Effect of regularized functions on the dynamic response of a clutch system using a high-order algorithm

The main objective of this work is to propose some regularization techniques for modeling contact actions in a clutch system and to solve the obtained nonlinear dynamic problem by a high-order algorithm. This device is modeled by a discrete mechanical system with eleven degrees of freedom. In several works, the discontinuous models of the contact actions are replaced by the smoothed functions using the hyperbolic tangent. We propose, in this work, to replace the discontinuous model by a regularized model with new continuous functions that permit us to search the solution under Taylor series expansion. This regularized model approaches better the discontinuous model than the model based on the smoothing functions, especially in the vicinity of the zone of singularities. To solve the equations of motion of discrete mechanical systems, we propose to use a high-order algorithm combining a time discretization, a change of variable based on the previous time, a homotopy transformation and Taylor series expansion in the continuation process. The results obtained by this modeling are compared with those computed by the Newton–Raphson algorithm.     

HIGH-ORDER SCHEMES IN BOUNDARY ELEMENT METHODS FOR TRANSIENT NON-LINEAR FREE SURFACE PROBLEMS

In this paper the boundary element method is applied to solve transient non-linear free surface flow problems formulated from potential theory. For the temporal evolution a high-order time-stepping procedure based on a truncated forward-time Taylor series expansion is compared with the classical Runge–Kutta technique. The numerical code for both two-dimensional and axisymmetric configurations has been successfully implemented. Emphasis in the paper is placed on describing the analytical development achieved by the use of Maple software. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids, 24: 1049–1072, 1997.     

Improved linear interpolation practice for finite‐volume schemes on complex grids

Methods for the computation of flow problems based on finite‐volume discretizations and pressure‐correction methods frequently require the interpolation of control volume face values from nodal values. The simple, often employed central differencing scheme (CDS) leads to a significant loss in accuracy when the numerical grid is non‐regular as it is usual when modelling complex geometries. An alternative technique based on a multi‐dimensional Taylor series expansion (TSE) is proposed, which preserves the CDS‐like sparsity pattern of the discrete system. While the TSE scheme computationally is only slightly more expensive than the CDS approach, it results in a significantly higher accuracy, where the difference increases with the grid irregularity. The method is investigated and compared to the CDS approach for some representative test cases. Copyright © 2002 John Wiley & Sons, Ltd.     

Taylor-Galerkin-based spectral element methods for convection-diffusion problems

Several explicit Taylor-Galerkin-based time integration schemes are proposed for the solution of both linear and non-linear convection problems with divergence-free velocity. These schemes are based on second-order Taylor series of the time derivative. The spatial discretization is performed by a high-order Galerkin spectral element method. For convection-diffusion problems an operator-splitting technique is given that decouples the treatment of the convective and diffusive terms. Both problems are then solved using a suitable time scheme. The Taylor-Galerkin methods and the operator-splitting scheme are tested numerically for both convection and convection-diffusion problems.     

Computing singular solutions to partial differential equations by Taylor series

The Taylor Meshless Method (TMM) is a true meshless integration-free numerical method for solving elliptic Partial Differential Equations (PDEs). The basic idea of this method is to use high-order polynomial shape functions that are approximated solutions to the PDE and are computed by the technique of Taylor series. Currently, this new method has proved robust and efficient, and it has the property of exponential convergence with the degree, when solving problems with smooth solutions. This exponential convergence is no longer obtained for problems involving cracks, corners or notches. On the basis of numerical tests, this paper establishes that the presence of a singularity leads to a worsened convergence of the Taylor series, but highly accurate solutions can be recovered by including a few singular solutions in the basis of shape functions.     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

具有动应力和动位移约束的离散变量结构形状优化设计方法

讨论了动应力、动位移约束下离散变量状优化设计问题。首先用拟静力算法,将结构惯性力极值作为静载荷施加到结构上,求得结构的动位移和动内力,然后将考虑动应力约束和动作移约束的离散变量结构优化设计问题化为静应力和静位移约束的优化问题。在求解过程中,将单元内力作了一阶近似,并将多约束问题转化为单约束问题,然后利用两类变量统一考虑的离散变量结构形状优化设计的综合算法进行求解。     

High-order divergence-free velocity reconstruction for free surface flows on unstructured Voronoi meshes

In this paper, we present an efficient semi-implicit scheme for the solution of the Reynolds-averaged Navier-Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi-Lagrangian approach, which requires an interpolation of the three-dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high-order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence-free velocity field. A comparison with an analogous reconstruction, which is not divergence-free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high-order spatial discretizations. The reconstructed velocity field is then used for evaluating high-order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi-implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth-order of accuracy in space is shown.     

Steady state response analysis for fractional dynamic systems based on memory-free principle and harmonic balancing

A semi-analytic approach is proposed to analyze steady state responses of dynamic systems containing fractional derivatives. A major purpose is to efficiently combine the harmonic balancing (HB) technique and Yuan–Agrawal (YA) memory-free principle. As steady solutions being expressed by truncated Fourier series, a simple yet efficient way is suggested based on the YA principle to explicitly separate the Caputo fractional derivative as periodic and decaying non-periodic parts. Neglecting the decaying terms and applying HB procedures result into a set of algebraic equations in the Fourier coefficients. The linear algebraic equations are solved exactly for linear systems, and the non-linear ones are solved by Newton–Raphson plus arc-length continuation algorithm for non-linear problems. Both periodic and triple-periodic solutions obtained by the presented method are in excellent agreement with those by either predictor–corrector (PC) or YA method. Importantly, the presented method is capable of detecting both stable and unstable periodic solutions, whereas time-stepping integration techniques such as YA and PC can only track stable ones. Together with the Floquet theory, therefore, the presented method allows us to address the bifurcations in detail of the steady responses of fractional Duffing oscillator. Symmetry breakings and cyclic-fold bifurcations are found and discussed for both periodic and triple-periodic solutions.     

A novel finite point method for flow simulation

A novel finite point method is developed to simulate flow problems. The mashes in the traditional numerical methods are supplanted by the distribution of points in the calculation domain. A local interpolation based on the properties of Taylor series expansion is used to construct an approximation for unknown functions and their derivatives. An upwind‐dominated scheme is proposed to efficiently handle the non‐linear convection. Comparison with the finite difference solutions for the two‐dimensional driven cavity flow and the experimental results for flow around a cylinder shows that the present method is capable of satisfactorily predicting the flow separation characteristic. The present algorithm is simple and flexible for complex geometric boundary. The influence of the point distribution on computation time and accuracy of results is included. Copyright © 2002 John Wiley & Sons, Ltd.     

Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering function

In this paper, a model of topology optimization with linear buckling constraints is established based on an independent and continuous mapping method to minimize the plate/shell structure weight. A composite exponential function (CEF) is selected as filtering functions for element weight, the element stiffness matrix and the element geomet-ric stiffness matrix, which recognize the design variables, and to implement the changing process of design variables from“discrete”to“continuous”and back to“discrete”. The buck-ling constraints are approximated as explicit formulations based on the Taylor expansion and the filtering function. The optimization model is transformed to dual programming and solved by the dual sequence quadratic programming algo-rithm. Finally, three numerical examples with power function and CEF as filter function are analyzed and discussed to demonstrate the feasibility and efficiency of the proposed method.     

Dynamic Model for LES Without Test Filtering: Quantifying the Accuracy of Taylor Series Approximations

The dynamic model for large-eddy simulation (LES) of turbulent flows requires test filtering the resolved velocity fields in order to determine model coefficients. However, test filtering is costly to perform in LES of complex geometry flows, especially on unstructured grids. The objective of this work is to develop and test an approximate but less costly dynamic procedure which does not require test filtering. The proposed method is based on Taylor series expansions of the resolved velocity fields. Accuracy is governed by the derivative schemes used in the calculation and the number of terms considered in the approximation to the test filtering operator. The expansion is developed up to fourth order, and results are tested a priori based on direct numerical simulation data of forced isotropic turbulence in the context of the dynamic Smagorinsky model. The tests compare the dynamic Smagorinsky coefficient obtained from filtering with those obtained from application of the Taylor series expansion. They show that the expansion up to second order provides a reasonable approximation to the true dynamic coefficient (with errors on the order of about 5% for c _s ²), but that including higher-order terms does not necessarily lead to improvements in the results due to inherent limitations in accurately evaluating high-order derivatives. A posteriori tests using the Taylor series approximation in LES of forced isotropic turbulence and channel flow confirm that the Taylor series approximation yields accurate results for the dynamic coefficient. Moreover, the simulations are stable and yield accurate resolved velocity statistics. Received 20 February 2001 and accepted 24 July 2001     

Fhree-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

Moving least squares reconstruction for sharp interface immersed boundary methods

We propose a new approach for reconstructing velocity boundary conditions in sharp-inerface immersed boundary (IB) methods based on the moving least squares (MLS) interpolation method. The MLS is employed to not only reconstruct velocity boundary conditions but also to calculate the pressure and velocity gradients in the vicinity of the immersed body, which are required in fluid structure interaction problems to obtain the force exerted by the fluid on the structure. To extend the method to arbitrarily complex geometries with nonconvex shaped boundaries, the visibility method is combined with the MLS method. The performance of the proposed curvilinear IB MLS (CURVIB-MLS) is demonstrated by systematic grid-refinement studies for two- and three-dimensional tests and compared with the standard CURVIB method employing standard wall-normal interpolation for reconstructing boundary conditions. The test problems are flow in a lid-driven cavity with a sphere, uniform flow over a sphere, flow on a NACA0018 airfoil at incidence, and vortex-induced vibration of an elastically-mounted cylinder. We show that the CURVIB-MLS formulation yields a method that is easier to implement in complex geometries and exhibits higher accuracy and rate of convergence relative to the standard CURVIB method. The MLS approach is also shown to dramatically improve the accuracy of calculating the pressure and viscous forces imparted by the flow on the body and improve the overall accuracy of FSI simulations. Finally, the CURVIB-MLS approach is able to qualitatively capture on relatively coarse grids important features of complex separated flows that the standard CURVIB method is able to capture only on finer grids.     

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method,a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper.Numerical characteristics of the scheme are studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node,the proposed scheme is explicit and can achieve arbitrary order of accuracy in space.Application examples for the convection- diffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given.It is found that the proposed compact scheme is not only simple to implement and economical to use,but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

两类变量综合处理的结构形状优化设计方法

本文针对截面变量为离散变量为连续变量的结构优化问题提出了一种优化设计的方法,首先将单元内力作一阶近似,利用凝聚函数多约束问题转化了单约束问题。在解过程中,把定义在连续区间上的形状变量看成是在一些离散以值的离工用变量,然后将两类变量统一考虑并利用相对差商法求解。将该算法应用于几个经典的结构优化算例,运算结果显示了该方法是可行的,优化结果也比较满意。     

An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters

This paper proposes two interval analysis methods, called the first-order interval parameter perturbation method (FIPPM) and the modified interval parameter perturbation method (MIPPM), for use in exterior acoustic field prediction when there are uncertainties in both the material properties and the external load. Interval variables are used to quantitatively describe the uncertain parameters in the face of limited information. The conventional first-order Taylor expansion and perturbation terms are employed in the FIPPM, while the MIPPM introduces modified Taylor series to approximate the non-linear interval matrix and vector. The high-order terms of the Neumann expansion are retained to calculate the interval matrix inverse. A numerical example is given by comparing the results with a Monte Carlo simulation to demonstrate the feasibility and effectiveness of the proposed methods at evaluating the sound pressure ranges in an exterior acoustic field.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

结构动力问题的高精度组合差分时程积分法

基于Taylor级数展开得到位移和加速度的中心差分格式,并结合速度的后差分格式,构造了一种求解结构动力问题的组合差分格式的时程积分算法,该算法为自起步的两步高精度算法。通过求解递推格式的传递矩阵及其特征值,对该算法的稳定性和精度进行了理论分析,结果表明,本文提出的算法虽属条件稳定,但其精度极高,具有周期延长率小、没有振幅衰减等优点。数值分析结果也证明本文提出的算法具有较高精度。