Evaluating the analytic solution to a three‐dimensional boundary value problem in elasticity using Shanks’ transformation

1994 ◽  
Vol 95 (4) ◽  
pp. 2022-2028
Author(s):  
Alexandria Oakes
Keyword(s):  
Boundary Value Problem ◽  
Analytic Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Dimensional Boundary

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

scholarly journals An approximate analytic solution of a particular boundary value problem

2001 ◽  
Vol 27 (8) ◽  
pp. 513-520
Author(s):  
Ugur Tanriver ◽  
Aravinda Kar
Keyword(s):  
Boundary Value Problem ◽  
Heat Conduction Equation ◽  
Analytic Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Welding Process ◽  
Approximate Analytic Solution ◽  
Analytic Expression ◽  
Steady State Heat ◽  
Laser Welding Process

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the wholex′y′-plane and finite inz′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution.

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

1999 ◽  
Vol 6 (6) ◽  
pp. 517-524
Author(s):  
M. Basheleishvili
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Mixture Theory ◽  
Potential Method ◽  
Fredholm Integral Equations ◽  
Infinite Domains ◽  
Elastic Mixture ◽  
Uniqueness Of The Solution ◽  
Dimensional Boundary

Abstract The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.

Sharp uniform bounds for steady potential fluid-Poisson systems

1997 ◽  
Vol 127 (3) ◽  
pp. 479-516 ◽  
Author(s):  
Irene M. Gamba
Keyword(s):  
Boundary Value Problem ◽  
Smooth Solution ◽  
Gas Flow ◽  
Boundary Value ◽  
Three Dimensional ◽  
Flow Models ◽  
Dissipation Term ◽  
Dimensional Boundary ◽  
Outflow Boundary ◽  
Steady Potential

We consider steady potential hydrodynamic-Poisson systems with a dissipation term (viscosity) proportional to a small parameter v in a two- or three-dimensional bounded domain. We show here that for any smooth solution of a boundary value problem which satisfies that the speed, denoted by |∇φv|, has an upper coarse bound , uniform in the parameter v, then a sharper, correct uniform bound is obtained: the viscous speed |∇φv| is bounded pointwise, at points x0 in the interior of the flow domain, by cavitation speed (given by Bernoulli's Law at vacuum states) plus a term of that depends on . The exponent is β = 1 for the standard isentropic gas flow model and β = 1/2 for the potential hydrodynamic Poisson system. Both cases are considered to have a γ-pressure law with 1<γ<2 in two space dimensions and 1 < γ< 3/2 in three space dimensions. These systems have cavitation speeds which take not necessarily constant values. In fact, for the potential hydrodynamic-Poisson systems, cavitation speed is a function that depends on the potential flow function and also on the electric potential.In addition, we consider a two-dimensional boundary value problem which has been proved to have a smooth solution whose speed is uniformly bounded. In this case, we show that the pointwise sharper bound can be extended to the section of the boundary ∂Ω\∂3Ω, where ∂3Ω is called the outflow boundary. The exponent β varies between 1 and 1/8 depending on the location of x0 at the boundary and on the curvature of the boundary at x0. In particular, our estimates apply to classical viscous approximation to transonic flow models.

Linear elastostatics

2020 ◽  
pp. 128-144
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
General System ◽  
Static Case ◽  
Field Equations ◽  
Basic Field ◽  
Linear Elastostatics ◽  
Dimensional Boundary ◽  
Uniqueness Of A Solution

In this chapter the basic field equations in terms of displacement, strain, and stress, and typical boundary conditions which are necessary to formulate a complete three-dimensional boundary-value-problem for linear elasticity in the static case (i.e., neglect of inertial terms) are stated. The uniqueness of a solution to such a boundary value problem is discussed. There are a number of alternate ways that one can approach the statement of an elastostatic boundary-value-problem. The first major approach is obtained by reducing the set of field equations by expressing them solely in terms of the displacement field --- the Navier equations --- while in the second major approach the general system of equations may be reformulated by eliminating the displacement and strain fields and casting the system solely in terms of the stress field --- the Beltrami-Michell equations. The special formulation of idealized two-dimensional boundary value problems is presented the two basic theories of plane strain and plane stress for isotropic materials are discussed.

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