Multi-connected Domains Research Articles

The inverse problem of potential theory in piecewise-smooth multiconnected domains

The inverse problem of potential theory in piecewise-smooth multiconnected domains

Adaptive grid generation from harmonic maps on Riemannian manifolds

In this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on Riemannian manifolds. The reliability of the method is assured by an existence and uniqueness theorem for one-to-one maps between multidimensional multiconnected domains. We formulate an adaptive Riemannian metric consistent with this theorem. Several examples demonstrating application of the developed procedure are provided.

First and second fundamental axisymmetric problems of elasticity theory for doubly-connected domains bounded by the surfaces of a sphere and a spheroid

The method of constructing solutions of the fundamental boundary-value problems for a homogeneous Lamé equation of multiconnected domains bounded by canonical surfaces of cylindrical and spheroidal coordinate systems described in /1/ is extended to domains with other geometry. The problems under consideration reduce to infinite systems of linear algebraic equations of the second kind with completely continuous opertors. A solution in the form of expansions in a small parameter for the problem of the hydrostatic pressure of a sphere with a centrally located spheroidal cavity is presented as an example.

Vorticity—velocity formulation in the computation of flows in multiconnected domains

AbstractThe thermofluid dynamic fields in two‐dimensional multiconnected domains are analysed by solving the Navier–Stokes equations with the Boussinesq approximation in the vorticity–velocity formulation. The need of an integral condition for the pressure to be single‐valued on each independent irreducible loop, in analogy with the ω–Ψ formulation, is demonstrated. The field equations are discretized by a finite difference technique and solved at the steady state via an alternating direction implicit method of a scalar type. Two test cases at low Reynolds and Rayleigh numbers are considered: the multiconnected driven cavity and an annulus with isothermal walls and stationary or rotating inner cylinder.

Boundary element method for 2D problems of ideal fluid flows with free boundaries

An approach is considered for application of an indirect BEM for solving 2D boundary value problems for ideal fluid flows in multiconnected domains. A setting is given as well as a technique for obtaining numerical solutions of direct, inverse and mixed boundary value problems. Numerical results of model and real problems are presented.

Optimal elastic domains of maximum stiffness

The problem of maximizing the stiffness (minimizing the work of the external forces) of an elastic domain of given volume is examined. Control is achieved by the shape of the domain /1–3/. A necessary Legendre condition is obtained as a result of investigating the second variation. The optimal solution can be found in the class of multiconnected elastic domains. The problem of increasing the connectedness is solved by using the Weierstrass necessary condition for a strong maximum. A dual problem is constructed to estimate the global maximum. An example is presented of the domain of maximum stiffness.

Fully developed flow in periodic multi-connected domains: Numerical and experimental study

The fluid-solid interaction force in two-dimensional multi-connected periodic domains can be determined by a local flow solution in a periodic microelement. Integration of Navier-Stokes equations in the microelement yields a unique solution in velocity field and in pressure gradient, but not in pressure itself, which is given by the sum of an unknown periodic term and an assigned linear term. The numerical solution of the equations of motion in general curvilinear coordinates is obtained by a finite difference technique, assuming periodic boundary conditions. Pressure drop measurements and flow visualization through arrays of cylinders in different configurations have been conducted for the validation of the proposed method. Comparisons of present numerical results with experimental data and other numerical solutions, for Reynolds number up to 800 and different geometrical configurations of cylinders, are presented and discussed.

Boundary integral techniques for multi-connected domains

Several boundary integral techniques are available for the computation of the solution to Laplace's equation in multi-connected domains. However, for cases where the domain is changing, such as incompressible, inviscid fluid flow with free surfaces, iterative methods are highly attractive. The paper describes one such formulation and tests it on circular and elliptic annuli. It is necessary to use interpolated quadrature points to maintain accuracy when regions of the annuli are thin.

On expansion of the possibilities of the integral transformation method in solving problems of mechanics

The applicability of the integral transform method to a boundary value problem is related to whether the boundary conditions of this problem in the variable with respect to which the transformation is performed agree with the boundary conditions of the corresponding Sturm—Liouville problem. A method is proposed for transferring the integral transform method over to the case of boundary value problems when there is no such agreement. Moreover, the range of variation of the variable with respect to which the transformation is performed is smaller than the range of determination of the corresponding Sturm-Liouville problem, and the boundary condition of the boundary value problem is mixed. A method is indicated for transferring the mentioned method to boundary value problems in multiconnected and complex domains whose contours are inscribed in the coordinate mesh under consideration. The method is illustrated in anti-plane and plane problems of elasticity theory.

Homogenization and bifurcation of perforated plates

This short paper aims to present an application of the homogenization method to a nonlinear situation. We show that the buckling and post buckling analysis of a multiperforated plate can be reduced to that of an homogeneous plate, the constitutive law of which is given. We use a nonclassical formulation of the Von Karman equations which is adapted to multiconnected domains and seems to be useful for others problems.