Boundary conditions Dirichlet

Figure 4. Same as Figure 3 for transverse (nonremovable) part of the ab initio 6rst-derivative coupling vector 6, obtained using the all-Dirichlet boundary conditions. 

As a first example of developments of the last Section, we now consider the electromagnetic field satisfying Dirichlet boundary conditions on parallel planes (metallic plates), normal to the -direction, at finite temperature. In this case, vl(a) is given by Eq. (25) with = — 1 (corresponding to the choice a = (/ , 0,0, i2L)) and Eq. (28) reduces to... 

In this plot, we can see that if we increase the pressure, the energy also will be increased but the rate of this increment will be different for each state. The results discussed for the PIAB model are particular situations of generalizations reported for systems confined with Dirichlet boundary conditions [2]. We must remember these results for further discussion through this chapter. Let us conclude this section with the remark that the state dependence of the effective pressure at the given value of Rc can be analogously understood in terms of the different electron densities and their derivatives at the boundaries. In most general case of atoms and molecules, scaled densities may have to be employed in order to include the excited states. In the next section, we present some basic results on such connections between wave function and electron density. 

The boundary conditions on ipni(r) are determined by the boundary conditions of R i(r). Because R,/(r) is finite in the origin, then i/rn/(0) = 0. Furthermore, as we have a potential wall of infinite height, similar to that found in the PIAB, the resulting wave function on the surface of this wall must vanish. Thus, we have the Dirichlet boundary conditions for this problem... 

The multiplying factor in equation (29) is chosen so that the Dirichlet boundary condition is satisfied for each orbital density, i.e., pk(r—R)—0. 

The mass diffusivity coefficient of isobutane blowing agent from LDPE foam was found using a onedimensional diffusion model of two concentric cylinders with Dirichlet boundary conditions. An average mass diffusivity coefficient was used to calculate the mass of isobutane remaining in the foam for different boundary conditions. The influence of temperature and additives on diffusion was also examined. The use of the mass diffusivity coefficient in assessing the flammability of PE foam in the post-extrusion period is discussed. 2 refs. USA... 

Stationary-state solutions Dirichlet boundary conditions... 

The simplest boundary conditions for the catalyst pellet are those for which the concentration and temperature at the edges of the slab are specified as being equal to the respective reservoir values. These Dirichlet boundary conditions then give... 

It is generally good practice to represent the boundary conditions in residual form, even though in many cases a simple Dirichlet boundary condition could be imposed directly and not included in the y vector. For example, take the burner-face temperature specified as T (z = 0) = Tb. The residual form yields... 

Dirichlet boundary condition, when the generic variable on the boundary assumes a known and constant value ... 

In this work, we use the confined atoms model, with hard walls, to estimate the pressure on confined Ca, Sr, and Ba atoms. With this approach, we will give an upper limit to the pressure, because it is well known that the Dirichlet boundary conditions give an overestimation to this quantity. By using this approach, we obtain the profiles of some electronic properties... 

The computation of the electronic structure for each Rc is by using the KS approach with a code designed to use Dirichlet boundary conditions. In this work, we use the Perdew and Wang exchange-correlation functional [33] within the local density approximation [34], Details about this code can be found in Ref. [9] and some applications are in Refs. [35-37],... 

Melt front node (d). These are the nodes that are temporarily on the free flow front during mold filling, and are therefore partially filled (0 < /< < 1). During that specific time step this node is assigned a zero pressure boundary condition, pd = 0 (essential or Dirichlet boundary condition). 

Since we dropped the last term in the equation, we are satisfying the adiabatic boundary condition (Neumann), q(L) = 0. On the other hand, we still must consider the Dirichlet boundary condition, T(0) = T0. Since the Neumann boundary conditions is automatically satisfied, while the Dirichlet must be enforced, in the finite element language they are usually referred to as natural and essential boundary conditions. 

For pressure, the number of nodes can be lower, due to the fact that, when applying velocity Dirichlet boundary conditions, the pressure remains unknown. For pressure we can write... 

The pressure Dirichlet boundary condition, for a fully developed velocity, can be written as... 

The Dirichlet boundary conditions apply to eqns (6) and (7) since external mass transfer is neglected. Finally, the dimensionless expression for the rate of advance of the interface is ... 

Such boundary conditions, specifying the values of the solution, are known as Dirichlet boundary conditions. The so-called Neumann boundary conditions, which define the derivative of the solution on the boundaries, form another important category, considered among others later in this chapter. 

It is apparent from the first and last rows of this matrix, that again the simple Dirichlet boundary conditions, Eq. (8-3), have been considered. Since X > 0, the matrix A is positive definite and diagonally dominant. For solving system (8-28), the very efficient Crout factorization method for linear systems with tri-diagonal matrix can be applied (see Press et al. 1986, Section 2.4). 

While the Cottrell system might be regarded as the simplest possible model with a Dirichlet boundary condition (that is, in which boundary concentrations are specified), the constant current case is the simplest possible for the Neumann boundary condition, in which a concentration gradient is specified at the boundary. This model can also be called the chronopotentiometric experiment since here, the current is given and it is the electrode potential that is measured against time. Mathematically this model is defined by the usual (2.33), here with the boundary conditions... 

Boundary value — A boundary value is the value of a parameter in a differential equation at a particular location and/or time. In electrochemistry a boundary value could refer to a concentration or concentration gradient at x = 0 and/or x = oo or to the concentration or to the time derivative of the concentration at l = oo (for example, the steady-state boundary condition requires that (dc/dt)t=oo = 0). Some examples (dc/ dx)x=o = 0 for any species that is not consumed or produced at the electrode surface (dc/dx)x=o = -fx=0/D where fx=o is the flux of the species, perhaps defined by application of a constant current (-> von Neumann boundary condition) and D is its diffusion coefficient cx=o is defined by the electrode potential (-> Dirichlet boundary condition) cx=oo, the concentration at x = oo (commonly referred to as the bulk concentration) is a constant. 

---