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.

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.

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.

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.

First application of large reactivity measurement through rod drop based on three-dimensional space–time dynamics

Abstract Reactivity measurement is an essential part of a zero-power physics test,which is critical to reactor design and development.The rod drop experimental technique is used to measure the control rod worth in a zero-power physics test.The conventional rod drop experimental technique is limited by the spatial effect and the difference between the calculated static reactivity and measured dynamic reactivity;thus,the method must be improved.In this study,a modified rod drop experimental technique that constrains the detector neutron flux shape function based on three-dimensional space–time dynamics to reduce the reactivity perturbation and a new method for calculating the detector neutron flux shape function are proposed.Correction factors were determined using Monte Carlo N-particle transport code and transient analysis code for a pressurized water reactor at the Ulsan National Institute of Science and Technology and Xi’an Jiaotong University,and a large reactivity of over 2000 pcm was measured using the modified technique.This research evaluated the modified technique accuracy,studied the influence of the correction factors on the modification,and investigated the effect of constraining the shape function on the reactivity perturbation reduction caused by the difference between the calculated neutron flux and true value,using the new method to calculate the shape function of the detector neutron flux and avoiding the neutron detector response function(weighting factor)calculation.

A Meshless Method for Retrieving Nonlinear Large External Forces on Euler-Bernoulli Beams

We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data,comprising two-end fixed,cantilevered,clamped-hinged,and simply supported conditions in this study.Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams;however,an effective numerical algorithm to solve these inverse problems is still not available.We cope with the homogeneous boundary conditions,initial data,and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions.The unknown nonlinear large external force can be recuperated via back-substitution of the solution into the nonlinear Euler-Bernoulli beam equation when we acquire the solution by utilizing the boundary shape function scheme and deal with a smallscale linear system to gratify an additional right-side boundary data.For the robustness and accuracy,we reveal that the current schemes are substantiated by comparing the recuperated numerical results of four instances to the exact forces,even though a large level of noise up to 50%is burdened with the overspecified conditions.The current method can be employed in the online real-time computation of unknown force functions in space-time for varied boundary supports of the vibrating nonlinear beam.

Three-Dimensional Meshfree Analysis of Interlocking Concrete Blocks for Step Seawall Structure

This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional(3D)complex geometry structures,which are the interlocking concrete blocks of step seawall.The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain.The 3D moving least square(MLS)approximation is applied to build the interpolation functions of unknowns.The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results.The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method(FEM)using linear shape functions on tetrahedral elements and the well-known commercial software,ANSYS.The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM.The studied meshfree method is effective in the analysis of static responses of complex geometry structures.The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes.

STIFFNESS EQUATION OF FINITE SEGMENT FOR FLEXIBLE BEAM-FORMED STRUCTURAL ELEMENTS

The finite segment modelling for the flexible beam-formed structural elements is presented, in which the discretization views of the finite segment method and the difference from the finite element method are introduced. In terms of the nodal model, the joint properties are described easily by the model of the finite segment method, and according to the element properties, the assumption of the small strain is only met in the finite segment method, i. e., the geometric nonlinear deformation of the flexible bodies is allowable. Consequently,the finite segment method is very suited to the flexible multibody structure. The finite segment model is used and the are differentiation is adopted for the differential beam segments. The stiffness equation is derived by the use of the principle of virtual work. The new modelling method shows its normalization, clear physical and geometric meanings and simple computational process.