Parallel tangents, method

Shah, B.V., Buehler, R.J., Kempthome, ft The Method of Parallel Tangents (PARTAN) for Finding an Optimum, Technical Report No. 2, Office of Naval Research, Contract NONR 530(05), Statistical Laboratory, Iowa State University, Ames, Apr. 1961 (revised Aug. 1962). 

Shah et al. (1964) extended these geometrical considerations to the multidimensional case, by obtaining a method called the PARTAN PARallel TANgent). A subsequent version of this method, named the continued PARTAN, is based on a gradient feature that easily allows the parallel tangents for finding the conjugate diameters to be evaluated. 

The procedure and methods for the MEP determination by the NEB and parallel path optimizer methods have been explained in detail elsewhere [25, 27], Briefly, these methods are types of chain of states methods [20, 21, 25, 26, 30, 31]. In these methods the path is represented by a discrete number of images which are optimized to the MEP simultaneously. This parallel optimization is possible since any point on the MEP is a minimum in all directions except for the reaction coordinate, and thus the energy gradient for any point is parallel to the local tangent of the reaction path. 

In the peak-tangent (P- T) method a common tangent is drawn to two neighboring maxima or minima, and the distance to the intermediate extremum value is measured parallel to the ordinate ( i, 2 h Fig 2-24). This method can be applied satisfactorily if a linear background is present, but most of the time it is better to check whether a higher order will give more exact results. 

We note that while the Newton method is the most robust and most widely used in nonlinear finite element software, it is also computationally expensive primarily due to the necessity to solve a system of linear equations. It also imposes considerable computer memory requirements since a global system matrix is used. This method also is not as easily parallelized as some other iterative methods. In order to achieve the optimal performance of the Newton method, it is crucial to calculate the tangent stiffness matrix that is indeed tangent or, in other words, is the derivative with respect to unknowns that are calculated very accurately. 

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