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Item STABILITY OF VISCOELASTIC FILAMENTS: COMPARISON OF CONSTITUTIVE MODELS(Department of Mathematical Sciences, 1999-07-23) Olagunju, David O.A thin filament model is used to analyze the stability of a viscoelastic thread subject to uniaxial stretching. Linear stability analysis is carried out for a number of different constitutive models, namely the Johnson{Segalman,Giesekus, Phan{Thien Tanner, and FENE{CR. Our analysis shows that stability is controlled by the competing effects of surface tension which is destabilizing and axial normal stress which is stabilizing. Numerical simulations of the model equations are used to check the prediction of linear analysis. Results obtained agree with experimental observations.Item Integral Representation in the Hodograph Plane for Compressible flow Problems(Department of Mathematical Sciences, 1999-12-07) Hanson, E.B.; Hsiao, George C.Compressible flow is considered in the hodograph plane. The fact that the equation for the stream function is linear there is exploited to derive a representation formula for the stream function, involving boundary data only, and a fundamental solution to the equation. For subsonic flow, an efficient algorithm for computation of the fundamental solution is also developed.Item Variational Methods for Boundary Integral Equations: Theory and Applications(Department of Mathematical Sciences, 1999-12-19) Hsiao, George C.Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. The later has become one of the most popular numerical schemes in recent years. In this expository paper, we discuss some of the essential features of the methods, their intimate relations with the variational formulations of the corresponding partial differential equations and recent developments with respect to applications in domain composition from both mathematical and numerical points of view.Item Recent Developments in Inverse Acoustic Scattering Theory(Department of Mathematical Sciences, 2000) Colton, David; Coyle, J.; Monk, Peter B.We survey some of the highlights of inverse scattering theory as it has developed over the past fifteen years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.Item Boundary Integral Methods in Low Frequency Acoustics(Department of Mathematical Sciences, 2000) Hsiao, George C.; Wendland, W.L.This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.Item Finite Element Method for Approximating Electro-Magnetic Scattering from a Conducting Object(Department of Mathematical Sciences, 2000) Kirsch, A.; Monk, Peter B.We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell’s equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincare-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.Item On the Convergence and Numerical Stability of the Second Waterman Scheme for Approximation of the Acoustic Field Scattered by a Hard Object(Department of Mathematical Sciences, 2000) Dallas, A.G.The numerical schemes of P.C. Waterman (J. Acoust. Soc. Am.45 (1969), 1417-1429), frequently referred to under the name of "the T-Matrix method," have formed the basis for many scattering computations in many settings. However, no successful analyses of the algorithms have been published, so the limitations on their range of applicability and numerical stability remain largely unknown; this is of particular importance because of the apparently inconsistent success achieved in numerical experiments. Here, we give an operator condition that guarantees the viability of the algorithm and mean-square convergence of the far-field patterns generated by the second Waterman scheme for the case of time-harmonic acoustic scattering by a hard obstacle; we prove further that the operator condition holds at least whenever the scattering obstacle is ellipsoidal. For the convergence proof, we also assume that the square of the wavenumber is not an interior Dirichlet eigenvalue for the negative Laplacianl in the contrary case, we show that the algorithm is at best numerically ill-coordinated. With this and previous experience in numerical applications, it appears that the performance of the algorithm is markedly shape-dependent; for certain obstacles, e.g., ellipsoids, instabilities are so localized in wavenumber that they are practically numerically irrelevant, while it is not clear whether the erratic results found in applications to various other shapes arise from a failure of convergence or form numerical instability.Item Hybrid Coupled Finite-Boundary Element Methods for Elliptical Systems of Second Order(Department of Mathematical Sciences, 2000) Hsiao, George C.; Schnack, E.; Wendland, W.L.In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied a priori within the subdomain elements. If the Trefftz elements are modelled with boundary potentials supported by the individual element boundaries, this defines the so–called macro–elements. These allow one to handle in particular situations involv-ing singular features such as cracks, inclusions, corners and notches providing a locally high resolution of the desired stress fields, in combination with a traditional global varia-tional FEM analysis. The global stiffness matrix is here sparse as the one in conventional FEM. In addition, with slight modifications, the macro–elements can be incorporated into standard commercial FEM codes. The coupling between the elements is modelled by using a generalized compatibility condition in a weak sense with additional elements on the skeleton. The latter allows us to relax the continuity requirements for the global displacement field. In particular, the mesh points of the macro–elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. We present here the formulation of the method and its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustration, we include some computational results in two– and three–dimensional elasticity.Item Toward the Direct Analytical Determination of the Pareto Optima of a Differentiable Mapping, I: Domains in Finite-Dimensional Spaces(Department of Mathematical Sciences, 2000) Dallas, A.G.The problem of locating the Pareto-optimal points of a differentiable mapping $F: {\mathcal M}^N \to {I\kern-.30em R}^n$ is studied, with the domain ${\cal M}^N$ a differentiable N-dimensional submanifold-without-boundary in a euclidean space ${I\kern-.30em R}^{N_{0}}$ and $N_0 \ge N \ge n$. The case in which the domain is the closure of a bounded, regular, open subset of ${I\kern-.30em R}^N$ is also discussed. The search is initiated from these observations: for a manifold-domain, (1) the image of any Pareto optimum lies in the boundary of the range of F; (2) a point of the boundary of the range of F that also lies in the range must be the image of a singular point of F, i.e., must appear amongst the singular values of the map. Further conditions are then needed to distinguish which of the singular values should be discarded because they belong to the interior of the range; local tests of this sort are given for the bicriterial case (n = 2). A search procedure based on the present developments can systematically determine all of the Pareto optima for sufficiently simple F. The conditions established here may be regarded as analogues of the classical ones for the determination of the global extrema of a real-valued differentiable function. The results proven are illustrated with single examples, including plots of the ranges, singular points, and singular values.Item Scattering of Time-Harmonic Electromagnetic Waves by Anisotropic Inhomongeneous Scatters or Impenetrable Obstacles(Department of Mathematical Sciences, 2000) Monk, Peter B.; Coyle, J.We investigate an overlapping solution technique to compute the scattering of time-harmonic electromagnetic waves in two dimensions. The technique can be used to compute waves scattered by penetrable anisotropic inhomogeneous scatterers or impenetrable obstacles. The major focus is on implementing the method using finite elements. We prove existence of a unique solution to the disctretized problem and derive an optimal convergence rate for the scheme, which is verified numerical by examples.Item Basis Properties of Traces and Normal Derivatives of Spherical-Separable Solutions of the Helmholtz Equation(Department of Mathematical Sciences, 2000) Dallas, A.G.The classical solutions of the Heimholtz equation resulting from the separation-of-variables procedure in spherical coördinates are frequently used in one way or another to approximate other solutions. In particular, traces and/or normal derivatives of certain sequences of these spherical-separable solutions are commonly used as trial-and-test-functions in Galerkin procedures for the approximate solution of boundary-operator problems arising from the reformulation of exterior or interior boundary-value problems and set on the boundary Γ of the domain where a solution is wanted. While the completeness properties of these traces and normal derivatives in the usual Hilbert space L2( Γ) are well known, their basis properties are not. We show that such sequences of traces or normal derivatives of the outgoing spherical-separable solutions form bases for L2( Γ) only when Γ is a sphere centered at the pole of the spherical solutions; corresponding results are given for the entire solutions, accounting for the possibility of an interior eigenvalue. We identify other Hilbert spaces, connected with the far-field pattern, for which these functions do provide bases. We apply the results to discuss some aspects of the Waterman schemes for approximate solutions of scattering problems (the so-called “T-matrix method”), including the previous article of KRISTENSSON, RAMM, and STRÖM (J. Math. Phys.24 (1983), 2619-2631) on the convergence of such methods.Item Accurate Discretisation of a Nonlinear Micromagnetic Problem(Department of Mathematical Sciences, 2000) Monk, Peter B.; Vacus, O.In this paper we propose a finite element discretization of the Maxwell-Landau-Lifchitz-Gilbert equations governing the electromagnetic field in a ferromagnetic material. Our point of view is that it is desirable for the discrete problem to possess conservation properties similar to the continuous system. We first prove the existence of a new class of Liapunov functions for the continuous problem, and then for a variational formulation of the continuous problem. We also show a special continuous dependence result. Then we propose a family of mass-lumped finite element schemes for the problem. For the resulting semi-discrete problem we show that magnetization is conserved and that semi-discrete Liapunov functions exist. Finally we show the results of some computations that show the behavior of the fully discrete Liapunov functions.Item Mathematical Foundations for the Boundary-Field Equation Methods in Acoustic and Electromagnetic Scattering(Department of Mathematical Sciences, 2000) Hsiao, George C.The essence of the boundary-field equation method is the reduction of the boundary value problem under consideration to an equivalent nonlocal boundary value problem in a bounded domain by using boundary integral equations. The latter can then be treated by the standard variational method including its numerical approximations. In this paper, various formulations of the nonlocal boundary value problems will be given for the Helmholtz equation as well as for the time-harmonic Maxwell equations. Emphasis will be placed upon the variational formulation for the method and mathematical foundations for the solution procedure. Some numerical experiments are included for a model problem in electromagnetic scattering.Item Domain Decomposition Methods via Boundary Integral Equations(Department of Mathematical Sciences, 2000-12-21) Hsiao, George C.; Steinbach, O.; Wendland, W.L.Domain decomposition methods are designed to deal with coupled or transmission problems for partial differential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of different discretization schemes. In general, the coupled problem is reduced to the Schur complement equation on the skeleton of the domain decomposition. Boundary integral equations are used to describe the local Steklov-Poincare operators which are basic for the local Dirichlet-Neumann maps. Using different representations of the Steklov-Poincare operators we formulate and analyze various boundary element methods employed in local discretization schemes. We give sufficient conditions for the global stability and derive corresponding a priori error estimates. For the solution of the resulting linear systems we describe appropriate iterative solution strategies using both local and global preconditioning techniques.Item Reformulating a Boundary-Integral Equation in Three Dimensions as as Integral-Operator Problem in a Plane Region(Department of Mathematical Sciences, 2001) Dallas, A.G.Motivated by a desire to simplify the design of numerically stable and efficient approximation schemes for boundary-operator problems, we develop a framework in which an integral equation on the boundary of a domain ${I\kern-.30em R}^3$ can be systematically reformulated as an integral-operator problem set in a region in the plane; some geometric restrictions are imposed on the shape of the (smooth) boundary. When the plane region is chosen to be a rectangle, the necessary Sobolev-space structures can be handled numerically rather easily in the new simpler geometry, in contrast to the situation on the original boundary. Moreover, familiar trial- and test- functions can then be employed in the construction of approximate solutions of the reformulated problems. We show for two examples how a well-posed problem can be transferred from the domain-boundary setting to the plane-region setting. We describe a numerical implementation of these ideas to a lower-dimensional example involving the approximate solution of a first-kind integral equation associated with the Helmholtz equation that is originally set on the boundary of a domain in ${I\kern-.30em R}^2$.Item Hydrodynamics of Bounded Vertical Film with Nonlinear Surface Properties(Department of Mathematical Sciences, 2001) Heidari, A.H.; Braun, Richard J.; Hirsa, A. H.; Snow, S.A.; Naire, S.The drainage of a thin liquid film with an insoluble monolayer down a vertical wall is studied. Lubrication theory is used to develop a model where the film is pinned at the top with a given thickness and the film drains into a bath at the bottom. A nonlinear equation of state is used for the surface tension and the surface viscosity is a nonlinear function of the surfactant concentration; these are appropriate for some aqueous systems. The three partial differential equations are solved via discretization in space and then solving the resulting differential algebraic system. Results are described for a wide range of parameters, and the conditions under which the free surface is immobilized are discussed.Item High Order Vortex Methods With Deforming Elliptical Gaussian Blobs 1: Derivation and Validation(Department of Mathematical Sciences, 2001) Rossi, Louis F.This manuscript introduces a new vortex method based on elliptical Gaussian basis functions. Each basis function translates, nutates, elongates and spreads through the action of the local flow field and fluid viscosity. By allowing elements to deform, the method captures the effects of local flow deviations with a fourth order spatial accuracy. This method uses a fourth order asymptotic approximation to the Biot-Savart integrals for elliptical Gaussian vorticity distributions to determine velocity and velocity derivatives. A robust adaptive refinement procedure reconfigures elements that spread beyond the specified resolution. The high order convergence rate is verified by comparing calculations with the vortex method to exact solutions in a variety of controlled experiments.Item A Model for Anisotropic Epitaxial Lateral Overgrowth(Department of Mathematical Sciences, 2001) Khenner, M.; Braun, Richard J.; Mauk, M.G.The model for anisotropic crystal growth on a substrate covered by a mask material with a periodic series of parallel long trenches where the substrate is exposed to the vapor phase if developed. The model assumes that surface diffusion and deposition flux are the main mechanisms of the growth, and that the three key surface quantities (energy, mobility and adatom diffusivity) are anisotropic with either four- or six-fold symmetry. A geometrical approach to the motion of crystal surface in two dimensions is adopted and nonlinear evolution equations are solved by a finite-difference method. The model allows the direct computation of the crystal surface shape and the study of effects due to finite mask thickmness. As in experiments, lateral overgrowth of crystal onto the mask if found, as well as comparable crystal shapes; the anisotropy of the surface mobility is found to play the dominant role in the shape selection. The amount of the overgrowth and the shapes can be effectively controlled by orienting the fast and slow growth directions with respect to the substrate.Item Numerical Experiments with Isometric Mapping and Back-Projection for Conditioning Families in a Simple Sobolev Space(Department of Mathematical Sciences, 2001) Dallas, A.G.Numerically stable Galerkin procedures can be constructed by ensuring that the families of trial- and test-functions are well conditioned in the respective Hilbert spaces between which the operator is an (appropriate) isomorphism. We explain an idea for constructing a family that may be well conditioned in a given Sobolev space of nonzero fractional order from a family that is well conditioned in the corresponding zero-order space, by using a naturally occurring isometric operator followed by projection back onto the original subspace. Effectively, the construction results in "preconditioning matrices," to be used in transforming the original Galerkin matrix to produce new ones which may be of much smaller condition number. The underlying geometric setting must be sufficiently simple, so that the Sobolev structures can me "manipulated numerically." While we have not yet proven the well-conditioning of the constructed families, the use of the scheme is illustrated numerically in applications to the approximate solution of a first-kind integral equation arising in two-dimensional acoustic scattering, where a pronounced stabilizing effect is observed.Item A High Order Langrangian Scheme for Flow Through Unsaturated Porous Media(2001) Rossi, Louis F.A new high order Lagrangian method uses moving basis functions to represent the moisture content in an unsaturated porous material. The basis functions are localized elliptical Gaussians that can move, spread, elongate and rotate under the action of the local velocity, velocity deviations, and diffusivity with corrections for local material properties. The velocity and its deviations are calculated from the local moisture potential and its derivatives which can be obtained from published experiments. This numerical technique is natu- rally adaptive in the sense that computational effort is expended only where there is moisture and nowhere else, and this method is capable of capturing infiltration instabilities in the wetting front observed by experimentalists and predicted by linear stability analysis. This method borrows many ideas from high Reynolds number vortex methods and other applications of Lagrangian schemes to nonlinear partial differential equations.