Mesoscopic aspects of the computational homogenization with meshless modeling for masonry material

The multiscale homogenization scheme is becoming a diffused tool for the analysis of heterogeneous materials as masonry since it allows dealing with the complexity of formulating closed-form constitutive laws by retrieving the material response from the solution of a unit cell (UC) boundary value problem (BVP). The robustness of multiscale simulations depends on the robustness of the nested macroscopic and mesoscopic models. In this study, specific attention is paid to the meshless solution of the UC BVP under plane stress conditions, comparing performances related to the application of linear displacement or periodic boundary conditions (BCs). The effect of the geometry of the UC is also investigated since the BVP is formulated for the two simpliest UCs, according to a displacement-based variational formulation assuming the block indefinitely elastic and the mortar joints as zero-thickness elasto-plastic interfaces. It will be showed that the meshless discretization allows obtaining some advantages with respect to a standard FE mesh. The influence of the UC morphology as well as the BCs on the linear and nonlinear UC macroscopic response is discussed for pure modes of failure. The results can be constructive in view of performing a general Fe·Meshless or Meshless² analysis.     

An algorithm for stress and mixed control in Galerkin-based FFT homogenization

A new algorithm is proposed to impose a macroscopic stress or mixed stress/deformation gradient history in the context of nonlinear Galerkin-based fast Fourier transform homogenization. The method proposed is based on the definition of a modified projection operator in which the null frequencies enforce the type of control (stress or strain) for each component of either the macroscopic first Piola stress or the deformation gradient. The resulting problem is solved exactly as the original variational method, and it does not require additional iterations compared to the strain control version, neither in the linear iterative solver nor in the Newton scheme. The efficiency of the proposed method is demonstrated with a series of numerical examples, including a polycrystal and a particle-reinforced hyperelastic material.     

An implicit BEM formulation for gradient plasticity and localization phenomena

The aim of this paper is to discuss a boundary element formulation for non‐linear structural problems involving localization phenomena. In order to overcome the well‐known mesh dependency observed in local plasticity, a gradient plasticity model is used. An implicit boundary element formulation is proposed and the underlying consistent tangent operator defined. This formulation is based on the classical displacement and strain integral representations combined with an integral representation of the plastic multiplier. First numerical examples are presented to illustrate the application of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Lippmann-Schwinger solvers for the computational homogenization of materials with pores

We show that under suitable hypotheses on the nonporous material law and a geometric regularity condition on the pore space, Moulinec-Suquet's basic solution scheme converges linearly. We also discuss for which derived solvers a (super)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key technical argument relies on a specific subspace on which the homogenization problem is nondegenerate, and which is preserved by iterations of the basic scheme. Our line of argument is based in the nondiscretized setting, and we draw conclusions on the convergence behavior for discretized solution schemes in FFT-based computational homogenization. Also, we see how the geometry of the pores' interface enters the convergence estimates. We provide computational experiments underlining our claims.     

Lippmann-Schwinger solvers for the explicit jump discretization for thermal computational homogenization problems

We present a variational formulation and a Lippmann-Schwinger equation for the explicit jump discretization of thermal computational homogenization problems, together with fast and memory-efficient matrix-free solvers based on the fast Fourier transform (FFT). Wiegmann and Zemitis introduced the explicit jump discretization for volumetric image-based computational homogenization of thermal conduction. In contrast to Fourier and finite difference-based discretization methods classically used in FFT-based homogenization, the explicit jump discretization is devoid of ringing and checkerboarding artifacts. Originally, the explicit jump discretization was formulated as the discrete equivalent of a boundary integral equation for the jump in the temperature gradient. The resulting equations are not symmetric positive definite, and thus solved by the BiCGSTAB method. Still, the numerical scheme exhibits stable convergence behavior, also in the presence of pores. In this work, we exploit a reformulation of the explicit jump system in terms of harmonically averaged conductivities. The resulting system is intrinsically symmetric positive definite and admits a Lippmann-Schwinger formulation. A seamless integration into existing FFT-based software packages is ensured. We demonstrate our improvements by numerical experiments.     

On boundary conditions for homogenization of volume elements undergoing localization

This paper presents an adaption of periodic boundary conditions (BC), which is termed tessellation BC. While periodic BC restrict strain localization zones to obey the periodicity of the microstructure, the proposed tessellation BC adjust the periodicity frame to meet the localization zone. Thereby, arbitrary developing localization zones are permitted. Still the formulation is intrinsically unsusceptible against spurious localization. Additionally, a modification of the Hough transformation is derived, which constitutes an unbiased criterion for the detection of the localization zone. The behavior of the derived formulation is demonstrated by various examples and compared with other BC. It is thereby shown that tessellation BC lead to a reasonable dependence of the effective stress on the localization direction. Furthermore, good convergence of stiffness values with increasing size of the representative volume element is shown as well as beneficial characteristics in use with strain softening material.     

Static and dynamic analysis of micro beams based on strain gradient elasticity theory

The static and dynamic problems of Bernoulli-Euler beams are solved analytically on the basis of strain gradient elasticity theory due to Lam et al. The governing equations of equilibrium and all boundary conditions for static and dynamic analysis are obtained by a combination of the basic equations and a variational statement. Two boundary value problems for cantilever beams are solved and the size effects on the beam bending response and its natural frequencies are assessed for both cases. Two numerical examples of cantilever beams are presented respectively for static and dynamic analysis. It is found that beam deflections decrease and natural frequencies increase remarkably when the thickness of the beam becomes comparable to the material length scale parameter. The size effects are almost diminishing as the thickness of the beam is far greater than the material length scale parameter.     

Implicit iterative finite element scheme for a strain gradient crystal plasticity model based on self-energy of geometrically necessary dislocations

In this study, an implicit iterative finite element scheme is developed for the strain gradient theory of single-crystal plasticity that accounts for the self-energy of geometrically necessary dislocations (GNDs). This strain gradient theory belongs to the Gurtin framework for viscoplastic single-crystals. The self-energy of GNDs gives a specific form of energetic higher-order stresses. An implicit finite element equation is obtained for solving a set of homogenization equations. The developed scheme is employed to analyze a model grain, and is verified by comparison with the analytical estimation derived by Ohno and Okumura (2007) [4]. The computational efficiency of the scheme and the incremental stability are discussed. Furthermore, it is shown that the developed scheme is available and applicable to different types of higher-order stresses including energetic and dissipative terms.     

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.     

Finite element analysis and algorithms for single‐crystal strain‐gradient plasticity

We provide optimal a priori estimates for finite element approximations of a model of rate‐independent single‐crystal strain‐gradient plasticity. The weak formulation of the problem takes the form of a variational inequality in which the primary unknowns are the displacement and slips on the prescribed slip systems, as well as the back‐stress associated with the vectorial microstress. It is shown that the return mapping algorithm for local plasticity can be applied element‐wise to this non‐local setting. Some numerical examples illustrate characteristic features of the non‐local model. Copyright © 2012 John Wiley & Sons, Ltd.     

基于运动一致性的视频对象分割方法研究

视频运动对象检测和分割是图像处理中最具挑战性的问题之一。针对目前大部分分割算法相当复杂而且计算量大的问题，提出了一种基于运动一致性的视频对象分割方法。该方法从MPEG压缩码流中提取运动矢量场来分割视频对象，首先对运动矢量场进行滤波和校正，然后进行全局运动补偿得到对象的绝对运动矢量场，最后采用K-means聚类算法对运动矢量场进行聚类分析从而分割出感兴趣的视频运动对象。MPEG标准测试序列的试验结果证明，该方法是有效的。     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

Variational projection methods for gradient crystal plasticity using Lie algebras

A computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation. While such gradient terms reproduce the size effects exhibited in experiments, incorporating derivatives of the plastic strain yields a nonlocal constitutive model. Rather than applying mixed methods, we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field. Crucially, the projection utilizes the mapping between Lie groups and algebras in order to preserve essential physical properties, such as orthogonality of the plastic rotation tensor. Following the projection, the plastic strain field is directly differentiated to yield the Nye tensor. Additionally, an augmentation scheme is introduced within the global Newton iteration loop such that the computed Nye tensor field is fed back into the stress update procedure. Effectively, this method results in a fully implicit evolution of the constitutive model within a traditional displacement‐based formulation. An elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference. A series of numerical tests are performed for several element types in order to assess the robustness of the method, with emphasis placed upon polycrystalline domains and multi‐axis loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational damage mechanics for composite materials based on mathematical homogenization

This paper is aimed at developing a non‐local theory for obtaining numerical approximation to a boundary value problem describing damage phenomena in a brittle composite material. The mathematical homogenization method based on double‐scale asymptotic expansion is generalized to account for damage effects in heterogeneous media. A closed‐form expression relating local fields to the overall strain and damage is derived. Non‐local damage theory is developed by introducing the concept of non‐local phase fields (stress, strain, free energy density, damage release rate, etc.) in a manner analogous to that currently practiced in concrete [1, 2], with the only exception being that the weight functions are taken to be C⁰ continuous over a single phase and zero elsewhere. Numerical results of our model were found to be in good agreement with experimental data of 4‐point bend test conducted on composite beam made of Blackglas™/Nextel 5‐harness satin weave. Copyright © 1999 John Wiley & Sons, Ltd.     

Morphology and orientation evolution of Cu6Sn5 grains on (001)Cu and (011)Cu single crystal substrates under temperature gradient

The morphology and orientation evolution of Cu6Sn5 grains formed on (001)Cu and (011)Cu single crystal substrates under temperature gradient (TG) were investigated.The initial orientated prism-type Cu6Sn5 grains transformed to non-orintated scallop-type after isothermal reflow.However,the Cu6Sn5 grains with strong texture were revealed on cold end single crystal Cu substrates by imposing TG.The Cu6Sn5 grains on (001)Cu grew along their c-axis parallel to the substrate and finally merged into one grain to form a fully IMC joint,while those on (011)Cu presented a strong texture and merged into a few dominant Cu6Sn5 grains showing about 30° angle with the substrate.The merging between neighboring Cu6Sn5 grain pair was attributed to the rapid grain growth and grain boundary migration.Accordingly,a model was put forward to describe the merging process.The different morphology and orientation evolutions of the Cu6Sn5 grains on single crystal and polycrystal Cu substrates were revealed based on crystallographic relationship and Cu flux.The method for controlling the morphology and orientation of Cu6Sns grains is really benefitial to solve the reliability problems caused by anisotropy in 3D packaging.     

Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity

The paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains. Starting from the classical local continuum formulation, which fails to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. The model is based on an assumption for the Helmholtz free energy incorporating the gradient of the internal history variable, a yield condition and the postulate of maximum dissipation resulting in an associated structure. As a result the driving force conjugated to the hardening evolution is identified as the quasi‐non‐local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non‐linear programming. Moreover, different discretization techniques are proposed in order to compare the FE‐performance in local plasticity with the advocated gradient formulation both for hardening and softening. Copyright © 2001 John Wiley & Sons, Ltd.     

Incorporating the grain boundary misorientation effects on slip activity into crystal plasticity

The role of grain boundary misorientation angle (GBMA) distribution on slip activity in a high-manganese austenitic steel was investigated through experiments and simulations. Crystal plasticity simulations incorporating the GBMA distribution and the corresponding dislocation–grain boundary interactions were conducted. The computational analysis revealed that the number of active slip systems decreased when GBMA distribution was taken into account owing to the larger volume of grain boundary–dislocation interactions. The current results demonstrate that the dislocation–grain boundary interactions significantly contribute to the overall hardening, and the GBMA distribution constitutes a key parameter dictating the slip activity.     

温度梯度对碲锌镉单晶体生长过程热应力场的影响

采用单晶位错研究的热弹性模型,计算模拟了垂直布里奇曼法碲锌镉单晶生长过程中的热应力场,研究了炉膛温度梯度对晶体内热应力的影响.计算结果表明:径向上晶体边缘与坩埚壁接触位置处的热应力远大干晶体中心处的热应力;轴向上晶体底部位置的热应力远大于晶体顶部的热应力.在晶体底部边缘与坩埚接触的位置出现最大热应力值σmax.当炉膛温度梯度从5K/cm增加到20K/cm,晶体内的热应力显著提高,σmax从41.83MPa增加到79.88MPa;当温度梯度超过20K/cm进一步增加时,晶体内的热应力增加很少,σmax仅增加了约5.3%.     

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.