Multiscale stochastic finite element modeling of random elastic heterogeneous materials

In Xu et al. (Comput Struct 87:1416–1426, 2009) a novel Green-function-based multiscale stochastic finite element method (MSFEM) was proposed to model boundary value problems involving random heterogeneous materials. In this paper, we describe in detail computational aspects of the MSFEM explicitly across macro–meso–micro scales. Different numerical algorithms are introduced and compared in terms of numerical accuracy and convergence.     

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells

Following the approach developed for rods in Part 1 of this paper (Pimenta et al. in Comput. Mech. 42:715–732, 2008), this work presents a fully conserving algorithm for the integration of the equations of motion in nonlinear shell dynamics. We begin with a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, allowing for an extremely simple update of the rotational variables within the scheme. The weak form is constructed via non-orthogonal projection, the time-collocation of which ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that general hyperelastic materials (and not only materials with quadratic potentials) are permitted in a totally consistent way. Spatial discretization is performed using the finite element method and the robust performance of the scheme is demonstrated by means of numerical examples.     

Multi-scale nonlinear modelling of sandwich structures using the Arlequin method

The paper presents a combination of the Arlequin Method (AM) and the Asymptotic Numerical Method (ANM) for studying nonlinear problems related to the mechanical behavior of sandwich composite structures. The Arlequin Method is a multi-scale method in which different models are crossed and glued to each other. The ANM is an alternative method which falls into the category of numerical perturbation techniques. By introducing the power series expansions into the equilibrium equation, the nonlinear problem is transformed into a sequence of linear problems and solved by the standard finite element method. Compared to other classical solvers (Newton–Raphson Method, Modified Newton–Raphson Method), ANM offers a considerable interest in the computation time and reliability. To validate this method, the AM is combined with the ANM to simulate the local damage of 2D–2D and 2D–2D-coupled sandwich beams. The simulation results are compared to a reference solution calculated from a 2D beam without any coupling. In case of the 2D–2D-coupled sandwich beam, the simulation shows a good agreement with the reference solution for both the local damage and the deformation at the loaded point. However, in case of 2D–1D-coupled sandwich beam, the simulation deviate from the reference solution due to the constant thickness of the 1D zig-zag element used to model the 1D zone of the sandwich beam.     

Solving hyperelastic material problems by asymptotic numerical method

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton–Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton–Raphson method for problems involving hyperelastic structures with large strains and instabilities.     

A subspace iteration algorithm for the eigensolution of large structures subject to non‐classical viscous damping

This paper describes an efficient and numerically stable algorithm for accurately computing the solutions of the quadratic eigenproblem associated to non‐proportionally viscously damped structures characterized by symmetric matrices. Combining the simultaneous inverse iteration with a generalized Rayleigh–Ritz analysis, the proposed procedure is well suited for extracting the subset of the lowest natural frequencies, the corresponding subcritical damping ratios and the mode shapes of large dissipative systems with non‐classical viscous damping. The iterative process exploits the specific nature of non‐proportionally damped structures, takes full advantage of the banded configuration of the structural matrices involved in the eigenproblem, avoids the computation of the left eigenvectors and circumvents the use of complex algebra owing to a unitary transformation strategy. An academic test case and an industrial numerical example are presented to highlight the effectiveness of the algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

A non-local criterion for modelling unbalanced woven ply laminates with stress concentrations

A non-local ply scale criterion [Hochard C, Lahellec N, Bordreuil C. A ply scale non-local fibre rupture criterion for CFRP woven ply laminated structures. Compos Struct 2007;80:321–26] was previously developed for predicting the failure of balanced woven ply structures with stress concentrations. This non-local criterion was based on the mean values determined over a Fracture Characteristic Volume (FCV) corresponding to a cylinder with a circular area and the same thickness as the ply. This non-local approach along with a ply scale continuum damage behavioural model was implemented in the ABAQUS Finite Element Code. The behavioural model was developed from a classical Continuum Damage Mechanics (CDM) model [Ladevèze P. A damage computational method for composite structures. Comput Struct 1992;44:79–87]. In the present study, this approach was extended to the case of unbalanced woven ply. The FCV approach and the CDM behavioural model are presented and comparisons are made between the experimental data and the modelling predictions obtained on plates with open holes, notches and saw cuts.     

Influence diagnostics in a general class of beta regression models

In this article, we consider the issue of assessing influence of observations in the general class of beta regression models introduced by Simas et al. (Comput. Stat. Data Anal. 54:348–366, 2010), which is very useful in situations in which the response is restricted to the standard unit interval (0,1). Our results generalize those in Espinheira et al. (Comput. Stat. Data Anal. 52:4417–4431, 2008a; J. Appl. Stat. 35:407–419, 2008b), which only apply to linear beta regression models. We define some residuals, and a Portmanteau test for serial correlation. Further, some influence methods, such as the global, local, and total local influence of an individual and generalized leverage, are discussed. Moreover, we also derive the normal curvatures of local influence under various perturbation schemes. Finally, simulation results and an application to real data show the usefulness of our results.     

An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation

We investigate the numerical implementation of the alternating iterative algorithm originally proposed by Kozlov et al. (Comput Math Math Phys 31:45–52) for the Cauchy problem associated with the two-dimensional modified Helmholtz equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method.     

热过屈曲梁振动的解析解

该文导出了面内热载荷作用下, 梁在其过屈曲构形附近微幅振动的解析解。首先基于经典梁理论, 推导了控制轴向和横向变形的基本方程。然后, 将2 个非线性方程化为一个关于横向挠度的四阶非线性积分-微分方程。假设梁的振幅以及由此引起的附加应变为无限小, 另设其响应为谐振, 则该非线性积分-微分方程将化为两组耦合的微分方程:一组控制非线性静态响应;另一组就是叠加于梁屈曲构形之上的线性振动方程。直接求解这些问题, 可以得到梁热过屈曲构形以及固有频率的解析解, 这些解是外加热载荷的函数。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

PeliGRIFF,a parallel DEM-DLM/FD direct numerical simulation tool for 3D particulate flows

The problem of particulate flows at moderate to high concentration and finite Reynolds number is addressed by parallel direct numerical simulation. The present contribution is an extension of the work published in Computers & Fluids 38:1608 (2009), where systems of moderate size in a 2D geometry were examined. At the numerical level, the suggested method is inspired by the framework established by Glowinski et al. (Int J Multiph Flow 25:755, 1999) in the sense that their Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) formulation and their operator-splitting idea are employed. In contrast, particle collisions are handled by an efficient Discrete Element Method (DEM) granular solver, which allows one to consider both smoothly (sphere) and non-smoothly (angular polyhedron) shaped particles. From a computational viewpoint, a basic though efficient strategy has been developed to implement the DLM/FD method in a domain decomposition/distributed fashion. To achieve this goal, the serial code, GRIFF (GRains In Fluid Flow; see Comput Fluids 38:1608–1628, 2009) is upgraded to fully MPI capabilities. The new code, PeliGRIFF (Parallel Efficient Library for GRains in Fluid Flow) is developed under the framework of the fully MPI open-source platform PELICANS. The parallel computing capabilities of PeliGRIFF offer new perspectives in the study of particulate flows and indeed increase the number of particles usually simulated in the literature. Solutions to address new issues raised by the parallelization of the DLM/FD method and assess the scalable properties of the code are proposed. Results on the 2D/3D sedimentation of a significant collection of isometric polygonal/polyhedral particles in a Newtonian fluid with collisions are presented as a validation test and an illustration of the class of particulate flows PeliGRIFF is able to investigate.     

Perturbation technique and method of fundamental solution to solve nonlinear Poisson problems

We show in this work that the Asymptotic Numerical Method (ANM) combined with the Method of Fundamental Solution (MFS) can be a robust algorithm to solve the nonlinear Poisson problem. The ANM transforms the nonlinear problem into a sequence of linear ones which can be solved by MFS. This last method consists of approximating the solution of the linear Poisson problem by a linear combination of fundamental solutions. Some examples are presented to show the efficiency of the proposed method.     

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.     

On necessary and sufficient conditions for differential flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-B?cklund equivalence, introduced in Fliess et al. (IEEE Trans Automat Contr 44(5):922–937, 1999), to this implicit context and focus attention on Lie-B?cklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \fracddt{\frac{d}{dt}} , whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.     

Parametric and internal resonances of in-plane accelerating viscoelastic plates

This paper analytically and numerically investigates the nonlinear vibration in parametric and internal resonances of in-plane accelerating viscoelastic plates subjected to plane stresses. An approximate nonlinear plate theory was developed under the Kirchoff assumptions. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean axial speed. The governing equation with the associated boundary conditions is derived from the generalized Hamilton principle and the Kelvin constitutive relation. The method of multiple scales is applied to establish the solvability conditions in principal parametric and internal resonances. The steady-state responses are predicted in three possible patterns: trivial, single-mode, and two-mode solutions. The stabilities of the steady-state responses are determined based on the Routh-Hurwitz criterion. The effects of the mean in-plane translating speed, the in-plane translating speed fluctuation amplitude, the viscosity coefficient, and the nonlinear coefficient on the steady-state responses are examined. The differential quadrature schemes are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the trivial and single-mode solutions of the steady-state responses.     

Boundary element–minimal error method for the Cauchy problem associated with Helmholtz-type equations

An iterative procedure, namely the minimal error method, for solving stably the Cauchy problem associated with Helmholtz-type equations is introduced and investigated in this paper. This method is compared with another two iterative algorithms previously proposed by Marin et al. (Comput Mech 31:367–377, 2003; Eng Anal Bound Elem 28:1025–1034, 2004), i.e. the conjugate gradient and Landweber–Fridman methods, respectively. The inverse problem analysed in this study is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method for both two-dimensional Helmholtz and modified Helmholtz equations.     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

干摩擦阻尼叶片的界面约束力描述及振动响应求解

发展了一种描述复杂运动状态下界面约束力的三维干摩擦接触数值模型,该模型通过在接触界面建立多个摩擦接触点对得到多点分布的界面约束力,可以描述界面的粘滞-滑动共存状态和法向接触正压力不均匀分布。在该模型中还考虑了界面动静摩擦系数的不同和界面的各向异性。采用三维干摩擦接触数值模型和高阶谐波平衡法,计算了某真实围带阻尼结构汽轮机叶片在复杂激励下的非线性振动响应。计算出叶片振动响应在一个运动周期出现多个局部极值,呈含多谐波的周期函数。     

Exact solutions for free vibrations of orthotropic rectangular Mindlin plates

Exact closed-form solutions are obtained for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method although it is difficult to solve them. The plates have two opposite edges simply supported and all possible combinations of classical boundary conditions at the other two edges. The exact solutions of orthotropic rectangular Mindlin plates are compared with those of isotropic ones and their differences are discussed. The exact solutions are validated through both mathematical proof and numerical comparisons with available p-Ritz solutions and the differential quadrature finite element method solutions calculated by the authors.     

Transient three-dimensional contact problems: mortar method. Mixed methods and conserving integration

The present work deals with the development of an energy-momentum conserving method to unilateral contact constraints and is a direct continuation of a previous work (Hesch and Betsch in Comput Mech 2011, doi:) dealing with the NTS method. In this work, we introduce the mortar method and a newly developed segmentation process for the consistent integration of the contact interface. For the application of the energy-momentum approach to mortar constraints, we extend an approach based on a mixed formulation to the segment definition of the mortar constraints. The enhanced numerical stability of the newly proposed discretization method will be shown in several examples.