Computational homogenization of periodic beam-like structures

This paper is concerned with the computation of the effective elastic properties of periodic beam-like structures. The homogenization theory is used and leads to an equivalent anisotropic Navier–Bernoulli–Saint-Venant beam. The overall behavior is obtained from the solution of basic cell problems posed on the three-dimensional period of the structure and solved using three-dimensional finite element implementation.This procedure is first applied to two corrugated zigzag and sinus beams subjected to in-plane loading. Next, the axial elastic properties of a stranded ‘6 + 1’ wire-cable are computed. The effective properties values obtained appear to be very close to analytical reference results showing the efficiency of the approach.     

A homogenization theory for elastic-viscoplastic materials with misaligned internal structures

In this study, a homogenization theory for non-linear time-dependent materials is rebuilt for periodic elastic-viscoplastic materials with misaligned internal structures, by employing a unit cell defined for the aligned structure as an analysis domain. For this, it is shown that the perturbed velocity fields in such materials possess periodicity in the directions of misaligned unit cell arrangement. This periodicity is used as a novel boundary condition for unit cell analysis to rebuild the homogenization theory. The resulting theory is able to deal with arbitrary misalignment using the same unit cell, avoiding not only geometry and mesh generation of a unit cell for every misalignment, but also the influence of mesh dependence. To verify the theory, an elastic-viscoplastic analysis of plain-woven glass fiber/epoxy laminates with misaligned internal structures is performed. It is shown that the misalignment of internal structures affects viscoplastic properties of the plain-woven laminates both macroscopically and microscopically.     

A homogenization theory for time-dependentnonlinear composites with periodic internal structures

A homogenization theory for time-dependent deformation such as creep andviscoplasticity of nonlinear composites with periodic internal structures is developed. To beginwith, in the macroscopically uniform case, a rate-type macroscopic constitutive relation betweenstress and strain and an evolution equation of microscopic stress are derived by introducing twokinds of Y-periodic functions, which are determined by solving two unit cell problems.Then, the macroscopically nonuniform case is discussed in an incremental form using thetwo-scale asymptotic expansion of field variables. The resulting equations are shown to beeffective for computing incrementally the time-dependent deformation for which the history ofeither macroscopic stress or macroscopic strain is prescribed. As an application of the theory,transverse creep of metal matrix composites reinforced undirectionally with continuous fibers isanalyzed numerically to discuss the effect of fiber arrays on the anisotropy in such creep.     

Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis

In this study, using a virtual work equation, a micro-/macro-kinematic relation and a linearized constitutive relation, a boundary value problem is fully implicitly formulated to determine perturbed displacement increment fields in elastoplastic unit cells for two-scale analysis. It is shown that this implicit homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increment fields in unit cells, and that the boundary value problem formulated provides a computational algorithm which is versatile for initial setting of strain increment fields. The computational algorithm developed is then examined by performing a two-scale analysis of a holed plate with an elastoplastic micro-structure, subjected to tensile loading. This demonstrates that the convergence in iteratively solving the implicit homogenization problem strongly depends on the initial setting of strain increment fields in unit cells.     

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in  and , and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.     

风与柔性结构流固耦合作用的预处理方法研究

针对强耦合方法求解风与柔性结构流固耦合作用时,大量计算资源都耗费在对强耦合方程求解中这一弊端,本文研究了强耦合方程的预处理求解方法。在风与柔性结构流固耦合作用的强耦合整体方程的基础上,将时空离散和线性化后的类似结构方程看成是鞍点问题,首先推导得到了类似结构方程的预处理矩阵;再基于此推导出了强耦合整体方程的预处理矩阵。首先采用预处理方法对经典二维流固耦合问题进行了计算,验证了提出的预处理矩阵的正确性;然后对风与三维膜结构的流固耦合作用进行了分析,评估了所提出预处理方法的相关计算参数。计算结果表明,所提出的预处理方法可使强耦合整体方程的求解在计算精度和计算效率上都得到较大提升,证明本文提出的预处理方法适用于风与柔性结构的流固耦合分析。     

Pressure and time treatment for Chebyshev spectral solution of a Stokes problem

To investigate the influences of time scheme, pressure treatment and initial conditions in incompressible fluid dynamics, a Stokes problem is solved numerically on a slab geometry within the framework of spectral approximation in space. Four algorithms are examined: splitting schemes, influence matrix method, penalty formulation and pseudo-spectral space-time technique. It is shown that splitting schemes are less accurate than the other processes. Furthermore, the initial field should respect a compatibility condition to avoid singularities at the initial time. If it is not possible to build such a compatible field, the numerical procedure has to present good damping properties at the first steps of the time integration.     

Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial–temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity.     

The spectral problem of Poiseuille flow

For Poiseuille flow the Orr-Sommerfeld equation is solved exactly. Regular solutions are obtained, thereby eigenvalue equation can be analyzed analytically and explicitly. The bifurcation solutions will be discussed subsequently.     

On a formulation for a multiscale atomistic-continuum homogenization method

The homogenization method is used as a framework for developing a multiscale system of equations involving atoms at zero temperature at the small scale and continuum mechanics at the very large scale. The Tersoff–Brenner Type II potential [Physical Review Letters 61(25) (1988) 2879; Physical Review B 42 (15) (1990) 9458] is employed to model the atomic interactions while hyperelasticity governs the continuum. A quasistatic assumption is used together with the Cauchy–Born approximation to enforce the gross deformation of the continuum on the positions of the atoms. The two-scale homogenization method establishes coupled self-consistent variational equations in which the information at the atomistic scale, formulated in terms of the Lagrangian stiffness tensor, concurrently feeds the material information to the continuum equations. Analytical results for a one dimensional molecular wire and numerical experiments for a two dimensional graphene sheet demonstrate the method and its applicability.     

Periodic beam-like structures homogenization by transfer matrix eigen-analysis: A direct approach

The paper deals with a direct approach to homogenize lattice beam-like structures via eigen- and principal vectors of the state transfer matrix. Since the girders unit cells transmit two bending moments, one given by the axial forces, the other originated by nodal moments, the Timoshenko couple-stress beam is employed as substitute continuum. The main advantage of the method is the possibility of operating directly on the sub-partitions of the unit cell stiffness matrix. Closed form solutions for the Pratt and X-braced girders are achieved and used into the homogenization. Unit cells with more complex geometries are numerically addressed with direct approach, showing that the principal vector problem corresponds to the inversion of a well-conditioned matrix. Finally, a validation of the procedure is carried out comparing the predictions of the homogenized models with the outcomes of f.e. analyses performed on a series of girders.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.     

Elastic-plastic homogenization for layered composites

A general method for the homogenization of rigid plastic or elastic-plastic layered structures is proposed. This method is based on a hybrid formulation of the constitutive equation for each phase, thus reducing the homogenization procedure to a simple mixture law. This non-classical hybrid formulation is obtained for both the rigid-plastic and elastic-plastic models. The resulting homogenized behaviour is then illustrated by some examples with special emphasis on the resulting anisontropy and in particular on the off-axis tensile behaviour.     

Adaptive force density method for form-finding problem of tensegrity structures

A numerical method is presented for form-finding of tensegrity structures. Eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates. The equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation. A unique and non-degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates. A simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non-degenerate structure. Several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method.     

Reflection of bounded acoustic beams on a fluid-solid interface

The reflection of an acoustic beam onto a fluid-solid interface is studied under the assumption that the solid medium is viscoelastic. The incident beam is represented as a superposition of plane monochromatic homogeneous waves and its profile is assumed to be Gaussian shaped. The outcoming wave field at the interface, and away from it in the fluid, is numerically calculated for different values of the frequency and of the beam width.

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Sommario Si studia la riflessione di un fascio di onde acustiche sulla superficie di separazione fra un liquido ed un solido nell'ipotesi che il mezzo solido sia viscoelastico. Il fascio incidente ha un profilo gaussiano e viene rappresentato mediante la sovrapposizione di onde piane monocromatiche omogenee. Si calcola numericamente il profilo del fascio riflesso sia sull'interfaccia, sia lontano da essa nel fluido, per diversi valori della frequenza e della larghezza del fascio.

     

Double porosity in fluid-saturated elastic media: deriving effective parameters by hierarchical homogenization of static problem

We propose a model of complex poroelastic media with periodic or locally periodic structures observed at microscopic and mesoscopic scales. Using a two-level homogenization procedure, we derive a model coherent with the Biot continuum, describing effective properties of such a hierarchically structured poroelastic medium. The effective material coefficients can be computed using characteristic responses of the micro- and mesostructures which are solutions of local problems imposed in representative volume elements describing the poroelastic medium at the two levels of heterogeneity. In the paper, we discus various combinations of the interface between the micro- and mesoscopic porosities, influence of the fluid compressibility, or solid incompressibility. Gradient of porosity is accounted for when dealing with locally periodic structures. Derived formulae for computing the poroelastic material coefficients characterize not only the steady-state responses with static fluid, but are relevant also for quasistatic problems. The model is applicable in geology, or in tissue biomechanics, in particular for modeling canalicular-lacunar porosity of bone which can be characterized at several levels.