Numerical homogenization of N‐component composites including stochastic interface defects

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so defined is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the effective modules method to media random in the micro‐scale. To obtain such an approach the classical mathematical homogenization method is formulated for n‐component composite with random elastic components and implemented in the FEM‐based computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

A heuristic,dynamic programming-based approach for a two-dimensional cutting problem with defects

This paper deals with a two-dimensional cutting problem in which small rectangular items of given types are to be cut from a rectangular large object which contains several defects. It is assumed that the number of pieces of each small item type which can be cut from the large object is not limited. In addition, all cuts are restricted to be of the guillotine-type and the number of stages necessary to perform all cuts is not limited. Furthermore, no small item must overlap with a defective region. The objective is to maximize the value of the cut small items. For the solution of the above-described problem, a heuristic, dynamic programming-based approach is presented which overcomes the structural and computational limitations of previously-proposed methods. In the presence of defects, the representation of the defective regions and the definition of discretization sets are revisited. This allows for an improvement of the computational efficiency as well as of the storage space requirements for solving the given problem with any number of defects in this approach. We further analyze the computational complexity of the algorithm and identify the factors which affect its running time. The proposed method is evaluated by means of a series of detailed numerical experiments which are performed on problem instances extracted from the literature, as well as on randomly generated instances. The experiments do not only illustrate how the suggested method is able to identify optimal solutions of the test problem instances, but they also explain why already existing methods fail to do so. Furthermore, the computational results indicate that the proposed method, equipped with the newly-proposed discretization sets, is capable of efficiently generating a high percentage of optimal solutions to the corresponding problem with defects.     

Production scheduling optimisation with machine state and time-dependent energy costs

The increase of energy costs specially in manufacturing system encourages researchers to pay more attention to energy management in different ways. This paper investigates a non-preemptive single-machine manufacturing environment to reduce total energy costs of a production system. For this purpose, two new mathematical models are presented. The first contribution consists of an improvement of a mathematical formulation proposed in the literature which deals and deals with a scheduling problem at machine level to process the jobs in a predetermined order. The second model focuses on the generalisation of the previous one to deal simultaneously with the production scheduling at machine level as well as job level. So, the initial predetermined fixed sequence assumption is removed. Since this problem is NP-hard, an heuristic algorithm and a genetic algorithm based on the second model are developed to provide good solutions in reasonable computational time. Finally, the effectiveness of the proposed models and optimisation methods have been tested with different numerical experiments. In average, for small size instances which the mathematical model provides a solution in reasonable computational time, a gap of 2.2% for the heuristic and 1.82% for GA are achieved comparing to the exact method’s solution. These results demonstrate the accuracy and efficiency of both proposed algorithms.     

复合材料非定常温度场的有限元计算

本文借助复合材料的微观特性建立了宏观物理模型,并把空间有限元离散模型和时问-空间有限元离散模型应用于具有任意结构参数和边界条件的复合材料的非定常温度场的有限元计算。计算结果表明,采用集中质量热容矩阵可以得到比采用一致质量热容矩阵更高的计算精度;采用时间-空简有限元可以得到比采用空间有限元更高的计算精度。计算结果还表明,由于复合材料的导热性能极差,因此它是作为航天飞机和空间飞机的热保护装置的理想材料之一。      

Homogenization of transient heat transfer problems for some composite materials

The article presented is devoted to the homogenization of transient heat transfer problems in some composite materials. The mathematical model used in the FEM computation is based on the effective modules method introduced for periodic composites. The effective heat conductivity is calculated in the closed form; effective heat capacity and mass density for the composite are obtained by simple spatial averaging. Such a homogenization scheme makes it possible to significantly simplify the numerical analysis of transient heat phenomena in various types of composites. Computational experiments performed using symbolic mathematics show the variability of effective heat conductivity for 2D and 3D composites as a function of the reinforcement volume ratio, of composite components conductivity coefficients as well as of the probabilistic moments of material properties versus volume ratio. Finally, using the Finite Element Method program, the comparison of transient heat transfer problem for the real and homogenized composites models is carried out.     

Stochastic second-order BEM perturbation formulation

This paper presents the stochastic second order moment perturbation approach to the classical deterministic Boundary Element Method (BEM) formulation. Numerous applications of such a formulation in different problems of stochastic mechanics, especially in the field of computational modeling of structural defects in homogeneous and composite materials occurring randomly in solids and engineering structures, were the main reasons to introduce the proposed model. The stochastic boundary element method (SBEM) formulation of the general linear elasticity boundary value has been provided together with an appropriate discretization. The equations describing the expected values and the covariances of stress and strain tensors for points lying on the boundary and inside the region are considered. This set of equations constitutes a formal mathematical statement of the problem and is suitable for computational implementation.     

Multi-physics computational grains (MPCGs) for direct numerical simulation (DNS) of piezoelectric composite/porous materials and structures

Conceptually simple and computationally most efficient polygonal computational grains with voids/inclusions are proposed for the direct numerical simulation of the micromechanics of piezoelectric composite/porous materials with non-symmetrical arrangement of voids/inclusions. These are named “Multi-Physics Computational Grains” (MPCGs) because each “mathematical grain” is geometrically similar to the irregular shapes of the physical grains of the material in the micro-scale. So each MPCG element represents a grain of the matrix of the composite and can include a pore or an inclusion. MPCG is based on assuming independent displacements and electric-potentials in each cell. The trial solutions in each MPCG do not need to satisfy the governing differential equations, however, they are still complete, and can efficiently model concentration of electric and mechanical fields. MPCG can be used to model any generally anisotropic material as well as nonlinear problems. The essential idea can also be easily applied to accurately solve other multi-physical problems, such as complex thermal-electro-magnetic-mechanical materials modeling. Several examples are presented to show the capabilities of the proposed MPCGs and their accuracy.     

基于等参元的三维电阻抗成像的数值模拟和分析

对于电阻抗成像的数学模型,本文用等参元方法将对应的椭圆型方程离散化,把成像问题转化为非线性优化问题,给出了目标函数梯度及近似Hesee阵的计算公式；提出了伪单元刚度矩阵的概念,给出了利用其在迭代过程中的不变性来提高计算效率的方法：分别用BFGS校正拟Newton算法和Goldfeld修正Gauss-Newton算法对三维成像问题进行了一系列数值模拟实验,证实了算法的有效性,指出了其中存在的问题。     

Estimation of the modulus of the vector of displacements in the process of simultaneous formation of cracks in the composites

We propose a computational model of the emission of acoustic signals accompanying the fracture processes in composite materials caused by the formation of several cracks regarded as separate events of macrocrack initiation. We perform the numerical analysis of the components of the vector of displacements for some typical cases of the relative location of two simultaneously formed penny-shaped cracks. The analysis of the modulus of the vector of displacements reveals the direct proportionality between its maximum values and the total area of the formed defects that, as well as the inverse proportionality between the same quantities and the period of relaxation of stresses on the cracks lips.     

Effects of geometric nonlinearities on damage propagation in patched beam-plates subjected to pressure loading

The problem of edge debonding of patched beam-plates subjected to transverse pressure is examined using two related mathematical models; one which incorporates geometric nonlinearities and the other which neglects them. The models, developed in a prior study, present the energy release rates in self-consistent functional form and yield closed form analytical solutions for the specific problem of interest. Results of numerical simulations based on each model are presented in the form of debond growth paths and compared. The growth paths are subsequently presented with corresponding pre-growth load-deflection paths to further examine the differences resulting from each model. It is seen that significant discrepancies occur between the behaviors predicted by the two models, both with regard to the onset of damage propagation and with regard to the stability of the process, as well as with regard to the pre-growth behavior, demonstrating the critical influence of geometric nonlinearities on the phenomena of interest.     

复合材料胶接修复受损结构力学建模研究进展

复合材料胶接修复结构是一个包含损伤母板、聚合物胶粘剂层及复合材料补片的多层复杂体系,其力学模型主要有2种,即解析模型与数值模型.详细阐述了2种力学模型研究的发展过程及现状,介绍了国内外对该项研究的主要内容,并展望了其发展趋势.     

Experimental and computational investigations on fire resistance of GFRP composite for building façade

Composite materials such as glass fibre reinforced polymers (GFRPs) possess the advantages of high strength and stiffness, as well as low density and highly flexible tailoring; therefore, their potential in replacing conventional materials (such as concrete, aluminium and steel) in building façade has become attractive. This paper addresses one of the major issues that hinder the extensive use of composite structures in the high-rise building industry, which is the fire resistance. In this study, a fire performance enhancement strategy for multilayer composite sandwich panels, which are comprised of GFRP composite facets and polyethylene foam core, is proposed with the addition of environmentally friendly, fire retardant unsaturated polyester resins and gel-coats. A series of burning experimental studies including thermo-gravimetric analysis (TGA) and single burning item (SBI) are carried out on the full scale composite sandwich as well as on single constituents, providing information regarding heat release rate, total heat release, fire growth rate, and smoke production. Experimental results are compared with fire safety codes for building materials to identify the key areas for improvements. A fire dynamic numerical model has been developed in this work using the Fire Dynamics Simulator (FDS) to simulate the burning process of composite structures in the SBI test. Numerical results of heat production and growth rate are presented in comparison with experimental observations validating the computational model and provide further insights into the fire resisting process. Parametric studies are conducted to investigate the effect of fire retardant additives on the fire performance of the composite sandwich panel leading to optimum designs for the sandwich panel.     

金属预成形优化设计及凝聚函数方法

研究了金属预成形设计中的形状优化设计问题及相应的求解算法,并讨论了凝聚函数在这类问题中的应用。金属预成形设计在本质上是一类反问题,给出采用形状优化设计方法求解这类问题的数学模型,讨论了目标函数构造方法对数值计算收敛性的影响。提出采用凝聚函数将∞-范数形式的形状误差函数转化为光滑可微的目标函数,显著提高了求解以金属预成形设计为背景的优化问题的收敛性。采用流动模型描述金属材料高温下的变形过程,利用基于梯度的数学规划方法求解了金属预成形优化设计问题。数值算例验证了所提出方法的有效性。     

A mathematical approach to a forest-impact problem

In this article a mathematical model of forests impact on aquifers is proposed. This phenomenon is the lowering of the groundwater table under areas covered by trees. The model includes a boundary-value problem with contact and free-boundary conditions. A variational formulation of this problem, which is a quasi-variational inequality, is obtained. Its equivalence with the original problem is proved; existence and uniqueness results are obtained. A numerical example of the model is given.     

Stable numerical simulations of propagations of complex damages in composite structures under transverse loads

In this paper, to deal with complex damage propagations in various composite structures under quasi-static transverse loads, a numerical simulation methodology based on the three-dimensional (3D) finite element method (FEM) is proposed. In this numerical model, two categories of damage patterns existing in composite structures under transverse loads are tackled independently. First, a kind of stress-based criteria is adopted to deal with the first category, which includes various in-plane damages, such as fiber breakage, transverse matrix cracking, matrix crushing, etc. Second, a bi-linear cohesive interface model is employed to deal with the second category, i.e., interface damages, such as delaminations. Also, to overcome the numerical instability problem when using the cohesive model, a simple and useful technique is proposed. In this technique, the move-limit in the cohesive zone is built up to restrict the displacement increments of nodes in the cohesive zone of laminates after delaminations occurred. The effectiveness of this method is illustrated using a DCB example and its characteristic is discussed in detail. This numerical model is further applied to various composite structures, such as 2D laminated plates and 3D laminated shells under transverse loads. The results of the numerical simulations are compared with the experimental results and good agreements are observed. The obtained information is helpful for understanding the propagation mechanisms of various damages in composite structures.     

Higher order refined computational model with 12 degrees of freedom for the stress analysis of antisymmetric angle-ply plates – analytical solutions

Analytical formulations and solutions for the stress analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with twelve degrees of freedom already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. In addition, two higher order computational models, one with nine and the other with five degrees of freedom already available in the literature are also considered for comparison. The equations of equilibrium are obtained using Principle of Minimum Potential Energy (PMPE). Solutions are obtained in closed form using Navier’s technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already available in the literature. After establishing the accuracy of the solutions, numerical results with real properties using all the computational models are presented for the stress analysis of multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

Systematic study of thermal properties of CNT composites by the fast multipole hybrid boundary node method

Carbon nanotubes (CNTs) are predicted to possess superior heat conductivity, which makes the CNTs promising in development of fundamentally new composite material. With the current advancement in nanotechnology, it is possible to design materials with desired properties for specific applications. On the other hand, the overall properties of CNT composites are usually evaluated using a representative volume element (RVE) with a number of CNTs embedded. For realistic modeling, an RVE including a large number of CNTs, for example, tens or hundreds, is necessary. However, analysis of such an RVE using standard numerical methods faces two severe difficulties: discretization of the geometry into elements and the very large computational scale. In this paper, the first difficulty is alleviated by developing the hybrid boundary node method (HdBNM), which is a boundary-type meshless method. To overcome the second difficulty, a simplified mathematical model for thermal analysis of CNT analysis is first proposed, by which the size of the linear system can be reduced by nearly half. Then, the HdBNM is combined with the Fast Multipole Method (FMM) based on the model to further reduce the computational scale. A variety of RVEs containing different numbers of CNTs, from small to large scales, have been studied in an attempt to investigate the influence of CNT length, distribution, orientation and volume fraction on the overall thermal properties of the composites. Insights have been gained into the thermal behavior of the CNT composite material.     

Direct search solution of an inverse problem in elastoplasticity: Identification of cohesion,friction angle and in situ stress by pressure tunnel tests

The identification (or ‘calibration’ or ‘inverse’) problem with in this paper, can be outlined as follows. The ‘real system’ is a deep rock formation which can be regarded as isotropic and elastoplastic. A mathematical-numerical model, intended for the analysis of its response to excavations, rests on the assumption of an elastic-perfectly plastic Mohr–Coulomb constitutive law and of a homogeneous isotropic initial (in situ) stress state. The values of cohesion, friction angle and initial stress to be introduced in this model are identified by minimizing a measure of the discrepancy (error) between theoretical and experimental relationship (pressure vs. average diameter increase), concerning a standard pressure tunnel test carried out well inside the nonlinear range. For the error minimization process, two very general ‘search techniques’ are adopted and discussed from the computational standpoint: the flexible polyhedron (modified simplex) strategy and the alternating variable strategy in the Rosenbrock version. Both are found to be adequate for solving this inverse problem, when the mathematical model has to be use as a ‘black box’ in a purely numerical identification process.     

动力松弛法在复合材料叠层板非线性弯曲问题中的应用

动力松弛法是近十多年来提出并发展的一种数值方法。本文探讨了该法在复合材料叠层板非线性弯曲问题上的应用,推导了用于求解该问题的重要参数——虚拟密度的一般计算公式,用该公式对对称正交铺设及反对称角铺设叠层矩形板进行了数值计算,其结果与试验吻合得较好。