Consider, for example, a Hamiltonian which is separable into two terms, one involving coordinate and the other involving coordinate. If we assume that the total wavefunction can be written in the form , where and are eigenfunctions of and with eigenvalues and , then 54 55 56 57 Going back to our original problem, Eq.
We first invoke the Born-Oppenheimer approximation by recognizing that, in a dynamical sense, there is a strong separation of time scales between the electronic and nuclear motion, since the electrons are lighter than the nuclei by three orders of magnitude.
This can be exploited by assuming a quasi-separable ansatz of the form. Fitting polar wander paths , Phys. Earth planet. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Search Menu.
Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents Summary. A normal approximation to the Fisher distribution. Step 4. Gradient approximation. The simultaneous perturbation approximation of unknown gradient was as follows:.
Step 5. Updating the estimated value of. Updating the value of to a new value of was done by using stochastic algorithm standard formulae, as follows:. Applying the results obtained in the previous steps, then values in equations 9 and 10 were replaced for obtaining new values: and. The results were used to start the next iteration of the industrial process given in. Step 6. Iteration or termination.
The algorithm was terminated if there was a small change in MSE in several successive iterations or the maximum number of iterations had been rejected. When applying the experimental procedure mentioned in this example then optimal values were found for the three industrial processes' independent variables and based on the minimum value obtained from the MSE. Once these optimal values had been obtained, verification and validation runs were performed for each manufacturing process.
Tables 1 , 2 and 3 show the results obtained for the three industrial processes, respectively. The results show that a minimum MSE in the third iteration called 2 was obtained in process one since MSE increased in the fourth iteration named 3 and thus the algorithm was stopped, as mentioned in step 6. Table 2 shows the results for process two where a minimum MSE was obtained in the initial iteration called since there were significant increases in MSE in iterations two and three and thus the algorithm was stopped.
Based on the results obtained by applying SPSAA, simulation, verification and validation tests were carried out on the 34 replicas in each process evaluated.
Table 4 shows the average results for these experiments for validating the independent variables of temperature and time, and response variables 1 and 2. A modified stochastic approximation algorithm was proposed in this research project, working with second-order models to determine the optimal value of the variables involved in 3 industrial processes. Table 4 shows the results obtained.
It was thus concluded that this algorithm provided satisfactory results regarding processes 1 and 2 when working with a second-order model in which the response variables analysed came within satisfactory parameters for achieving product quality in terms of specified final moisture content and colour.
Variable response 1 for process 3 was below the established final moisture parameter while response variable 2 "colour" was certainly successful It was also concluded that this is a simple algorithm to apply, given that it did not require a deep understanding of a particular process or of the true functional relationship between the response variable and controllable factors and it was easy to use as it did not need not be operated by highly qualified personnel.
Future research will be aimed at evaluating process capability index to measure the process' ability or aptitude. The simultaneous perturbation stochastic approximation algorithm SPSAA will be evaluated by using central composite design CCD to analyse whether better efficiency can be achieved with regarding 3k designs.
The accuracy of these calculations was checked by conservation of flux and convergence tests and by verifying that they satisfied a rigorous theorem proven by Clarke and Thiele. These accurate results were then used to determine the range of validity of the adiabatic approximation of Thiele and Katz as applied to this model system by Clarke and Thiele.
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