Often it is desirable to generate simple polynomial expressions
that will approximate TCAD models at a given
input, over a limited region in the parameter space. The polynomial
expressions are typically several orders of magnitude faster than the
model equations. They are used in lieu of the original TCAD model in
applications where computational efficiency and/or continuity requirements
are critical. For instance, their substitute role in Monte Carlo analysis to
simulate statistical manufacturing variation is well
documented [95][65][37].
The new set of equations can be written as a standard model with no input variables. Furthermore, the task of generating the approximating polynomial for each model output is independent of the other outputs which are simultaneously calculated during model evaluation. In what follows, the discussion is restricted to one output:
The process is replicated for outputs in a multiple-output TCAD model.
A polynomial function in the parameters 
, 
, 
,
is a linear combination of products of the powers of 
's. It can be
written as:
where 
is the number of terms in the model, 
is the constant
coefficient multiplying the 
th term 
. Each is a product
of powers of 's:
where 
is the exponent of the parameter 
in the th term
product: 
, 
being a small positive number.
The order of the th term, 
, is calculated by summing the values
of its exponents:
Finally, the degree of a polynomial is equal to the highest order in its terms.
In practice small order (
) polynomials are used. Let:
be the vector of coefficients, and
be the matrix of exponents of all terms.
The model building algorithm objective is to determine the number of
terms and the values of the elements of 
and 
.
The first step consists of identifying the order of dependency between The
model output and each of the model parameters. This can be accomplished
using some initial screening algorithm [95][65][15]
or by user input based on expectation and/or knowledge of the underlying
physical theory. In general, the higher the order of
dependency the more time consuming the building process will be as more model
evaluations are required.
Based on the dependency information an experimental
run matrix is generated. Each row in the run matrix represent one
point in the model parameter space. By evaluating the TCAD model at each
of these points the required fitting data is generated.
The number of terms and the corresponding coefficients are then determined by
polynomial approximation with a minimum number of terms. The experimental
matrix generation and the polynomial approximation are described next.