Topological sensitivity of energy cost functional for wave-based defect identification

This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.     

基于NURBS插值的三维渐进结构优化方法

提出一种基于等几何控制点密度变量的三维双向渐进结构拓扑优化方法。在当前列式下,高阶NURBS基函数被同时用于CAD模型中NURBS实体片的几何场、位移场和温度场以及密度场插值,实现了几何模型、分析模型和优化模型的有效统一,确保了位移场、温度场及密度场的高阶连续性;详细推导了基于等几何控制点密度变量的三维渐进结构法模型及其灵敏度分析列式;最后几个典型的数值算例,包括最小柔顺性、热传导优化问题及三维结构自由振动的基频最大化问题,验证了本文方法的有效性。     

考虑结构自重的基于NURBS插值的3D拓扑描述函数法

在许多如大坝、桥梁等大型土木工程结构中,结构的自重是初始设计阶段必须考虑的重要载荷之一,因此研究自重载荷作用下的结构拓扑优化设计问题具有十分重要的意义.针对考虑自重载荷作用的拓扑优化问题所面临的主要困难,总结了现有处理考虑自重载荷的拓扑优化问题的三类主要方法;提出一种基于非均匀有理B样条(non-uniform rational B-splines,NURBS)基函数插值的拓扑描述函数方法,基于此方法研究了考虑设计依赖自重载荷作用的2D/3D结构优化设计问题.在列式下,高阶NURBS基函数被同时用于三维NURBS实体片中的几何场、位移场及设计变量场插值,实现了几何模型、分析模型和优化模型的有效统一,确保了位移场及设计变量场的高阶连续性;详细推导了基于NURBS基函数插值的考虑自重载荷作用的三维结构拓扑优化模型及其灵敏度列式,并采用移动渐进线方法(method of moving asymptotes,MMA)进行了优化求解;多个算例验证了方法的有效性和稳定性,结果表明,优化迭代过程稳健,收敛快,能够有效地克服自重载荷作用下连续体结构拓扑优化中经常遇到的低密度区域材料的寄生效应及目标函数的非单调性等问题.     

From imaging to material identification: A generalized concept of topological sensitivity

To establish a compact analytical framework for the preliminary stress-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic Green's function, obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a dissimilar elastic inclusion in a defect-free “reference” solid. The featured formula, consisting of an inertial-contrast monopole term and an elasticity-contrast dipole term, is shown to be applicable to a variety of reference solids (semi-infinite and infinite domains with constant or functionally graded elastic properties) for which the Green's functions are available. To deal with situations when the latter is not the case (e.g. finite reference bodies or those with pre-existing defects), an adjoint field approach is employed to derive an alternative expression for topological sensitivity that involves the contraction of two (numerically computed) elastodynamic states. A set of numerical results is included to demonstrate the potential of generalized topological derivative as an efficient tool for exposing not only the geometry, but also material characteristics of subsurface material defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at sampling location. Beyond the realm of non-invasive characterization of engineered materials, the proposed developments may be relevant to medical diagnosis and in particular to breast cancer detection where focused ultrasound waves show a promise of superseding manual palpation.     

An Apparent Interface Location as a Tool to Solve the Porous Interface Flow Problem

A new approach for solving the laminar flow problem above a porous medium is presented here, using an apparent interface for which both superficial velocity and intrinsic shear stress are continuous. The derivation of this approach is based on a detailed investigation of the Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38:2635–2646, 1995a) jump condition and its sensitivity to the value of β (the jump condition coefficient) and to an error in the interface location. While the value of the jump condition coefficient is highly sensitive to the interface location, the new apparent interface approach does not require an a priori information about the location of the interface. This approach can be easily used knowing only one measurable parameter—the maximum velocity or the flow rate. Validation of the apparent interface approach against measurements from other works shows that it can be successfully used to predict the velocity profile for different geometrical models.     

An inverse problem to estimate temperature dependent heat capacity under convection processes

Solutions of the heat capacity versus temperature in a one-dimensional slab have been studied for different types of dependency (lineal, sinusoidal, piece-wise and rectangular) under boundary conditions of natural and forced convection on both sides of the slab. The input data of this inverse problem are the temperature history ("measurements") at a particular location within the slab, obtained by adding a specified random error to the set of temperatures which are the solution of the direct problem. No prior information is used as regards the temperature-dependent functional forms of the unknown heat capacity. In all cases, a piece-wise function is used to approach the solution. Using a programming routine that minimises a classical predefined functional, successive stretches of this piece-wise function are obtained step by step by (i) fixing its length and (ii) increasing or decreasing its slope. The Network Simulation Method is used to solve both the direct and inverse problems. No mathematical manipulations of the finite-difference differential equations are required by the programmer, since they are contained in the computer code used in the method. The basic network for the inverse problem, which is basically the same as for the direct problem, is easy to design and has very few devices. Several examples are shown to prove the accuracy and effectiveness of the proposed method.     

An unsteady single‐phase level set method for viscous free surface flows

The single‐phase level set method for unsteady viscous free surface flows is presented. In contrast to the standard level set method for incompressible flows, the single‐phase level set method is concerned with the solution of the flow field in the water (or the denser) phase only. Some of the advantages of such an approach are that the interface remains sharp, the computation is performed within a fluid with uniform properties and that only minor computations are needed in the air. The location of the interface is determined using a signed distance function, and appropriate interpolations at the fluid/fluid interface are used to enforce the jump conditions. A reinitialization procedure has been developed for non‐orthogonal grids with large aspect ratios. A convective extension is used to obtain the velocities at previous time steps for the grid points in air, which allows a good estimation of the total derivatives. The method was applied to three unsteady tests: a plane progressive wave, sloshing in a two‐dimensional tank, and the wave diffraction problem for a surface ship, and the results compared against analytical solutions or experimental data. The method can in principle be applied to any problem in which the standard level set method works, as long as the stress on the second phase can be specified (or neglected) and no bubbles appear in the flow during the computation. Copyright © 2006 John Wiley & Sons, Ltd.     

地震动随机场投影展开法

本文发展了随机场投影展开方法,对地震动随机场进行变界近似解析展开。地震动采用自功率谱与具有高斯指数衰减的相关函数模型。本文方法只须求解一次随机场特征值问题,具有简单和展开误差小的特点,非常适应结构多点随机输入分析。     

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.     

D2车轮钢原始组织对滑动磨损性能的影响

采用MRH-5A型环块磨损试验机对D2车轮钢及U71Mn钢轨钢采取对摩方式进行滑动磨损试验,研究原始组织对D2车轮钢滑动磨损性能的影响. 结果表明:以回火索氏体(TS)为原始组织的D2车轮钢比片状珠光体组织(P)+先共析铁素体(F)的D2车轮钢具有更好的耐磨性能. P+F和TS表面磨损机制均以磨粒磨损和黏着磨损为主,而P+F表面磨损更严重且伴随大块白层剥落现象. TS塑性变形层更薄,其内的铁素体细化成纳米晶,粒状渗碳体不发生剪切变形,主要以溶解为主,不易形成较厚的白层,不发生大块剥落现象,提高耐磨性能.      

基于等厚干涉的轴向空间相干度分析及其在表面形状测量中的可行性探讨

从等厚子干涉的解释出发，推导出了二维空间的相干方程。基于空间相干方程，利用SLM产生了一个新的空间非相干光源。同时提出了新的相移方案，并结合这个新的相移方案，实现了轴向空间距离的探测和干涉条纹的相移。分析了空间方向角度相干方程的影响，并给出了实验验证。通过对非相干光源的参数和特定的空间距离以及空间方向角的匹配，重建了一个具有标准段差的实际物体的表现形状。为光学表面形状测量提供了新的思路。     

三维扰动波的非平行边界层稳定性研究

导出了三维扰动波的原始变量形式的抛物化稳定性方程（PSE）,研究了三维空间模态TS波的非平行边界层稳定性问题.采用了法向四阶紧致格式,以提高计算精度.通过给出不会导致奇性的坐标变换、修改外边界条件以及克服平行流初始值的瞬态影响和推进步长的限制,保证了计算的数值稳定.用补全元素带状矩阵法求解块三对角矩阵,大大提高了速度.计算结果清楚地显示了三维扰动波的演化过程和非平行性对边界层稳定性的影响,特别是,观察到非平行性对三维扰动波的影响,有时会使其稳定性出现逆转的现象.还研究了逆压梯度的作用.算例的结果与其他结果符合良好.     

Effect of geometry upon the performance of a thin film ferroelectric capacitor

The finite element method is used to investigate the performance of a ferroelectric random access memory as a function of its geometry. Performance is characterised by the charge versus electric field relation, and the sensitivity of performance to geometry is explored. The primary geometric variables are the dimensions of a prismatic two-dimensional (2D) island of ferroelectric material, and the edge inclination angle caused by the etching process along the sides of the island. The performance of the two-dimensional ferroelectric device is compared to those of an unsupported ferroelectric thin film and of a ferroelectric film bonded to a substrate.     

Three-dimensional exact analysis of piezoelectric laminated plates via a sampling surfaces method

A paper focuses on the use of the efficient approach to three-dimensional (3D) exact solutions of electroelasticity for piezoelectric laminated plates. This approach is based on the new method of sampling surfaces (SaS) developed recently by the authors. We introduce inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the plate and choose displacements of these surfaces as basic plate variables. Such an idea permits the representation of the proposed piezoelectric plate formulation in a very compact form. This fact gives the opportunity to derive the 3D exact solutions of electroelasticity for thick and thin piezoelectric laminated plates with a specified accuracy utilizing a sufficient number of SaS, which are located at interfaces and Chebyshev polynomial nodes.     

A three-dimensional,higher-order,elasticity-based micromechanics model

The three-dimensional (3D) version of a new homogenization theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] is presented. The 3D theory utilizes a higher-order, elasticity-based cell model (ECM) analysis for a periodic array of 3D unit cells. The unit cell is discretized into eight subregions or subcells. The displacement field within each subcell is approximated by a (truncated) eigenfunction expansion of up to fifth order. The governing equations are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through the given order of the subcell displacement fields. The specified governing equations are valid for any type of constitutive model used to describe the behavior of the material in a subcell. The specialization of the theory to lower orders and to two-dimeinsions (2D) and to the exact one-dimensional (1D) theory is discussed.Since the proposed 3D homogenization theory correctly reduces to both 2D and 1D the validation process applied to the 2D theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] directly applies to the current formulation. Additional comparisons of the predicted responses obtained from the 3D ECM theory with existing published results are conducted. The good agreement obtained shows that the current theory represents a viable 3D homogenization tool. The improved agreement between the current theory results and published results as compared to the comparison of the MOC results and the published results is due to the correct incorporation of the coupling effects between the local fields. Additional results showing the convergence behavior of the average fields as a function of the order of the analysis is presented. These results show that the 1st order theory may not accurately predict the local averages but that consistent and converged behavior is obtained using the higher order ECM theories.The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, three-dimensional microstructures.     

Improved ALE mesh velocities for complex flows

A key choice in the development of arbitrary Lagrangian‐Eulerian solution algorithms is how to move the computational mesh. The most common approaches are smoothing and relaxation techniques, or to compute a mesh velocity field that produces smooth mesh displacements. We present a method in which the mesh velocity is specified by the irrotational component of the fluid velocity as computed from a Helmholtz decomposition, and excess compression of mesh cells is treated through a noniterative, local spring‐force model. This approach allows distinct and separate control over rotational and translational modes. The utility of the new mesh motion algorithm is demonstrated on a number of 3D test problems, including problems that involve both shocks and significant amounts of vorticity.     

Analyses of wrinkled and slack membranes through an error reproducing mesh-free method

A mesh-free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper. A modification of the wrinkling theory, originally proposed by Roddeman et al. (1987) [Roddeman, D.G., Drukker, J., Oomens, C.W.J., Janssen, J.D., 1987, The wrinkling of thin membranes: Part I—theory; Part II—numerical analysis. ASME J. Appl. Mech. 54, 884–892.], is proposed to study the behaviour of an isotropic membrane under the mixed state of stress (taut, wrinkled and slack). Using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous (and possibly non-differentiable) tensor, the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely. While the deformation gradient need not be everywhere-differentiable in a wrinkled membrane, it is argued that the fictive non-wrinkled (non-slack) surface may be looked upon as an everywhere-taut surface in the limit as the minor (and major) principal tensile stresses over the wrinkled (slack) portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere-differentiable tensors. The weighted residual from the governing equations are presently solved via a mesh-free method, where the entire domain is discretized only by a set of grid points. A non-uniform-rational-B-spline (NURBS) based error reproducing kernel method (ERKM) has been used to approximate the field variable over the domain. The first step in the method is to approximate a function and its derivatives through NURBS basis functions. However, since NURBS functions neither reproduce any polynomial nor interpolate the grid points (also referred to as control or nodal points), the approximated functions result in uncontrolled errors over the domain including the grid points. Accordingly the error functions in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Several numerical examples on wrinkled and/or slack membranes are also provided.     

A study of conditions for dislocation nucleation in coarser-than-atomistic scale models

We perform atomistic simulations of dislocation nucleation in defect free crystals in 2 and 3 dimensions during indentation with circular (2D) or spherical (3D) indenters. The kinematic structure of the theory of Field Dislocation Mechanics (FDM) is shown to allow the identification of a local feature of the atomistic velocity field in these simulations as indicative of dislocation nucleation. It predicts the precise location of the incipient spatially distributed dislocation field, as shown for the cases of the Embedded Atom Method potential for Al and the Lennard–Jones pair potential. We demonstrate the accuracy of this analysis for two crystallographic orientations in 2D and one in 3D. Apart from the accuracy in predicting the location of dislocation nucleation, the FDM based analysis also demonstrates superior performance than existing nucleation criteria in not persisting in time beyond the nucleation event, as well as differentiating between phase boundary/shear band and dislocation nucleation. Our analysis is meant to facilitate the modeling of dislocation nucleation in coarser-than-atomistic scale models of the mechanics of materials.     

A two-dimensional,higher-order,elasticity-based micromechanics model

A new, two-dimensional (2D) homogenization theory is proposed. The theory utilizes a higher-order, elasticity-based cell model (ECM) analysis. The material microstructure is modeled as a 2D periodic array of unit cells where each unit cell is discretized into four subregions (or subcells). The analysis utilizes a (truncated) eigenfunction expansion of up to fifth order for the displacement field in each subcell. The governing equations for the theory are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through an order consistent with the order of the subcell displacement field. The formulation is carried out independently of any specified constitutive models for the behavior of the individual phases (in the sense that the general governing equations hold for any constitutive model). The fifth order theory is subsequently specialized to a third order theory. Additionally, the higher order analyzes reduce to a theory equivalent to the original 2D method of cells (MOC) theory when all higher order terms are eliminated. The proposed 2D theory is the companion theory to an equivalent 3D theory [T.O. Williams, A three-dimensional, higher-order, elasticity-based micromechanics model, Int. J. Solids Struc., in press].Comparison of the predicted bulk and local responses with published results indicates that the theory accurately predicts both types of responses. The high degree of agreement between the current theory results and published results is due to the correct incorporation of the coupling effects between the local fields.The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, two-dimensional microstructures.     

Initiation and propagation of complex 3D networks of cracks in heterogeneous quasi-brittle materials: Direct comparison between in situ testing-microCT experiments and phase field simulations

We provide the first direct comparisons, to our knowledge, of complex 3D micro cracking initiation and propagation in heterogeneous quasi-brittle materials modelled by the phase field numerical method and observed in X-ray microtomography images recorded during in situ mechanical testing. Some material parameters of the damage model, including the process zone (internal) length, are identified by an inverse approach combining experimental data and 3D simulations. A new technique is developed to study the micro cracking at a finer scale by prescribing the local displacements measured by digital volume correlation over the boundary of a small sub-volume inside the sample during the numerical simulations. The comparisons, performed on several samples of lightweight plaster and concrete, show a remarkable quantitative agreement between the 3D crack morphology obtained by the model and by the experiments, without any a priori knowledge about the location of the initiation of the cracks in the numerical model. The results indicate that the crack paths can be predicted in a fully deterministic way in spite of the highly random geometry of the microstructure and the brittle nature of its constituents.