Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

Fuzzy-Model-Based Finite Frequency Fault Detection Filtering Design for Two-Dimensional Nonlinear Systems

This article studies the fault detection filtering design problem for Roesser type two-dimensional(2-D)nonlinear systems described by uncertain 2-D Takagi-Sugeno(T-S)fuzzy models.Firstly,fuzzy Lyapunov functions are constructed and the 2-D Fourier transform is exploited,based on which a finite frequency fault detection filtering design method is proposed such that a residual signal is generated with robustness to external disturbances and sensitivity to faults.It has been shown that the utilization of available frequency spectrum information of faults and disturbances makes the proposed filtering design method more general and less conservative compared with a conventional nonfrequency based filtering design approach.Then,with the proposed evaluation function and its threshold,a novel mixed finite frequency H_(∞)/H_(-)fault detection algorithm is developed,based on which the fault can be immediately detected once the evaluation function exceeds the threshold.Finally,it is verified with simulation studies that the proposed method is effective and less conservative than conventional non-frequency and/or common Lyapunov function based filtering design methods.

Atlantic blocking events in a simplified nonlinear baroclinic model for local finite-amplitude wave activity

为研究北大西洋阻高的形成机制,本文在局部有限振幅波活动(LWA)框架下进行了一系列数值实验.采用的数值模型能显式地描绘出两种重要的大气内部动力过程,即非线性纬向位涡通量和Rossby波包传播.模拟结果显示,这两种动力学过程均是形成大西洋阻高的重要机理.具体地,非线性纬向位涡通量和Rossby波包传播,分别是大西洋阻高南部和北部LWA形成的主导因子.因此,本研究综合了前人关于大西洋阻高的研究成果,为其形成机理提供了新的认识.

Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.

Nonlinear anisotropic finite element analysis of liquefiable tunnel–sand–pile interaction under seismic excitation

Nonlinear time‐history analysis can be used to determine the liquefiable behaviors of the tunnel-sand-pile interaction(TSPI)model with the consideration of sand anisotropy.This study presents the nonlinear response of the TSPI model with the existence of liquefaction under seismic excitation.The analysis reveals that tunnel and pile behave as isotropic elements,while sand shows isotropic,orthotropic,and anisotropic characteristics.Three constitutive models including UBC3D‐PLM(two yield surfaces associated with the hardening rule),NGI‐ADP(yielding with associated plastic potential function),and a user‐specified constitutive model are adopted to evaluate the isotropic,orthotropic,and anisotropic behaviors of sand.On this basis,two finite element‐based codes(ETABS 18.1.1 and Plaxis 3D)are used to evaluate sand behaviors and responses.Responses of the tunnel,sand,pile,and excess pore pressure ratio are recorded in the interaction zone by varying the pile diameter,tunnel diameter,and tunnel-pile clearance.Compared with the orthotropic and isotropic conditions,lower variations of results are found in the anisotropic condition,except for the case of generation of excess pore pressure.In addition,the present reanalysis results are in agreement with previous analytical and case study results,which further indicates the effectiveness of the finite element‐based numerical codes.

Finite-Time H∞ Control of Switched Nonlinear Systems under State-Dependent Switching

This paper investigates the finite-time H∞ control problem of switched nonlinear systems via state-dependent switching and state feedback control. Unlike the existing approach based on time-dependent switching strategy, in which the switching instants must be given in advance, the state-dependent switching strategy is used to design switching signals. Based on multiple Lyapunov-like functions method, several criteria for switched nonlinear systems to be finite-time H∞ control are derived. Finally, a numerical example with simulation results is provided to show the validity of the conclusions.

Event-Triggered Finite-Time H∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.

ALE FINITE ELEMENT ANALYSIS OF NONLINEAR SLOSHING PROBLEMS BASED ON F LUID VELOCITY POTENTIAL

Base d on fluid velocity potential, an ALE finite element formulation for the analysi s of nonlinear sloshing problems has been developed. The ALE kinemat ical description is introduced to move the computational mesh independently of f luid motion, and the container fixed noninertial coordinate system is employed to establish the governing equations so that the mesh is needed to be updated in this coordinate system only. This leads to a very simple mesh moving algorithm which makes it easy to trace the motion of the moving boundaries and the free su rface without producing undesirable distortion of the computational mesh. The fi nite element method and finite difference method are used spacewise and timewise , respectively. A numerical example involving either forced horizontal oscillati on or forced pitching oscillation of the fluid filled container is presented to illustrate the effectiveness and the robustness of the method. In additi on, this work can be extended for the fluid structure interaction problems.

New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

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.

Nonlinear Bend Stiffener Analysis Using A Simple Formulation and Finite Element Method

Flexible marine risers are commonly used in deepwater floating systems. Bend stiffeners are designed to protect flexible risel against excessive bending at the connection with the hull. The structure is usually analyzed as a cantilever beam subjected to an inclined point load. As deflections are large and the bend stiffener material exhibits nonlinear stress-strain characteristics, geometric and material nonlinearities are important considerations. A new approach has been developed to solve this nonlinear problem. Its main advantage is its simplicity; in fact the present method can be easily implemented on a spreadsheet. Finite element analysis using ABAQUS is performed to validate the method. Solid elements are used for the bend stiffener and flexible pipe. To simulate the near inextensibility of flexible risers, a simple and original idea of using truss elements is proposed. Through a set of validation studies the present method is found to be in a good agreement with the finite element analysis. Further, parametric studies are performed by using both methods to identify the key parameters and phenomena that are most critical in design. The most important finding is that the common practice of neglecting the internal steel sleeve in the bend stiffener analysis is non-conservative and therefore needs to be reassessed.

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.

Nonlinear Finite Element Analysis of Ocean Cables

This study has focused on developing numerical procedures for the dynamic nonlinear analysis of cable structures subjected to wave forces and ground motions in the ocean. A geometrically nonlinear finite element procedure using the isoparametric curved cable element based on the Lagrangian formulation is briefly summarized. A simple and accurate method to determine the initial equilibrium state of cable systems associated with self-weights, buoyancy and the motion of end points is presented using the load incremental method combined with penalty method. Also the Newmark method is used for dynamic nonlinear analysis of ocean cables. Numerical examples are presented to validate the present numerical method.

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element （FE） model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near （a） the boundary of a circular hole, （b） the boundary of an elliptic hole and （c） the tip of a crack, have been analyzed using the proposed nonlinear finite element method （FEM）. The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.

A PRIORI L_2 ERROR ESTIMATES FOR A NONLINEAR PARABOLIC SYSTEM BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH MIXED FINITE ELEMENT PROCEDURE

A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure.

Equivalent low-order angular flux nonlinear finite difference equation of MOC transport calculation

The key issue in accelerating method of characteristics(MOC)transport calculations is in obtaining a completely equivalent low-order neutron transport or diffusion equation.Herein,an equivalent low-order angular flux nonlinear finite difference equation is proposed for MOC transport calculations.This method comprises three essential features:(1)the even parity discrete ordinates method is used to build a low-order angular flux nonlinear finite difference equation,and different boundary condition treatments are proposed;(2)two new defined factors,i.e.,the even parity discontinuity factor and odd parity discontinuity factor,are strictly defined to achieve equivalence between the low-order angular flux nonlinear finite difference method and MOC transport calculation;(3)the energy group and angle are decoupled to construct a symmetric linear system that is much easier to solve.The equivalence of this low-order angular flux nonlinear finite difference equation is analyzed for two-dimensional(2D)pin,2D assembly,and 2D C5G7 benchmark problems.Numerical results demonstrate that a low-order angular flux nonlinear finite difference equation that is completely equivalent to the pin-resolved transport equation is established.