Lagrange's Method Of Undetermined Multipliers Research Articles

Shape optimization method for strength design problem of microstructures in a multiscale structure

AbstractIn this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.

Shape optimization method for strength design problem of microstructures in a multiscale structure

In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed; minimization of the maximum stress of the microstructure and minimization of the maximum stress of the macrostructure. The homogenization method is used to bridge the macrostructure and the microstructures, and also to calculate the local stress in the microstructures. By replacing the maximum value of the stress with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability on the maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem with the area constraint including the whole microstructures. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructures, while reducing the objective function and maintaining the smooth design boundaries. In the numerical examples, the optimal shapes obtained for the maximum local stress minimization of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.

Simultaneous optimization of multiple DVAs for multi-layered structure

The design theory of dynamic absorbers is a classic research area in the vibration engineering. In contrast, there remain many problem settings where rigorous formulations are not widely known. This paper deals with a minimization of a maximum transmissibility when multiple dynamic absorbers are installed in a multi-layered structure. For the optimal solution, the author has found that the necessary conditions can be expressed in terms of equations via the submatrix of Jacobian, and optimization can be achieved using the Newton-Raphson method. This is suitable for the case that the numerical solution confirms that the heights of all resonance points coincide at the optimal solution. In this paper, it is shown that this formulation can be explained in the context of Lagrange's undetermined multiplier method.

A novel systematic multi-objective optimization to achieve high-efficiency and low-emission waste polymeric foam gasification using response surface methodology and TOPSIS method

Gasification is one of the most important thermochemical processes to convert a solid fuel to energy carriers of a gaseous product called syngas. This process has been well addressed in the literature for biomass and several valuable researches have been performed on plastic waste gasification. However, it is the first effort for a systematic comprehensive investigation and multi-objective optimization of gasification process for waste polymeric foams using response surface methodology and TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) approach. Air and steam waste rigid polyurethane foam gasifications were modeled using coupled method of Gibbs minimization free energy and Lagrange method of undetermined multipliers, and then, validated. Effects of key features consisting gasification temperature and moisture content in both air and steam gasification types and equivalence ratio in air and steam to waste foam ratio in steam types were thoroughly studied on gas composition and energy and hydrogen efficiencies. Analysis of variance was employed for recognizing the most effective parameters on air and steam gasification performances. The results revealed that hydrogen and energy efficiencies of air gasification at multi-objective optimum conditions were 42.68 % and 89.58 %, respectively, and these values were 64.02 % and 96.52 % for steam gasification type. Air gasification of waste rigid polyurethane foam produced 3.13 g of CO2 emission at optimum state; however, its value was 10.02 g for steam gasification type. A multi-criteria decision analysis based on TOPSIS method was utilized for selecting the best gasification type in different scenarios and the findings indicated that air gasification had better performance compared to steam type due to its low emissions.

Polyethylene waste gasification syngas analysis and multi-objective optimization using central composite design for simultaneous minimization of required heat and maximization of exergy efficiency

Polyethylene contributes most to post-consumer plastic waste worldwide and gasification is one of the thermo-chemical methods applied to plastic waste. In this study, the gasification of polyethylene waste was modeled using the Gibbs free energy minimization and Lagrange method of undetermined multipliers, and was then validated. A central composite design was employed to assess and optimize the polyethylene waste gasification. The findings revealed that hydrogen production was significantly improved by 48% by raising the steam to polyethylene waste ratio according to the water–gas shift and reforming reactions. Hydrogen production and exergy efficiencies were enhanced by increasing temperature. Regression models produced from the central composite design had high accuracy for estimating the heat required and exergy efficiency with R2 values of 91.6% and 96.8%, respectively. Multi-objective optimization was performed to minimize the heat required and maximize exergy efficiency. A steam to polyethylene waste ratio of 1 and temperature of 700 °C were the optimum conditions. Required heat of 213.7 kJ and exergy efficiency of 95.37% were the responses at the optimum state.

Gibbs energy minimization using Lagrange method of undetermined multipliers for electrochemical and thermodynamic modeling of a MCFC with internal steam reforming

Precise determination of the chemical equilibrium state is important for electrochemical and thermodynamic modeling of a molten carbonate fuel cell (MCFC). One method for determining the equilibrium state of a MCFC with internal reforming is to compute the total Gibbs free energy of the system (G) and minimize it by adjusting the molar amounts of each substance. Multi-dimensional optimization algorithms can be used for this purpose. The Lagrange method of undetermined multipliers as a powerful analytical solution is used in the present work to estimate chemical equilibrium state. Under this condition, the effects on voltage losses, output voltage, stack power, efficiency and molar portions of the exhaust chemical species at the anode and cathode are investigated, separately of such parameters as current density, reaction temperature, stack pressure as well as fuel and carbon dioxide utilization ratios. A simulation of a MCFC stack performance by the Lagrange method at a specified state, results in an efficiency of 50.3% and output power of 554.3 kW.

Co-generation of electricity and heating using a SOFC-ScCO2 Brayton cycle-ORC integrated plant: Investigation and multi-objective optimization

A co-generation plant consisting of a solid oxide fuel cell, a supercritical carbon dioxide Brayton cycle, an organic Rankine cycle, and a domestic heat recovery was modeled based on the zero-dimensional approach. The minimization of Gibbs free energy and Lagrange method of undetermined multipliers were applied to find the unknown moles in the reactions. The effects of the current density, the fuel utilization ratio, and the anode recycle ratio were investigated on the plant performance. Net power, heated water and fuel flow rates, energy and exergy efficiencies, and CO2 emission were considered as outputs of the co-generation plant. The contributions of the parameters were determined using analysis of variance results and half-normal plot of the standardized effects. The regression models of the outputs were achieved with high credibility. The findings indicated that the current density had the highest contribution to plant performance. The multi-objective optimization showed that the current density of 7787.879 A/m2 and the anode recycle ratio of 0.424 is the optimum condition. In this state, the electrical exergy efficiency, the heated water flow rate, and the CO2 co-generation emission were 43.01%, 2769 gr/s, and 213.6 kg/(MW.h), respectively.

Multi-objective optimization and decision analysis of a system based on biomass fueled SOFC using couple method of entropy/VIKOR

A power generation system including solid oxide fuel cell (SOFC), gasifier, and high-temperature sodium heat pipes was modeled using the Lagrange method of undetermined multipliers. The model validation was performed in comparison with the data available in the literature. The results showed that the presented model had a high validity. The parametric analysis showed that there was an optimum value for current density to achieve the highest net power but the exergy efficiency was reduced gradually with the current density. Response surface methodology (RSM) was utilized for multi-objective optimization purpose. The analysis of variance (ANOVA) was performed and different regression models with high validity were presented for power (R2 = 93.40%) and exergy efficiency (R2 = 96.22%). The one-objective optimization was performed using the interaction effect of parameters derived by ANOVA. The multi-objective optimization results indicated that the optimum power was achieved at relatively low exergy efficiencies and the optimum exergy efficiency was accompanied by low powers. A stepwise multi-objective optimization procedure was conducted and an optimum region was recognized. The performance of the system was assessed from energy, exergy, and environmental viewpoints for different biomasses including pine sawdust, groundnut shell, rice straw, rice husk, eucalyptus, sunflower shell, and sugarcane bagasse. Multi-criteria decision making (MCDM) analysis was implemented in order to select the biomass with the best performance. The cold gas efficiency, energy efficiency, exergy destruction rate, CO2 emission, and fuel flow rate were selected as different criteria. The MCDM results revealed that eucalyptus had the best performance.

Development of an efficient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation

The absolute nodal coordinate formulation (ANCF) is often used for the large rotation and deformation analysis. In the ANCF, absolute nodal coordinates and global slopes are utilized as the element nodal coordinates. Accordingly, it gives a constant and symmetric mass matrix. On the other hand, elastic forces are generally expressed by highly nonlinear terms. Therefore, one of the important topics in the implementation of the ANCF is to derive mathematical descriptions of the elastic forces which can be calculated efficiently. This study focuses on development of efficient calculation procedures for the ANCF beam element under the assumption of the Euler–Bernoulli beam theory. In particular, we focus on the fact that the elastic forces can be described in simpler forms with the element coordinates. It implies that appropriate choices of the coordinate systems for the inertia and the elastic forces could contribute to the development of efficient calculation procedures. In the present formulation, the strain and the kinetic energies are expressed as functions of the element and the global coordinates, respectively. Then algebraic constraints regarding the relations between the global and the element coordinate systems are introduced by means of the Lagrange method of undetermined multipliers. These constraints are used for eliminating the redundant degrees of freedom from a standpoint of consistency in the formulation. Therefore, this method can be categorized as an augmented formulation technique. The equations of motions of this constrained system are derived by the method for constrained Hamiltonian system, namely Dirac’s method with the Poisson bracket formalism. In order to evaluate the proposed method, it is applied to the large deformation problem in the plane case. The equivalence of the previous approach and the proposed method is discussed by a comparison of the two. In addition, its computational performance is investigated.

Estimating the vector electric field using monostatic, multibeam incoherent scatter radar measurements

AbstractAn algorithm has been developed to image the local structure in the convection electric field using multibeam incoherent scatter radar (ISR) data. The imaged region covers about 4° in magnetic latitude and 8° in magnetic longitude for the specific geometry considered (that of the Poker Flat ISR). The algorithm implements the Lagrange method of undetermined multipliers to regularize the underdetermined problem posed by the radar measurements. The error on the reconstructed image is estimated by mapping the mathematical form to a Bayesian estimate and observing that the Lagrangian method determines an effective a priori covariance matrix from a user‐defined regularization metric. There exists a unique solution when the average measurement error is smaller than the average measurement amplitude. The algorithm is tested using synthetic and real data and appears surprisingly robust at estimating the divergence of the field. Future applications include imaging the current systems surrounding auroral arcs in order to distinguish physical mechanisms.

Chirality dependence of quantum thermal transport in carbon nanotubes at low temperatures: A first-principles study

We study the transport properties of single-walled carbon nanotubes (SWCNTs) using the nonequilibrium Green's function method based on first-principles calculations. We compared three SWCNTs with different chiralities (3, 3), (5, 0), and (4, 2), and found that the thermal conductance varies significantly with the chirality, especially at low temperatures. Such differences are attributed to the dependence on the chirality of the frequency of the lowest optical mode and phonon-phonon interaction with the semi-infinite leads. To obtain accurate low-vibrational frequencies, a force constant correction based on the Lagrange undetermined multiplier method was employed. The phonon-phonon interaction was analyzed in terms of the projection of the phonon coupling with the semi-infinite leads onto the normal modes of the center region. Our result indicates that high optical mode frequency and weak phonon coupling on the armchair (3, 3) SWCNT are the origin of the long quantized plateau found in the experimental thermal conductance.

A Gibbs energy minimization method for constrained and partial equilibria

The conventional Gibbs energy minimization methods apply elemental amounts of system components as conservation constraints in the form of a stoichiometric conservation matrix. The linear constraints designate the limitations set on the components described by the system constituents. The equilibrium chemical potentials of the constituents are obtained as a linear combination of the component-specific contributions, which are solved with the Lagrange method of undetermined multipliers. When the Gibbs energy of a multiphase system is also affected by conditions due to immaterial properties, the constraints must be adjusted by the respective entities. The constrained free energy (CFE) minimization method includes such conditions and incorporates every immaterial constraint accompanied with its conjugate potential. The respective work or affinity-related condition is introduced to the Gibbs energy calculation as an additional Lagrange multiplier. Thus, the minimization procedure can include systemic or external potential variables with their conjugate coefficients as well as non-equilibrium affinities. Their implementation extends the scope of Gibbs energy calculations to a number of new fields, including surface and interface systems, multi-phase fiber suspensions with Donnan partitioning, kinetically controlled partial equilibria, and pathway analysis of reaction networks.

Discussion on: “Why Is Resorting to Fate Wise? A Critical Look at Randomized Algorithms in Systems and Control”

Precise determination of the chemical equilibrium state is important for electrochemical and thermodynamic modeling of a molten carbonate fuel cell (MCFC). One method for determining the equilibrium state of a MCFC with internal reforming is to compute the total Gibbs free energy of the system (G) and minimize it by adjusting the molar amounts of each substance. Multi-dimensional optimization algorithms can be used for this purpose. The Lagrange method of undetermined multipliers as a powerful analytical solution is used in the present work to estimate chemical equilibrium state. Under this condition, the effects on voltage losses, output voltage, stack power, efficiency and molar portions of the exhaust chemical species at the anode and cathode are investigated, separately of such parameters as current density, reaction temperature, stack pressure as well as fuel and carbon dioxide utilization ratios. A simulation of a MCFC stack performance by the Lagrange method at a specified state, results in an efficiency of 50.3% and output power of 554.3 kW.

Thermodynamic equilibrium analysis of combined carbon dioxide reforming with partial oxidation of methane to syngas

The chemical equilibrium analysis on combined CH 4-reforming with CO 2 and O 2 (combined CORM–POM) has been conducted by total Gibbs energy minimization using Lagrange's undetermined multiplier method. The equilibrium compositions of the combined CORM–POM process were considerably influenced by CH 4:CO 2:O 2 feed ratios and operating temperatures. Methane oxidation reaction occurred predominantly at low temperatures, while the CO 2 conversion was strongly influenced by the O 2/CH 4 feed ratio. The addition of O 2 to the CORM process improved the CH 4 conversion, H 2 and H 2O yields and also the H 2/CO product ratio at the expense of CO 2 conversion and CO yield. Accordingly, the optimal equilibrium conditions for the CH 4:CO 2:O 2 ratio were within the range of 1:0.8:0.2–1:1:0.2 and a minimum requirement temperature of 1000 K.

Hydraulically Efficient Power-Law Channels

A power-law channel is a generalized form of a channel and includes parabolic and triangular cross sections. For an exponent m<0.5 in the power law, the relative wetted perimeter has been estimated from a series expansion truncated to four terms. For values of the exponent m 0.5 the relative wetted perimeter has been estimated using an appropriate non-linear interpolation expression. A table to estimate relative wetted perimeter based on these expressions is presented for design purposes. With these expressions for relative wetted perimeter, and using the Lagrange method of undetermined multipliers, for any given maximum side slope, the area and/or wetted perimeter is minimized subject to the equality constraint of a uniform flow (Mannings) equation. Using this technique, for any given side slope, the exponent of the power-law channel can be determined and hydraulically efficient power-law channels can be designed. Optimized power-law channels are compared with trapezoidal and parabolic channels. The existing parabolic design of the Pehur High Level Canal, Pakistan is compared with an optimum power-law channel.

Calculation of Entropy Production in Process Models

A thermochemical method is presented by which multiphase processes can be simulated with concurrent calculation of the Gibbs energy of the reactive mixture during a chemical change. Algorithmic constraints are set for the overall reaction kinetics when the Lagrange method of undetermined multipliers is used to minimize the Gibbs energy of the multicomponent system. Consequently, the chemical change is calculated as a series of successive “virtual” states, which follow the extent of the overall reaction. From the Gibbs energy of these intermediate states, other thermodynamic quantities for the changing chemical system can be deduced, and the entropy production of the process can be calculated. A particular process model was developed for two isothermal aqueous systems and for a nonisothermal high-temperature process. The entropy production in the reactive systems is presented, and the validity of the simulation models is assessed in terms of their time-dependent Gibbs energy and entropy profiles.

Estimation of excess enthalpy for binary systems

Based on the author’s molecular thermodynamic model for solution structure, a new equation with three parameters for correlating the excess enthalpy of binary solutions is proposed. We make use of the group statistical theory, Lagrange undetermined multiplier method and a pseudo-chemical reaction mechanism. Tested by 6059 experimental data in 282 kinds of binary systems, the new model shows the characteristics of convenience and wide applicability. The accuracy of the new equation has an advantage over the Redlich–Kister equation [J. Chem. Ind. Eng. (China) 49 (1998) 103] and the Shen–Liu equation proposed recently [J. Chem. Ind. Eng. (China) 51 (2000) 181], both of which have three parameters too. As for the SSF equation with four parameters, the accuracy of it and the new equation is equivallent, some times the latter is more accurate. The new equation is still tenable in the limiting condition of x→0 and 1.

Lorentz-covariantquantization of massless non-Abelian gauge fields in theHamiltonian path-integral formalism

The Lorentz-covariant quantization performed in the Hamiltonianpath-integral formalism for massless non-Abelian gauge fields has beenachieved. In this quantization, the Lorentz condition, as a constraint, mustbe introduced initially and incorporated into the Yang-Mills Lagrangian bythe Lagrange undetermined multiplier method. In this way, it is found thatall Lorentz components of a vector potential have their correspondingconjugate canonical variables. This fact allows us to defineLorentz-invariant Poisson brackets and carry out the quantization in aLorentz-covariant manner.

Exploring a rheonomic system

A simple and illustrative rheonomic system is explored in the Lagrangian formalism. The difference between the Jacobi integral and the energy is highlighted. A sharp contrast with remarks found in the literature is pointed out. The non-conservative system possesses a Lagrangian that is not explicitly dependent on time and consequently there is a Jacobi integral. The Lagrange undetermined multiplier method is used as a complement to obtain a few interesting conclusions.

Efficient recursive algorithm for inverse dynamics

An efficient recursive computational scheme for the robot manipulator inverse dynamics is outlined. The algorithm is a new computational approach and is based on the application of the Gibbs-function. The solution of the inverse dynamic problem is formulated as a problem in quadratic programming, and the recursive equations for the computation of the driving torques are derived from the necessary conditions of minimum usage of the Lagrange method of undetermined multipliers. The algorithm involves the recursive computations of links' angular velocities, angular accelerations and linear accelerations expressed in moving coordinate systems. Further simplification can be performed by writing the vector equations as equations of coordinates. The reduced computational scheme is presented by a Pascal program detail. The computational complexity of the algorithm is analyzed.