and produces a maximally smooth image among all those with the same coefficients and boundary conditions. It recovers each damaged block by minimizing the intersample variation within the block and across the block boundary. The optimal solution is achievable through two linear transformations, where the transform matrices depend on the loss pattern and can be calculated in advance. The reconstruction of contiguously damaged blocks is accomplished iteratively using the previous solution as the boundary conditions in each new step. This technique is applicable to any unitary block-transform and is effective for recovering the DC and low-frequency coefficients. When applied to still image coders using the discrete cosine transform (DCT), high quality images are reconstructed in the absence of many DC and low-frequency coefficients over spatially adjacent blocks. When the damaged blocks are isolated by block interleaving, satisfactory results have been obtained even when all the coefficients are missing. >

KIVA3. A KIVA Program With Block-Structured Mesh for Complex Geometries

Abstract: KIVA3 is a computer program for the numerical calculation of transient, two and three-dimensional, chemically reactive flows with sprays. It is an extension of the earlier KIVA2, and uses the same numerical solution procedure and solves the same set of equations. KIVA3 differs in that it uses a block-structured mesh with connectivity defined through indirect addressing. The departure from a single rectangular structure in logical space allows complex geometries to be modeled with significantly greater efficiency because large regions of deactivated cells are no longer necessary. Cell-face boundary conditions permit greater flexibility and simplification in the application of boundary conditions.

Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations

Abstract: Three-dimensional, incompressible Euler calculations of the interaction of perturbed anti-parallel vortex tubes using a variety of smooth initial profiles in bounded domains with bounded initial vorticity is discussed. It will be shown that trends towards either exponential, non-singular growth of the peak vorticity or power law, singular behavior can be strongly dependent on details of the initial conditions. A numerical method that uses symmetries and additional resolution in the direction and location of maximum coin-pression is used to simulate periodic boundary conditions in all directions. For the initial condition that yields singular type behavior the growth of the peak vorticity roughly obeys (t c − t)−1 and the growth of the strain along the vorticity at this point obeys (t c − t)−γwhere γ ≈ 1.

Heat pulse experiments revisited

Abstract: This is a review of heat propagation — theory and experiment — in dielectric solids at low temperatures where the phenomenon of second sound occurs. The review does not merely present a list of the various explanations of the observed phenomena. Rather it views them as special cases of a unified theory which is formulated within the framework of extended thermodynamics of phonos. Field equations are derived by averaging over the phonon-Boltzmann equation and initial and boundary value problems are solved. Thus it became possible to achieve a full explanation of the observations of the heat-pulse experiments in which ballistic phonons, second sound and ordinary heat conduction compete.

Boundary Mixing and Arrested Ekman Layers: Rotating Stratified Flow Near a Sloping Boundary

Abstract: We are concerned here with the behavior of a rotating, stratified fluid near a sloping rigid boundary, with boundary conditions of zero normal buoyancy flux and no slip. Although this is an interesting fluid dynamical problem in its own right, we have been motivated by two major, and at first sight disparate, topics in physical oceanography. The first, known as "boundary mixing", is concerned with how turbulent mixing at the sloping sides of the density-stratified ocean affects the stratification in the interior. The second topic involves the way in which the combination of strati-

Large-Eddy Simulation of the Convective Boundary Layer: A Comparison of Four Computer Codes

Abstract: To test the consistency of large-eddy simulation we have run four existing large-eddy codes for the same case of the convective atmospheric boundary layer. The four models differ in various details, such as: the subgrid model, numerics and boundary conditions.

On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

Abstract: The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.

Boundary K-matrices for the six vertex and the n(2n-1)An-1 vertex models

Abstract: Boundary conditions compatible with integrability are obtained for two-dimensional models by solving the factorizability equations for the reflection matrices K+or-( theta ). For the six vertex model the general solution depending on four arbitrary parameters is found. For the An-1 models all diagonal solutions are found. The associated integrable magnetic Hamiltonians are explicitly derived.

A systematic and efficient method of computing normal modes for multilayered half-space

Abstract: SUMMARY We present a systematic and efficient approach for computing the dispersion curves (i.e. phase velocities for a given frequency) as well as the eigenfunctions of normal modes in a multilayered half-space medium. Our approach is superior to previous approaches in the following aspects. First, it is a simple and self-contained algorithm for simultaneously determining both the phase velocities and the corresponding eigenfunctions. From the basic principle that the normal modes are nun-trivial solutions of the free elastodynamic equation under appropriate boundary conditions, we naturally derive the phase velocities and eigenfunctions. Second, because we use the reflection/transmission coefficients of Luco & Apsel’s version (Luco & Apse1 1983), which intrinsically exclude the growth terms, our algorithm not only exhibits the physical mechanism of the formation of normal modes, i.e. constructive interference, as Kennett’s did (1979,1983), but also is numerically more stable for high-frequency cases. Furthermore, we derive a high-frequency asymptotic solution of the fundamental Rayleigh mode. Therefore, our approach can provide efficient and accurate solutions in any frequency range, and it is expected to be a powerful and useful tool for simulating the LR and RR phases and complete seismograms at regional distances.