The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.     

Vibration analysis of beams with non‐local foundations using the finite element method

In this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

基于时域响应灵敏度分析的板结构损伤识别

提出了一种基于响应灵敏度分析的有限元模型修正法,对平板结构的局部损伤进行识别。在正问题研究中,将结构的局部损伤模拟为板结构单元杨氏模量的减少,建立了板结构的有限元动力学方程,利用直接积分法获得了结构强迫振动响应。在损伤识别反问题中,基于响应灵敏度分析,直接利用结构的动态响应进行有限元模型修正和损伤识别。算例表明,本文方法能有效识别板类结构的局部损伤,具有需要测点数目少,损伤识别精度高,对模拟的测量噪声不大敏感的优点。     

橡胶隔振器高频动态特性的计算方法

该文采用一种由Mooney-Rivlin模型和多个Maxwell模型叠加组成的非线性粘弹性本构模型,用于计算橡胶隔振器的高频动态特性。该文给出了在时域和频域范围内拟合本构模型中粘弹性参数的方法,利用拟合得到的本构模型参数,对某款橡胶悬置跨点动态特性进行计算,并与实验结果进行对比。该文还建立了橡胶隔振器等效力学模型,分析了原点动刚度和跨点动刚度的区别,分析表明:使用跨点动态特性测试法可消除测试中附加惯性力的影响,适用于橡胶隔振器高频动特性的测试;同时,该文搭建了橡胶隔振器有限元模型,分别用于分析其跨点动刚度与原点动刚度,并将分析结果与实验结果进行对比,分析结果验证了有限元模型和力学模型的正确性。除此之外,该文还分析对比了时域（松弛、蠕变）和频域（简谐动态试验）拟合粘弹性参数方法的优缺点。     

基于有限元分析的复杂结构弹性振动传递函数建模

为了解决复杂结构的弹性振动建模问题,提出了一种基于有限元分析的弹性振动传递函数建模方法。首先建立能够准确反映结构动力学特性的有限元模型;然后根据输入参数和辨识算法类型的不同,分别从时域和频域两方面进行传递函数建模,其中时域辨识建模以PRBS信号的瞬态响应结果为辨识数据进行时域参数辨识,频域辨识建模以结构的频率响应函数为辨识数据进行频域参数辨识;最后以“时域建模频域验证,频域建模时域验证”的方法检验传递函数的精度。通过建立悬臂梁、某运载火箭和某弹体局部结构的传递函数模型,证明了该方法的可行性和有效性,为该方法在工程实践中的应用提供参考。     

A cohesive finite element for quasi-continua

In this paper, a cohesive finite element method (FEM) is proposed for a quasi-continuum (QC), i.e. a continuum model that utilizes the information of underlying atomistic microstructures. Most cohesive laws used in conventional cohesive FEMs are based on either empirical or idealized constitutive models that do not accurately reflect the actual lattice structures. The cohesive quasi-continuum finite element method, or cohesive QC-FEM in short, is a step forward in the sense that: (1) the cohesive relation between interface traction and displacement opening is now obtained based on atomistic potentials along the interface, rather than empirical assumptions; (2) it allows the local QC method to simulate certain inhomogeneous deformation patterns. To this end, we introduce an interface or discontinuous Cauchy–Born rule so the interfacial cohesive laws are consistent with the surface separation kinematics as well as the atomistically enriched hyperelasticity of the solid. Therefore, one can simulate inhomogeneous or discontinuous displacement fields by using a simple local QC model. A numerical example of a screw dislocation propagation has been carried out to demonstrate the validity, efficiency, and versatility of the method. An erratum to this article can be found at     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

考虑接头作用的全复合材料桁架结构多尺度分析

为提高全复合材料桁架分析的精度和效率,引入结构多尺度有限元思想,对接头进行精细化建模,通过建立两点位移约束实现不同尺度模型连接,从而将接头模型嵌入宏观桁架模型,并针对具体制备工艺赋予桁架材料属性。为验证多尺度模型的优势,同时进行全复合材料桁架实验,以及分别基于全部梁单元模型和全部实体单元模型的有限元分析。对比相关模型的计算精度与效率,结果表明多尺度模型能够较好地兼顾计算精度与效率。该文针对全复合材料桁架的结构多尺度有限元建模方法,可精确分析全复合材料桁架承载性能,并且能够提供有效的局部信息,可用于分析其他包含复杂细节构造的大尺度复合材料结构。     

Strategies for finding the design point in non-linear finite element reliability analysis

An essential step in FORM, SORM and importance sampling reliability methods is the determination of the so-called design point. This point is the solution of a constrained optimization problem in the outcome space of the random variables, which is commonly solved by an iterative, gradient-based search algorithm. In solving this problem in the context of non-linear finite element reliability analysis, two serious impediments are encountered: (a) for certain material models, the constraint function may have a discontinuous gradient, leading to failure of the search algorithm to converge. (b) The search algorithm may generate trial points too far in the failure domain, where the finite element code fails to produce a result due to lack of numerical convergence. In this paper, remedying strategies are developed for both impediments. The first impediment is addressed by using smooth or smoothed material models, including a smoothed bi-linear model, a Bouc–Wen model and a generalized plasticity model. This is complemented by a proof that sudden elastic unloading does not give rise to gradient discontinuities. The second impediment is addressed by modifying or introducing search algorithms that prevent the trial points from overshooting into the failure domain. Numerical examples are used to demonstrate the two impediments and effectiveness of the proposed remedies.     

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.     

A feed-back approach to error control in finite element methods: application to linear elasticity

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.     

Circuit theoretical approach to couple two-dimensional finiteelement models with external circuit equations

This paper presents a new method to couple two-dimensional finite element models with circuit equations. The method is based on handling of the finite element model as a circuit theoretical multiport element. This multiport element is treated in the same way as ordinary nonlinear circuit elements within the Newton-Raphson iteration of the circuit equations. The method has been utilized in a simulator program for analyzing power electronic drives in the time domain. The electrical machine is modeled by the two-dimensional finite element method. The power electronic circuit and the connections of the windings of the machine may have an arbitrary topology which is given by a net-list file (SPICE-type input file). The applicability of the method is investigated with two example cases which are verified by measurements. According to tests, the method proposed provides an effective and reliable way to construct simulators including finite element modeling     

Recovery of equilibrium on star patches using a partition of unity technique

A technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two‐dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

二维平纹机织复合材料弹性性能预测的域分解方法

为了预测二维平纹机织复合材料的弹性性能, 提出了基于有限元重合网格法的域分解方法。域分解方法与传统代表体元法的有限元建模不同, 前者不再建立精细的纤维与基体模型, 而是分别建立二维平纹机织复合材料单胞的整体域与纤维域, 整体域是真实基体体积与纤维体积的叠加, 两区域网格独立剖分, 互不影响。采用MSC. Nastran中的多节点约束在纤维节点与基体节点之间建立位移协调来模拟纤维和基体单元的位移函数关系, 实现了纤维域和基体域的耦合计算。研究表明, 域分解方法大大简化了机织复合材料细观力学建模的复杂性, 降低了建模时间, 采用域分解方法预测的二维平纹机织复合材料弹性常数与试验值吻合较好, 充分说明了该预测模型与方法的正确性。研究了不同纤维体积分数下, 域分解方法预测二维平纹机织复合材料的弹性常数的变化趋势, 结果表明, 随纤维体积分数增加, 模量呈上升趋势, 泊松比呈降低趋势。     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid multiscale finite element/peridynamics method

A hybrid method is presented that uses a representative volume element-based multiscale finite element technique combined with a peridynamics method for modeling fracture surfaces. The hybrid method dynamically switches from finite element computations to peridynamics based on a damage criterion defined on the peridynamics grid, which is coincident with the nodes of the finite element mesh. Nodal forces are either computed by the finite element method or peridynamics, as appropriate. The multiscale finite element method used here is a representative volume element-based approach so that inhomogeneous local scale material properties can be derived using homogenization. In addition, automatic cohesive zone insertion is used at the local scale to model fracture initiation. Results demonstrate that local scale flaw distributions can alter fracture patterns and initiation times, and the use of cohesive zone insertion can improve accuracy of crack paths.     

Interval boundary element method in the presence of uncertain boundary conditions,integration errors,and truncation errors

In engineering, most governing partial differential equations for physical systems are solved using finite element or finite difference methods. Applications of interval methods have been explored in finite element analysis to model systems with parametric uncertainties and to account for the impact of truncation error on the solutions. An alternative to the finite element method is the boundary element method. The boundary element method uses singular functions to reduce the dimension of the domain by transforming the domain variables to boundary variables. In this work, interval methods are developed to enhance the boundary element method for considering causes of imprecision such as uncertain boundary conditions, truncation error, and integration error. Examples are presented to illustrate the effectiveness and potential of an interval approach in the boundary element method.     

Multiscale coupling using a finite element framework at finite temperature

This paper presents the formulation and application of a multiscale methodology that couples three domains using a finite element framework. The proposed method efficiently models atomistic systems by decomposing the system into continuum, bridging, and atomistic domains. The atomistic and bridging domains are solved using a combined finite element–molecular mechanics simulation where the system is discretized into atom/nodal centric elements based on the atomic scale finite element method. Coupling between the atomistic domain and continuum domain is performed through the bridging cells, which contain locally formulated atoms whose displacements are mapped to the nodes of the bridging cell elements. The method implements a temperature‐dependent potential for finite temperature simulations. Validation and demonstration of the methodology are provided through three case studies: displacement in a one‐dimensional chain, stress around nanoscale voids, and fracture. From these studies differences between multiscale and fully atomistic simulations were very small with the simulation time of the proposed methodology being approximately a tenth of the time of the fully atomistic model. Copyright © 2012 John Wiley & Sons, Ltd.