Boundary conditions, numerical

The two-dimensional distinct element code, UDEC (Itasca, 2000) is applied for the modeling of mechanical and hydraulic behaviours. UDEC simulates the response of a fractured rock mass represented as an assemblage of discrete deformable blocks, subjected to the mechanical stress and hydraulic pressure boundary condition. Numerical experiment consists of 1) generation of a Discrete Fracture Network (DFN) as a geometric model, 2) application of various stress conditions, and 3) application of fluid boundary condition and calculation of equivalent permeability. 

A detailed theoretical background and the techniques of NEMD calculations are given in the monograph of Evans and Morriss. More recent algorithms and applications can be found in review articles. Useful information about NEMD is also provided in the monograph by Hoover. The basic methods of equilibrium MD (e.g., periodic boundary conditions, numerical integrators) are described in the monograph by Allen and Tildesley. ... 

It is not yet known how T (ro) behaves in the vicinity of a solid boundary, when translational invariance is broken. The compact support of the weighting function A suggests that is a local correction and therefore largely independent of macroscopic boundary conditions. Numerical simulations with periodic boundary conditions (Sect 4.6) show that g is independent of system size and fluid viscosity. The weak system-size dependence reported in [140] is entirely accounted for by the difference between the periodic Green s function [165] and the Oseen tensor. Thus, in a periodic unit cell of length L, (292) requires a correction of order 1/L [19],... 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Often in numerical calculations we detennine solutions g (R) that solve the Scln-odinger equations but do not satisfy the asymptotic boundary condition in (A3.11.65). To solve for S, we rewrite equation (A3.11.65) and its derivative with respect to R in the more general fomi ... 

Numerical solution of this set of close-coupled equations is feasible only for a limited number of close target states. For each N, several sets of independent solutions F.. of the resulting close-coupled equations are detennined subject to F.. = 0 at r = 0 and to the reactance A-matrix asymptotic boundary conditions,... 

Renardy, M., 1997. Imposing no boundary condition at outflow why does it work Int. J. Numer. Methods Fluids TA, 413-417. 

Sani, R. L. and Gresho, P. M., 1994. Resume and remarks on the Open Boundary Condition Mini-symposium, Int. J. Numer. Fluids 18, 983-1008,... 

Tanner, R.I. 2000. Engineering Rheology, 2nd edti, Oxford University Press, Oxford. Taylor, C., Ranee, J. and Medwell, J. O., 1985. A note on the imposition of traction boundary conditions when using FEM for solving incompressible flow problems. Comnmn. Appl. Numer. Methods 1, 113-121. 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

These four equations, using the appropriate boundary conditions, can be solved to give current and potential distributions, and concentration profiles. Electrode kinetics would enter as part of the boundary conditions. The solution of these equations is not easy and often involves detailed numerical work. Electroneutrahty (eq. 28) is not strictly correct. More properly, equation 28 should be replaced with Poisson s equation... 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

With these two-point boundary conditions the dispersion equation, Eq. (23-50), may be integrated by the shooting method. Numerical solutions for first- and second-order reaciions are plotted in Fig. 23-15. 

B. Engquist and A. Majda, Absorbing Boundary Conditions for the Numerical Simulation of Waves, Math. Comput. 31, No. 139 (1977). 

The original particle mesh (P3M) approach of Hockney and Eastwood [42] treats the reciprocal space problem from the standpoint of numerically solving the Poisson equation under periodic boundary conditions with the Gaussian co-ion densities as the source density p on the right-hand side of Eq. (10). Although a straightforward approach is to... 

Numerous attempts have been made to develop hybrid methodologies along these lines. An obvious advantage of the method is its handiness, while its disadvantage is an artifact introduced at the boundary between the solute and solvent. You may obtain agreement between experiments and theory as close as you desire by introducing many adjustable parameters associated with the boundary conditions. However, the more adjustable parameters are introduced, the more the physical significance of the parameter is obscured. 

Numerical simulation of hood performance is complex, and results depend on hood design, flow restriction by surrounding surfaces, source strength, and other boundary conditions. Thus, most currently used method.s of hood design are based on experimental studies and analytical models. According to these models, the exhaust airflow rate is calculated based on the desired capture velocity at a particular location in front of the hood. It is easier... 

This equation, taken in conjunction with appropriate boundary conditions, can be solved analytically or numerically to give the ideal flow field. 

To start the numerical solution, initial and boundary conditions must be specified. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

In order to illustrate these ideas, consider the simplest case of a It/ box containing only L = 2 = 8 cells (in C numeration starts always with 0) and with periodic boundary conditions ... 

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