Math Technical Report Series

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A 2+1 Dimensional Insoluble Surfactant Model for a Vertical Draining Free Film
(Department of Mathematical Sciences, 2002-05-09) Naire, S.; Braun, Richard J.; Snow, S.A.
A 2 + 1 dimensional mathematical model is constructed to study the evolution of a vertically-oriented thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite surface viscosity, on its free surface. Lubrication theory for this free film results in four coupled nonlinear partial differential equations (PDEs) describing the free surface shape, the surface velocities and the surfacant transport, at leading order. Numerical experiments are performed to understand the stability of the system to perturbations across the film. In the limit of large surface viscosities, the evolution of the free surface is that of a rigid film. In addition, these large surface viscosities act as stabilizing factors due to their energy dissipating effect. An instability is seen for the mobile case; this is caused by a competition between gravity and the Marangoni effect. The behavior observed from this model qualitatively matches some structures observed in draining film experiments
A1-L1 Phase Boundaries and Anisotropy via Multiple-Order-Parameter Theory for an FCC Alloy
(Department of Mathematical Sciences, 2003) Tanoglu, G.B.; Braun, Richard J.; Cahn, J.W.; McFadden, G.B.
The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an FCC lattice is studied as a function of their orientation. Using a recently-developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L12 and L10) such as occurs in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L10 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find the interfacial energy, γ, as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman ‘ξ-vector’ formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is “transversely isotropic” with respect to the tetragonal axis of the L10 phase. We verify the model’s consistency with the Gibbs adsorption equation.
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.
Achieving High-Order Convergence Rates with Deforming Basis Functions
(Department of Mathematical Sciences, 2003) Rossi, Louis F.
This article studies the use of moving, deforming elliptical Gaussian basis functions to compute the evolution of passive scalar quantities in a two-dimensional, incompressible flow field. We compute an evolution equation for the velocity, rotation, extension and deformation of the com- putational elements as a function of flow quantities. We find that if one uses the physical flow velocity data calculated from the basis function centroid, the method has only second order spatial accuracy. However, by computing the residual of the numerical method, we can determine adjustments to the centroid data so that the scheme will achieve fourth-order spatial accuracy. Simulations with nontrivial flow parameters demonstrate that the methods exhibit the properties predicted by theory.
Approximations in Canonical Electrostatic MEMS Models
(Kluwer Academic Publishers (see: http://www.sherpa.ac.uk/romeo.php for publisher's conditions for archiving in an institutional repository), 2004-09-28) Pelesko, John A.; Driscoll, Tobin A.
The mathematical modeling and analysis of electrostatically actuated micro- and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored.
Axisymmetric Finite Element Solution of Non-isothermal Parallel-plate Flow
(Department of Mathematical Sciences, 2004-11-19) Zhang, Shangyou; Olagunju, David O.
Steady non-isothermal parallel-plate flow of a Newtonian fluid with a temperature dependent viscosity is considered. The viscosity is modelled by a Nahme type law. We apply axisymmetric Q k finite elements to the coupled nonlinear system to obtain numerical solutions for a wide range of parameters.
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.
Boundary Element Methods – An Overview
(Department of Mathematical Sciences, 2004) 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. This paper gives an overview of the method from both theoretical and numerical point of view. It summaries the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.
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.
Bounded Film Evolution with Nonlinear Surface Properties
(Department of Mathematical Sciences, 2001-09-12) Debisschop, C.A.; Braun, Richard J.; Snow, S.A.
We study the evolution of a Newtonian free surface of a thin film above a solid wall. We consider the case in which the horizontal solid is covered by a non-wetting fluid and an insoluble monolayer of surfactant is present on the fluid-air interface. We pose a model that incorporates a variety of interfacial effects: van der Waals forces, variable surface tension and surface viscosity. The surface tension and surface viscosity depend nonlinearly on the surfactant concentration. Using lubrication theory we obtain a leading order description of the shape and velocity of the fluid-air interface, and the surfactant concentration, in the form of coupled nonlinear partial differential equations. A linear stability analysis reveals that the wavenumber that characterizes the marginal state is independent of the presence of the surfactant and the nonlinearity of the surface properties. We solve the 1+1-dimensional system numerically to obtain the spatio-temporal evolution of the free surface in the nonlinear regime, and observe the progression to rupture.
A Comparative Study of Lagrangian Methods Using Axisymmetric and Deforming Blobs
(Department of Mathematics, 2003) Rossi, Louis F.
This paper presents results from a head-to-head comparison of two Lagrangian methods for solutions to the two-dimensional, incompressible convection-diffusion equations. The first Lagrangian method is an axisymmetric core spreading method using Gaussian basis functions. The second method uses deforming elliptical Gaussian basis functions. Previous results show that the first method has second-order spatial accuracy and the second method has fourth-order spatial accuracy. However, the deforming basis functions require more computational effort per element, so this paper examines computational performance as well as overall accuracy. The test problem is the deformation and diffusion of ellipsoidal distribution of scalar with an underlying flow field that has closed circular streamlines. The test suite includes moderate, high and infinite Peclet number problems. The results indicate that the performance tradeoff for the sample flow calculation occur at modest problem sizes, and that the fourth-order method offers distinct advantages as a general approach for challenging problems.
Computing Eigenmodes of Elliptical Operators Using Radial Basis Functions
(Department of Mathematical Sciences, 2003-03-21) Platte, Rodrigo B.; Driscoll, Tobin A.
Radial basis function (RBF) approximations have been successfully used to solve boundary-value problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Special attention is given to the Laplacian operator in two dimensions. We include techniques to avoid degradation of the solution near the boundaries and corner singularities. Numerical results compare favorably to basic finite element methods.
Cyclic Relative Difference Sets and Their p-Ranks
(Department of Mathematical Sciences, 2002) Chandler, D.B.; Xiang, Qing
By modifying the constructions in [10] and [15], we construct a family of cyclic ((q 3k − 1)/(q − 1), q − 1, q 3k − 1 , q 3k − 2 ) relative difference sets, where q = 3 e . These relative difference sets are “liftings” of the difference sets constructed in [10] and [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q = 3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.
The Direct and Inverse Scattering Problems for Partially Coated Obstacles
(Department of Mathematical Sciences, 2002) Cakoni, Fioralba; Colton, David; Monk, Peter B.
We consider the direct and inverse scattering problems for partially coated obstacles. To this end, we first use the method of integral equations of the first kind to solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet-impedance boundary conditions on the Lipschitz boundary of the scatterer D. We then use the linear sampling method to solve the inverse scattering problem of determining D from a knowledge of the far field pattern of the scattered field. Numerical examples are given showing the performance of the linear sampling method in this case.
The Direct and Inverse Scattering Problems for Partially Coated Obstacles
(Department of Mathematical Sciences, 2002) Monk, Peter B.
The time harmonic Maxwell's equations for a lossless medium are neither elliptic or definite. Hence the analysis of numerical schemes for these equations presents some unusual difficulties. In this paper we give a simple proof, based on the use of duality, for the convergence of edge finite element methods applied to the cavity problem for Maxwell's equations. The cavity is assumed to be a general Lipschitz polyhedron, and the mesh is assumed to be regular but not quasi-uniform.
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.
Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: divergence.
(Department of Mathematical Sciences, 2004-11-19) Borwein, J. M.; Luke, D. Russell
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters [4, 3, 2]. A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in [2], we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions.
Dynamics of Random Continued Fractions
(Department of Mathematical Sciences, 2004-11-16) Borwein, J. M.; Luke, D. Russell
We study a generalization of a continued fraction of Ramanujan with random coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain stochastic difference equations and the stability of random dynamical systems. We determine the convergence properties of stochastic difference equations and so divergence of their corresponding continued fractions.
Eigenvalue stability of radial basis function discretizations for time-dependent problems
(Department of Mathematical Sciences, 2005) Platte, R.B.; Driscoll, Tobin A.
Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collo- cation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.
Error Analysis of a Finite Element-Integral Equation Scheme for Approximating the Time-Harmonic Maxwell System
(Department of Mathematical Sciences, 2002) Hsiao, George C.; Monk, Peter B.; Nigam, N.
In 1996 Hazard and Lenoir suggested a variational formulation of Maxwell's equations using an overlapping integral equation and volume representation of the solution. They suggested a numerical scheme based on this approach, but no error analysis was provided. In this paper, we provide a convergence analysis of an edge finite element scheme for the method. The analysis uses the theory of collectively compact operators. It's novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation.

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