Natural boundary conditions

One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Chang, M. W., and B. A. Finlayson, Inf. J. Num. Methods Eng. 15, 935-942 (1980), and Finla-son, B. A. (1992). The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

Let us discuss the four main types of boundary conditions reflecting, absorbing, periodic, and the so-called natural boundary conditions that are much more widely used than others, especially for computer simulations. 

Natural Boundary Conditions. If the Markov process is considered in infinite interval, then boundary conditions at oo are called natural. There are two possible situations. If the considered potential at +oo or —oo tends to —oo (infinitely deep potential well), then the absorbing boundary should be supposed at Too or —oo, respectively. If, however, the considered potential at Too or — oo tends to Too, then it is natural to suppose the reflecting boundary at Too or —oo, respectively. 

More specifically, the condition that the probability flux at the boundaries is zero and the condition that the mean mixture-fraction vector is constant in a homogeneous flow lead to natural boundary conditions (Gardiner 1990) for the mixture-fraction PDF governing equation. 

This is the so-called natural boundary condition to the Fokker-Planck equation (Gardiner 1990). 

Note that w r=o = 0 is a natural boundary condition based on the other natural boundary condition that C r=o is finite. This one-dimensional diffusion problem can be solved using the method of separation of variables and Fourier series discussed in Section 3.2.7, and the specific solution can be found in Appendix A3.2.4c as... 

The particular problem illustrated here is only representative. There are many variations that could be solved using the same approaches. Certainly changing boundary conditions could have a major effect on the solutions. For example, instead of fixing the rod temperature, a more natural boundary condition might be a zero temperature gradient. By... 

The FFPE (19) contains the generalized friction constant r a and the generalized diffusion constant Ka, of dimensions [r]a] = s -2 and [i a] = cm2 s . The physical origin of these fractional dimensions will be explained in the next section. In what follows, we assume natural boundary conditions, that is, lim i-nx) W(x, t) = 0. The FFPE (19) describes a physical problem, where the system is prepared at to = 0 in the state W(x, 0). 

Before we proceed to our discussion of global stiffness matrix storage schemes, we will discuss the last aspect of the finite element implementation, namely, the application of the boundary conditions. As discussed earlier, the natural boundary conditions are imbedded in the finite element equation system - it is implied that every boundary node without an... 

The Neumann (natural) boundary condition qx = 0 is automatically satisfied. The above system of algebraic equations can easily be solved to give T) = 200, T2 = 275 and I s = 300. A comparison between the analytical finite element solutions is shown in Fig. 9.7. As can be seen, the agreement is excellent. 

Once the boundary conditions are applied, the pressure field can be solved using the appropriate matrix solving routines. Note that for mold filling problems, there is a natural boundary condition that satisfies no flow across mold boundaries or shear edges, dp/dn = 0. Once the pressure field has been solved, it is used to perform a mass balance using eqn. (9.144) or (9.145). Once the flowrates across nodal control volume boundaries are known, a simulation program updates the nodal control volume fill factors using... 

A special treatment has to be applied to the central gridpoint n = 0. The singularity arising in the second term is overcome by imposing the natural boundary condition that the first order derivative vanishes at n = 0. By introducing a fictitious gridpoint r0 = ri - h, this condition may be approximated by the second order centered-differ-ence scheme ... 

Stokes equations, Dirichlet and Neumann, or essential and natural, boundary conditions may be satisfied by different means. 

A linked grouping of neighbouring atoms (a concept defined more fully in the following Section) as well as individual atoms are bounded by a zero-flux surface. Such a surface may be used to define a Wigner—Seitz cell in a solid, a solute in a solution, or a molecule in a molecular crystal. The zero-flux surface, eqn (2.9), is the natural boundary condition for a system defined in real space and such a surface can always be used to define the physically... 

One may obtain eqn (5.61a or b) as the Euler equation in the variation of without imposing any prescribed boundary conditions on >j(r), the change or variation of ip r). This is accomplished by introducing the natural boundary conditions (Courant and Hilbert 1953). The necessary condition for to be stationary is that its first variation as given in eqn (5.59)... 

The trial functions representing variations in are given by eqn (5.69) and substitution of (r) for into n] yields fi]- At the point of variation, = and fJ] equals fi]. The variations 5ij/ and dij/ are not given prescribed values on any of the boundaries, including the boundary of the subsystem. Instead only the natural boundary condition, that V,t/ nj and Vji/ n, together with ij/ and, vanish on all infinite boundaries, will be invoked. The functional [(, fi] is to be varied not only with respect to however, but also with respect to the surface defining the subsystem fJ. Only by having the surface itself considered to be a function of

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The non-vanishing of the flux of a quantum mechanical current in the absence of a magnetic field is what distinguishes the mechanics of a subsystem from that of the total system in a stationary state. The flux in the current density will vanish through any surface on which i// satisfies the natural boundary condition, Vi/ n = 0 (eqn (5.62)), a condition which is satisfied by a system with boundaries at infinity. Thus, for a total system the energy is stationary in the usual sense, 5 [i/ ] = 0, and the usual form of the hyper-virial theorem is obtained with the vanishing of the commutator average. 

Variation of the action integral with respect to T, subject to the natural boundary condition that V, I -n = 0 on the surface at infinity and the condition that the variations in T vanish at the time end-points, yields as the equation of motion the Euler-Lagrange equation of the variation. This statement of Schrodinger s equation, the one appropriate for use when the system is in the presence of electric and magnetic fields, is... 

The solutions obtained are not completely independent of the rest of the space and this manifests itself through the boundary conditions For an unconfined system we mean that the radial coordinate is completely unrestricted but natural boundary conditions still exist. For example, in the usual treatments we require the wavefunctions to be finite at r = 0 and to be asymptotically zero as r -> oo. For confined systems one or both of these boundary conditions may be changed. 

These conditions are known as the natural boundary conditions. In either case, the optimal function satisfies the Euler-Lagrange equation. 

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