Multi time-step integrators

M. Klisinski, A. Mostrom, On stability of multi-time step integration algorithm, J. Eng. Mech. ASCE, 124 (1998) 783. 

Note that Eqs. (5.4) and (5.5) are coupled via the mean fields and the matrix elements. In practice, the EOMs are decoupled during an update time step, by the so called constant mean-field approach. Nevertheless, the evaluation of the EOMs (5.4) and (5.5) is rather costly as the calculation of the mean fields and matrix elements involves multi-dimensional integrals over all physical DOF. A key ingredient of the MCTDH algorithm therefore is that the Hamiltonian operator can be expressed in terms of products of low-dimensional terms such that... 

In a multistep integration method, the update rule also depends upon the values ofx at times in the immediate past, up to a horizon length ih- Imphcit multi-step integration methods are of the general form... 

The experimental data is concentration-time data, and integrated rate expressions give the observed rate constants. The technique can be extended to multi-step equilibria and to equilibrium reactions with consecutive steps. For multi-step equilibria, analysis results in a sum of exponentials, which may or may not reduce to first order. 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

The ability to accurately model die pattern evolution as discussed in this paper provides a solution applicable to the ran by run control of multi-product patterned wafers [13]. As shown in Fig. 10, a feedback control loop incorporating the integrated density and step-height pattern dependent model was developed. For each device type, an appropriate set of model parameters (including effective blanket rate BR and planarization length) were determined. The model for the effective blanket rate includes a term Delta(n) that is updated on each run n to track the tool drift in rate over time due to pad and consumable wear ... 

Creating a safety culture takes time. It is frequently a multi-year process. A series of continuous process improvement steps can be followed to create a safety culture. Management and employee commitment are hallmarks of a true safety culture where safety is an integral part of daily operations [7]. 

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