A NEW MATHEMATICAL MODEL FOR FINITE-DIFFERENCE SOLUTION OF THREE-DIMENSIONAL MIXED-BOUNDARY-VALUE ELASTIC PROBLEMS

2005 ◽  
Vol 02 (01) ◽  
pp. 99-126 ◽  
Author(s):  
M. ZUBAER HOSSAIN ◽  
S. REAZ AHMED ◽  
M. WAHHAJ UDDIN
Keyword(s):  
Finite Difference ◽  
Potential Function ◽  
Solid Mechanics ◽  
Present Model ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Computational Time ◽  
Comparison Of The Results

This paper describes a new mathematical formulation, specifically suitable for finite-difference analysis of stresses and displacements of three-dimensional mixed-boundary-value elastic problems. Earlier, mathematical models of elasticity were very deficient in handling three-dimensional practical stress problems. In the present model, a new scheme of reduction of unknowns is used to formulate the three-dimensional problem in terms of a single potential function, defined in terms of the three displacement components. Compared to the conventional models, the present model provides numerical solution of higher accuracy in a shorter period of computational time. The application of the potential function formulation is demonstrated here through a number of classical problems of solid mechanics, and the results are compared with the available solutions in the literature. The comparison of the results establishes the rationality of the present approach.

Finite-difference techniques for a harmonic mixed boundary problem having a reentrant boundary

1971 ◽  
Vol 323 (1553) ◽  
pp. 271-276 ◽  
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem ◽  
Mixed Boundary Problem ◽  
Finite Difference Techniques

The proof of uniform convergence of a family of finite-difference solutions to the exact solution is outlined for a harmonic mixed boundary-value problem in a rectangle containing a slit. Finite-difference results in the neighbourhood of the tip of the slit are given for reference.

STEADY FLOW OF FLUIDS WITH SHEAR-DEPENDENT VISCOSITY UNDER MIXED BOUNDARY VALUE CONDITIONS IN POLYHEDRAL DOMAINS

2000 ◽  
Vol 10 (05) ◽  
pp. 629-650 ◽  
Author(s):  
C. EBMEYER
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Steady Flow ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Partial Differential ◽  
Shear Dependent Viscosity ◽  
Polyhedral Domain ◽  
Dependent Viscosity

In this paper the system of partial differential equations [Formula: see text] is studied, where e is the symmetrized gradient of u, and T has p-structure for some p<2 (e.g. div T is the p-Laplacian and p<2). Mixed boundary value conditions on a three-dimensional polyhedral domain are considered. Ws,p-regularity (s=3/2-ε) of the velocity u and Wr,p′-regularity of the pressure π are proven.

On mixed boundary value problems for the Helmholtz equation

1977 ◽  
Vol 77 (1-2) ◽  
pp. 65-77 ◽  
Author(s):  
R. Kress ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Theoretic Approach ◽  
Mixed Boundary Value Problems

SynopsisExistence and uniqueness theorems are obtained for a class of mixed boundary value problems associated with the three-dimensional Helmholtz equation. In this context the boundary of the region of interest is assumed to consist of the union of a finite number of disjoint, closed, bounded Lyapunov surfaces on some of which are imposed Dirichlet conditions whilst Neumann conditions are imposed on the remainder. An integral equation method is adopted throughout. The required boundary integral equations are generated by a modified layer theoretic approach which extends the work of Brakhage and Werner [1] and Leis [2, 3].

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

1997 ◽  
Vol 4 (3) ◽  
pp. 243-258
Author(s):  
T. Burchuladze ◽  
Yu. Bezhuashvili
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Singular Integrals ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Mixed Boundary Conditions ◽  
Theory Of Elasticity ◽  
Classical Solutions

Abstract We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [Kupradze, Gegelia, Basheleishvili, and Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, 1979, Russian original, 1976–Mikhlin, Multi-dimensional singular integrals and integral equations, 1962], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

scholarly journals Numerical Modeling of the Process of Thermoplastic Deformation of Transversally Isotropic Parallelepipeds

2020 ◽  
Vol 9 (4) ◽  
pp. 1484-1491
Keyword(s):  
Finite Difference ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Displacement Vector ◽  
Boundary Value ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Discrete Analogue ◽  
First Approximation ◽  
Transversally Isotropic

The paper proposes a modified version of the iterative method for numerically solving a three-dimensional uncoupled boundary-value problem that describes the process of thermoplastic deformations of a transversely isotropic parallelepiped. A discrete analogue of the boundary value problem is compiled on the basis of the finite-difference method. A recurrent finite-difference relation is written which allows one to find the desired components of the displacement vector in combination with the iterative method. It is assumed that, at a first approximation, the values of the sought displacements in the internal nodes are trivial. The essence of the method is demonstrated by solving the thermoplastic boundary-value problem for a transversely isotropic parallelepiped. The proposed method can be applied to solve related problems of dynamic thermoplasticity.

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