Large Number Of Nonlinear Equations Research Articles

Adaptively exploring the feature space of flowsheets

AbstractSimulation and optimization of chemical flowsheets rely on the solution of a large number of nonlinear equations. Finding such solutions can be supported by constructing machine learning‐based surrogate models, relating features and outputs by simple, explicit functions. In order to generate training data for those surrogate models computationally efficiently, schemes to adaptively sample the feature space are mandatory. In this article, we present a novel family of utility functions to favor an adaptive, Bayesian exploration of the feature space in order to identify regions that are convergent and fulfill customized inequality constraints. Moreover, points close to the Pareto‐optimal domain with respect to conflicting objectives can be identified, serving as good start values for a multicriteria optimization of the flowsheet. The benefit is illustrated by small toy‐examples as well as by industrially relevant chemical flowsheets.

Endogenous Social Interactions with Unobserved Networks

Abstract We present a model of endogenous network formation to recover unobserved social networks using only observable outcomes. We propose a novel equilibrium concept that allows for a sharp characterization of equilibrium behaviour and that yields a unique prediction under testable conditions. While the equilibrium is characterized by a large number of non-linear equations, we show that it can be efficiently employed to recover the networks by an appropriately designed approximate Bayesian computation method. We apply the model to recover the network of social links between lawmakers in the U.S. Congress using data from the 109th to 113th legislatures. We show that social connections are important for legislators’ productivities, and we identify some of the key determinants of network centralities in the U.S. Congress.

Seismic performance analysis of a wind turbine tower subjected to earthquake and ice actions.

Sea ice is one of the main loads acting on a wind turbine tower in areas prone to icing, and this threatens safe working life of the wind turbine tower. In our study, a simplified calculated model of ice, wind turbine tower, and water dynamic interaction under earthquake action was proposed, which could avoid to solve a large number of nonlinear equations. Then, the seismic behaviour of the wind turbine tower with and without the influence of sea ice was investigated, and we found that the influence of the greater mass of the sea ice on the seismic response of a wind turbine tower should be considered when the wind turbine tower is designed in an area with thick ice. With the influence of the most unfavourable ice mass, the deformation and energy dissipation capacity of the wind turbine tower are decreased, and the wall thickness or stiffening rib thickness should be increased to improve the seismic performance and ductility of the wind turbine tower; the shear force and bending moment increased significantly on the wind turbine tower, and the shear force changes at the bottom of the wind turbine tower and position of action of the sea ice: attention should be paid to the wind turbine tower design at these positions. Finally, we conducted the shaking table test, and verified the rationality of our proposed simplified model.

HDK: Toward High-Performance Deep-Learning-Based Kirchhoff Analysis

The Kirchhoff law is one of the most widely used physical laws in many engineering principles, e.g., biomedical engineering, electrical engineering, and computer engineering. One challenge of applying the Kirchhoff law to real-world applications at scale lies in the high, if not prohibitive, computational cost to solve a large number of nonlinear equations. Despite recent advances in leveraging a convolutional neural network (CNN) to estimate the solutions of Kirchhoff equations, the low performance is still significantly hindering the broad adoption of CNN-based approaches. This paper proposes a high-performance deep-learning-based approach for Kirchhoff analysis, namely HDK. HDK employs two techniques to improve the performance: (i) early pruning of unqualified input candidates and (ii) parallelization of forward labelling. To retain high accuracy, HDK also applies various optimizations to the data such as randomized augmentation and dimension reduction. Collectively, the aforementioned techniques improve the analysis speed by 8× with accuracy as high as 99.6%.

Mathematical modeling of HIV-like particle assembly in vitro

In vitro, the recombinant HIV-1 Gag protein can generate spherical particles with a diameter of 25–30 nm in a fully defined system. It has approximately 80 building blocks, and its intermediates for assembly are abundant in geometry. Accordingly, there are a large number of nonlinear equations in the classical model. Therefore, it is difficult to compute values of geometry parameters for intermediates and make the mathematical analysis using the model. In this work, we develop a new model of HIV-like particle assembly in vitro by using six-fold symmetry of HIV-like particle assembly to decrease the number of geometry parameters. This method will greatly reduce computational costs and facilitate the application of the model. Then, we prove the existence and uniqueness of the positive equilibrium solution for this model with 79 nonlinear equations. Based on this model, we derive the interesting result that concentrations of all intermediates at equilibrium are independent of three important parameters, including two microscopic on-rate constants and the size of nucleating structure. Before equilibrium, these three parameters influence the concentration variation rates of all intermediates. We also analyze the relationship between the initial concentration of building blocks and concentrations of all intermediates. Furthermore, the bounds of concentrations of free building blocks and HIV-like particles are estimated. These results will be helpful to guide HIV-like particle assembly experiments and improve our understanding of the assembly dynamics of HIV-like particles in vitro.

MATHEMATICAL MODELS OF COGENERATION STEAM TURBINES BASED ON EXPERIMENTAL CHARACTERISTICS OF TURBINE STAGES AND COMPARTMENTS

The article covers approaches to building the mathematical models of cogeneration steam turbine units. The objective of this work is the development, justifi cation and practical implementation of methods of mathematical modeling of cogeneration steam turbines for solving research problems and improving the energy effi ciency of the CHP. The use of normative characteristics to solve these tasks, which require the calculation of variable modes, in most cases turns out to be unjustifi ed, due to a number of their signifi cant shortcomings. It is shown that the application of the developed mathematical models based on experimentally determined fl ow and power characteristics allows to solve various practical tasks arising during operation. We have given an example of constructing a computational mathematical model of a T-50-12.8 type turbine. The method of construction of this model is described, which consists of a large number of nonlinear equations (over 50), and an enlarged block diagram of calculations for this turbine (using the modernized Newton method with a step change of the iteration process in several variables) is provided. The method of calculating an arbitrary turbine compartment or a stage using some experimental fl ow and power characteristics is given as part of the general algorithm. A method for calculating the LPH-1 regenerative heater is also given as an example. It is shown how similar models of other equipment can be developed on the basis of the described mathematical model. The article provides examples of the use of mathematical models of turbines for solving problems of determining the most economical and safe operating modes and preventing ineffi cient solutions. The obtained results allow to determine the optimal operating modes of real CHP equipment and calculating their absolute and comparative energy efficiency.

A method for calculation of oil supply to the parts of units of agricultural machinery

Force lubrication systems are widely used in engines and transmission units of agricultural machinery; the oil in them is supplied to consumers by the action of oil pumps, thus reducing the nonproductive friction losses and providing the necessary thermal conditions for parts. For calculation of lubrication systems when carrying out the design and further development works, mathematical models are used, representing the complex systems consisting of large number of nonlinear equations. Such systems require the development of special solution methods. A solution method of system of linear and nonlinear equations describing the oil movement through the channels of force lubrication systems should provide the simplicity, unambiguity and high accuracy of calculations. The analysis of mutual influence of the calculated quantities is performed. The three main dependencies are pointed out: the coefficient of friction losses for each of the supply line sections is a function of the volumetric flow rate of oil; the oil density is a function of the pressure in input-output section of the supply line; the pressure by the action of centrifugal forces is a function of the oil density at the input of rotating section. Based on the performed analysis, a solution method for mathematical model of force lubrication system satisfying the suggested requirements is developed. To validate the method, the system of planetary transmission lubrication is calculated; the total volumetric flow rate of oil in the process of its operation under various modes is measured. The obtained results show the good match of calculated and measured values of the volumetric flow rate of oil, which confirms the accuracy of the proposed solution method for system of equations and the possibility of its use for calculation of lubrication systems.

Comment on “Comment on ‘Superposition of elliptic functions as solutions for a large number of nonlinear equations”’ [J. Math. Phys. 56, 084101 (2015)

The assertion in a theorem that the superposition of two particular Jacobian elliptic functions is not a solution of the nonlinear Schrodinger equation is demonstrated to be invalid.

Response to “Comment on ‘Superposition of elliptic functions as solutions for a large number of nonlinear equations’” [J. Math. Phys. 56, 084101 (2015)

In a recent paper, Zhang and Li [J. Math. Phys. 56, 084101 (2015)] have doubted our claim that whenever a nonlinear equation has solutions in terms of the Jacobi elliptic functions cn(x, m) and dn(x, m), then the same nonlinear equation will necessarily also have solutions in terms of dn(x,m)±mcn(x,m). We point out the flaw in their argument and show why our assertion is indeed valid.

Comment on “Superposition of elliptic functions as solutions for a large number of nonlinear equations” [J. Math. Phys. 56, 032104 (2015)

By using the method of planar dynamical systems, we solve exactly a nonlinear Schrödinger (NLS) equation discussed by Khare and Saxena [J. Math. Phys. 56, 032104 (2015)], and give the exact explicit parametric representations of all the traveling wave solutions.

The initial value problem for the mathematical model of a multiple-input multiple-output flexible parallel mechanism

The cable-net support structure of the Five-hundred-meter Aperture Spherical Telescope (FAST) is a multiple-input multiple-output (MIMO) flexible parallel mechanism. The mathematical model of the cable-net structure is set up with the catenary equation, which produces a large number of non-linear equations. Finding the initial value plays an important role in the working state of the FAST. In this paper, a novel method is proposed to solve the equations of the model. This method uses the solution of the (n−1)-th layer cable-net as the initial value of the n-th layer cable-net. The n-layer cable-net equations are then solved simultaneously and the tension of each cable and the coordinate of each node in the cable-net can be obtained. With the introduction of the symmetry of cable-net, the number of unknown parameters in the equations is reduced to about one-tenth of the whole cable-net. The initial value of the cable-net support structure of the FAST is provided to show the validity of the method. The results of this study are of practical value for the FAST's new type of flexible cable driving mechanism design and control method.

Study on the working state of the five hundred-meter aperture spherical telescope using step-wise assignment method

The working state of the five hundred-meter aperture spherical telescope (FAST) is solved using the step-wise assignment method. In this paper, the mathematical model of the cable-net support structure of the FAST is set up by the catenary equation. There are a large number of nonlinear equations and unknown parameters of the model. The nonlinear equations are solved by using the step-wise assignment method. The method is using the analytical solutions of the cable-net equations of one working state as the initial value for the next working state, from which the analytical solutions of the nonlinear equations of the cable-net for each working state of the FAST and the tension and length of each driving cable can be obtained. The suggested algorithm is quite practically well suited to study the working state of the cable-net structures of the FAST. Also, the working state analysis result of the cable-net support structure of a reduced model of the cable-net structure reflector for the FAST is given to verify the reliability of the method. In order to show the validity of the method, comparisons with another algorithm to set the initial value are presented. This method has an important guiding significance to the further study on the control of the new type of flexible cable driving mechanism, especially the FAST.

Superposition of elliptic functions as solutions for a large number of nonlinear equations

For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ4, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn2(x, m), it also admits solutions in terms of \documentclass[12pt]{minimal}\begin{document}${\rm dn}^2(x,m) \pm \sqrt{m} {\rm cn}(x,m) {\rm dn}(x,m)$\end{document} dn 2(x,m)±m cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.

Linear superposition for a class of nonlinear equations

We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$, then it also admits solutions in terms of their sum as well as difference, i.e. $\dn(x,m) \pm \sqrt{m}\, \cn(x,m)$. Further, we also show that whenever a nonlinear equation admits a solution in terms of $\dn^2(x,m)$, it also has solutions in terms of $\dn^2(x,m) \pm \sqrt{m}\, \cn(x,m)\, \dn(x,m)$ even though $\cn(x,m)\, \dn(x,m)$ is not a solution of that nonlinear equation. Finally, we obtain similar superposed solutions in coupled theories.

A Simplified Approach Based Phase Angle for Tooth Contact Analysis of Planetary Gear Trains

Planetary gear trains have many advantages in applications. In these advantages, quiet noise and slight vibration may be contribute to low Transmission Error (TE), which can be calculated by Tooth Contact Analysis (TCA). However, to obtain numerical solution of the meshing equations based on TCA for planetary gear trains is very difficult because of a large number of nonlinear equations and unknowns. A simplified method utilizing planet phase angle is investigated and the numerical solution of planetary gear trains TE by TCA can be realized in this paper.

Optimal Design of Shell-and-Tube Heat Exchangers Using Particle Swarm Optimization

In this paper, the shell-and-tube heat exchangers design is formulated as an optimization problem and solved with particle swarm optimization (PSO). The objective is to minimize the global cost including area cost and pumping cost or just area minimization, depending on data availability, rigorously following the standards of the Tubular Exchanger Manufacturers Association and respecting pressure drops and fouling limits. Given fluids temperatures, flow rates, physical properties (density, heat capacity, viscosity, and thermal conductivity), pressure drop and fouling limits, and area cost data, the proposed methodology calculates the optimal mechanical and thermal-hydraulic variables. The Bell−Delaware method is used for the shell-side calculations. Some literature cases are studied and results show that in this type of problem, with a very large number of nonlinear equations, the PSO algorithm presents better results, avoiding local minima.

Solution of ODE u″ + p(u)(u′)2 + q(u) = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc, are reduced to an integrable ODE expressed by u″ + p(u)(u′)2 + q(u) = 0 whose general solution can be given. Furthermore, combining complete discrimination system for polynomial, the classifications of all single travelling wave solutions to these equations are obtained. The equation u″ + p(u)(u′)2 + q(u) = 0 includes the equation (u′)2 = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.

Simulation model for predicting the concentration of reactant parameters in water distribution systems

Water quality although acceptable when it leaves the source or the treatment plant may deteriorate through the distribution network. The concentration of non-conservative, or reactant constituents in the distribution system, such as chlorine, nitrates, THM etc., is affected both by mixing at the nodes of the network and by chemical or biological processes occurring in the stream flow through the pipes. In complex water supply systems which consist of large number of pipes and nodes the quality simulation of non-conservative constituents needs remarkable capacity of internal computer memory and run time, due to large number of nonlinear equations (Shamir & Howard, 1991). The present article applies steady-state approach to modeling the pipes network as reactors network, assuming first and second order reactions, which meet the kinetics of most reactions in the bulk stream of water distribution and supply systems. The model which has been developed in this work enables the simulation of large water systems with many pipes and nodes as simple structure of plug flow and backmixed reactors with a specific residence time distribution. Thus, controlling the transportation of quality parameters or tracking the movement of accidental contamination within the distribution systems becomes to be much more applicable objective.

Efficient scalar quantization of exponential and Laplacian random variables

This paper presents solutions to the entropy-constrained scalar quantizer (ECSQ) design problem for two sources commonly encountered in image and speech compression applications: sources having the exponential and Laplacian probability density functions. We use the memoryless property of the exponential distribution to develop a new noniterative algorithm for obtaining the optimal quantizer design. We show how to obtain the optimal ECSQ either with or without an additional constraint on the number of levels in the quantizer. In contrast to prior methods, which require a multidimensional iterative solution of a large number of nonlinear equations, the new method needs only a single sequence of solutions to one-dimensional nonlinear equations (in some Laplacian cases, one additional two-dimensional solution is needed). As a result, the new method is orders of magnitude faster than prior ones. We show that as the constraint on the number of levels in the quantizer is relaxed, the optimal ECSQ becomes a uniform threshold quantizer (UTQ) for exponential, but not for Laplacian sources. We then further examine the performance of the UTQ and optimal ECSQ, and also investigate some interesting alternatives to the UTQ, including a uniform-reconstruction quantizer (URQ) and a constant dead-zone ratio quantizer (CDZRQ).

Parallel Processing for Large-Scale Nonlinear Control Experiments in Economics

In general, the econometric models relevant for pur poses of evaluating economic policy contain a large number of nonlinear equations. Therefore, in applying optimal control techniques, computational difficulties are encountered. This paper presents the most common al gorithm for computing nonlinear control problems and investigates the degree to which vector processing and parallel processing can facilitate optimal control experiments.