Mead Method Research Articles

Determination of the Kinetics of the Ethoxylation of Octanol in Homogeneous Phase

A two-dimensional computational fluid dynamics model was utilized to test different kinetic models for the description of the anionic polymerization of octanol with ethylene oxide in a microreactor. The reaction was performed as a continuous reaction under elevated pressure and temperatures in a one-phase system. The kinetic parameters were determined with numerical methods by reducing the deviation to experimental data based on the Nelder–Mead method. Four different reaction models with one, two, three, and four different rates for the first propagation steps were tested. The best agreement with the experimental data was found for the four rate model, with a prediction accuracy close to the experimental error. The gathered data suggests an increasing reaction rate for the first four propagation steps, which is in agreement with the Weibull–Tornquist effect [Weibull and Tornquist. Berichte vom VI. Internationalen Kongress fur Grenzflachenaktive Stoffe, Zurich, vom 11. bis 15. September 1972; Carl Hanser V...

Design of reduced search space strategy based on integration of Nelder–Mead method and pattern search algorithm with application to economic load dispatch problem

This paper presents a novel hybrid technique for the solution of economic load dispatch problem with valve point loading effect using Nelder–Mead (NM) simplex method and pattern search (PS) algorithm. Strength of globalized NM optimization algorithm has been employed to explore the search space for near optimal solution, and PS algorithm is used in combination with a search space reduction strategy, incorporating the principles of selection and stochastic reproduction, to fine-tune the result. The proposed technique has been applied to three different systems having 3, 13 and 40 generating units to demonstrate the application for small to large load dispatch set-up. The efficacy of the design scheme is established from comparison of the results with the state-of-the-art solvers, and it is found that the proposed scheme gives the best result in terms of mean cost while the average computational time is less than most of the reported methods.

Определение значений граничных концентраций растворенных в масле газов методом минимального риска

The article describes a method for determining the limit concentrations of gases dissolved in oil at which the minimal possible risk is achieved in case of making erroneous decisions. The limit concentrations of gases dissolved in oil are suggested to be determined by minimizing the average risk function for multidimensional distributions using the Nelder—Mead method. The average risk values accompanied by using the limit gas concentrations stipulated by the well-known international and national procedures are subjected to a comparative analysis. The effect the values of distribution law parameters, the flaw occurrence probabilities, and the erroneous decision cost ratios have on the limit concentration values is investigated. It has been revealed that the limit gas concentration values at which the minimal average risk is achieved are not constant, but vary depending on the values of gas distribution laws’ parameters for healthy transformers and for transformers containing developing flaws, on the values of erroneous decision costs, and on the probabilities of defective and flawless state of transformers. It follows from the above-mentioned findings that the limit gas concentration values at which the minimal risk is achieved should be determined taking into account the above-mentioned factors.

A sharp interface method for coupling multiphase flow, heat transfer and multicomponent mass transfer with interphase diffusion

Mixing of partially miscible fluids plays an important role in many physical and chemical processes. The modeling complexities lie in the tight coupling of the multiphase flow, heat transfer and multicomponent mass transfer, as well as diffusions across the phase interface. We present a sharp interface method for modeling such process. The non-ideal equation of state is used to compute the fluid properties such as density, fugacity and enthalpy, and to predict phase equilibrium composition. The phase interface location is tracked using the phase propagation velocity. A third-order one-sided finite difference scheme using a variable grid size according to the interface location is utilized to discretize the partial derivatives immediately next to the interface, while a second-order central scheme is used for the bulk of fluids. An optimization method, the Nelder–Mead method, is applied to search for (1) the phase compositions on both sides of the interface, and (2) the phase propagation velocity based on the coupling of the multicomponent phase equilibrium and the species' balance across the interface. The temperature at the interface is determined by the energy balance. Numerical results are used to demonstrate the convergence of our method and show its capability to simulate the mixing of multicomponent partially miscible fluids.

Construction of pore network models for Berea and Fontainebleau sandstones using non-linear programing and optimization techniques

Pore network models (PNM) of Berea and Fontainebleau sandstones were constructed using nonlinear programming (NLP) and optimization methods. The constructed PNMs are considered as a digital representation of the rock samples which were based on matching the macroscopic properties of the porous media and used to conduct fluid transport simulations including single and two-phase flow. The PNMs consisted of cubic networks of randomly distributed pores and throats sizes and with various connectivity levels. The networks were optimized such that the upper and lower bounds of the pore sizes are determined using the capillary tube bundle model and the Nelder–Mead method instead of guessing them, which reduces the optimization computational time significantly. An open-source PNM framework was employed to conduct transport and percolation simulations such as invasion percolation and Darcian flow. The PNM model was subsequently used to compute the macroscopic properties; porosity, absolute permeability, specific surface area, breakthrough capillary pressure, and primary drainage curve. The pore networks were optimized to allow for the simulation results of the macroscopic properties to be in excellent agreement with the experimental measurements. This study demonstrates that non-linear programming and optimization methods provide a promising method for pore network modeling when computed tomography imaging may not be readily available.

Optimizing the Heck–Matsuda Reaction in Flow with a Constraint-Adapted Direct Search Algorithm

The optimization of a palladium-catalyzed Heck–Matsuda reaction using an optimization algorithm is presented. We modified and implemented the Nelder–Mead method in order to perform constrained optimizations in a multidimensional space. We illustrated the power of our modified algorithm through the optimization of a multivariable reaction involving the arylation of a deactivated olefin with an arenediazonium salt. The great flexibility of our optimization method allows to fine-tune experimental conditions according to three different objective functions: maximum yield, highest throughput, and lowest production cost. The beneficial properties of flow reactors associated with the power of intelligent algorithms for the fine-tuning of experimental parameters allowed the reaction to proceed in astonishingly simple conditions unable to promote the coupling through traditional batch chemistry.

Nature-inspired computational intelligence integration with Nelder–Mead method to solve nonlinear benchmark models

In the present study, nature-inspired computing technique has been designed for the solution of nonlinear systems by exploiting the strength of particle swarm optimization (PSO) hybrid with Nelder–Mead method (NMM). Fitness function based on least square approximation theory is developed for the systems, while optimization of the design variables is performed with PSO, an efficient global search method, refined with NMM for rapid local convergence. Sixteen variants of the proposed hybrid scheme PSO-NMM have been evaluated on five benchmark nonlinear systems, namely interval arithmetic benchmark model, kinematic application model, neurophysiology problem, combustion model and chemical equilibrium system. Reliability and effectiveness of the proposed solver have been validated after comparison with the results of statistical analysis based on massive data generated for sufficiently large number of independent executions.

A directed tabu search method for solving controlled Volterra integral equations

This paper proposes a pseudo-spectral scheme for obtaining approximate optimal control and state. This computational scheme represents the solution of the optimal control problem (OCP) by an mth degree Lagrange interpolating polynomial, using Legendre nodes. Then, OCP of non-linear Volterra integral equation is transformed into an optimization problem. Directed tabu search (DTS) method is utilized to derive the solutions of the optimal control and state as well as the optimal value of the objective function. In the DTS method, two neighborhood-local search strategies based on Nelder–Mead method and adaptive pattern search are applied. In addition, a tabu list with anti-cycling rules, the so-called tabu regions and semi-tabu regions are used. In addition, diversification and intensification search schemes are employed. DTS is able to converge to the global optimum solutions of a set of numerical examples. Therefore, some good balance between the diversification and intensification ensures a faster and efficient convergence to get quality solutions. Numerical examples are provided which confirm the reliability and efficiency of the proposed method. Moreover, a comparison is made with optimal solutions obtained by the other numerical methods in the literature.

A hybrid cuckoo search algorithm with Nelder Mead method for solving global optimization problems.

Cuckoo search algorithm is a promising metaheuristic population based method. It has been applied to solve many real life problems. In this paper, we propose a new cuckoo search algorithm by combining the cuckoo search algorithm with the Nelder–Mead method in order to solve the integer and minimax optimization problems. We call the proposed algorithm by hybrid cuckoo search and Nelder–Mead method (HCSNM). HCSNM starts the search by applying the standard cuckoo search for number of iterations then the best obtained solution is passing to the Nelder–Mead algorithm as an intensification process in order to accelerate the search and overcome the slow convergence of the standard cuckoo search algorithm. The proposed algorithm is balancing between the global exploration of the Cuckoo search algorithm and the deep exploitation of the Nelder–Mead method. We test HCSNM algorithm on seven integer programming problems and ten minimax problems and compare against eight algorithms for solving integer programming problems and seven algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

Optimization Design of PID Controller Parameters Based on Nelder Mead Algorithm for Stepper Motors

This paper presents a design of optimized PID controller for stepper motor system used in robotics. Currently, PID controller has been used to operate in stepper motor system because its structure is simpler compared to others. However, the issue of tuning and designing PID controller adaptively and efficiently is still open. This paper presents an improved PID controller efficiency from tuning by Nelder Mead method. The parameters of PID controller shall be obtained from the Nelder Mead optimization procedure. Errors between desired magnitude response and actual magnitude response are calculated by using the Integral of Absolute Error (IAE). The proposed Nelder Mead based PID design method is simpler, more efficient and effective than the existing traditional methods included Ziegler Nichols, PSOGSA and Genetic algorithm (GA). Simulation result shows that the performance of PID controller using this proposed method is better than traditional methods and resistant to disturbance.

Modeling of Dispersive Materials Using Dispersion Models for FDTD Application

A nonlinear optimization algorithm based on the Nelder Mead method is used to characterize the frequency dependent permittivity of 15 materials based on Lorentz, modified Lorentz, Drude and Lorentz-Drude models. The optimized model parameters are used to calculate the complex relative permittivity of each material and compare it with experimental data. In each case, a very good match is found between the optimized and experimental data over a long wavelength range. Comparative study of the models used for each material is performed based on accuracy, wavelength range of applicability and computational efficiency. The parameters presented can be used for computer simulation of electromagnetic wave phenomena involving these materials.

Evolving a Nelder-Mead Algorithm for Optimization with Genetic Programming.

We used genetic programming to evolve a direct search optimization algorithm, similar to that of the standard downhill simplex optimization method proposed by Nelder and Mead ( 1965 ). In the training process, we used several ten-dimensional quadratic functions with randomly displaced parameters and different randomly generated starting simplices. The genetically obtained optimization algorithm showed overall better performance than the original Nelder-Mead method on a standard set of test functions. We observed that many parts of the genetically produced algorithm were seldom or never executed, which allowed us to greatly simplify the algorithm by removing the redundant parts. The resulting algorithm turns out to be considerably simpler than the original Nelder-Mead method while still performing better than the original method.

Fractional Order PI Controller Design for Non-Monotonic Phase Systems

Abstract In this paper, a fractional order proportional-integral (FOPI) controller is proposed for controlling the non-monotonic decreasing phase DC-buck regulator system. The parameters of FOPI controller are optimized by Nelder's-Mead (NM) method. The FOPI controller provides fast closed-loop performance as well as improves the robust properties of the system in time and frequency domain. The controller preserves the monotonic phase between the desired bandwidth and improves the characteristic performance of the system. The proposed method is validated by comparing the time domain as well as frequency domain characteristics with the integer order proportional-integral (IOPI) controllers tuned using various techniques.

Controlling of Local search Methods’ Parameters in Memetic Algorithms Using the Principles of Simulated Annealing

In simulated annealing the probability of transition to a state with worse value of objective function is guided by a cooling schedule. The more iterations are spent, the more strict the acceptance probability function becomes. In the end of the optimization process, the probability of transfer to worse state approaches zero. In this paper the principles of cooling schedules are used to control the parameters of local search methods in the memetic algorithm. The memetic algorithm in this paper is a combination of genetic algorithm, Hooke-Jeeves method, Nelder-Mead simplex method and Dai-Yuan version of nonlinear conjugate gradient method. The controlled parameter of Hooke-Jeeves method is the radius r, for Nelder Mead method the size of edge of the simplex and the length of step for nonlinear conjugate gradient method.

Effective methods of parameter identification for creep models with account of III stage

The account of tertiary (third) stage of creep is important for the failure analysis of structures. The correct description of third stage of creep is based on taking into account of damage accumulation process. An identification of the parameters of coupled creep-damage model leads to the optimization problem of finding the global minimum of a functional, having a complex, non-linear structure and a large number of local minimums. The Nelder–Mead method is applied to the solution to the multiparameter identification problem. The effectiveness and robustness of the applied approach are illustrated on examples of mechanical and civil engineering.

Determining the mechanical characteristics of nonlinearly elastic materials by vibrational tests

The equations of free vibration of a nonlinear system are solved numerically by the fourth-order Runge–Kutta method on the basis of vibrational tests of a rigidly fixed PVC cantilever beam. The solution is optimized by the Nelder–Mead method in terms of the key parameters. The dissipation and nonlinearity of the material are determined.

Use of depth-first search and direct search methods to optimize even-aged stand management: a case study involving maritime pine in Asturias (northwest Spain)

Maritime pine (Pinus pinaster Ait.) is one of the most important timber species in Asturias and more generally in northwest Spain. A dynamic growth model has recently been developed for this species and region, allowing computation of the merchantable volume by two alternative methods: a disaggregation system and a stand volume ratio function. The model enables optimization of the management schedule for the species by modifying the rotation age and the number, intensity, and timing of thinning operations. The two methods of volume estimation were compared in optimization by using the depth-first search (DFS) method, and both were found to provide similar results. Because the stand volume ratio function is computationally much more efficient, it was used in the next step, in which five direct search methods were tested: Hooke and Jeeves method (HJ); differential evolution (DE); particle swarm optimization (PS); evolution strategy (ES); Nelder and Mead method (NM); the last four are population-based methods. The HJ and DE methods yielded the highest values of the objective function, slightly outperforming the results of DFS in most cases, which proved to be about 100 times slower than HJ and 30 times slower than DE. DE was more stable than HJ (standard deviation was €30.6·ha−1for HJ and €8.8·ha−1for DE) and was therefore used for subsequent evaluation of the effects of site quality, stem density, and discount rate on the optimal management schedule. Rotation age and timing of thinnings both decreased as site quality and discount rate increased. The optimal management schedules often included three heavy thinning operations. The pseudo-code of the optimization methods tested is provided in an Appendix.

Simulation and evaluation of different statistical functions for describing lag time distributions of a bacterial growth curve

The objective of this study was to simulate the lag time distribution of individual bacterial cells in a population incubated under an isothermal condition. Spores of Clostridium sporogenes PA 3679 inoculated to ground beef were incubated at 37 °C to develop a growth curve and determine its apparent lag time. The lag times of individual cells were simulated using normal, lognormal, Gumbel, gamma, Weibull, and exponential distributions. The growth process was modeled using the McKellar model. A 4th order Runge–Kutta method was used to solve the McKellar model in conjunction with the Nelder–Mead method. The objective of the numerical analysis was to search for the parameters in each lag time distribution model and the kinetic parameters (specific growth rate and mean lag time) of the growth model such that a minimized residual sum of squared errors (RSS) was found.Although different in shapes, the simulation results showed that the performance of normal, lognormal, Gumbel, and gamma distributions was practically the same when used to fit the growth curve, yielding the same estimates of initial and stationary phase bacterial concentrations, specific growth rate, and mean lag time as well as RSS. The lognormal, Gumbel, and gamma distributions of the lag times were slightly skewed to the right of the mean. The standard deviations of the lag times with these distributions were relatively small, suggesting that cells may leave the lag phase almost simultaneously. The Weibull and exponential distributions were also suitable for describing the distribution of lag times, but with larger standard deviations. In general, all these statistical distributions were found suitable for simulating the distribution of lag times of individual cells in a population.

ClamR: A statistical evaluation of isotopic and temperature records in sclerochronologic studies

Isotope sclerochronology has been widely used to reconstruct paleoclimate from the hard part remains of a broad range of taxa from many different environmental settings and geological and cultural contexts. Oxygen isotope (δ 18 O) time series derived from these ultra high-resolution (daily, seasonal, annual) records have been used to estimate annual averages, seasonal ranges, and seasonal extremes. These time series, however, inherently include uncertainties such as unconstrained environmental noise, changes in seasonal growth rates resulting in weighting annual temperature estimates toward seasons of maximum growth, growth cessation, and reliance on the most extreme δ 18 O values in the time series to estimate seasonal temperature range, etc. To better quantify uncertainties in paleoclimate reconstructions using sclerochronologic data, we developed an open-source R package, ClamR. The ClamR evaluation tool improves upon the previous best-fit sinusoid approach by introducing additional statistical computation and analysis, including Discrete Fourier Transform to initialize parameters for the sinusoidal function, Nelder–Mead method to optimize parameters, jackknife error estimation, and window size calculation for sinusoidal smoothing. It can automatically calculate and plot statistical errors for oxygen isotope or temperature data series and provide statistical estimation of temperature and isotopic variables such as annual averages, seasonal ranges, and seasonal extremes. Moreover, it allows evaluation of the influence of environmental and/or physiological noise added to the climate signal. Therefore, ClamR enables more comprehensive and justified comparison between paleoclimate reconstructions and modern instrument records and ultimately advances our interpretation of seasonal isotope records from biogenic hard parts. • We developed ClamR to quantify uncertainties in sclerochronological studies. • We examine ClamR with oxygen isotope records of mollusc shells and otoliths. • ClamR affords a measure of assurance in making paleoclimate interpretations.

High-speed railway scheduling based on user preferences

This paper proposes an optimization model for high-speed railway scheduling. The model is composed of two sub-models. The first is a discrete event simulation model which represents the supply of the railway services whereas the second is a constrained logit-type choice model which takes into account the behaviour of users. This discrete choice model evaluates the attributes of railway services such as the timetable, fare, travel time and seat availability (capacity constraints) and computes the High-Speed railway demand for each planned train service.A hybridisation of a Standard Particle Swarm Optimisation and Nealder–Mead methods has been applied for solving the proposed model and a real case study of the High-Speed corridor Madrid–Seville in Spain has been analysed. Furthermore, parallel computation strategies are used to speed up the proposed approach.