Wavelet–Galerkin solutions for one-dimensional partial differential equations

In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies¹) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund² and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.     

Finite difference analysis for developing laminar flow in circular tubes applied to forced and combined convection

The complete two-dimensional partial differential equations for developing laminar flow in a circular tube have been treated by a finite difference analysis. Property variation with temperature, especially that of viscosity, is allowed for in a flexible manner. The continuity and momentum equations, and then the energy equations, are solved by direct elimination at each axial step, and marching procedure used in the axial direction. A new technique is that the stepwise energy balance is rigidly satisfied throughout by using it as a constituent equation in place of the ‘explicit’ wall thermal boundary condition normally used. The analysis predicts the complete developing hydrodynamic and thermal fields, together with friction factors and heat transfer coefficients. It has been tested for a range of fluid velocity and thermal boundary conditions and for various fluids, including high viscosity oils, water and air. Data for constant wall heat flux have already been published. ^1,2 Predictions for constant wall temperature presented here are for forced and combined convection and are compared with experimental data of Test³ and Zeldin and Schmidt⁴.     

An ablation problem in a finite medium with time‐dependent heat flux

Abstract

The objective of this work is to develop an approximate analytical solution for the transient ablation problem in a finite medium. The medium is subjected to time‐dependent boundary heat flux, i. e., q₀=at^P , and with this refined integral technique, the complicated nonlinear problem is reduced to an initial value problem, which is then solved by the Runge‐Kutta method. Results are more accurate than with the classical heat balance integral method and also indicate that the dependence of the solution on the assumed temperature profile is much weaker than is the case with the classical integral method.     

Finite elements for an optimal control problem governed by a parabolic equation

A stationary variational formulation of the necessary conditions for optimality is derived for an optimal control problem governed by a parabolic equation and mixed boundary conditions. Then a mixed finite element model with elements in space and time is utilized to solve a simple numerical example whose analytical and finite difference solutions are given elsewhere. Numerical results show that the proposed method with C^° continuity elements constitutes a powerful numerical technique for solution of optimal control problems of distributed parameter systems.     

NON-SINGULAR SOMIGLIANA STRESS IDENTITIES IN ELASTICITY

The paper presents two fully equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is C^1,α continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at corners and edges. These new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity. The resulting regularized and finite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to C^1,α continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for corners is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.     

The global element method for stationary advective problems

We extend the variation principle used in the global element method for self-adjoint elliptic problems¹, to problems containing advective terms. Used with the global element formalism, the extended principle yields a uniform framework for treating advective or boundary layer problems, with the attractive feature that the implicit treatment of interface conditions between elements yields an effective decoupling between ‘boundary layer’ and ‘interior’ parts of the solution. As a numerical example, we solve the one-dimensional model problem of Christie and Mitchell³, obtaining high accuracy for Peclet numbers up to 10⁶ with no sign of instability. These results suggest that, given a suitable choice of global elements, the decoupling is very effective in damping the oscillations found in standard finite difference or finite element treatment.     

ELASTOHYDRODYNAMIC LUBRICATION ANALYSIS OF LAYERED LINE CONTACT BY THE BOUNDARY ELEMENT METHOD

This paper presents a numerical routine to compute the contact characteristics of elastomer layered cylinders lubricated by isoviscous liquids. The indentation of the elastic layer is calculated from boundary integral equations which are solved by linear and quadratic boundary element methods for a finite plane model and a circular representation of the junction. The hydrodynamic equation is also transformed into a boundary integral equation and solved by Simpson's rule. Some factors which possibly affect numerical accuracy are examined. Examples for finite plane and circular layer are analysed with reference to parameters for printing press roller contact, in which results are obtained for the indentation, film thickness and liquid pressure, as well as internal stresses through the simultaneous solution of the elasticity and hydrodynamic equations. The results show that high precision is easily achieved and the method is efficient for such layered problems.     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

Structure of boundary conditions: The super-normal interface

We derive the general form of hydrodynamic boundary conditions for the planar super-normal interface. The results are applied to⁴He-II-solid, and to superfluid³He-vessel wall.     

A mixed boundary value problem for the unsteady Stokes system in a bounded domain in Rn

A mixed boundary value problem for the unsteady Stokes system is studied from the point of view of the theory of hydrodynamic potentials. Existence and uniqueness results as well as boundary integral representations of the classical solution are given for bounded domains having compact but not connected boundaries of class C^1,α (0<α≤1).     

The numerical calculation of turbulent boundary layer development

A mathematical model of the two-dimensional, incompressible turbulent boundary layer in arbitrary pressure gradient described and solved in earlier work¹ is solved by an implicit finite difference method. Comparisons of skin friction coefficient and momentum thickness values for the two methods for four test cases of different character indicate that the present method is a useful alternative to the method of characteristics which was used in previous work. One feature of the present method is that allowance is made for the logarithmic character of the velocity profile when constructing one of the finite difference expressions.     

A fast multipole accelerated method of fundamental solutions for potential problems

The fast multipole method (FMM) is a very effective way to accelerate the numerical solutions of the methods based on Green's functions or fundamental solutions. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million unknowns on a desktop computer. The method of fundamental solutions (MFS), also called superposition or source method and based on the fundamental solutions but without using integrals, has been studied for several decades along with the BEM. The MFS is a boundary meshless method in nature and offers more flexibility in modeling of a problem. It also avoids the singularity of the kernel by placing the source at some auxiliary points off the problem domain. However, like the traditional BEM, the conventional MFS also requires O(N²) operations to compute the system of equations and another O(N³) operations to solve the system using direct solvers, with N being the number of unknowns. Combining the FMM and MFS can potentially reduce the operations in formation and solution of the MFS system, as well as the memory requirement, all to O(N). This paper is an attempt in this direction. The FMM formulations for the MFS is presented for 2D potential problem. Issues in implementation of the FMM for the MFS are discussed. Numerical examples with up to 200,000 DOF's are solved successfully on a Pentium IV PC using the developed FMM MFS code. These results clearly demonstrate the efficiency, accuracy and potentials of the fast multipole accelerated MFS.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

A fixed domain method for diffusion with a moving boundary

Summary The one-dimensional diffusion equation for a region with one fixed boundary and one unknown moving boundary is transformed to a non-linear equation on a fixed region by using the moving boundary position as the time variable. The boundary velocity becomes a second dependent variable, with dependence only on the new time variable. An implicit finite difference scheme, marching in time, is applied to a problem with known analytic solution to demonstrate the computing speed and accuracy of this approach, and also to a problem solved previously by variable time step methods. This transformation reduces any parabolic or elliptic system of equations on a domain with moving boundary, or with unknown free surface in two space variables, to a non-linear fixed domain system which has advantages for computation.     

Application of the sweep method in solving one-dimensional equations of blood flow and pulse wave propagation in the arterial vascular system

A hydrodynamic model of blood flow in the arterial system of elastic vessels has been considered and an algorithm for its calculation based on the numerical integration of one-dimensional nonstationary hydrodynamical equations by the finite difference method is proposed. The given algorithm reduces the considered problem to a system of nonlinear algebraic equations solved by the Newton iteration method. Within the framework of this method, the linearized system of algebraic equations for the dendratic structure of vessels has been solved with the use of the sweep method. Comparison of the results of calculations with the literature data has shown that they agree with the blood flow characteristics observed in vivo during a cardiac cycle, as well as with the experimental time dependences of the blood pressure and circulation rate in vessels.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Solution of the Pontriagin–Vitt equation for the moments of time to first passage of the randomly accelerated particle by the finite element method

The Pontriagin–Vitt equation governing the mean of the time of first passage of a randomly accelerated particles has been studied extensively by Franklin and Rodemich.¹ In their paper is presented the analytic solution for the two-sided barrier problem and solutions by several finite difference procedures. This note demonstrates solution of the problem by a Petrov–Galerkin finite element method using upstream weighting functions,² shown to give rapidly convergent results. In addition, the equation is generalized to include higher statistical moments, and solutions for the first few ordinary moments are reported.     

Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack

In this paper a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r ^-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poisson's ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.