Lagrange Multiplier Technique Research Articles

A variational method for post-buckling analysis of end-supported nanorods under self-weight with surface stress effect

In this paper, the formulation of post-buckling of end-supported nanorods under self-weight was developed by the variational method. The surface stress effect was considered following the surface elasticity theory of Gurtin–Murdoch. The variational formulation involving the strain energy in the bulk material, the strain energy of the surface layer, and the potential energy due to self-weight was expressed in terms of the intrinsic coordinates. The variational formulation was accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method was used to derive a system of nonlinear equations resulting from the stationary of the total potential energy, and then, Newton–Raphson iterative procedure was applied to solve this system of equations. The post-buckled configurations of nanorods under self-weight due to various boundary conditions were presented and demonstrated that the variational formulation expressed in terms of intrinsic coordinate is highly recommended for post-buckling analysis of end-supported nanorods. In addition, the surface stress effect significantly influenced the post-buckling response of nanorods and exhibited higher stiffness in comparison with nanorods without surface stress. The model formulation presented in this study is of special interest in the design and application of advanced technological devices.

A Lagrange Multiplier-based Technique within the Nonlinear Finite Element Method in Cracked Columns

In this research, two energy-based techniques, called Lagrange multiplier and conversion matrix, are applied to involve crack parameters into the non-linear finite element relations of Euler-Bernoulli beams made of functionally graded materials. The two techniques, which divide a cracked element into three parts, are implemented to enrich the secant and tangent stiffness matrices. The Lagrange multiplier technique is originally proposed according to the establishment of a modified total potential energy equation by adding continuity conditions equations of the crack point. The limitation of the conversion matrix in involving the relevant non-linear equations is the main motivation in representing the Lagrange multiplier. The presented Lagrange multiplier is a problem-solving technique in the cracked structures, where both geometrical nonlinearity and material inhomogeneity areas are considered in the analysis like the post-buckling problem of cracked functionally graded material columns. Accordingly, some case-studies regarding the post-buckling analysis of cracked functionally graded material columns under mechanical and thermal loads are used to evaluate the results.

Iterative analytic approximation to one‐dimensional nonlinear reaction–diffusion equations

The paper is concerned with a class of nonlinear reaction–diffusion equations with a dissipating parameter. The problem is singularly perturbed from a mathematical perspective. Solutions of these problems are known to exhibit multiscale character. There are narrow regions in which the solution has a steep gradient. To approximate the multiscale solution, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. We obtain closed‐form analytic approximation to nonlinear boundary value problems through iteration. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville–Green transforms. The idea of the paper is to overcome the well‐known difficulties associated with numerical methods. Two test examples are taken into account, and rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.

Lightweight sandwich and composite beam analysis using improved higher-order theory with respect to strain energy fidelity in ply-wise approach

ABSTRACT This approach present improved higher-order layer-wise theory for the investigation of flawlessly wedged sandwich-composite beams with general laminate configurations. Our analysis incorporates the continuity assumption of interlaminar shear stresses and in-plane and flexural displacements between interfaces. In addition, interlaminar shear stresses are constrained using the Lagrange multiplier technique by introducing new unknown variables. These unknown variables are expressed with interlaminar strain energy, assuming that the strain energy is continuous throughout the overall thickness of the beam. To govern the newly introduced and other unknown variables, the total potential energy (TPE) is minimised using variational calculus. The numerical analysis results show that our approach provides enhanced accuracy to examine sandwich-composite beams.

Friction-induced backward rub of rotors in non-annular clearances: Experimental observations and numerical analysis

During backward rub in a turbomachine, rotor is compelled by frictional forces into a severe precessional sliding motion inside stationary parts such as seals or journal bearings in opposite direction of rotor's rotation. Backward rub occurs with high frequency and high amplitude of vibration and can deteriorates rubbing surfaces just for few moments. This paper presents and investigates the idea of clearance profile alteration from annular form to non-annular 3-lobed form in order to affect the reverse rub's characteristics. For this purpose, a test setup consisting of a lightly damped rotor-stator system is designed and built in SUT's condition-monitoring laboratory for testing the efficacy of the idea. Degree of deviation of the clearance from the annular state is defined by “preload factor”. Different test specimens were fabricated for realizing clearances with different preload factors. Test results showed that the backward frequency and the rotor's vibration amplitude are effectively reduced in sufficiently high preload values. For interpreting the observed results, a numerical model is constructed to precisely capture the physics underneath the attenuation of the rub. A contact model for the rotor and the lobed stator is developed based on unilateral contact constraint and friction condition. Lagrange multiplier technique is implemented along with direct linearization of the constrained equation of motion for conserving the frictional energy in the discrete time domain. The numerical solution is obtained by an implicit time marching algorithm for different friction coefficients and preload factors. Good agreement is observed between the numerical and experimental results. Both results indicate that the non-annular 3-lobed clearances can mitigate the backward rub by retarding the tangential friction force transmission between rotor and stator and thus by reducing backward frequency and amplitude of vibration.

SDP-IGD: An Iterative Power Allocation Technique for Cluster-Based Multihop Vehicular Communications

This paper addresses the formulation of power allocation for cluster based multihop vehicular relaying communications in order to improve the network quality-of-services (QoS). The cluster-to-cluster (C2C) channel is completely depend on the power gain of vehicle-to-vehicle (V2V) channel that increases the difficulties of the power allocation problem. To meet this goal, we have derived a global channel obtained by joint V2V and C2C cooperation. The most challenging aspect is however the joint V2V and C2C power allocation of the global channel due to the association of its multi-variables objective function. To solve this problem, we apply an alternative optimization technique to make the global problem into a series of sub-problems. We then apply a semi-definite programming (SDP)-based iterative gradient descent (SDP-IGD) power allocation to assign power in each relay. The SDP-IGD power allocation algorithm has been extended from Lagrange Multiplier (LM) that provides better performance than separate LM technique. The minimum mean-square error (MMSE)-decision feedback equalizer (MMSE-DFE) is designed at the receiver to improve the signal detection capability under exact channel state information (CSI) in all forward-and-backward links. Finally, simulation results confirm that our proposed solution performs significantly superior compared to other solutions found in the literature.

Determining the peak spatial and resultant accelerations of tall buildings tested in the wind tunnel

This paper presents a frequency domain approach to determine the expected peak spatial (x, y, torsion) and resultant accelerations of tall buildings tested in the wind tunnel. Auto-spectra and cross-spectra are computed from the generalized force time series determined from measured wind tunnel data. Spectral analysis is used to determine the auto- and cross-spectra of the modal responses from which the covariance matrix of the modal responses is calculated. A Lagrangian multiplier technique is then used in conjunction with the mode shapes to determine the peak expected spatial responses in the x, y, and torsion directions, as well as the peak resultant acceleration. The method is applied to three buildings that were tested in the wind tunnel and range from straightforward mode shapes, to complicated mode shapes with a high degree of correlation among degrees of freedom. To evaluate the proposed model, time domain analysis is conducted using the time series of loading determined from the wind tunnel and using artificially-generated wind loading. In all cases considered, the peak spatial responses determined using the proposed method agree with the peak responses observed from the time domain analysis. The proposed method therefore provides a novel, simple, robust, and mathematically justifiable method to determine peak accelerations for buildings subjected to random wind-induced vibrations.

LPI performance optimisation strategy for a JRCS

In this study, the problem of the low probability of intercept (LPI) performance optimisation for a joint radar-communications system (JRCS) is investigated, which can simultaneously estimate surveillance channel parameters from the target returns and decode the received communications signals. The primary purpose of the proposed LPI performance optimisation strategy is to improve the LPI performance of a JRCS by optimising the energy spectral density of radar waveform design and the communications power allocation while maintaining a predetermined mutual information threshold for parameter estimation and a certain communications data rate for information delivery. The traditional isolated sub-band (TISB) situation, radar isolated sub-band (RISB) situation, and communications isolated sub-band (CISB) situation are analysed. Subsequently, the technique of Lagrange multipliers and the Karush-Kuhn-Tuckers optimality conditions are employed to solve the resulting optimisation problems. Moreover, the successive interference cancellation method is adopted to process the received radar-communications compound signal. Finally, several numerical simulations are conducted to demonstrate the theoretical calculations and to validate the effectiveness of the proposed LPI performance optimisation strategy. It is also shown that the achievable LPI performance in RISB and CISB situations is much better than that in TISB situation.

Robust incentive Stackelberg strategy for Markov jump linear stochastic systems via static output feedback

In this study, a robust static output feedback (SOF) incentive Stackelberg game for a Markov jump linear stochastic system governed by Ito differential equations with multiple leaders and multiple followers is investigated. The existence conditions for the SOF incentive Stackelberg strategies are derived in terms of the solvability of a set of higher-order cross-coupled stochastic algebraic Lyapunov type equations (CCSALTEs). A classical Lagrange multiplier technique is employed to solve the CCSALTEs; therefore, the solution of the bilinear matrix inequality, which is a common NP-hard problem when designing SOF strategies, is not required. A heuristic algorithm is developed based on the CCSALTEs. In particular, it is shown that a robust convergence is guaranteed by combining the Krasnoselskii–Mann iterative algorithm with a new convergence condition. The performance of the proposed algorithm is discussed and a simple practical example is provided to demonstrate the effectiveness of the proposed algorithm and the SOF incentive Stackelberg strategies.

Optimization Methods for Redundancy Allocation in Large Systems

This paper addresses the issue of optimal allocation of spare modules in large series-redundant systems in order to obtain a required reliability under cost constraints. Both cases of active and standby redundancy are considered. Moreover, for a subsystem with standby redundancy, two cases are examined: in the first case, all the spares are maintained in cold state (cold standby redundancy) and, in the second one, to reduce the time needed to put a spare into operation when the active one fails, one of the spares is maintained in warm conditions. To solve this optimization problem, for the simpler case of active redundancy an analytical method based on the Lagrange multipliers technique is first applied. Then the results are improved by using Pairwise Hill Climbing, an original fine-tuning algorithm. An alternative approach is an innovative evolutionary algorithm, RELIVE, in which an individual lives for several generations and improves its fitness based on local search. These methods are especially needed in case of very large systems.

Optimal Mode Selection in D2D Communication With Deadline Constraint: A CMDP Perspective

Device to device (D2D) communication is a potential technology in the next generation communication system. We study the mode selection problem in a scenario where the D2D communication has a deadline constraint while the cellular user requires a minimum rate. We formulate the mode selection problem as a Constrained Markov Decision Process (CMDP). By adopting the Lagrangian multiplier technique, we first transform the CMDP into a non-constrained MDP problem. Then, we propose an optimal stationary mode selection algorithm combining the value iteration algorithm and the sub-gradient updating of the Lagrangian multiplier. Further, we propose an optimal randomized mode selection algorithm. Moreover, we contribute on deriving the closed-form policy selection probability. Simulation results have verified the optimality of the proposed algorithm.

Numerical Solution of Fractional Optimal Control Problems with Inequality Constraint Using the Fractional-Order Bernoulli Wavelet Functions

This paper studies the fractional optimal control problems (FOCPs) with inequality constraints. Using the Caputo definition, an optimization method based on a set of basis functions, namely the fractional-order Bernoulli wavelet functions (F-BWFs), is proposed. The solution is expanded in terms of the F-BWFs with unknown coefficients. In the first step, we convert the inequality conditions to equality conditions. In the second step, we use the operational matrix (OM) of fractional integration and the product OM of F-BWFs, with the help of the Lagrange multipliers technique for converting the FOCPs into an easier one, described by a system of nonlinear algebraic equations. Finally, for illustrating the efficiency and accuracy of the proposed technique, several numerical examples are analysed and the results compared with the analytical or the approximate solutions obtained by other techniques.

Approximate analytical solution of the linear and nonlinear multi-pantograph delay differential equations

In this paper, a combination of Laplace transform and variational iteration method are applied to get an approximate analytic solution for the multi-pantograph delay with higher order differential equations. Lagrange multiplier technique is constructed a correction functional which obtained by using Laplace transform with the variational theory. Numerical studies for the application of the present method for the considered problems are given and graphically illustrated, our proposed method is compared favorably with other methods. The simplicity and efficiency of the method.

Parametric Optimization of Exergy Efficiency in Solar Thermoelectric Generators

This paper presents the optimization of the exergy efficiency of a solar thermoelectric generator (STEG) with respect to its hot-side temperature, length-weighted current density, thermoelement area ratio, length of the thermoelectric (TE) legs, and the thermal concentration ratio (CR). Lagrange multiplier technique was employed to perform the optimization. It was found that optimal hot-side temperatures exist at which maximum exergy efficiencies of STEGs are attained. In the absence of optical concentration, but with thermal CRs as large as 600–700, these exergy efficiencies were as much as 6–8% for a hot-side temperature range of 175–277 °C. The optimal hot-side temperatures were found to be independent of both the TE device geometry and the thermal CR, but varied with the TE device’s dimensionless figure of merit, as well as the properties of the STEG’s optical component and absorber. Moreover, optimal combination of these parameters gives the configuration that optimizes the efficiency. For instance, for a Bi2Te3-based TE material, with ZTm = 1, and under vacuum conditions, the optimal configuration is: a solar surface absorber area of 1300 mm2; cross-sectional area of 1 mm2 and height of 1 mm for both p and n legs; and an emittance of 0.05, giving a conversion efficiency of about 7%. So with the optimized configuration, complicated optical systems (which usually include tracking mechanisms) may be avoided, and the required amounts of TE material minimized. This ultimately results in substantial reductions in material, manufacturing and system costs.

Numerical study of nonlinear 2D optimal control problems with multi-term variable-order fractional derivatives in the Atangana-Baleanu-Caputo sense

Abstract This paper introduces a new category of nonlinear 2D optimal control problems (OPCs) with multi-term variable-order (V-O) fractional derivatives in their dynamical systems. The fractional derivatives are considered in the sense of Atangana-Baleanu-Caputo. A computational procedure based on the Chebyshev cardinal functions (CCFs) and the Lagrange multipliers technique is developed for the approximate solution of such problems. The presented method simplify these complex problems by expanding the state and control functions in terms of the CCFs. The applicability of the formulated approach is investigated through forth numerical examples. The obtained results manifest the high precision of the proposed algorithm.

First-principles calculations of anharmonic and deuteration effects on the photophysical properties of polyacenes and porphyrinoids.

A new method for calculating internal conversion rate constants (k[combining low line]IC), including anharmonic effects and using the Lagrangian multiplier technique, is proposed. The deuteration effect on k[combining low line]IC is investigated for naphthalene, anthracene, free-base porphyrin (H2P) and tetraphenylporphyrin (H2TPP). The results show that anharmonic effects are important when calculating k[combining low line]IC for transitions between electronic states that are energetically separated (ΔE) by more than 20 000-25 000 cm-1. Anharmonic effects are also important when ΔE < 20 000-25 000 cm-1 and when the accepting modes are X-H stretching vibrations with a frequency larger than 2000 cm-1. The calculations show that there is mixing between the S1 and S2 states of naphthalene induced by non-adiabatic interactions. The non-adiabatic interaction matrix element between the S1 and S2 states is 250 cm-1 and 50 cm-1 for the normal and fully deuterated naphthalene structure and this difference significantly affects the estimated fluorescence quantum yield. Besides aromatic hydrocarbons H2P and H2TPP, the k[combining low line]IC rate constant is also calculated for pyrometene (PM567) and tetraoxa[8]circulene (4B) with a detailed analysis of the effect of the vibrational anharmonicity.

A Modified Differential Beamforming and Its Application for DOA Estimation of Low Frequency Underwater Signal

Conventional beamforming (CBF) will lose its directivity once the signal frequency is much lower than the designed frequency of the array. In this paper, a modified differential beamforming (MDBF) method is proposed to deal with the underwater acoustic signal at low frequency band and then its application for direction of arrival (DOA) estimation with deconvolution technique is given. The MDBF proposed in this paper is a variation of the conventional differential beamforming (DBF) method and it constructs an optimization problem with an additional robustness constraint. The solution of MDBF solved via Lagrange multiplier technique owns a closed-form similar to that of DBF except two additional loading terms. Simulations with planar arrays show that for the frequency band where CBF owns an almost omnidirectional beampattern, MDBF outperforms DBF in robustness at the cost of some directivity loss. The application of MDBF for DOA estimation is demonstrated with both simulated and experimental data. The results show that the proposed MDBF outperforms both DBF and CBF in DOA estimation at low frequency band.

Applying Massively Parallel Computing to Multiscale Ka Dual-Band Transmit-Array Analysis Using FETI-2LM

Transmit-arrays (TAs) are a popular cost-effective solution for high-gain antennas at millimeter waves. The design of these antennas relies on the fine tuning of the subwavelength unit cells that compose the aperture. The intricacy of the unit cells increases as new features are implemented, such as dual-band operation and wide-angle beam steering, making this antenna even more computationally challenging. In this article, a high-gain (25 dBi @ 20 GHz and 28 dBi @ 30 GHz) multiscale Ka dual-band TA for beam-steering applications is analyzed using a massively parallel implementation on multicores clusters of the finite element tearing and interconnecting method, with a two Lagrange multiplier (FETI-2LM) technique. The main contribution of this article is the implementation of 3-D nonperiodic grids with subdomains of different scales, that is suitable for proper modeling of different regions of the antenna (horn feed, lens cells, and air region). An automatic batch file procedure is proposed for the TA meshing, allowing the limited set of constitutive unit cells of the TA to be meshed separately. Additionally, we are using a block-Krylov strategy to efficiently capture the TA beam-steering capability, without restarting from scratch the interface problem for each feed position. Since other finite element method results were not accessible due the size of the problem, the FETI-2LM numerical results provided in this article are compared with other multilevel fast multipole methods and transmission line modeling method of CST Microwave Studio solvers, as well as with the measured results of the corresponding Ka dual-band prototype. The computational infrastructure has been used only a fraction of its capacity. Therefore, a much higher gain design (40 dBi) can be assessed, opening a new realm of applicability of TAs in the space segment usually occupied by reflector-based solutions.

Robust Incentive Stackelberg Strategy for Markov Jump Delay Stochastic Systems via Static Output Feedback

A static output feedback (SOF) incentive Stackelberg game (ISG) for a continuous-time Markov jump delay stochastic system (MJDSS) is discussed. The existence conditions on the SOF incentive Stackelberg strategy set are established in terms of the solvability of a set of higher-order cross-coupled stochastic algebraic Lyapunov-type equations (CCSALTEs). A classical Lagrange-multiplier technique is used to derive the CCSALTEs, thereby avoiding having to solve the bilinear matrix inequalities (BMIs), a well-known NP-hard problem in designing the SOF strategy. A heuristic algorithm is proposed to solve CCSALTEs such that convergence is attained by applying the Krasnoselskii-Mann (KM) iterative algorithm. A simple numerical example demonstrates the efficiency of the SOF incentive Stackelberg strategy.

Novel Proportionate Adaptive Filters with Coefficient Vector Reusing

In this paper, a new family of adaptive filtering algorithms is presented, which aims to combine the small misalignment resulting from the reuse of past weight vectors with the fast convergence arising from the proportionate adaptation and logarithmic cost functions. This family of algorithms is obtained as a solution to a deterministic constrained optimization problem, by using the Lagrange multipliers technique, which differs from the traditionally employed stochastic gradient technique. Two special cases are proposed, namely the improved mu-law proportionate least mean logarithmic square with reuse of coefficients (IMPLMLS-RC) algorithm and the improved mu-law proportionate least logarithmic absolute difference with reuse of coefficients (IMPLLAD-RC) algorithm. An energy conservation relationship is established, which can be employed to perform stochastic transient analyses of the proposed algorithms. Simulations in system identification and active noise control applications show the advantages of the IMPLMLS-RC and IMPLLAD-RC algorithms over the traditional LMS and LAD, and the recently proposed LMLS and LLAD, with respect to both steady-state performance and robustness against impulsive noise.