Topology Optimization for Harmonic Excitation Structures with Minimum Length Scale Control Using the Discrete Variable Method

Continuumtopology optimization considering the vibration response is of great value in the engineering structure design.The aimof this studyis toaddress the topological designoptimizationof harmonic excitationstructureswith minimumlength scale control to facilitate structuralmanufacturing.Astructural topology design based on discrete variables is proposed to avoid localized vibration modes,gray regions and fuzzy boundaries in harmonic excitation topology optimization.The topological design model and sensitivity formulation are derived.The requirement of minimum size control is transformed into a geometric constraint using the discrete variables.Consequently,thin bars,small holes,and sharp corners,which are not conducive to the manufacturing process,can be eliminated from the design results.The present optimization design can efficiently achieve a 0–1 topology configuration with a significantly improved resonance frequency in a wide range of excitation frequencies.Additionally,the optimal solution for harmonic excitation topology optimization is not necessarily symmetric when the load and support are symmetric,which is a distinct difference fromthe static optimization design.Hence,one-half of the design domain cannot be selected according to the load and support symmetry.Numerical examples are presented to demonstrate the effectiveness of the discrete variable design for excitation frequency topology optimization,and to improve the design manufacturability.

APPROACH FOR LAYOUT OPTIMIZATION OF TRUSS STRUCTURES WITH DISCRETE VARIABLES UNDER DYNAMIC STRESS, DISPLACEMENT AND STABILITY CONSTRAINTS

A mathematical model was developed for layout optimization of truss structures with discrete variables subjected to dynamic stress, dynamic displacement and dynamic stability constraints. By using the quasi-static method, the mathematical model of structure optimization under dynamic stress, dynamic displacement and dynamic stability constraints were transformed into one subjected to static stress, displacement and stability constraints. The optimization procedures include two levels, i.e., the topology optimization and the shape optimization. In each level, the comprehensive algorithm was used and the relative difference quotients of two kinds of variables were used to search the optimum solution. A comparison between the optimum results of model with stability constraints and the optimum results of model without stability constraint was given. And that shows the stability constraints have a great effect on the optimum solutions.

Explicit Solutions for a （2＋1）-Dimensional Toda Lattice with Two Discrete Variables

An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems is found. As an application, explicit solutions of the （2＋1）-dimensional Toda equation with two discrete variables are obtained.

A METHOD FOR TOPOLOGICAL OPTIMIZATION OF STRUCTURES WITH DISCRETE VARIABLES UNDER DYNAMIC STRESS AND DISPLACEMENT CONSTRAINTS

A method for topological optimization of structures with discrete variables subjected to dynamic stress and displacement constraints is presented. By using the quasistatic method, the structure optimization problem under dynamic stress and displacement constraints is converted into one subjected to static stress and displacement constraints. The comprehensive algorithm for topological optimization of structures with discrete variables is used to find the optimum solution.

Mapped Finite Element Discrete Variable Representation

Efficient numerical solver for the SchrSdinger equation is very important in physics and chemistry. The finite element discrete variable representation （FE-DVR） was first proposed by Rescigno and Mc-Curdy [Phys. Rev. A 62, 032706 （2000）] for solving quantum-mechanical scattering problems. In this work, an FE-DVR method in a mapped coordinate was proposed to improve the efficiency of the original FE-DVR method. For numerical demonstration, the proposed approach is applied for solving the electronic eigenfunctions and eigenvalues of the hydrogen atom and vibrational states of the electronic state 3E＋ of the Cs2 molecule which has long-range interaction potential. The numerical results indicate that the numerical efficiency of the original FE-DVR has been improved much using our proposed mapped coordinate scheme.

Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer(BO)approximation,nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface(PES).However,this single PES picture should be reconsidered if a conical intersection(CI)is present,although the energy is well below the CI.The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation,i.e.,the diagonal BO correction(DBOC)and the geometric phase(GP),which are divergent at the CI.At the same time,there are cusps in the adiabatic PESs.Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation.A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation(DVR)method.We examine the numerical accuracy of the Sinc DVR method for solving the Schrodinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation.The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points,without special treatment of the divergence of the DBOC and the GP.The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP,whose accurate form usually is not easy to obtain.

STUDIFS ON OPTIMAL TOPOLOGY DES IGN OF STRUCTURES WITH DISCRETE VARIABLES

Some problems in the optimal topology design of structures with discrete variables are studied in this paper.The problem of a model of discrete optimization is discussed and a neglected fact that discrete optimum design may be controlled by the discreteness of sizing variables and global con- straints is pointed out.A heuristic algorithm for solving discrete topology optimization problems of trusses and frames is proposed.

A Mixed-Variable Experimental Design Method

Mixed-variable problems are inevitable in engineering. However, few researches pay attention to discrete variables. This paper proposed a mixed-variable experimental design method (ODCD): first, the design variables were divided into discrete variables and continuous variables;then, the DVD method was employed for handling discrete variables, the LHD method was applied for continuous variables, and finally, a Columnwise-Pairwise Algorithm was used for the overall optimization of the design matrix. Experimental results demonstrated that the ODCD method outperforms in terms of the sample space coverage performance.

An equilibrium multi-objective optimum design for non-circular clearance hole of disk with discrete variables

An Equilibrium Multi-objective Optimization Model（EMOM） with self-regulated weighting factors has been proposed for the optimum design of non-circular clearance hole on the front flange of turbine disk. In the ‘‘equilibrium design＂, both the stress decrease around the hole and the least hole＇s profile variation are considered, which balances two ambivalent design goals. Specific discrete variables are applied to realize the standardization design in the optimization process, in which a Surrogate Genetic Coding Algorithm（SGCA） is introduced, and a special check module is used to get rid of repeated fitness evaluation of the samples. The method offers an equilibrium design for the non-circular clearance hole of the turbine disk with great accuracy and efficiency.

Discrete Optimization on Unsteady Pressure Fluctuation of a Centrifugal Pump Using ANN and Modified GA

Pressure fluctuation due to rotor-stator interaction in turbomachinery is unavoidable,inducing strong vibration in the equipment and shortening its lifecycle.The investigation of optimization methods for an industrial centrifugal pump was carried out to reduce the intensity of pressure fluctuation to extend the lifecycle of these devices.Considering the time-consuming transient simulation of unsteady pressure,a novel optimization strategy was proposed by discretizing design variables and genetic algorithm.Four highly related design parameters were chosen,and 40 transient sample cases were generated and simulated using an automatic program.70%of them were used for training the surrogate model,and the others were for verifying the accuracy of the surrogate model.Furthermore,a modified discrete genetic algorithm(MDGA)was proposed to reduce the optimization cost owing to transient numerical simulation.For the benchmark test,the proposed MDGA showed a great advantage over the original genetic algorithm regarding searching speed and effectively dealt with the discrete variables by dramatically increasing the convergence rate.After optimization,the performance and stability of the inline pump were improved.The efficiency increased by more than 2.2%,and the pressure fluctuation intensity decreased by more than 20%under design condition.This research proposed an optimization method for reducing discrete transient characteristics in centrifugal pumps.

Hybrid Genetic Algorithm for Engineering Structural Optimization with Dis crete Variables

Aiming at the phenomenon of discrete variables whic h generally exists in engineering structural optimization, a novel hybrid genetic algorithm (HGA) is proposed to directly search the optimal solution in this pape r. The imitative full-stress design method (IFS) was presented for discrete struct ural optimum design subjected to multi-constraints. To reach the imitative full -stress state for dangerous members was the target of IFS through iteration. IF S is integrated in the GA. The basic idea of HGA is to divide the optimization t ask into two complementary parts. The coarse, global optimization is done by the GA while local refinement is done by IFS. For instance, every K generations, th e population is doped with a locally optimal individual obtained from IFS. Both methods run in parallel. All or some of individuals are continuously used as initial values for IFS. The locally optimized individuals are re-implanted into the current generation in the GA. From some numeral examples, hybridizatio n has been discovered as enormous potential for improvement of genetic algorit hm. Selection is the component which guides the HGA to the solution by preferring in dividuals with high fitness over low-fitted ones. Selection can be deterministi c operation, but in most implementations it has random components. "Elite surviv al" is introduced to avoid that the observed best-fitted individual dies out, j ust by selecting it for the next generation without any random experiments. The individuals of population are competitive only in the same generation. There exists no competition among different generations. So HGA may be permitted to h ave different evaluation criteria for different generations. Multi-Selectio n schemes are adopted to avoid slow refinement since the individuals have si milar fitness values in the end phase of HGA. The feasibility of this method is tested with examples of engineering design wit h discrete variables. Results demonstrate the validity of HGA.

Binary discrete method of topology optimization

The numerical non-stability of a discrete algorithm of topology optimization can result from the inaccurate evaluation of element sensitivities. Especially, when material is added to elements, the estimation of element sensitivities is very inaccurate, even their signs are also estimated wrong. In order to overcome the problem, a new incremental sensitivity analysis formula is constructed based on the perturbation analysis of the elastic equilibrium increment equation, which can provide us a good estimate of the change of the objective function whether material is removed from or added to elements, meanwhile it can also be considered as the conventional sensitivity formula modified by a non-local element stiffness matrix. As a consequence, a binary discrete method of topology optimization is established, in which each element is assigned either a stiffness value of solid material or a small value indicating no material, and the optimization process can remove material from elements or add material to elements so as to make the objective function decrease. And a main advantage of the method is simple and no need of much mathematics, particularly interesting in engineering application.

A COMBINAT0RIAL ALGORITHM FOR THE DISCRETE OPTIMIZATION OF STRUCTURES

The definition of local optimum solution of the discrete optimization is first given.and then a comprehensive combinatorial algorithm is proposed in this paper. Two-leveloptimum method is used in the algorithm. In the first level optimization, anapproximate local optimum solution X is found by using the heuristic algorithm,relative difference quotient algorithm. with high computational efficiency and highperformance demonstrated by the performance test of random samples. In the secondlevel, a mathematical model of (- 1, 0, 1) programming is established first, and then itis changed into (0, 1) programming model. The local optimum solution X will befrom the (0. 1) programming by using the delimitative and combinatorial algorithm orthe relative difference quotient algorithm. By this algorithm, the local optimumsolution can be obtained certainly, and a method is provnded to judge whether or notthe approximate optimum solution obtained by heuristic algorithm is an optimumsolution. The above comprehensive combinatorial algorithm has higher computationalefficiency.

Power supply for generating frequency-variable resonant magnetic perturbations on the J-TEXT tokamak

To further research the response of the tearing mode（TM） to dynamic resonant magnetic perturbation（DRMP） on the J-TEXT tokamak, a modified series resonant inverter power supply（MSRIPS） with a function of discrete variable frequency is designed for DRMP coils in this study. The MSRIPS is an AC–DC–AC converter, including a phase-controlled rectifier, an LC filter, an insulated gate bipolar transistor（IGBT） full bridge, a matching transformer, three resonant capacitors with different capacitance values, and three corresponding silicon controlled rectifier（SCR） switches. The function of discrete variable frequency is realized by switching over different resonant capacitors with corresponding SCR switches while matching the corresponding driving frequency of the IGBT full bridge. A detailed switching strategy of the SCR switch is put forward to obtain sinusoidal current waveform and realize current waveform smooth transition during frequency conversion. In addition, a resistor and thyristor bleeder is designed to protect the SCR switch from overvoltage. Manufacturing of the MSRIPS is completed, and the MSRIPS equipment can output current with an amplitude of 1.5 kA when its working frequency jumps among different frequencies. Moreover, the current waveform is sinusoidal and can smoothly transition during frequency conversion. Furthermore, the transition time when the current amplitude rises from zero to a steady state is less than 2 ms during frequency conversion. By using the MSRIPS, the expected discrete variable frequency DRMP is generated, and the phenomenon of the TM being locked to the discrete variable frequency DRMP is observed on the J-TEXT tokamak.

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for cor- rectly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian （J=0） in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OC10 molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.

Optimization Design of RC Ribbed Floor System Using Eagle Strategy with Particle Swarm Optimization

The eagle strategy algorithm is combined with particle swarm optimization in this paper.The new algorithm,denoted as the ES-PSO,is implemented by interfacing Etabs structural analysis codes.ES-PSO is used to optimize the RC ribbed floor system,including floor and underground garage roof.By considering the effects of reinforcement,the principle of virtual work is applied to calculate the deflections of components.Construction cost is taken as the objective function and the constraint conditions are required to satisfy.Accordingly,the optimal layout,the optimal sections of the beams and slabs and the corresponding reinforcements are obtained for different column grids.In this investigation,the RC ribbed floor system is optimized according to the Chinese standard,whose column grids are 8.4 m and 8.4 m.The performance of the ES-PSO algorithm is good enough,which can be applied to practical engineering.The paper can also provide a basis for subsequent optimization design of monolithic structures.

New model for deducing directional extrema based on multivariate extremum statistical theory

The parameters of principal and directional extrema in a marine environment are important in marine engineering design, especially for appropriate construction of oceanic platforms and other structures. When designing wave walls and break water structures, the orientation of the breakwater or seawall depends mainly on the direction of the strongest waves. However, the strength of the breakwater and the elevation of the seawall depend on the magnitude of the biggest wave height of the strongest waves. Thus, identification of directional extrema plays an important role in the design of wave factors. When calculating the directional extremum, different materials may require different specific computational methods, yet few theoretical studies have been conducted in this field of research. Based on multivariate extremnm statistical theory, this paper utilizes a discrete random variable to build a joint probability model compounded by a discrete random variable and a multivariate continuous random variable. Furthermore, this paper provides the first investigation on the theories and methodologies to deduce wave directional extrema. The results provide tools for both creating the calculation method of the directional extremum value and providing the rational directional extremum parameters for marine engineering design.

DISCRETE CHARACTERIZATION OF THE PALEY-WIENER SPACE WITH SEVERAL VARIABLES

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted by Bπ,p(Rn), 1≤p<∞, i.e., for 1<p<∞, Bπ,p(Rn) is isomorphic to lp(Zn), and for p=1, Bπ,1(Rn) is isomorphic to the discrete Hardy space with several variables, which is denoted by H(Zn).

Geom/G_1,G_2/1/1 REPAIRABLE ERLANG LOSS SYSTEM WITH CATASTROPHE AND SECOND OPTIONAL SERVICE

This paper studies a single server discrete-time Erlang loss system with Bernoulli arrival process and no waiting space. The server in the system is assumed to provide two different types of services, namely essential and optional services, to the customer. During the operation of the system, the arrival of the catastrophe will break the system down and simultaneously induce customer to leave the system immediately. Using a new type discrete supplementary variable technique, the authors obtain some performance characteristics of the queueing system, including the steady-state availability and failure frequency of the system, the steady-state probabilities for the server being idle, busy, breakdown and the loss probability of the system etc. Finally, by the numerical examples, the authors study the influence of the system parameters on several performance measures.

Adaptive Stabilization of Neural Control for Robot Trajectory Tracking

A neural network (NN) based adaptive control law is proposed for the tracking control of an n link robot manipulator with unknown dynamic nonlinearities. Basis function like nets are employed to approximate the plant nonlinearities, and the bound on the NN reconstruction error is assumed to be unknown. The proposed NN based adaptive control approach integrates an NN approach with an adaptive implementation of discrete variable structure control with a simple estimation law to estimate the upper bound on the NN reconstruction error and an additional control input to be updated as a function of the estimate. Lyapunov stability theory is used to prove the uniform ultimate boundedness of the tracking error.