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.

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.

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.

Digital Image Correlation Using Specific Shape Function for Stress Intensity Factor Measurement

The stress intensity factor (SIF) is a critical parameter associated with the fracture behaviour of materials. In this paper, we select the displacement function around a crack tip as the shape function of the digital image correlation (DIC), which makes it possible to directly calculate the SIF by the correlation scheme. Moreover, we use a non-rectangular subset, which can reduce the influence of plastic deformation and crack width on the DIC measurement accuracy. We measured the SIF of a mode I crack in a super-hard aluminium alloy specimen to verify the performance of the proposed method. Our experimental results show that a DIC with a specific shape function can be used to accurately and efficiently calculate the SIF. Furthermore, we also present a practical application of our proposed method for determining the SIF, crack propagation angle and crack tip displacement. © 2017, Tianjin University and Springer-Verlag Berlin Heidelberg.

Design and additive manufacturing of bionic hybrid structure inspired by cuttlebone to achieve superior mechanical properties and shape memory function

Lightweight porous materials with high load-bearing,damage tolerance and energy absorption(EA)as well as intelligence of shape recovery after material deformation are beneficial and critical for many applications,e.g.aerospace,automobiles,electronics,etc.Cuttlebone produced in the cuttlefish has evolved vertical walls with the optimal corrugation gradient,enabling stress homogenization,significant load bearing,and damage tolerance to protect the organism from high external pressures in the deep sea.This work illustrated that the complex hybrid wave shape in cuttlebone walls,becoming more tortuous from bottom to top,creates a lightweight,load-bearing structure with progressive failure.By mimicking the cuttlebone,a novel bionic hybrid structure(BHS)was proposed,and as a comparison,a regular corrugated structure and a straight wall structure were designed.Three types of designed structures have been successfully manufactured by laser powder bed fusion(LPBF)with NiTi powder.The LPBF-processed BHS exhibited a total porosity of 0.042% and a good dimensional accuracy with a peak deviation of 17.4μm.Microstructural analysis indicated that the LPBF-processed BHS had a strong(001)crystallographic orientation and an average size of 9.85μm.Mechanical analysis revealed the LPBF-processed BHS could withstand over 25000 times its weight without significant deformation and had the highest specific EA value(5.32 J·g^(−1))due to the absence of stress concentration and progressive wall failure during compression.Cyclic compression testing showed that LPBF-processed BHS possessed superior viscoelastic and elasticity energy dissipation capacity.Importantly,the uniform reversible phase transition from martensite to austenite in the walls enables the structure to largely recover its pre-deformation shape when heated(over 99% recovery rate).These design strategies can serve as valuable references for the development of intelligent components that possess high mechanical efficiency and shape memory capabilities.

A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of the governing Helmholtz equation.In this paper we are concerned with fast solver for the system generated by the method in[1].We propose a new preconditioner for such system,which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in[13].In our numerical experiments,this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations,and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

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.

Star-shaped Differentiable Functions and Star-shaped Differentials

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.

TORSIONAL IMPACT RESPONSE OF A PENNY-SHAPED CRACK IN A FUNCTIONAL GRADED STRIP

The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived.And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and (strip's) highness on the dynamic fracture behavior. Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip's highness.

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.

Influence of Inclusion Shape on Thermoelasto-Plastic Optimun Design of Ceramic Metal Functionally Graded Materials

A nonlinear finite element method is applied to observe how inclusion shape influence the thermal response of a ceramic-metal functionally graded material (FGM). The elastic and plastic behaviors of the layers which are two-phase isotropic composites consisting of randomly oriented elastic spheroidal Inclusions and a ductile matrix are predicted by cc mean field method. The prediction results show that inclusion shape has remarkable influence on the overall behavior of the composite. The consequences of the thermal response analysis of the FGM are that the response is dependent on inclusion shape and its composition profile cooperatively and that the plastic behavior of each layer should be taken into account in optimum design of a ceramic-metal FGM.

PENALTY FUNCTION METHOD OF CONTINUUM SHAPE OPTIMIZAION

The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.

THE APPLICATION OF SPLINE FUNCTION IN THE GEOMETRIC REPRODUCTION OF LOG SHAPE

The paper introduces a method to get three-dimensional reproduction of log shape by adopting Spline Function in fitting the curve of the finite log data. The method has ad-vantages of higher accuracy, less acquired data, easier to use, etc. Making use of high-precision drawing function of computer, the graphs of log geometric shape in different visual angles can be achieved easily with this method. It also provided a firm foundation for the determination of optimum saw cutting scheme.

3D analysis of functionally graded material plates with complex shapes and various holes

In this paper, the basic formulae for the semi-analytical graded FEM on FGM members are derived. Since FGM parameters vary along three space coordinates, the parameters can be integrated in mechanical equations. Therefore with the parameters of a given FGM plate, problems of FGM plate under various conditions can be solved. The approach uses 1D discretization to obtain 3D solutions, which is proven to be an effective numerical method for the mechanical analyses of FGM structures. Examples of FGM plates with complex shapes and various holes are presented.

Evaluation of the Effect of Sigmoid-Shaped Interventricular Septum on Left Ventricular Systolic Function in Patients with Essential Hypertension by Two-Dimensional Speckle Tracking Echocardiography

Objective: To evaluate left ventricular regional and global systolic function by measuring left ventricular longitudinal strain (LS) in hypertensive patients with sigmoid-shaped interventricular septum (SIS) by two-dimensional speckle tracking (2D-STE);in order to explore whether the sigmoid-shaped interventricular septum affects the left ventricular systolic function in patients with hypertension. Methods: Routine echocardiographic parameters were measured in 30 hypertensive patients with SIS (SIS group) and 30 hypertensive patients without SIS (non-SIS group). The left ventricular segments and global LS were measured by 2D-STE, and the two sets of parameters were compared. Results: The value of the thickness of the basal segment of the interventricular septum (IVSBT), the thickness of the middle segment of the interventricular septum (IVSMT) and the ratio of the basal segment of the ventricular septum to the middle segment of the interventricular septum (IVSBT/IVSMT) in SIS group was higher than that in non-SIS group. However, the value of left ventricular outflow tract diameter (LVOTD) in SIS group was lower than that in non-SIS group. There was a significant difference between the two groups (all P Conclusion: SIS affects left ventricular regional systolic function of patients with hypertension. 2D-STE can early evaluate left ventricular longitudinal systolic function in hypertensive patients with SIS.

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.