非均质材料动力分析的广义多尺度有限元法

自然界和工程中的大部分材料都具有多尺度特征,当考察尺度小到一定程度后,都将表现出非均质性.针对非均质材料的动力问题,提出了一种广义多尺度有限元方法,其基本思想是利用静态凝聚法以及罚函数法构造能够反映单元内部材料非均质特性的多尺度位移基函数.与传统扩展多尺度有限元法中的基函数构造方式不同,广义多尺度有限元法的基函数无需通过在子网格域上多次求解椭圆问题得到,而可直接通过矩阵运算获得.其主要步骤如下:利用数值基函数将一个非均质单胞等效为一个宏观单元,进而形成整个结构的等效刚度矩阵,并得到宏观网格的节点位移,最后再次利用数值基函数得到微观尺度上的位移结果.该广义多尺度有限元法是扩展多尺度有限元法的一种新的拓展,可模拟具有更加复杂几何的非均质单胞的力学行为.通过数值算例,模拟了非均质材料的静力问题、广义特征值问题以及瞬态响应问题,计算结果表明:在边界条件一样的情况下,广义多尺度有限元法的计算结果与传统有限元的计算结果保持高度一致.与传统有限元相比,该方法在保证计算精度的同时极大地提高了计算效率.研究结果表明,广义多尺度有限元法能够很好地模拟非均质单胞的力学行为,具有良好的工程应用潜力.      

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

材料非线性微-宏观分析的多尺度方法研究

介绍并比较了近年来在材料非线性微-宏观分析多级数值方法方面的研究工作. 针 对考虑材料内摩擦接触的颗粒材料多尺度计算问题，建立一种基于数值技术的多级分析方 法. 方法的特点是在对材料进行微观分析的基础上建立宏观材料的多尺度非线性数值本构模 型. 而对材料弹塑性多级分析问题，建立了基于转换场技术的算法，采用近似技术建立非线 性分析的本征应变矩阵，使方法具有表达简单与实现方便的特点. 给出了数值算例， 通过比较说明了方法的正确性与有效性.     

二维类桁架材料结构弹塑性分析

类桁架材料所构成结构的弹塑性行为的精确建模分析保证非常耗时, 为了 在保证精度的前提下提高此类问题的求解效率, 本文利用类桁架材料基本构件长细比 较大的特点，将材料单胞简化为桁架模型. 考虑到微单胞空间分布的周期性，基于 数值均匀化理论提出了类桁架材料结构的宏微观两 级弹塑性求解格式. 原问题转化为宏观上一个非线性弹性连续体计算问题和微观上多个小规 模桁架系统的弹塑性计算问题. 两个数值算例分别考虑了简单加载，非单调加载，规则宏观 结构和具有非完整单胞的较复杂宏观结构等问题. 与实际结构计算结果在精度和时间等方面 的比较验证了求解格式的有效性. 最后还探讨了算法的适用范围.     

非均质饱和多孔介质弹塑性动力分析的广义耦合扩展多尺度有限元法

针对非均质饱和多孔介质弹塑性动力问题分析提出了一种广义耦合扩展多尺度有限元方法。首先,提出了基于细尺度等效刚度阵的粗尺度单元数值基函数构造方法,并给出了构造数值基函数的一般公式,所构造的耦合数值基函数有效考虑了动力相关效应与固液之间的耦合效应。其次,针对弹塑性非线性问题迭代求解,给出了基于摄动方法的位移与孔隙压强降尺度计算修正方案。最后,针对材料的强非均质特征,利用多节点粗单元技术来提高多尺度有限元方法的计算精度。通过与基于精细网格的传统有限元分析结果对比,验证了本文所提出方法的有效性与高效性。     

多尺度材料模拟的自适应有限元方法研究

研究了材料模拟中一类新型耦合多尺度的自适应有限元方法. 采用微观分子动力学耦合宏观有限元的桥尺度方法来模拟材料破坏的前期行为,其中宏观有限元计算推广到了一般非结构三角形网格. 材料破坏形成后,停止微观尺度的计算,它的进一步发展和演化通过一个宏观模型来描述,采用自适应有限元方法来求解这一宏观模型. 其中,后验误差估计的基础是变分多尺度理论,即自适应网格加密是基于粗尺度上残差分布和细尺度上单元Green's函数. 计算中采用了破坏准则来模拟材料的断裂. 数值实验表明了方法的有效性.      

周期性点阵桁架材料力学性能分析的一种新的多尺度计算方法

论文提出一种周期性点阵桁架材料力学性能分析的新的多尺度方法.方法的主要思想是通过数值构造能反映周期性桁架材料单胞内部非均质性的多尺度基函数,从而在大尺度上求得单胞的等效刚度阵,大大减小了模型计算量.通过引入基函数的耦合附加项,以考虑多维矢量场问题不同方向问的耦合作用.数值研究表明,采用线性边界条件构造基函数有时会产生较...     

基于自适应网格的材料渗透系数设计

采用基于单元(结点)密度为设计变量进行结构和材料的拓扑优化设计时,有限元网格的密度对优化设计有很大影响. 在以渗透系数为目标进行材料微结构设计时,为了较好地描述单胞中的流固边界,需要将单胞划分为很小的网格,进一步增加了有限元计算和优化分析的规模. 为了降低计算规模, 研究了基于自适应网格的逆均匀化方法,以最大化各向同性等效渗透系数为目标,进行材料微结构设计. 优化迭代过程中,对单胞中流固界面处的网格进行自适应加密,降低优化问题的计算规模. 采用这一算法,对不同初始密度分布得到的单胞优化结果虽然不同,但具有相同的材料微结构,一定程度上说明了该方法的有效性.      

一类多孔固体的等效偶应力动力学梁模型

一维多孔固体结构可采用等效连续介质梁模型来研究其动力学行为. 当类梁结构的高度尺寸和多孔固体单胞结构尺寸相近时,等效模型的力学行为会产生尺寸效应现象. 等效经典模型由于不包含尺度参数而无法描述尺寸相关特点,而广义连续介质力学模型则可以准确地考虑尺寸效应的影响. 基于偶应力理论,对一类单胞含有圆形孔洞的周期性多孔固体类梁结构,给出了分析其横向自由振动的等效连续介质铁木辛柯梁模型. 通过对单胞分析,在应变能等价和几何平均的意义下,定义了等效偶应力介质的材料常数. 利用已有的材料常数,推导了等效铁木辛柯梁的动力学微分方程. 将实际多孔固体结构进行完全的动力学有限元离散计算,所获得的解作为精确解以检验等效梁模型所获得的频率和振型的精度. 振型的比较借助于模态置信准则矩阵方法. 大量算例表明,等效偶应力铁木辛柯梁模型在频率和振型两方面均具有较高的计算精度. 重点研究了单胞孔径的相对大小、类梁结构高度与单胞尺寸比以及类梁结构长高比对等效梁模型精度的影响. 在此基础上,偏保守地建议了多孔固体类梁结构自振分析方法.      

裂缝介质渗流多尺度混合有限元数值模拟研究

针对裂缝介质具有多尺度特点,建立了Darcy/Stokes-Brinkman多尺度耦合模型,采用多尺度混合有限元方法,对裂缝介质渗流问题进行了研究.阐述了多尺度混合有限元方法的基本原理,并推导得到Darcy/Stokes-Brinkman方程的多尺度混合有限元计算格式.数值计算结果表明,大尺度Darcy模型能够捕捉到小尺度上裂缝网络渗流特征;与网格粗化、传统有限元方法相比,多尺度混合有限元方法的基函数具有能反映单元内参数变化的优点,在保证计算精度的同时能够减少计算量,对于裂缝油藏具有良好的适用性.     

A variational asymptotic micromechanics model for predicting thermoelastic properties of heterogeneous materials

A variational asymptotic micromechanics model has been developed for predicting effective thermoelastic properties of composite materials, and recover the local fields within the unit cell. This theory adopts essential assumptions within the concept of micromechanics, achieves an excellent accuracy, and provides a unified treatment for 1D, 2D, and 3D unit cells. This theory is implemented using the finite element method into the computer program, VAMUCH, a general-purpose micromechanics analysis code. Several examples are used to validate the theory and the code. The results are compared with those available in the literature and those produced by a commercial finite element package.     

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.     

基于扩展多尺度有限元法的含液闭孔材料拓扑优化

基于扩展多尺度有限元方法,提出了含液闭孔结构多尺度拓扑优化方法。该多尺度优化方法旨在研究含液闭孔胞元布局对整体含液闭孔结构力学性能的影响。首先针对含液闭孔结构的整体结构柔顺性问题,采用类似SIMP模型对结构的宏观粗网格等效刚度阵进行插值,建立含液闭孔结构柔顺性的拓扑优化列式;其次,针对含液闭孔材料能够利用胞体内部液体腔体积增量产生变形的特性,提出含液闭孔材料柔性机构的概念,并以结构指定位置方向输出位移为目标,建立液体体积膨胀作用下的含液闭孔柔性机构多尺度拓扑优化数学模型。本文基于自主软件平台SiPESC完成了程序研发,并通过数值算例验证了所提出的拓扑优化方法的有效性。     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then 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 limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials

Asymptotic homogenization (AH) is a general method for predicting the effective coefficient of thermal expansion (CTE) of periodic composites. It has a rigorous mathematical foundation and can give an accurate solution if the macrostructure is large enough to comprise an infinite number of unit cells. In this paper, a novel implementation algorithm of asymptotic homogenization (NIAH) is devel-oped to calculate the effective CTE of periodic composite materials. Compared with the previous implementation of AH, there are two obvious advantages. One is its implemen-tation as simple as representative volume element (RVE). The new algorithm can be executed easily using commercial finite element analysis (FEA) software as a black box. The detailed process of the new implementation of AH has been provided. The other is that NIAH can simultaneously use more than one element type to discretize a unit cell, which can save much computational cost in predicting the CTE of a complex structure. Several examples are carried out to demonstrate the effectiveness of the new implementation. This work is expected to greatly promote the widespread use of AH in predicting the CTE of periodic composite materials.     

Materially and geometrically non-linear woven composite micro-mechanical model with failure for finite element simulations

A computational micro-mechanical material model of woven fabric composite material is developed to simulate failure. The material model is based on repeated unit cell approach. The fiber reorientation is accounted for in the effective stiffness calculation. Material non-linearity due to the shear stresses in the impregnated yarns and the matrix material is included in the model. Micro-mechanical failure criteria determine the stiffness degradation for the constituent materials. The developed material model with failure is programmed as user-defined sub-routine in the LS-DYNA finite element code with explicit time integration. The code is used to simulate the failure behavior of woven composite structures. The results of finite element simulations are compared with available test results. The model shows good agreement with the experimental results and good computational efficiency required for finite element simulations of woven composite structures.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

考虑材料设计变量的热-固耦合结构的优化设计

从材料-结构协同设计的角度研究了热-固耦合结构的优化设计问题,将决定结构材料性质的细观参数与结构宏观几何参数作为设计变量,利用均匀化方法推导了细观设计变量灵敏度显式计算式,并结合耦合场有限元方程构造了耦合场设计变量灵敏度计算式;提出了材料-结构协同设计的三种优化设计模型.利用结构响应最小优化模型对算例进行了计算,比较了宏观设计变量优化和材料-结构协同设计优化的效果.计算结果显示,材料-结构协同优化设计可以取得较单一宏观设计变量更好的优化效果.     

涂层结构中温度场的边界元解

涂层结构由于其优良的物理化学性能而备受人们关注,但受其厚度尺寸的影响,涂层材料中物理量的数值计算一直是工程中的难点。边界元法分析涂层结构时,难点在于涂层子域的数值分析,边界量计算既涉及奇异积分又涉及几乎奇异积分。本文基于间接规则化边界积分方程,准确高效地计算奇异边界积分。针对计算边界量及内点物理量时涉及的几乎奇异积分,采用一类非线性变量替换法,有效地改善了被积函数的震荡特性,从而消除了积分核的几乎奇异性。通过采用二次单元逼近几何边界,使得高效准确地计算超薄的涂层结构成为可能。