On the convergence rate of the Kačanov scheme for shear-thinning fluids

We explore the convergence rate of the Kačanov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contracts along the sequence generated by the iteration. In addition, an a posteriori computable contraction factor is proposed, which improves, on finite-dimensional Galerkin spaces, previously derived bounds on the contraction factor in the context of the power-law model. Significantly, this factor is shown to be independent of the choice of the cut-off parameters whose use was proposed in the literature for the Kačanov iteration applied to the power-law model. Our analytical findings are confirmed by a series of numerical experiments.

Generating Set Search Method with Optimal Bind Tolerance for Solving the Economic Load Dispatch Problem of Thermal Stations

A non-derivative direct search approach called Generating Set Search (GSS) algorithm with varying bind tolerance to solve non-convex Economic Load Dispatch problem of the thermal stations in Nigeria is presented. A complete poll was carried out with initial mesh size of 1.0, expansion factor of 2.0 and contraction factor of 0.5. The binding tolerance was varied from 100 – 2200 with an increment of 100. The stopping criteria were based on the following: mesh tolerance of 0.000001, maximum iteration of 1500 and maximum function evaluation of 30000. The Economic Load Dispatch of 2500 MW, 3000 MW, 3500 MW and 4000 MW produced optimal solutions at binding tolerances of 500, 600, 1100, and 1600 respectively. The economic cost (measured in quantity of fuel) obtained for the dispatch of 2500 MW, 3000 MW, 3500 MW and 4000 MW were 83577.6936190168 MMBTU/hr, 83577.6936667599 MMBTU/hr, 83577.6937160183 MMBTU/hr and 83577.694264612 MMBTU/hr respectively. The evaluations carried out on the function in order to obtain the best solution were 1484, 5709, 6895 and 7556 for 2500 MW, 3000 MW, 3500 MW and 4000 MW of load respectively. Although the optimal bind tolerances had more iterations and evaluations, these can be traded off for the best solutions offered.<br>

Generating Set Search Method with Optimal Bind Tolerance for Solving the Economic Load Dispatch Problem of Thermal Stations

A non-derivative direct search approach called Generating Set Search (GSS) algorithm with varying bind tolerance to solve non-convex Economic Load Dispatch problem of the thermal stations in Nigeria is presented. A complete poll was carried out with initial mesh size of 1.0, expansion factor of 2.0 and contraction factor of 0.5. The binding tolerance was varied from 100 – 2200 with an increment of 100. The stopping criteria were based on the following: mesh tolerance of 0.000001, maximum iteration of 1500 and maximum function evaluation of 30000. The Economic Load Dispatch of 2500 MW, 3000 MW, 3500 MW and 4000 MW produced optimal solutions at binding tolerances of 500, 600, 1100, and 1600 respectively. The economic cost (measured in quantity of fuel) obtained for the dispatch of 2500 MW, 3000 MW, 3500 MW and 4000 MW were 83577.6936190168 MMBTU/hr, 83577.6936667599 MMBTU/hr, 83577.6937160183 MMBTU/hr and 83577.694264612 MMBTU/hr respectively. The evaluations carried out on the function in order to obtain the best solution were 1484, 5709, 6895 and 7556 for 2500 MW, 3000 MW, 3500 MW and 4000 MW of load respectively. Although the optimal bind tolerances had more iterations and evaluations, these can be traded off for the best solutions offered.<br>

Removing Thermostat Distortions of Protein Dynamics in Constant-Temperature Molecular Dynamics Simulations

Molecular dynamics simulations are widely used to determine equilibrium and dynamic properties of proteins. Nearly all simulations nowadays are carried out at constant temperature, with a Langevin thermostat among the most widely used. Thermostats distort protein dynamics, but whether or how such distortions can be corrected has long been an open question. Here we show that constant-temperature simulations with a Langevin thermostat dilate protein dynamics and present a correction scheme to remove the dynamic distortions. Specifically, ns- scale time constants for overall rotation are dilated significantly but sub-ns time constants for internal motions are dilated modestly, while all motional amplitudes are unaffected. The correction scheme involves contraction of the time constants, with the contraction factor a linear function of the time constant to be corrected. The corrected dynamics of eight proteins are validated by NMR data for rotational diffusion and for backbone amide and side-chain methyl relaxation. The present work demonstrates that, even for complex systems like proteins with dynamics spanning multiple timescales, one can predict how thermostats distort protein dynamics and remove such distortions. The correction scheme will have wide applications, facilitating force-field parameterization and propelling simulations to be on par with NMR and other experimental techniques in determining dynamic properties of proteins.

An iterative method for elliptic problems with rapidly oscillating coefficients

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.

Improved Bat Algorithm Based on Multipopulation Strategy of Island Model for Solving Global Function Optimization Problem

The bat algorithm (BA) is a heuristic algorithm that globally optimizes by simulating the bat echolocation behavior. In order to improve the search performance and further improve the convergence speed and optimization precision of the bat algorithm, an improved algorithm based on chaotic map is introduced, and the improved bat algorithm of Levy flight search strategy and contraction factor is proposed. The optimal chaotic map operator is selected based on the simulation experiments results. Then, a multipopulation parallel bat algorithm based on the island model is proposed. Finally, the typical test functions are used to carry out the simulation experiments. The simulation results show that the proposed improved algorithm can effectively improve the convergence speed and optimization accuracy.

Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

We consider an iteration method for solving an elliptic type boundary value problem {\mathcal{A}u=f}, where a positive definite operator {\mathcal{A}} is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator {\mathcal{A}_{0}} (inversion of {\mathcal{A}_{0}} is much simpler than inversion of {\mathcal{A}}), which can be viewed as a preconditioner for {\mathcal{A}}. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between {\mathcal{A}} and {\mathcal{A}_{0}}. For typical quasi-periodic structures, we establish simple relations that suggest an optimal {\mathcal{A}_{0}} (in a selected set of “simple” structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of {\mathcal{A}} admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter \frac{1}{\epsilon}.

Validation of a New Dynamic Muscle Fatigue Model and DMET Analysis

Automation in industries reduced the human effort, but still there are many manual tasks in industries which lead to musculo-skeletal disorder (MSD). Muscle fatigue is one of the reasons leading to MSD. The objective of this article is to experimentally validate a new dynamic muscle fatigue model taking cocontraction factor into consideration using electromyography (EMG) and Maximum voluntary contraction (MVC) data. A new model (Seth's model) is developed by introducing a co-contraction factor 'n' in R. Ma's dynamic muscle fatigue model. The experimental data of ten subjects are used to analyze the muscle activities and muscle fatigue during extension-flexion motion of the arm on a constant absolute value of the external load. The findings for co-contraction factor shows that the fatigue increases when co-contraction index decreases. The dynamic muscle fatigue model is validated using the MVC data, fatigue rate and co-contraction factor of the subjects. It has been found that with the increase in muscle fatigue, co-contraction index decreases and 90% of the subjects followed the exponential function predicted by fatigue model. The model is compared with other models on the basis of dynamic maximum endurance time (DMET). The co-contraction has significant effect on the muscle fatigue model and DMET. With the introduction of co-contraction factor DMET decreases by 25:9% as compare to R. Ma's Model.