Infinite Domain Problems Research Articles

Asymptotic results for nonlinear outdiffusion

Asymptotic methods are applied to the outdiffusion of a diffusant whose behaviour is governed by a power-law diffusivity. Both semi-infinite and finite domain problems are considered. The ‘fast’ diffusion (negative power-law) case is of particular interest, and the results are significantly different from those for the corresponding indiffusion problem.

An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain

For boundary value problems posed on unbounded domains it is often appropriate to impose a boundary condition at "infinity". For certain classes of boundary value problem obvious numerical difficulties can be avoided by truncating the unbounded domain and solving a sequence of finite domain problems instead. We introduce a novel technique which is straightforward to implement and which exploits information contained in this sequence in order to extrapolate to the unbounded case. The technique introduces a new and interesting application of a variety of convergence acceleration algorithms.

Use of infinite Elements for Potential Problems

Infinite elements is one way of solving infinite domain problems by the finite element method (FEM). Correct solutions are obtained only when certain constraints arc placed on the infinite elements. They are discussed in this paper.

Modified MWR approach: Application to agglomerative precipitation

AbstractA modified method of weighted residuals approach is presented that combines expansions based on limiting forms of the true solution to solve the population balance equation accurately for continuous agglomerative precipitation. A modified moment method is used that employs fractional moments and is stable for the infinite domain problem. Solutions are obtained for various values of the agglomeration parameter, and the results are fitted to Pade approximations providing simple relationships that can be used for data fitting, thus obviating further need to solve the population balance equation with a constant aggregation kernel. This method is applied to experimental data on the precipitation of nickel ammonium sulfate and calcium carbonate to yield kinetic parameters.

Source field superposition analysis of two‐dimensional potential problems

AbstractThe source field superposition method for problems governed by Laplace's equation involves representing the potential field in a given solution domain as a superposition of fields generated by a number of sources. These sources are located outside the solution domain, and in the case of two‐dimensional singly‐connected finite domain problems can be uniformly distributed around the perimeter of a circle enclosing the physical domain. Some test problems are used to make detailed comparisons with the boundary element method. The results show that remarkable accuracy can be achieved, often to five or six significant figures, with very little computational effort relative to other numerical methods. In contrast to the boundary element method, however, geometric corners require no special treatment, and high aspect ratio of the solution domain is not a significant limitation.

Infinite elements—theory and applications

The possibilities for solving numerically defined problems in infinite or semi-infinite domains by a combination of finite and infinite elements is considered. A review of the available infinite element formulations is made. A new infinite element technique is presented and tested for one- and two-dimensional problems.

Non-reflecting boundary conditions for elastic waves

An exact non-reflecting boundary condition is devised for time-harmonic two-dimensional elastodynamics in infinite domains. The domain is made finite by the introduction of a circular artificial boundary on which this exact condition is imposed. In the finite computational domain a finite element method is employed. Numerical examples are presented in which the accuracy and efficiency of the method using the exact non-local boundary condition are compared with those of methods based on approximate local boundary conditions. The method is also used to solve problems in large finite domains by reducing them to smaller domains. In addition, local boundary conditions are derived which are exact for waves with a limited number of angular Fourier components.

Analyses of waves in unbounded domains by use of direct combination of analytical solutions and finite elements.

An analysis of wave-motion problems in unbounded or infinite domains is presented. Analytical solutions that express outgoing waves from an artificial boundary such as a circle or a sphere can be obtained easily. These solutions may be transformed into discretized relations between values of velocity potential and volumetric flux on the artificial boundary. These relations are in the same forms as the FEM equations. They can be joined directly to finite elements within the artificial boundary in order to construct an equation system for the whole domain. Some problems of radiation and scattering are calculated numerically by this method. The so-called fictitious interior eigenvalue failure does not arise in this method.

Green’s tensors and associated potentials for electromagnetic waves in inhomogeneous material media

The explicit form of the Green’s tensor in unbounded inhomogeneous dielectric, magnetic and partly conducting material is obtained. The tensor is given explicitly in terms of two scalar Green’s function potentials, each of which obeys its own wave equation. It is found that the gradients of the constitutive parameters at the source of the radiating currents may have a strong effect on the multipolar field. For example, an electric dipole embedded in a uniaxially inhomogeneous source-zone is found to produce additional fields with the characteristics of electric quadrupole (l= 2,m= ± 1) and a magnetic dipole (l= 1,m= ± 1). I present a mathematical formulation of Huygens’ principle for inhomogeneous media in terms of the Green’s tensor and its associated tensor potentials, thus extending the use of the Green’s tensor to scattering problems in inhomogeneous finite domains.

A finite element method for large domains

A combined analytical and numerical method is devised to solve elliptic boundary value problems in large or infinite domains. First the domain is divided by an artificial boundary B into a small computational domain Ω and a large or infinite residual domain D. In D the problem is solved analytically, and from the solution an exact nonlocal relation between the solution and its derivatives on B is deduced. This relation is used as a boundary condition to complete the formulation of a problem in Ω which has exactly the same solution there as the original problem. Then a finite element formulation of this new problem in Ω is presented. The exact nonlocal boundary condition is given explicitly for Laplace's equation, for the equations of plane stress and plane strain in linear elastostatics, and for certain equations governing beams and axisymmetric cylindrical shells. The artificial boundary is chosen to be a sphere or a circle. The properties and computational implications of the boundary condition are discussed. Some numerical examples are presented, and the results are compared with those obtained by the standard finite element method using different approximate local boundary conditions on the artificial boundary.

Exact non-reflecting boundary conditions

An exact non-reflecting boundary condition is devised for use in solving the reduced wave equation in an infinite domain. The domain is made finite by the introduction of an artificial boundary on which this exact condition is imposed. In the finite domain a finite element method is employed. Although the boundary condition is non-local, that does not affect the efficiency of the computational scheme. Numerical examples are presented which show that the use of the exact non-local boundary condition yields results which are much more accurate than those obtained with various approximate local conditions. The method can also be used to solve problems in large finite domains by reducing them to smaller domains, and it can be adapted to other differential equations.

Wave field factorization for broadband acoustic signals

A framework has been demonstrated for factoring the Helmholtz equation, governing for narrow-band acoustic signals, into a pair of one-way equations [L. Fishman and J. J. McCoy, J. Math. Phys. 25, 285–296 (1984)]. This factoring requires a range-independent propagation environment. The factored Helmholtz equation has been demonstrated to provide a marching algorithm [L. Fishman and J. J. McCoy, Geophys. J. R. Astron. Soc. 80, 439–461 (1985)], which is a generalization of the split-step algorithm, the latter being frequently used for marching the solution of the ordinary parabolic wave equation. The extension of these results to broadband acoustic signals is considered. Two factorizations are considered and shown to apply to different initial/boundary value problems. The first factorization is based on a Fourier synthesis of that obtained for narrow-band signals. This applies for a forcing problem as a broadband time series acting across a source range plane. The second factorization obtains for the time coordinate of the wave equation. This applies for an infinite spatial domain problem for specified initial time data.

Boundary zone superposition method for linear and nonlinear dynamic analysis of infinite domain problems : Nelson, R B; Muki, Y Proc 3rd Conference on Dynamic Response of Structures, Los Angeles, March 31–April 2, 1986P732–744. Publ New York: ASCE, 1986

The method is applicable to problems where input loads are imposed from within the finite element model or to problems where disturbances outside the finite element model generate input data on the boundary. The method is general and not restricted by any special properties of the medium transmitting stress waves, nor to any frequency characteristics beyond those inherent in the definition of the finite element model. The method is valid for problems with confined nonlinearities, with linear behavior required only in the boundary zone surrounding the confined region. Several examples are presented to demonstrate the method.

Boundary zone superposition method for linear and nonlinear dynamic analysis of infinite domain problems : Nelson, R B; Muki, Y Proc 3rd Conference on Dynamic Response of Structures, Los Angeles, March 31–April 2, 1986P732–744. Publ. New York: ASCE, 1986

Boundary zone superposition method for linear and nonlinear dynamic analysis of infinite domain problems : Nelson, R B; Muki, Y Proc 3rd Conference on Dynamic Response of Structures, Los Angeles, March 31–April 2, 1986P732–744. Publ. New York: ASCE, 1986

Uniqueness of solutions to singular hyperbolic boundary-value problems in infinite domains

In a recent paper C.C. Travis has given a method of proof which is completely independent of the type of the partial differential equation being considered and allows the consideration of a class of nonstandard boundary conditions. In this paper we generalize his method and extend his analysis to consider a wide range of singular boundary value problems for hyperbolic type differential equations in infinite domains.

Nonlinear waves on liquid film surfaces—II. Bifurcation analyses of the long-wave equation

The long-wave equation for film flow in vertical pipes with and without interfacial stress is studied numerically. Two sets of boundary conditions are imposed—periodic and infinite, corresponding to monochromatic and multi-frequency waves. For the periodic boundary conditions, waves with increasingly many subharmonic modes bifurcate from the flat-film base state as the system parameter σ decreases below unity. Eventually, a strange attractor appears at σ ∼ 0.028. The chaotic solution for σ < 0.028 corresponds to the observed turbulent film surface near flooding conditions for systems with upward gas flow. The solutions of the infinite domain problem indicate that pulse-like initial profiles collapse into wave packets which propagate in both directions with speeds governed by the pulse heights. Interactions of these solitary waves are studied numerically.

Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems.

1. Heat Equation. 2. Method of Separation of Variables. 3. Fourier Series. 4. Vibrating Strings and Membranes. 5. Sturm-Liouville Eigenvalue Problems. 6. An Elementary Discussion of Finite Difference Numerical Methods for Partial Differential Equations. 7. Partial Differential Equations with at Least Three Independent Variables. 8. Nonhomogeneous Problems. 9. Green's Functions for Time-Independent Problems. 10. Infinite Domain Problems--Fourier Transform Solutions of Partial Differential Equations. 11. Green's Functions for Time-Dependent Problems. 12. The Method of Characteristics for Linear and Quasi-Linear Wave Equations. 13. A Brief Introduction to Laplace Transform Solution of Partial Differential Equations. 14. Topics: Dispersive Waves, Stability, Nonlinearity, and Perturbation Methods. Bibliography. Selected Answers to Starred Exercises. Index.

Surface integral finite element hybrid (SIFEH) method for fracture mechanics

AbstractAn effective surface integral and finite element hybrid (SIFEH) method has been developed to model fracture problems in finite plane domains. This hybridization by (incrementally) linear superposition combines the best features of both component methods. Finite elements are used to model the finite domain (and eventually nonlinearity), while continuous distributions of dislocations (resulting in surface integral equations) are used to model the fracture (i.e. displacement discontinuity). This method has been implemented in a computer program and results of representative problems are presented: these compare very well with known solutions and they demonstrate the computational advantages of SIFEH over other numerical methods (including the individual components).

A New Infinite Element for Finite Element Analysis of Three Dimensional Unbounded Field Problems

A simple and efficient infinite element has been developed for the finite element analysis of three dimensional field problems whose domain extends to infinity. The formulation is based on the assumed displacement hybrid finite element model. The present element has the following features: (1) Since it requires no interpolation nodes in the exterior region, both the number of total degrees of freedom and the bandwidth of the global stiffness matrix remain exactly the same as those of interior elements. (2) The generation of the element stiffness matrix requires neither the matrix inversion nor the integration for the direction extending to infinity. (3) The computational effort needed for the entire process is almost the same as that for a corresponding bounded problem. (4) It can be used in conjunction with conventional interior elements. Numerical results for several three dimensional infinite domain problems are presented.

Vertical infiltration of water into unsaturated soils with variable diffusivity and conductivity

A simple approximate analytical solution to the problem of vertical infiltration of water into unsaturated soils is presented. The soil-water diffusivity as well as the hydraulic conductivity are assumed to be exponential functions of the soil-water content. Constant-concentration conditions are assumed at the boundary. The infinite-domain problem is reduced to a finite-domain problem under the assumption of an abruptly terminating wetting front coinciding with the inflection point of the soil-water content profile. The location as well as the soil-water content at the inflection point are calculated and the result is employed for the approximate solution of the problem. Results are compared with others found in the existing literature. A discussion concludes the paper.