Finite element modelling on microstructure evolution during multi-pass hot compression for AZ31 alloys using incremental method

Based on the principle of piecewise linearization, the incremental forms of microstructure evolution models were integrated into the thermo-mechanical coupled finite element（FE） model to simulate nonlinear microstructure evolution during multi-pass hot deformation. This is an unsteady-state deformation where dynamic recrystallization（DRX）, meta-dynamic recrystallization（MDRX）, static recrystallization（SRX） and grain growth（GG） take place during hot deformation or deformation interval. The distributions of deformation and microstructure for cylindrical AZ31 sample during single-pass and double-pass hot compressions were quantitatively calculated and compared with the metallographic observation. It is shown that both the deformation and microstructure are non-uniformly distributed due to the presence of friction between the die and the flat end of sample. The average grain size and its standard deviation under the double-pass hot compression are slightly smaller than those under single-pass compression.The simulated average grain sizes agree well with the experiments, which validates that the developed FE model on the basis of incremental forms of microstructure evolution models is reasonable.

THE ADAPTIVE PARAMETER INCREMENTAL METHOD FOR THE ANALYSIS OF SNAPPING PROBLEMS

By the improvement of Riks’and Crisfield’s arc-length method,the adaptiveparameter incremental method is preasted for predicting the snapping response ofstructures. Its justification is fulfilled. Finally,the effectiveness of this method isdemonstrated by solving the snapping response of spherical caps subjected to centrallydistributed pressures.

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.

Incremental harmonic balance method for periodic forced oscillation of a dielectric elastomer balloon

Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent coefficients. Previous efforts toward this goal use largely the numerical integration method with the simple harmonic balance method as a supplement. The numerical integration and the simple harmonic balance method are inefficient for large parametric analysis or with difficulty in improving the solution accuracy. To overcome the weakness of these two methods,we describe formulations of the incremental harmonic balance(IHB) method for periodic forced solutions of such a unique system. Combined with an arc-length continuation technique, the proposed strategy can capture the whole solution branches, both stable and unstable, automatically with any desired accuracy.

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.

Approximation Maintenance When Adding a Conditional Value in Set-Valued Ordered Decision Systems

The integration of set-valued ordered rough set models and incremental learning signify a progressive advancement of conventional rough set theory, with the objective of tackling the heterogeneity and ongoing transformations in information systems. In set-valued ordered decision systems, when changes occur in the attribute value domain, such as adding conditional values, it may result in changes in the preference relation between objects, indirectly leading to changes in approximations. In this paper, we effectively addressed the issue of updating approximations that arose from adding conditional values in set-valued ordered decision systems. Firstly, we classified the research objects into two categories: objects with changes in conditional values and objects without changes, and then conducted theoretical studies on updating approximations for these two categories, presenting approximation update theories for adding conditional values. Subsequently, we presented incremental algorithms corresponding to approximation update theories. We demonstrated the feasibility of the proposed incremental update method with numerical examples and showed that our incremental algorithm outperformed the static algorithm. Ultimately, by comparing experimental results on different datasets, it is evident that the incremental algorithm efficiently reduced processing time. In conclusion, this study offered a promising strategy to address the challenges of set-valued ordered decision systems in dynamic environments.

Incremental-based Nonlinear Detection Algorithm for MIMO System

A novel incremental nonlinear detection algorithm is presented for Multiple-Input Multiple-Output （MIMO） system. In this algorithm, the received data at multiple receiver antennas are nonlinearly mapped and then summed with weights. The weight coefficients are incrementally computed to avoid direct computation of the inverse of a matrix, which greatly reduce the computational complexity. Simulation and comparison show that the proposed algorithm can obtain better performance of Bit Error Rate （BER） than linear Minimum Mean Square Error （MMSE）.

INCREMENTAL MICRO-MECHANICAL MODEL OF PLAIN WOVEN FABRIC

Warp yarns and weft yarns of plain woven fabric are the principal axes of mate- rial of fabric. They are orthogonal in their original con?guration, but are obliquely crisscross in deformed con?guration in general. In this paper the expressions of incremental components of strain tensor are derived, the non-linear model of woven fabric is linearized physically and its geometric non-linearity survives. The convenience of determining the total deformation is shown by the choice of the coordinate system of the principal axes of the material, with the convergence of the incremental methods illustrated by examples. This incremental model furnishes a basis for numerical simulations of fabric draping and wrinkling based on the micro-mechanical model of fabric.

Static and Dynamic Analysis of Mooring Lines by Nonlinear Finite Element Method

This study has focused on developing numerical procedures for the static and dynamic nonlinear analysis of mooring lines. A geometrically nonlinear finite element method using isoparametric cable element with two nodes is briefly presented on the basis of the total Lagrangian formulation. The static and dynamic equilibrium equations of mooring lines are established. An incremental-iterative method is used to determine the initial static equilibrium state of cable systems under the action of self weights, buoyancy and current. Also the Newmark method is used for dynamic nonlinear analysis of ocean cables. Numerical examples are presented to validate the present numerical method, and examine the effect of various parameters.

Force-Based Quadrilateral Plate Bending Element for Plate Using Large Increment Method

A force-based quadrilateral plate element( 4NQP13) for the analysis of the plate bending problems using large increment method( LIM) was proposed. The LIM, a force-based finite element method( FEM),has been successfully developed for the analysis of truss,beam,frame,and 2D continua problems. In these analyses,LIMcan provide more precise stress results and less computational time consumption compared with displacement-based FEM. The plate element was based on the Mindlin-Reissner plate theory which took into account the transverse shear effects.Numerical examples were presented to study its performance including accuracy and convergence behavior,and the results were compared with the results have been obtained from the displacementbased quadrilateral plate elements and the analytical solutions. The4NQP13 element can analyze the moderately thick plates and the thin plates using LIMand is free from spurious zero energy modes and free from shear locking for thin plate analysis.

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

Improved incremental transfer matrix method for nonlinear rotor-bearing system

Rotor system supported by nonlinear bearing such as squeeze film damper(SFD)is widely used in practice owing to its wide range of damping capacity and simplicity in structure.In this paper,an improved and effective Incremental transfer matrix method(ITMM)is first presented by combining ITMM and fast Fourier transform(FFT).Afterwards this method is applied to calculate the dynamic characteristics of a Jeffcott rotor system with SFD.The convergence dificulties incurred caused by strong nonlinearities of SFD has been dealt by adopting a control factor.It is found that for the more general boundary problems where the boundary conditions are not at input and output ends of a chain system,the supplementary equation is necessarily added.Additionally,the Floquet theory is used to analyze the stability and bifurcation type of the obtained periodic solution.The semi-analytical results,including the periodic solutions of the system,the bifurcation points and their types,are in good agreement with the numerical method.Furthermore,the involution mechanism of the quasi-periodic and chaotic motions near the first-order translational mode and the second order bending mode of this system is also clarified by this method with the aid of Floquet theory.

非均匀加筋板格极限受压强度计算建模研究

The aim of this paper is to develop computational models for the ultimate compressive strength analysis of stiffened plate panels with nonuniform thickness.Modeling welding-induced initial deformations and residual stresses was presented with the measured data.Three methods,i.e.,ANSYS finite element method,ALPS/SPINE incremental Galerkin method,and ALPS/ULSAP analytical method,were employed together with existing test database obtained from a full-scale collapse testing of steel-stiffened plate structures.Sensitivity study was conducted with varying the difference in plate thickness to define a representative(equivalent)thickness for plate panels with nonuniform thickness.Guidelines are provided for structural modeling to compute the ultimate compressive strength of plate panels with variable thickness.

Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain

The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the Rayleigh- Ritz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth, the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental （AVMI） factor is used. The results show good agreement with those from the Rayleigh-Ritz method.

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle.A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method.A high-dimensional model of the buckled beam is derived,concerning nonlinear coupling.The incremental harmonic balance(IHB)method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve,and the Floquet theory is used to analyze the stability of the periodic solutions.Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited.Bifurcations including the saddle-node,Hopf,perioddoubling,and symmetry-breaking bifurcations are observed.Furthermore,quasi-periodic motion is observed by using the fourth-order Runge-Kutta method,which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

Development of 2D Hybrid Equilibrium Elements in Large Increment Method

As a force-based finite element method (FEM), large increment method (LIM) has been developed in recent years. It has been shown that LIM provided prominent advantage of parallel computation with high efficiency and low time consumption for member structural system. To fully utilize its advantage in parallel computation, it is the time to extend LIM to 2D and 3D continua analysis. In this paper, a 2D finite element library with the capability of modeling arbitrary configurations is developed. Some illustrative numerical examples are solved by using the proposed library; the obtained results are compared with those obtained from both traditional displacement-based FEM and analytical solutions, which has clearly shown the advantages of LIM.

Development of the Large Increment Method in Analysis for Thin and Moderately Thick Plates

Many displacement-based quadrilateral plate elements based on Mindlin-Reissner plate theory have been proposed to analyze the thin and moderately thick plate problems. However, numerical inaccuracies of some elements appear since the presence of shear locking and spurious zero energy modes for thin plate problems. To overcome these shortcomings, we employ the large increment method(LIM) for the analyses of the plate bending problems, and propose a force-based 8-node quadrilateral plate(8NQP) element which is based on MindlinReissner plate theory and has no extra spurious zero energy mode. Several benchmark plate bending problems are presented to illustrate the accuracy and convergence of the plate element by comparing with the analytical solutions and displacement-based plate elements. The results show that the 8-node plate element produces fast convergence and accurate stress distributions in both the moderately thick and thin plate bending problems. The plate element is insensitive to mesh distortion and it can avoid the shear locking for thin plate analysis.

Linearized model for optimization of coupled electricity and natural gas systems

In this paper,a combined optimization of a coupled electricity and gas system is presented.For the electricity network a unit commitment problem with optimization of energy and reserves under a power pool,considering all system operational and unit technical constraints is solved.The gas network subproblem is a medium-scale mixed-integer nonconvex and nonlinear programming problem.The coupling constraints between the two networks are nonlinear as well.The resulting mixed-integer nonlinear program is linearized with the extended incremental method and an outer approximation technique.The resulting model is evaluated using the Greek power and gas system comprising fourteen gas-fired units under four different approximation accuracy levels.The results indicate the efficiency of the proposed mixed-integer linear program model and the interplay between computational requirements and accuracy.

Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems

As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for achieving high controlling precision is to search for an appropriate time delay.This paper proposes a simple yet effective approach,based on incremental harmonic balance method,to determine the optimal time delay in the delayed feedback controller.The time delay is adjusted within the iterative scheme provided by the proposed method,and finally converges to the period of the target UPO.As long as the optimal time delay is fixed,moreover,the attained solution makes it quite convenient to analyze its stability according to the Floquet theory,which further provides the effective interval of the feedback gain.

CHAOTIC ANALYSIS OF SUBHARMONIC RESONANT WAVES IN UNDAMPED AND DAMPED STRINGS

The chaotic phenomena of subharmonic resonant waves in undamped and damped strings are investigated in this paper. The model consistS of a constant-tension, stretched string whose partial differential equation is derived by taking into account its exact configuration. Simplification via a Taylor series expansion of the curvature term and then employing the Galerkin method, an ordinary differential equation for the nonlinear dyamics is obtained. For the undamped case, we can formulate the Hamiltonian energy form of the conservative string, under the influence of an external periodic excitation. This permits the subharmonic resonant condition for this system to be derived. We truncate the resulting infinite number of subharmonic resonant waves to just two waves by renormalizing the Hamiltonian energy function near the subharmonic resonant orbit of the system. Adopting the renormalization group technique for the nit6raction of the two subharmonic resonant waves, an approximate chaotic condition associated with the subharmonic resonance of this system is determined. For the case fo the damped string, the minimum condition for the bifurcation of the subharmonic resonant wave is computed using the incremental energy balance method. For model verification, we carried out numerical simulations and they show good agreement with our analytical results.