Energy step size

Unfortunately, discretization methods with large step sizes applied to such problems tend to miss this additional force term [3]. Furthermore, even if the implicit midpoint method is applied to a formulation in local coordinates, similar problems occur [3]. Since the midpoint scheme and its variants (6) and (7) are basically identical in local coordinates, the same problem can be expected for the energy conserving method (6). To demonstrate this, let us consider the following modified model problem [13] ... 

To be more precise, this error occurs in the limit /c — oo with Ef = 0(1) and step-size k such that k /ii = const. 3> 1. This error does not occur if Ef = 0 for the analytic problem, i.e., in case there is no vibrational energy in the stiff spring which implies V,. = U. 

Note that there are also variations in total energy which might be due to the so called step size resonance [26, 27]. Shown are also results for fourth order algorithm which gives qualitatively the same results as the second order SISM. This show that the step size resonances are not due to the low order integration method but rather to the symplectic methods [28]. 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

The Fe K-edge spectra were recorded in the transmission mode and a metallic iron foil spectrum was measured simultaneously with each sample spectrum for energy calibration. X-ray absorption spectra for each sample were collected from 7,520 to 8,470 eV, with a step size of 0.40 eV and acquisition times of ca. 68 s per sample. Measuring each sample, in turn, and repeating... 

The XPS data were acquired on a Physical Electronics model 5400 XPS system using a Mg anode. For survey spectra, the pass energy was 44.75 eV with a step size of 0.5 eV. The time per step was 50 msec. High resolution spectra were acquired with a pass energy of 35.75 eV and step size of 0.1 eV, The time per step was 50 msec. Thermogravimetric data were obtained on a Perkin-Elmer, Diamond Thermogravimetric/ Differential Thermal Analyzer (TG/DTA) with Pyris software, version 7.0-0.0110. 

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