Contrato por tiempo y materiales.

5.2 Recommendations

To get a better picture a priori of the final mesh, including all of the different

levels, more work is required in the area of error estimators. Perhaps through the

use of a better error estimator, improvements can be made in realizing more of the

potential storage savings.

The actual storage requirement can also be reduced by easing the uniformity

restrictions of the sub-domains. Since the contaminant plume is rounded and this

adaptive method ensures that each sub-domain is rectangular, it follows that some

of the added degrees of freedom are extraneous and do not enhance the quality

of the solution. Allowing non-rectangular grids would complicate the coding, but

perhaps a better algorithm could be developed that would make this more feasible.

As previously mentioned, error is introduced by this method in imposing bound¬

ary conditions on the sub-domains. This error could be reduced if better estimates

of the nodal values or derivatives normal to the boundaries were used. Setting

second-kind boundary conditions was chosen for use in this method in order to sim¬

plify the coding procedure. One approach that could be taken is to estimate the

derivatives normal to the boundaries by taking the average of the derivatives in the

appropriate direction across each of the two elements bordering a given boundary.

For the two problems used in the validation of the model, this was not attempted

since the derivatives at the boundaries were Hmited to very small numbers. Yet it

was shown that the error increases sharply as the derivatives at the boundaries are

allowed to increase. By using such an estimation procedure, perhaps the maximum

allowable derivative at the boundaries can be increased without significant loss of

accuracy. Another approach that could be taken is to use the values of the nodes on

the sub-domain boundaries as first-kind boundary conditions. The same procedure

would be used to determine the sub-domain boundaries, but instead of estimating

the derivatives, the values from the most current level solution could be used to impose first-kind boundary conditions.

The disadvantage of these estimation procedures is that they would require in¬ terpolation at the irregular nodes both spatially and temporally, which might cause problems for grids with higher degrees of irregularity. That is, some type of con¬ strained approximation would be necessary in order to impose boundary conditions at the nodes where no information is available at the previous level or previous time step. However, the error introduced by these approximations would be minimal if the nodal values along the boundaries were fairly constant, which is the case in the tails of the plume. Thus, by not allowing the sub-domains to shrink smaller than the plume, this approximation error might be minimized.

6 Appendices

6.1 Appendix A: The Finite Element Method - Gener^il

In the variational approach, the problem of finding a solution of the governing

differential equation is replaced by an equivalent variational problem that consists of

finding an unknown function that extremizes (makes stationary) a certain integral

quantity, subject to the prescribed boundary conditions. Such an integral is called a "functional" because it is a function of the unknown fimction. The two problem

statements are equivalent in that an exact solution of one is also the solution of the

other. This equivalence is demonstrated using the calculus of variations. To solve

the variational problem using the finite element method, the region V is discretized into m finite elements and let the subregion of a typical element e be denoted

V^. Within this subregion, D*, the unknown function u is approximated by a trial

function u, which is given by

n

u = ^NiUi (9)

where Ni are linearly independent functions selected a •priori and referred to as

"basis (or interpolation or shape) functions," u,- are values of u at the nodes of the

element, and n* is the number of nodes assigned to element e. By assuming that

the total functional is equal to the sum of the functionals for all individual elements

and then extremizing the total functional, we get a set of n^ equations for each

element, e, which can be expressed in matrix form and characterizes the behavior

of element e. After assembly of all the elements, the global matrix equation is obtained. This set of simultaneous algebraic equations is then solved to give values of u at each of the nodes in the region V. These values represent the approximating

solution to the variational problem. The preceding finite element approximation

may be regarded as a variant of the well-known classical Rayleigh-Ritz procedure

for obtaining an approximate solution of the variational problem. The distinction

between the Rayleigh-Ritz and finite element methods lies in the definition of basis

functions. In the Rayleigh-Ritz method, the basis functions are defined in the

entire region P, whereas in the FEM these functions are defined piecewise (element

by element). The basis functions for the Rayleigh-Ritz method are required to

satisfy the essential boundary conditions of the problem, whereas the piecewise

basis functions for the FEM need to satisfy certain continuity and completeness

conditions. Because the Rayleigh-Ritz method uses functions defined over the entire

region, it can be used only for regions of relatively simple geometric configuration. In

the FEM, the same geometric limitations exist, but only for each element. Because

these simply shaped elements can be assembled to represent exceedingly complex

geometries, the FEM method is a far more versatile tool than the classical Rayleigh-

Ritz method.

Practical problems axe frequently encountered for which the classical function-

als cannot be derived, or, in other words, the variational principles do not exist. For

these cases a more general approach in formulating the element matrix equations

has to be employed. The Galerkin method is one such approach and has been widely

used. It is a special case of the method of weighted residuals (MWR). Considering

a continuum problem governed by the differential equation

L(u)-/ = 0 (10)

in the region V enclosed by the boundary B., we can approximate the unknown

function u by a trial function, u, of the form given by Equation (9) . Because the