Closed-form Fundamental Solution Research Articles

Generalized fundamental solutions for unsteady viscous flows.

A number of closed-form fundamental solutions for generalized unsteady Oseen and Stokes flows associated with arbitrary time-dependent translational and rotational motions have been developed. These solutions are decomposed into two parts corresponding to a longitudinal wave and a transversal wave. As examples of application, the hydrodynamic forces acting on a sphere and on a circular cylinder translating in an unsteady rotating flow field at low Reynolds numbers are calculated using the generalized fundamental solutions.

Boundary integral equation solution of three-dimensional elastostatic problems in transversely isotropic solids using closed-form displacement fundamental solutions

The boundary integral equation method is used for the solution of three-dimensional elastostatic problems in transversely isotropic solids using closed-form fundamental solutions. The previously published point force solutions for such solids were modified and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.

Boundary integral equation solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form displacement fundamental solutions

The boundary integral equation method is used for the solution of three-dimensional elastostatic problems in transversely isotropic solids using closed-form fundamental solutions. The previously published point force solutions for such solids were modified and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.

The fundamental solution for shallow circular cylindrical shells: Part II: numerical computation and software

The explicit closed form fundamental solution of the elastostatic shallow circular cylindrical shell derived in Part I by Chen et al. [Int. J. Eng. Sci. to appear] is realized numerically. Major difficulties exist in approximating integrals of Hankel functions with irrational functions as arguments on the two-dimensional domain of definition. By analyzing the asymptotic characteristics of those integrals, we decompose the first quadrant of the ( x, s)-plane into five regions, and apply appropriate exact, series-truncation, or quadrature algorithms on each region to compute numerical values of the three components of the fundamental solution. Comparisons of errors are tabulated and graphics of solutions are illustrated.

Boundary element analysis of anisotropic bodies under thermo-mechanical body force loadings

Efficient boundary element formulations are presented for the analysis of anisotropic media due to common loadings such as inertial and centrifugal forces, and known temperature. The study is made comprehensive by considering both two- and three-dimensional behaviors. The anisotropy considered is of the most general form, viz. monoclinic for plane stress/strain behavior and the one with 21 independent elastic constants for the general three-dimensional case. While closed-form implicit fundamental solutions are employed for two-dimensional analysis, those for three-dimensional analysis are numerically computed. The body forces are accounted for via particular integrals so that only boundary discretization is necessary.

Elastic fields in two joined transversely isotropic solids due to concentrated forces

This paper presents closed-form fundamental solutions for the elastic fields in two joined dissimilar transversely isotropic semi-infinite solids induced by concentrated forces. Two types of concentrated forces are considered in the paper. One is the forces concentrated at a point and the other is the forces uniformly distributed along a circle in the interior of the two joined dissimilar solids. The fundamental solutions are expressed in the forms of elementary harmonic functions and the complete elliptic integrals for the concentrated point forces and the concentrated ring forces, respectively. Numerical results show that the anisotropic bi-material properties have a significant effect on the singularities of the elastic fields when the forces are located at the bi-material interface.

Modal analysis of elastic-plastic plate vibrations by integral equations

A direct boundary element method for the vibration problems of thne elastic-plastic plates is presented. Dynamic fundamental solutions of a suitably shaped finite domain are used in modal form. The series Green's functions are separated into a quasistatic and a dynamic part. Often the series of the quasistatic part can be written in a faster converging form than the equivalent modal series. Analytical integration in the vicinity of the singularity is performed on the closed form fundamental solutions of the infinite domain, and only the non-singular differences from the actual Green's functions are represented in series form. This paper gives a general formulation of this method for Kirchhoff plates on an arbitrary elastic foundation. After integration, the resulting algebraic equations are arranged in a form most convenient for a time-stepping analysis of inelastic response. This rearrangement has to be performed only once, if the time step is kept constant. Constitutive equations are integrated by an implicit backward Euler scheme for plane stress. Applications are shown for impacted circular plates on several different foundations.

DUAL RECIPROCITY BOUNDARY ELEMENT METHOD FOR THE TIME‐DEPENDENT STOKES FLOWS

This paper presents a boundary element formulation for the transient Stokes equations in which the well known closed form fundamental solution to the steady Stokes equations is employed and the time derivative is taken to the boundary with dual reciprocity method. This approach has the advantage of simplicity of formulation and implementation in relation to the alternative boundary element schemes previously presented. In addition in this paper the dual reciprocity method is presented in a more formal mathematical way using well established interpolation theories which guarantee the convergence of the method. Results are presented for a series of three‐dimensional internal problems in which the accuracy of the method is shown.

A unified approach to optimal image interpolation problems based on linear partial differential equation models

The unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed-form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal-spatial. The unknown image is reconstructed from a finite set of sampled data in such a way that a mean-square error is minimized by first expressing the solution in terms of the reproducing kernel of a related Hilbert space, and then constructing this kernel using the fundamental solution of an induced linear partial differential equation, or the Green's function of the corresponding self-adjoint operator. It is proved that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first- or second-order linear partial differential operator. An efficient method for obtaining the corresponding closed-form fundamental solutions (or Green's functions) of the operators is presented. A computer simulation demonstrates the reconstruction procedure.

Free vibrations of elastically restrained non-uniform plates

A simple and efficient method is presented to examine the Levy-type solutions for the free vibration of elastically restrained rectangular plates, resting on a non-uniform elastic Winkler foundation under the action of constant in-plane forces. Both the thickness of the plate and stiffness of the elastic foundation are uniform along one axis and variable along the other axis. The frequency equation is derived and concisely expressed in terms of the four fundamental solutions of the governing characteristic differential equation. If the closed form fundamental solutions of the system are not available, then approximate fundamental solutions can be obtained through a numerical algorithm which is shown to be efficient, convenient and accurate. The results are compared with those in the literature. Finally, the influence of the elastically restrained boundary conditions on the natural frequencies is investigated.

On linear ordinary differential equations with functionally commutative coefficient matrices

A system of linear ordinary differential equations x ̇ = A (t) x , t ∈ I , is called semiproper if A (t) A (τ)= A (τ) A (t) , t , τ ∈ I [also known as functional commutativity of A (t) ]. It is known that a semiproper system has a closed-form fundamental solution matrix X A (t)=exp∫ t A (τ)dτ , where the matrix exponential is defined by the power series exp(·)=Σ ∞ k=1 (·) k k! . Therefore the problem of solving a semiproper system amounts to that of finding a finite-form expression for the matrix exponential. Based on some recent results obtained by the authors for decomposing semiproper matrix functions, a systematic approach is developed for finding a finite-form analytical solution for the entire family of semiproper systems. This solution is then used to derive a number of important and practical stability criteria for semiproper systems. Applications of the new results are also discussed.

Applying BEM to modelling of uptaking pollutants in clouds or fog drops

A one-dimensional coupled diffusion and chemical reaction steady-state problem in spherical geometry has been considered. This type of problem is often encountered in atmospheric chemistry when dealing with uptaking of gaseous pollutants in cloud or fog drops. A closed form fundamental solution of the governing equation has been obtained. Making use of this solution and following the standard BEM approach, the boundary value problem under consideration has been transformed into a nonlinear integral equation. The discretization of this equation resulted in a set of nonlinear algebraic equations solved by the Newton-Raphson solver. For specific, linear cases the results obtained have been compared with analytical solutions available in the literature. Good accuracy has been observed. Numerical examples dealing with two coupled nonlinear equations describing diffusion with chemical reaction of two species have been included.

Analysis of non-uniform beam vibration

In this paper a systematic development of the solution theory for the non-uniform Bernoulli-Euler beam vibration, including both forced and free vibrations, with general elastically restrained boundary conditions, is presented. The frequency equation and dynamic forced response, which is shown in closed integral form, are concisely expressed in terms of the fundamental solutions of the system. If the exact, closed form fundamental solutions are not available, then approximate fundamental solutions can be obtained through a simple and efficient numerical method. The present analysis can also be applied to the vibrational analysis of a beam with viscous and hysteretic damping.

Free vibrations of a non-uniform beam with general elastically restrained boundary conditions

A simple and efficient method is presented to study the problem of free vibration of a non-uniform Bernoulli-Euler beam with general elastic restraints at boundary points. With one set of particularly chosen fundamental solutions of the system, the frequency equation is derived and expressed in concise form. If the closed form fundamental solutions of the system are not available, the approximate fundamental solutions can be obtained through a newly developed algorithm which is shown to be efficient, convenient and accurate. Finally, several limiting cases of the general system are examined and some examples are presented to illustrate the validity and accuracy of the proposed method.