Effects of the Similarity Model in Finite-Difference LES of Isotropic Turbulence Using a Lagrangian Dynamic Mixed Model

A Lagrangian dynamic formulation of the mixed similarity subgrid (SGS) model for large-eddy simulation (LES) of turbulence is proposed. In this model, averaging is performed over fluid trajectories, which makes the model applicable to complex flows without directions of statistical homogeneity. An alternative version based on a Taylor series expansion (nonlinear mixed model) is also examined. The Lagrangian models are implemented in a finite difference code and tested in forced and decaying isotropic turbulence. As comparison, the dynamic Smagorinsky model and volume-averaged formulations of the mixed models are also tested. Good results are obtained, except in the case of low-resolution LES (32³) of decaying turbulence, where the similarity coefficient becomes negative due to the fact that the test-filter scale exceeds the integral scale of turbulence. At a higher resolution (64³), the dynamic similarity coefficient is positive and good agreement is found between predicted and measured kinetic energy evolution. Compared to the eddy viscosity term, the similarity or the nonlinear terms contribute significantly to both SGS dissipation of kinetic energy and SGS force. In order to dynamically test the accuracy of the modeling, the error incurred in satisfying the Germano identity is evaluated. It is found that the dynamic Smagorinksy model generates a very large error, only 3% lower than the worst-case scenario without model. Addition of the similarity or nonlinear term decreases the error by up to about 50%, confirming that it represents a more realistic parameterization than the Smagorinsky model alone.     

Time series prediction of nonlinear ship structural responses in irregular seaways using a third-order Volterra model

To predict the nonlinear structural responses of a ship traveling through irregular waves, a third-order Volterra model was applied based on the given irregular data. A nonlinear wave–body interaction system was identified using the nonlinear autoregressive with exogenous input (NARX) technique, which is one of the most commonly used nonlinear system identification schemes. The harmonic probing method was applied to extract the first-, second- and third-order frequency response functions of the system. To achieve this, a given set of time history data of both the irregular wave excitation and the corresponding midship vertical bending moment for a certain sea state was fed into the three-layer perceptron neural network. The network parameters are determined based on the supervised training. Next, the harmonic probing method was applied to the identified system to extract the frequency response function of each order. While applying the harmonic probing method, the nonlinear activation function (i.e., the hyperbolic tangent function) was expanded into a Taylor series for harmonic component matching. After the frequency response functions were obtained, the structural responses of the ship under an arbitrary random wave excitation were easily calculated with rapidity using a third-order Volterra series. Additionally, the methodology was validated through the in-depth analysis of a nonlinear oscillator model for a weak quadratic and cubic stiffness term, whose analytic solutions are known. It was confirmed that the current method effectively predicts the nonlinear structural response of a large container carrier under arbitrary random wave excitation.     

Instability conditions due to structural nonlinearities in regenerative chatter

Chatter is an instability condition in machining processes characterized by nonlinear behavior, such as the presence of limit cycles, jump phenomenon, subcritical Hopf and period doubling bifurcations. Although the use of nonlinear techniques has provided a better understanding of chatter, neither a unifying model nor an exact solution has yet been developed due to the intricacy of the problem. This work proposes a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. An approximate solution is derived by using the method of multiple scales. In addition, a qualitative analysis of the effect of the nonlinear parameters on the stability of the system is performed. The structural cubic term gives a better representation of the nonlinear behavior, whereas the square term represents a distant attractor in the stability chart. Instability due to subcritical Hopf bifurcations is established in terms of the eigenvalues of the model in normal form. An important contribution of this analysis is the representation of hysteresis in terms of new lobes within the conventional stability limits, useful in restoring stability. This analysis leads to a further understanding of the nonlinear behavior of regenerative chatter.     

Application of the TVD scheme to the nonlinear instability analysis of a capillary jet

In the past, when either the perturbation‐type method or direct‐simulation approach was used to analyse capillary jets, the governing equations, which are parabolic in time and elliptic in space, were simplified or linearized. In the present study, the convective derivative term and a full, nonlinear form of the capillary pressure term are retained in the governing equations to investigate nonlinear effects on the break‐up of capillary jets. In this work, the TVD (i.e. total variation diminishing) scheme with flux‐vector splitting is applied to obtain the solutions of the system of nonlinear equations in a matrix form. Numerical results show that the present nonlinear model predicts longer jet break‐up lengths and slower growth rates for capillary jets than the previous linear model does. Comparing with other measurements from past literatures, the nonlinear results are consistent with the experimental data and appear more accurate than the linear analysis. In the past, the classic perturbation‐type analyses assumed constant growth rates for the fundamental and all harmonic components. By contrast, the present model is able to capture the local features of growth rates, which are not spatially and temporally constant. Copyright © 2006 John Wiley & Sons, Ltd.     

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

The purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L² norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.     

Excitation-Reshaping-Induced Chaotic Escape from a Potential Well

The chaotic escape of a nonlinearly damped oscillator excited by a periodic string of symmetric pulses from a cubic potential well is investigated. Analytical (Melnikov analysis) and numerical results show that chaotic escapes are typically induced over a wide range of parameters by hump-doubling of an external excitation which is initially formed by a periodic string of single-humped symmetric pulses. The role of a nonlinear damping term, proportional to the nth power of the velocity, on the hump-doubling scenario is also discussed.     

A Simplified Optimal Control Method for Homoclinic Bifurcations

A simplified optimal control method is presented for controlling or suppressing homoclinic bifurcations of general nonlinear oscillators with one degree-of-freedom. The simplification is based on the addition of an adjustable parameter and a superharmonic excitation in the force term. By solving an optimization problem for the optimal amplitude coefficients of the harmonic and superharmonic excitations to be used as the controlled parameters, the force term as the controller can be designed. By doing so, the control gain and small optimal amplitude coefficients can be obtained at lowest cost. As the adjustable parameter decreases, a gain of some amplitude coefficient ratio is increased to the highest degree, which means that the region where homoclinic intersection does not occur will be enlarged as much as possible, leading to the best possible control performance. Finally, it is shown that the theoretical analysis is in agreement with the numerical simulations on several concerned issues including the identification of the stable and unstable manifolds and the basins of attraction.     

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

IntroductionThepreciseintegrationmethod(PIM) [1],whichwasproposedforsolvingstructuraldynamicequations.Thismethodissimplerandpossesseshigherprecision .Forlinearsteadystructuraldynamicsystems,itsnumericalresultsattheintegrationpointsarealmostequaltothatoftheexactsolutioninmachineaccuracy .InthepreciseintegrationmethodforsolvingPDEs,theequationsshouldbediscretizedinthephysicalspaceforobtainingthesystemofODEsintime ,whichisoftenexecutedbythefinitedifferencemethodorthefiniteelementmethod .Inrec…     

A sum of squares approach to robust PI controller synthesis for a class of polynomial multi-input multi-output nonlinear systems

This paper presents a new algorithmic method to design PI controller for a general class of nonlinear polynomial systems. Design procedure can take place on certain or uncertain nonlinear model of plant and is based on sum of squares optimization.The so-called density function is employed to formulate the design problem as a convex optimization program in the sum of squares form. Robustness of design is guaranteed by taking parametric uncertainty into account with an approach similar to that of generalized ${\mathcal {S}}$ -Procedure. Validity and applicability of the proposed methods are verified via numerical simulations. The method presented here for PI controller design is not based on local linearization and works globally. Derived stability conditions overcome several drawbacks seen in previous results, such as depending on a linearized model or a stable model. Furthermore, employing sum of squares technique makes it possible to derive stability conditions with least conservatism and directly design controller for polynomial affine nonlinear systems.     

Robust high-order semi-implicit backward difference formulas for Navier-Stokes equations

Taylor-Hood finite elements provide a robust numerical discretization of Navier-Stokes equations (NSEs) with arbitrary high order of accuracy in space. To match the accuracy of the lowest degree Taylor-Hood element, we propose a very efficient time-stepping methods for unsteady flows, which are based on high-order semi-implicit backward difference formulas (SBDF) and the inclusion of grad -div term in the NSE. To mitigate the impact on the numerical accuracy (in time) of the extrapolation of the nonlinear term in SBDF, several variants of nonlinear extrapolation formulas are investigated. The first approach is based on an extrapolation of the nonlinear advection term itself. The second formula uses the extrapolation of the velocity prior to the evaluation of the nonlinear advection term as a whole. The third variant is constructed such that it produces similar error on both velocity and pressure to that with fully implicit backward difference formulas methods at a given order of accuracy. This can be achieved by fixing one-order higher than usually done in the extrapolation formula for the nonlinear advection term, while keeping the same extrapolation formula for the time derivative. The resulting truncation errors (in time) between these formulas are identified using Taylor expansions. These truncation error formulas are shown to properly represent the error seen in numerical tests using a 2D manufactured solution. Lastly, we show the robustness of the proposed semi-implicit methods by solving test cases with high Reynolds numbers using one of the nonlinear extrapolation formulas, namely, the 2D flow past circular cylinder at Re=300 and Re = 1000 and the 2D lid-driven cavity at Re = 50 000 and Re = 100 000. Our numerical solutions are found to be in a good agreement with those obtained in the literature, both qualitatively and quantitatively.     

Model reduction for reacting flow applications

A model reduction approach based on Galerkin projection, proper orthogonal decomposition (POD), and the discrete empirical interpolation method (DEIM) is developed for chemically reacting flow applications. Such applications are challenging for model reduction due to the strong coupling between fluid dynamics and chemical kinetics, a wide range of temporal and spatial scales, highly nonlinear chemical kinetics, and long simulation run-times. In our approach, the POD technique combined with Galerkin projection reduces the dimension of the state (unknown chemical concentrations over the spatial domain), while the DEIM approximates the nonlinear chemical source term. The combined method provides an efficient offline–online solution strategy that enables rapid solution of the reduced-order models. Application of the approach to an ignition model of a premixed H₂/O₂/Ar mixture with 19 reversible chemical reactions and 9 species leads to reduced-order models with state dimension several orders of magnitude smaller than the original system. For example, a reduced-order model with state dimension of 60 accurately approximates a full model with a dimension of 91,809. This accelerates the simulation of the chemical kinetics by more than two orders of magnitude. When combined with the full-order flow solver, this results in a reduction of the overall computational time by a factor of approximately 10. The reduced-order models are used to analyse the sensitivity of outputs of interest with respect to uncertain input parameters describing the reaction kinetics.     

Analysis and control of nonlinear systems with DC terms

Although nonlinear systems with a DC term are quite often found, very little work has been done about the important effects that the DC term produces on the nonlinear system response. Some work can be found based on NARMAX model identification—this work identifies the system by NARMAX models and from them obtains the Higher-order Frequency Response Functions (HFRFs). From the HFRFs, the effects of the DC term on the nonlinear system can be predicted.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

A massively parallel GPU‐accelerated model for analysis of fully nonlinear free surface waves

We implement and evaluate a massively parallel and scalable algorithm based on a multigrid preconditioned Defect Correction method for the simulation of fully nonlinear free surface flows. The simulations are based on a potential model that describes wave propagation over uneven bottoms in three space dimensions and is useful for fast analysis and prediction purposes in coastal and offshore engineering. A dedicated numerical model based on the proposed algorithm is executed in parallel by utilizing affordable modern special purpose graphics processing unit (GPU). The model is based on a low‐storage flexible‐order accurate finite difference method that is known to be efficient and scalable on a CPU core (single thread). To achieve parallel performance of the relatively complex numerical model, we investigate a new trend in high‐performance computing where many‐core GPUs are utilized as high‐throughput co‐processors to the CPU. We describe and demonstrate how this approach makes it possible to do fast desktop computations for large nonlinear wave problems in numerical wave tanks (NWTs) with close to 50/100 million total grid points in double/single precision with 4 GB global device memory available. A new code base has been developed in C++ and compute unified device architecture C and is found to improve the runtime more than an order in magnitude in double precision arithmetic for the same accuracy over an existing CPU (single thread) Fortran 90 code when executed on a single modern GPU. These significant improvements are achieved by carefully implementing the algorithm to minimize data‐transfer and take advantage of the massive multi‐threading capability of the GPU device. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of MLPG in Large Deformation Analysis

Two-dimensional large deformation analysis of hyperelastic and elasto-plastic solids based on the Meshless Local Petrov–Galerkin method (MLPG) is presented. A material configuration based the nonlinear MLPG formulation is introduced for the large deformation analysis of both path-dependent and path-independent materials. The supports of the MLS approximation functions cover the same sets of nodes during material deformation, thus the shape function needs to be computed only in the initial stage. The multiplicative hyperelasto-plastic constitutive model is adopted to avoid objective time integration for stress update in large rotation. With this constitutive model, the computational formulations for path-dependent and path-independent materials become identical. Computational efficiency of the nonlinear MLPG method is discussed and optimized in several aspects to make the MLPG an O(N) algorithm. The numerical examples indicate that the MLPG method can solve large deformation problems accurately. Moreover, the MLPG computations enjoy better convergence rate than the FEM under very large particle distortion.The project supported by the National Natural Science Foundation of China (10472051). The English text was polished by Keren Wang.     

泊松白噪声激励下强非线性系统的半解析瞬态解

自然界与工程中都普遍存在着随机扰动,且大多数呈现出固有的非高斯性质,若采用高斯激励建模可能会导致巨大的误差.泊松白噪声作为一种典型且重要的非高斯激励模型,已引起了广泛的关注.目前,泊松白噪声激励下系统的动态特性分析主要集中于稳态响应的研究,而针对瞬态响应的求解难度仍较大,需进一步发展.本文引入径向基神经网络,提出了一种泊松白噪声激励下单自由度强非线性系统瞬态响应预测的高效半解析方法.首先将广义Fokker-Plank-Kolmogorov (FPK)方程的瞬态解表示为一组含时变待定权值系数的高斯径向基神经网络;然后采用有限差分法离散时间导数项,并结合随机取样技术构造含时间递推式的损失函数;最后通过拉格朗日乘子法使得损失函数最小化获得时变最优权值系数.作为算例,探究了两个经典强非线性系统,并采用蒙特卡罗模拟方法对解析结果加以验证.结果表明:本文方法所获得的瞬时概率密度函数与蒙特卡罗模拟数据吻合地较好,并且算法具备较高的计算效率.在系统响应的整个演化过程中,本文所提方法能够非常有效地捕捉到系统响应在各个时刻下的复杂非线性特征.此外,本文方法所获得的高精度半解析瞬态解,不仅可作为基准解检验其...     

Natural convection in a rotating anisotropic porous layer with internal heat generation

In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.     

Finite memory quadratic Volterra model for the response prediction of a slender marine structure under a Morison load

A quadratic Volterra model with a finite nonlinear memory effect was introduced and applied to the time series prediction of a slender marine structure exposed to the Morison load. First, the unknown nonlinear single-input–single-output dynamic system was identified using the nonlinear autoregressive with exogenous input (NARX) technique based on the prepared datasets of the wave elevation and system response, which was obtained by running nonlinear time domain analysis for a certain short term sea state. The structure of NARX was designed in such a way that the linear part had infinite memory, whereas the nonlinear part had finite memory of a certain length. Second, the frequency domain Volterra kernels, both linear and quadratic, were derived analytically by applying the harmonic probing method to the identified system. To derive the frequency response functions, the sigmoidal function used in NARX to realize the nonlinear relationship between the input and output was expanded to polynomials based on the Taylor series expansion, so that the harmonics of same frequencies were easily matched between the input and output. Finally, the time series of the system response under arbitrarily given short term sea states were predicted using the quadratic Volterra series. The proposed methodology was used to predict the nonlinear dynamic response of a 2-dimentional free standing catenary riser exposed to a random ocean wave load, and the comparison between the prediction and simulation results was made on the probability distribution of the maximum excursion of riser top. The results show that the proposed methodology can successfully capture the nonlinear effects of the dynamic response of a slender marine structure induced by the quadratic term of the Morison formula.     

Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.