2.3 Empirical Model Building



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2.3 Empirical Model Building

Often it is desirable to generate simple polynomial expressionsgif 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.




next up previous contents index
Next: 2.3.1 Run Matrix Generation Up: 2 Mathematical Considerations Previous: 2.2.2 Sequential Quadratic Programming



Martin Stiftinger
Tue Aug 1 19:07:20 MET DST 1995