Interior Penalty Function Research Articles

Iterative eigenvalue method coupled with RSM for structural acoustic analysis and optimization of the CLD-damped orthotropic steel deck

In the present work, an iterative eigenvalue method (IEM) is proposed and combined with the boundary element method (BEM) to perform the structural acoustical analysis of the orthotropic steel deck (OSD) damped with constrained layer damping (CLD), through which the frequency-dependent properties of the viscoelastic core can be accurately considered. Then, a series of hammer tests are carried out in a semi-anechoic chamber to verify the acoustic prediction method. It shows that the simulated structural noises basically agree with the test results, and the overall deviations at the field points M2 and M8 are merely 2.4 and 2.3 dB, respectively. Subsequently, taking the thicknesses of the damping layer and the constraining layer, as well as the material properties of the constraining layer as design variables, a central composite experiment is designed and combined with response surface methodology (RSM) to fit the relationship between analysis objectives and design variables. With the objective of minimizing the acoustic radiation power, a program for finding the optimal values of design variables is developed based on the interior penalty function method. Through iterative computations, the optimal points are numerically determined as t 1 = 2.3 mm and t 2 = 0.3 mm for the thicknesses of the damping layer and the constraining layer, respectively. Steel alloy is found to be the optimal material for the constraining layer. After optimizing CLD treatment, the overall acoustical power level can be reduced from 97.7 dB to 93.1 dB and the overall sound pressure levels (SPLs) at the field points M2 and M8 can be reduced from 89.2 dB to 83.5 and 76.9 dB to 70.3 dB, respectively, indicating that the optimization of CLD design parameters can significantly improve acoustic performance of the CLD-damped structures.

An interior penalty function method for solving fuzzy nonlinear programming problems

In this article, we investigate fuzzy interior penalty function method for solving fuzzy nonlinear programming problems (FNLPP) based on a new fuzzy arith-metic, unconstrained optimization, and fuzzy ranking on the parametric form of triangular fuzzy numbers (TFN). The main objective of this paper is to solve constrained fuzzy nonlinear programming problems using interior penalty func-tions (IPF) by converting it into unconstrained optimization problems. We prove an important lemma and a convergence theorem for the interior penalty functions method. Interior penalty function techniques favor sites near the boundary of the feasible region in the interior. We present a numerical example of the suggested method and compare the results to those produced by existing methods.

A Two-Phase Constraint-Handling Technique for Constrained Optimization

A two-phase constraint-handling technique is integrated into the evolutionary algorithms to solve constrained optimization problems (called TPDE) in this article. In phase one, denoted as the exploration phase, an exterior penalty function method with the dynamic penalty coefficients is developed to compare any two candidate solutions, which aims to push the population into the feasible region. To reduce the computational burden, in phase two, denoted as the exploitation phase, an interior penalty function method with the dynamic penalty coefficients is developed, which enhances the search ability by using the information of constraints in feasible solutions. During the optimization process, differential evolution is adopted as the search algorithm to produce the offspring population. Experiment results on four benchmark test suites, namely, IEEE CEC 2006, IEEE CEC 2010, IEEE CEC 2017, and IEEE CEC 2020, indicate that TPDE is competitive with other popular algorithms.

A hybrid IMSE-FE-BE method coupled with RSM for vibro-acoustic analysis and optimization of an I-shaped steel beam damped with constrained layer damping

In this paper, a method for calculating vibration and noise from the I-shaped steel beam damped with constrained layer damping (CLD) was proposed based on iterative modal strain energy (IMSE) method and finite element (FE)-boundary element method (BEM). Then, a series of hammer tests were carried out to obtain the typical vibro-acoustic response samples of a CLD-damped beam. It is shown that the calculated noise and vibration agreed well with the measured results, indicating that the proposed method is effective and accurate. Then, the response surface methodology (RSM) was used to analyze the influence of CLD-structural parameters on the sound radiation characteristics and natural frequency of the damped beam. With the objective of minimizing the acoustic radiation power, a program for finding the optimal CLD-structural parameters was developed based on interior penalty function method (IPFM). Finally, the acoustic response comparisons between the original bare beam and the optimized CLD-damped beam were performed. It shows that the overall sound power level was decreased from 98.12 dB to 80.47 dB, and the overall sound pressure level at measuring point S3-3 was reduced from 78.86 dB to 61.07 dB, demonstrating that the optimization of CLD-structural parameters can significantly improve the acoustic performance of CLD-structures.

On the Bridge Between Exterior and Interior Penalty Method

In this paper, we present methods of penalty functions for solving constrained optimization problems. The methods that we characterize presently, attempt to approximate a constrained optimization problem with an unconstrained, one and then apply standard search techniques such as the exterior penalty function method and the interior penalty method to get solutions. The paper that follows assure exterior penalty methods recognizing that interior penalty function methods incarnate the same principles.

Game theory approach for optimum design of an aged structure with multiple objectives

A mathematical procedure based on the concept of game theory is described for evaluating an optimal strategy for increasing the residual life of an aged structure subject to cyclic loading. To increase the residual life of the structure, it is proposed to attach additional sections with the critical members. A computational procedure for estimating residual life of structure based on Paris equation for crack growth propagation with number of cycle is described. The decision making problem is formulated as constrained optimization problem having two objectives of conflicting nature. The design variables are taken as the cross-sectional area of attached sections in the structure. Geometric constraints in the form of lower and upper bounds on the design variables and constraint on stress developed are considered in the formulation. Two objectives considered in the present investigation are: minimization of the mass of attached sections and maximization of the extra residual life of structure due to mass attachment. The design problem is formulated as a two-person game and the Nash non-cooperative solution is evaluated for irrational play to determine the starting point of the game. For the cooperative game, a supercriterion is formulated for the overall benefit of the players. The game is terminated when an optimal trade-off between the objectives is reached with the maximization of the supercriterion. Two different optimization methods, namely interior penalty function method and grid search method have been used for supercriterion maximization. The methodology is demonstrated by solving a problem of practical interest.

Passenger inflow control with hierarchical coordination for overloaded metro lines

Due to the congestion in urban rail transit during peak hours, this study proposes a hierarchical coordinated passenger inflow control model for overloaded metro lines from three dimensions, namely line, station and ticket gate. The hierarchical structure divides the system into three control layers and the model is established using the linear programming method. An improved particle swarm optimisation (PSO) algorithm, combining the interior penalty function and the basic PSO algorithm, is used to find the optimal objective value. In addition, the objective function with the penalty function to transform the constrained optimisation problem into an unconstrained optimisation problem is developed to solve the model. Finally, a real-world instance with operation data of the Beijing Metro Line 5 is implemented to demonstrate the performance and effectiveness of the proposed approaches.

A double-layer optimization model for flatness control of cold rolled strip

The flatness control system consists of two subsystems. One is the feedback control subsystem which is used to implement the closed-loop flatness control. Another is usually referred to as feedforward control subsystem in the field of strip flatness control. To realize the cooperative control of roll bending between two subsystems, a feedforward-–feedback coordination control strategy based on double-layer optimization is proposed to obtain the optimal adjustments of actuators. Firstly, a global optimization model of actuator adjustment is established by the overall modeling, and an improved interior penalty function method is developed and utilized to solve this model subsequently. On this basis, an optimal allocation model is proposed to achieve a global optimal allocation of actuator adjustments between feedforward control and feedback control. Numerical experiments and applications show that this new control strategy and the corresponding models are effective in the flatness control process of cold rolling mill.

Energy-Latency Tradeoff for Energy-Aware Offloading in Mobile Edge Computing Networks

Mobile edge computing (MEC) brings computation capacity to the edge of mobile networks in close proximity to smart mobile devices (SMDs) and contributes to energy saving compared with local computing, but resulting in increased network load and transmission latency. To investigate the tradeoff between energy consumption and latency, we present an energy-aware offloading scheme, which jointly optimizes communication and computation resource allocation under the limited energy and sensitive latency. In this paper, single and multicell MEC network scenarios are considered at the same time. The residual energy of smart devices’ battery is introduced into the definition of the weighting factor of energy consumption and latency. In terms of the mixed integer nonlinear problem for computation offloading and resource allocation, we propose an iterative search algorithm combining interior penalty function with D.C. (the difference of two convex functions/sets) programming to find the optimal solution. Numerical results show that the proposed algorithm can obtain lower total cost (i.e., the weighted sum of energy consumption and execution latency) comparing with the baseline algorithms, and the energy-aware weighting factor is of great significance to maintain the lifetime of SMDs.

Shape and cross-section optimization of plane trusses subjected to earthquake excitation using gradient and Hessian matrix calculations

ABSTRACTThis article describes a second-order shape and cross-section optimization method of plane truss subjected to earthquake excitation. The method is based on gradient and Hessian matrix calculation. First, the first and second derivatives of dynamic response with respect to design variables are calculated based on the Newmark method. Second, the inequality time-dependent constraint problem is converted into a sequence of appropriately formed unconstrained problems using the integral interior penalty function method. Then, the gradient and Hessian matrix of the integral interior penalty function are computed. Third, Marquardt's method is employed to solve the unconstrained problems. Finally, the new approach is validated through several case studies. The results show that the new optimization method is an efficient and effective approach for minimum weight design of truss structures.

An Interior Penalty Method for the Generalized Method of Moments

The generalized method of moments (GMM) is a technique to discretize integral equations that permits integration of different types of basis functions as well as different geometric descriptions using a partition of unity framework. While accuracy and efficacy of the method have been demonstrated, the integration quadratures required to compute the inner products are often high, as they have to respect spatial variation of the integrand. To overcome this problem, we introduce an interior penalty function method within the GMM framework. The penalty formulation yields solutions that are smooth and accurate both in surface currents and fields with significantly lower numerical quadrature orders than would be required for the uncompensated operator. To demonstrate and analyze the method, we conduct an analytical and numerical investigation of the properties of the penalty method applied to the 2-D TE <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</sup> electric field integral equation (EFIE) operator.

Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

Modified maximum spacings method for generalized extreme value distribution and applications in real data analysis

This paper analyzes weekly closing price data of the S&P 500 stock index and electrical insulation element lifetimes data based on generalized extreme value distribution. A new estimation method, modified maximum spacings (MSP) method, is proposed and obtained by using interior penalty function algorithm. The standard error of the proposed method is calculated through Bootstrap method. The asymptotic properties of the modified MSP estimators are discussed. Some simulations are per- formed,whichshowthattheproposedmethodisnotonlyavailableforthewholeshape parameter space, but is also of high efficiency. The benchmark risk index, value at risk (VaR), is evaluated according to the proposed method, and the confidence interval of VaR is also calculated through Bootstrap method. Finally, the results are compared with those derived by empirical calculation and some existing methods.

Interior penalty functions and duality in linear programming

Logarithmic additive terms of barrier type with a penalty parameter are included in the Lagrange function of a linear programming problem. As a result, the problem of searching for saddle points of the modified Lagrangian becomes unconstrained (the saddle point is sought with respect to the whole space of primal and dual variables). Theorems on the asymptotic convergence to the desired solution and analogs of the duality theorems for the arising optimization minimax and maximin problems are formulated.

The optimal production-run time for a stock-dependent imperfect production process

This paper develops an inventory model for a hypothesized volume flexible manufacturing system in which the production rate is stock-dependent and the system produces both perfect and imperfect quality items. The demand rate of perfect quality items is known and constant, whereas the demand rate of imperfect (non-conforming to specifications) quality items is a function of discount offered in the selling price. In this paper, we determine an optimal production-run time and the optimal discount that should be offered in the selling price to influence the sale of imperfect quality items produced by the manufacturing system. The considered model aims to maximize the net profit obtained through the sales of both perfect and imperfect quality items subject to certain constraints of the system. The solution procedure suggests the use of ?Interior Penalty Function Method? to solve the associated constrained maximization problem. Finally, a numerical example demonstrating the applicability of proposed model has been included.

Use of barrier functions for optimal correction of improper problems of linear programming of the 1st kind

Possibilities are investigated of the combined use of interior and exterior penalty functions for discovering the generalized (approximate) solutions of improper problems of linear programming of the 1st kind. Diagrams of algorithms, theorems of convergence, results of numerical experiments are presented.

The Optimum Design on the Roller Chain Drive of the Machine for Supplementing the Traditional Chinese Pills

The roller chain drive of the machine for supplementing traditional Chinese pills has movement characteristics of gear drive and belt drive, and its movement is a little complex, so interior point penalty function method is applied in the optimum design of chain. First of all, the optimizing mathematical model is built by applying the interior point penalty function method, and the design variables, the target function and constraints are proposed, and then the optimal solution for key parameters of drive chain is obtained by the iterative, which will provide the reference and basis for the research on big drive power and the least rows number of drive chain.

Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If the control of the initial problem satisfies strict complementarity conditions for its Hamiltonian except for a finite number of times, the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem. Our results provide several types of efficiency measures of the penalization technique: error estimations of the control for $L^s$ norms ($s$ in $[1,+\infty]$), error estimations of the state and the adjoint state in Sobolev spaces $W^{1,s}$ ($s$ in $[1,+\infty)$) and also error estimates for the value function. For the $L^1$ norm and the logarithmic penalty, the optimal results are given. In this case we indeed establish that the penalized control and the value function errors are of order $O(\eps|\log\eps|)$.

Research on Topology Optimization Method for Continuum Structure under Geometric Constraints

A new topology optimization model with holes’ geometric constraints for continuum structure is presented. It is solved by an evolutionary optimization method based on interval relaxation, in which the problem is divided into two subproblems of topology optimization process and size/shape optimization process. The optimal topology of structure can be found gradually by introducing interval relaxation factor to adjust holes’ size constraints, delete noneffective holes and by generating new holes based on the sensitivity analysis of objective function. Interior penalty-function method is employed as an optimization technique for the size/shape optimization of the structure corresponding to the topology. When the holes’ size bounds are the actual values, the optimal solution is the smallest objective function structure in the various topologies. Thus realizes the holes’ geometric size and location design, topology design and layout design together. The optimization results of example shows the method proposed is of good effectiveness and engineering applicability.

Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data

This paper studies the application of extreme value statistics (EVS) theory on analysis for stock data, based on interior penalty function algorithm and Bootstrap methods. The generalized Pareto distribution (GPD) models are considered in analyzing the closing price data of Shanghai stock market. The maximum likelihood estimates (MLEs) are obtained by using the interior penalty function algorithm. Correspondingly, the bias and standard errors of MLEs, and the hypothesis test on the shape parameter are concerned through Bootstrap methods. Some simulations are performed to demonstrate the efficacy of parameter estimation and the power of the test. The estimates of the tail index in this paper are compared with those obtained via classical methods. At last, the model is diagnosed by numerical and graphical methods and the Value-at-Risk (VaR) is estimated.