Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.

Shape control on probability density function in stochastic systems

A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.

Improved shape hardening function for bounding surface model for cohesive soils

A shape hardening function is developed that improves the predictive capabilities of the generalized bounding surface model for cohesive soils, especially when applied to overconsolidated specimens. This improvement is realized without any changes to the simple elliptical shape of the bounding surface, and actually reduces the number of parameters associated with the model by one.

The choosing of reproducing kernel particle shape function with mathematic proof

Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain analytic solutions. So it is necessary to give numerical solutions. The reproducing kernel particle （RKP） method is based on the Carlerkin Meshless method. According to the Sobolev space and Fourier transform, the RKP shape function is mathematically proved in this paper.

Modeling unbalanced rotor system with continuous viscoelastic shaft by frequency-dependent shape function

A reduced-order dynamic model for an unbalanced rotor system is developed, taking the coupling between torsional and lateral vibrations into account. It is assumed that a shaft is regarded as a continuous viscoelastic shaft with unbalanced and small deformation properties. The equations of motion for the torsional and lateral vibrations are derived using Lagrange's approach with the frequency-dependent shape function. The rotor torsional vibration is coupled with the lateral vibrations by unbalance elements in a way of excitations. Simulation and experiment results show clearly that the torsional vibration has strong impact on the rotor lateral vibrations, and it causes subharmonic and superharmonic excitations through unbalance elements, which leads to the superharmonic resonances in the lateral vibrations. This model with low-order and high accuracy is suitable for rotor dynamic analysis in real time simulation as well as for active vibration control syntheses.

ON THE SELECTION OF SHAPE FUNCTION SPACES OF TRIANGULAR PLATE ELEMENTS

Using the method of undetermined coefficients, we construct a set of shape function spaces of nine-node triangular plate elements converging for any meshes, which generalize Spect's element and Veubeke's element.

ON SHAPE PRESERVING EXTENSION AND APPROXIMATION OF FUNCTIONS

We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,and gives many new shapes of functions.The main results show that the shape preserving approxi- mation has close relation to the shape preserving extension.One of our main results shows that if f∈C(Ω)has some shape defined by our definition,then f can be uniformly approximated by polynomials P_n ∈p_n(n∈N)which have the same shape in Ω,and the degree of the ap- proximation is Cω(f,n^(-β))with constants C,β>0.

Formation mechanism of inherent spatial heterogeneity of microstructure and mechanical properties of NiTi SMA prepared by laser directed energy deposition

Ni51Ti49 at.%bulk was additively manufactured by laser-directed energy deposition(DED)to reveal the microstructure evolution,phase distribution,and mechanical properties.It is found that the localized remelting,reheating,and heat accumulation during DED leads to the spatial heterogeneous distribution of columnar crystal and equiaxed crystal,a gradient distribution of Ni4Ti3 precipitates along the building direction,and preferential formation of Ni4Ti3 precipitates in the columnar zone.The austenite transformation finish temperature(Af)varies from-12.65℃(Z=33 mm)to 60.35℃(Z=10 mm),corresponding to tensile yield strength(σ0.2)changed from 120±30 MPa to 570±20 MPa,and functional properties changed from shape memory effect to superelasticity at room temperature.The sample in the Z=20.4 mm height has the best plasticity of 9.6%and the best recoverable strain of 4.2%.This work provided insights and guidelines for the spatial characterization of DEDed NiTi.

基于CST参数化方法气动优化设计研究

翼型及机翼优化设计中,设计变量的个数对优化算法的收敛速度及代理模型的精度有很大的影响。因此,在精确描述翼型的同时,发展较少设计变量的翼型参数化方法对翼型优化设计有着重要的意义。本文基于CST(class function/shape function transformation)翼型参数化方法对Kriging模型的预测精度进行研究,并采用改进的粒子群优化算法构建气动优化设计系统。某亚声速机翼单点减阻设计及超临界翼型的稳健性设计表明该系统具有较高的设计质量,方法可靠,有较高的工程应用前景。

Derivative Derivative-Based Envelope Design Technique for Wideband Envelope Tracking Power Amplifier with Digital Predistortion

A novel envelope design for an envelope tracking(ET)power amplifier(PA)based on its derivatives is proposed,which can trade well off between bandwidth reduc-tion and tracking accuracy.This paper theoretically analyzes how to choose an envelope design that can track the original envelope closely and reduce its bandwidth,and then demonstrates an example to validate this idea.The generalized memory polynomial(GMP)model is applied to compensate for the nonlinearity of ET PA with the proposed envelope design.Experiments are carried out on an ET system that is operated with the center fre-quency of 3.5 GHz and excited by a 20 MHz LTE signal,which show that the proposed en-velope design can make a good trade-off between envelope bandwidth and efficiency,and satisfactory linearization performance can be realized.

Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

Roll System and Stock's Multi-parameter Coupling Dynamic Modeling Based on the Shape Control of Steel Strip

The existence of rolling deformation area in the rolling mill system is the main characteristic which dis- tinguishes the other machinery. In order to analyze the dynamic property of roll system＇s flexural deformation, it is necessary to consider the transverse periodic movement of stock in the rolling deformation area which is caused by the flexural deformation movement of roll system simul- taneously. Therefore, the displacement field of roll system and flow of metal in the deformation area is described by kinematic analysis in the dynamic system. Through intro- ducing the lateral displacement function of metal in the deformation area, the dynamic variation of per unit width rolling force can be determined at the same time. Then the coupling law caused by the co-effect of rigid movement and flexural deformation of the system structural elements is determined. Furthermore, a multi-parameter coupling dynamic model of the roll system and stock is established by the principle of virtual work. More explicitly, the cou- pled motion modal analysis was made for the roll system. Meanwhile, the analytical solutions for the flexural defor- mation movement＇s mode shape functions of rolls are discussed. In addition, the dynamic characteristic of the lateral flow of metal in the rolling deformation area has been analyzed at the same time. The establishment ofdynamic lateral displacement function of metal in the deformation area makes the foundation for analyzing the coupling law between roll system and rolling deformation area, and provides a theoretical basis for the realization of the dynamic shape control of steel strip.

Parametric geometry representations for wind turbine blade shapes

Based on the Joukowsky transformation and Theodorsen method, a novel parametric function （shape function） for wind turbine airfoils has been developed. The airfoil design space and shape control equations also have been studied. Results of the analysis of a typical wind turbine airfoil are shown to illustrate the evaluation process and to demonstrate the rate of convergence of the geometric characteristics. The coordinates and aerodynamic performance of approximate airfoils is rapidly close to the baseline airfoil corresponding to increasing orders of polynomial. Comparison of the RFOIL prediction and experimental results for the baseline airfoil generally show good agreement. A universal method for three-dimensional blade integration-＂ Shape function/Distribution function＂ is presented. By changing the parameters of shape function and distribution functions, a three dimensional blade can be designed and then transformed into the physical space in which the actual geometry is defined. Application of this method to a wind turbine blade is presented and the differences of power performance between the represented blade and original one are less than 0. 5%. This method is particularly simple and convenient for bodies of streamline forms.

Pull-in instability analyses for NEMS actuators with quartic shape approximation

The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force is investigated. A new quartic polynomial is proposed as the shape function of the beam during the deflection, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, and the design parameters are preserved in the evaluated formulas. The analytic expressions are derived for the tip deflection and pull-in parameters of the cantilever beam. The micro-electromechanical system （MEMS） cantilever actuators and freestanding nano-actuators are considered as two special cases. It is proved that the proposed method is convenient for the analyses of the effects of the surface, the Casimir force, and the fringing field on the pull-in parameters.

Analysis of dynamic stress intensity factors of thick-walled cylinder under internal impulsive pressure

Dynamic stress intensity factors are evaluated for thick-walled cylinder with a radial edge crack under internal impulsive pressure. Firstly, the equation for stress intensity factors under static uniform pressure is used as the reference case, and then the weight function for a thick-walled cylinder containing a radial edge crack can be worked out. Secondly, the dynamic stresses in uncracked thick-walled cylinders are solved under internal impulsive pressure by using mode shape function method. The solution consists of a quasi-static solution satisfying inhomogeneous boundary conditions and a dynamic solution satisfying homogeneous boundary condi- tions, and the history and distribution of dynamic stresses in thick-walled cylinders are derived in terms of Fourier-Bessel series. Finally, the dynamic stress intensity factor equations for thick-walled cylinder containing a radial edge crack sub- jected to internal impulsive pressure are given by dynamic weight function method. The finite element method is utilized to verify the results of numerical examples, showing the validity and feasibility of the proposed method.

An Improved Method of Structure Damage Diagnosis for Jacket Platforms

In the exploitation of ocean oil and gas, many offshore structures may be damaged due to the severe environment, so an effective method of diagnosing structural damage is urgently needed to locate the damage and evaluate its severity. Genetic algorithms have become some of the most important global optimization tools and been widely used in many fields in recent years because of their simple operation and strong robustness. Based on the natural frequencies and mode shapes of the structure, the damage diagnosis of a jacket offshore platform is attributed to an optimization problem and studied by using a genetic algorithm. According to the principle that the structural stiffness of a certain direction can be greatly affected only when the brace bar in the corresponding direction is damaged, an improved objective function was proposed in this paper targeting measurement noise and the characteristics of modal identification for offshore platforms. This function can be used as fitness function of a genetic algorithm, and both numerical simulation and physical model test results show that the improved method may locate the structural damage and evaluate the severity of a jacket offshore platform satisfactorily while improving the robustness of evolutionary searching and the reliability of damage diagnosis.

SOLUTION OF DIFFERENT HOLES SHAPE BORDERS OF FIBRE REINFORCED COMPOSITE PLATES BY INTEGRAL EQUATIONS

Accurate boundary conditions of composite material plates with different holes are founded to settle boundary condition problems of complex holes by conformal mapping method upon the nonhomogeneous anisotropic elastic and complex function theory. And then the two stress functions required were founded on Cauchy integral by boundary conditions. The final stress distributions of opening structure and the analytical solution on composite material plate with rectangle hole and wing manholes were achieved. The influences on hole-edge stress concentration factors are discussed under different loads and fiber direction cases, and then contrast calculates are carried through FEM.

Mass-Stiffness Templates for Cubic Structural Elements Dedicated to Professor Karl Stark Pister for his 95th birthday

1 This paper considers Lagrangian finite elements for structural dynamics constructed with cubic displacement shape functions.The method of templates is used to investigate the construction of accurate mass-stiffness pairs.This method introduces free parameters that can be adjusted to customize elements according to accuracy and rank-sufficiency criteria.One-and two-dimensional Lagrangian cubic elements with only translational degrees of freedom(DOF)carry two additional nodes on each side,herein called side nodes or SN.Although usually placed at the third-points,the SN location may be adjusted within geometric limits.The adjustment effect is studied in detail using symbolic computations for a bar element.The best SN location is taken to be that producing accurate approximation to the lowest natural frequencies of the continuum model.Optimality is investigated through Fourier analysis of the propagation of plane waves over a regular infinite lattice of bar elements.Focus is placed on the acoustic branch of the frequency-vs.-wavenumber dispersion diagram.It is found that dispersion results using the fully integrated consistent mass matrix(CMM)are independent of the SN location whereas its lowfrequency accuracy order is O(κ8),whereκis the dimensionless wave number.For the diagonally lumped mass matrix(DLMM)constructed through the HRZ scheme,two optimal SN locations are identified,both away from third-points and of accuracy order O(κ8).That with the smallest error coefficient corresponds to the Lobatto 4-point integration rule.A special linear combination of CMM and DLMM with nodes at the Lobatto points yields an accuracy of O(κ10)without any increase in the computational effort over CMM.The effect of reduced integration(RI)on both mass and stiffness matrices is also studied.It is shown that singular mass matrices can be constructed with 2-and 3-point RI rules that display the same optimal accuracy of the exactly integrated case,at the cost of introducing spurious modes.The optimal SN location in two-dimensional,bicubic,isoparametric plane stress quadrilateral elements is briefly investigated by numerical experiments.The frequency accuracy of flexural modes is found to be fairly insensitive to that position,whereas for bar-like modes it agrees with the one-dimensional results.

A NOVEL ELLIPSOIDAL ACOUSTIC INFINITE ELEMENT

A novel ellipsoidal acoustic infinite element is proposed. It is based a new pressure representation, which can describe and solve the ellipsoidal acoustic field more exactly. The shape functions of this novel acoustic infinite element are similar to the (Burnett's) method, while the weight functions are defined as the product of the complex conjugates of the shaped functions and an additional weighting factor. The code of this method is cheap to generate as for 1-D element because only 1-D integral needs to be numerical. Coupling with the standard finite element, this method provides a capability for very efficiently modeling acoustic fields surrounding structures of virtually any practical shape. This novel method was deduced in brief and the conclusion was kept in detail. To test the feasibility of this novel method efficiently,in the examples the infinite elements were considered,excluding the finite elements relative. This novel ellipsoidal acoustic infinite element can deduce the analytic solution of an oscillating sphere. The example of a prolate spheroid shows that the novel infinite element is superior to the boundary element and other acoustic infinite elements. Analytical and numerical results of these examples show that this novel method is feasible.

Water Wave Scattering by a Nearly Circular Cylinder Submerged Beneath an Ice-cover

Assuming linear theory, the two-dimensional problem of water wave scattering by a horizontal nearly circular cylinder submerged in infinitely deep water with an ice cover modeled as a thin-elastic plate floating on water, is investigated here. The cross-section of the nearly circular cylinder is taken as r=a（1＋δC（θ））, where a is the radius of the corresponding circular cross-section of the cylinder, δ is a measure of small departure of the cross-section of the cylinder from its circularity and C（θ） is the shape function. Using a simplified perturbation technique the problem is reduced to two independent boundary value problems up to first order in δ. The first one corresponds to water wave scattering by a circular cylinder submerged in water with an ice-cover, while the second problem describes wave radiation by a submerged circular cylinder and involves first order correction to the reflection and transmission coefficients. The corrections are obtained in terms of integrals involving the shape function. Assuming a general Fourier expansion of the shape function, these corrections are evaluated approximately. It is well known that normally incident wave trains experience no reflection by a circular cylinder submerged in infinitely deep water with an ice cover. It is shown here that the reflection coefficient also vanishes up to first order for some particular choice of the shape function representing a nearly circular cylinder. For these cases, full transmission occurs, only change is in its phase which is depicted graphically against the wave number in a number of figures and appropriate conclusions are drawn.