Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Damage tolerance of fully orthotropic laminates in compression

The compression after impact (CAI) strength of fully orthotropic composite laminates with up to 21 plies is presented, as analysed by an existing strip model. Candidate layups, which can be symmetric, anti-symmetric or non-symmetric, are preselected to exhibit no elastic coupling response, with manufacturing rules applied. These criteria, along with the use of a simple surrogate sublaminate buckling model, were chosen to allow analysis of all feasible laminates in the design space without excessive computation time. Results indicate that although the inclusion of non-symmetric layups in the design space does not give benefits with respect to maximum achievable damage tolerance, these laminates can exhibit damage tolerance close to that of an anti-symmetric design for some ply counts, and better than symmetric solutions in most cases. It is also noted that in some instances increasing the number of plies in a laminate can actually reduce the highest achievable threshold load for damage tolerance, as a result of the large influence Poisson’s ratio has on sublaminate buckling. Average errors in the surrogate model were low in all cases, with maximum non-conservative errors less than 1%. The surrogate buckling model reduced computational time by over 99% when compared to the fully exhaustive search.     

A novel 3D mixed finite‐element model for statics of angle‐ply laminates

Analysis of angle‐ply laminates becomes critical and computationally involved because of the presence of extension–shear coupling. A refined three‐dimensional, mixed, 18‐node finite element (FE) model has been developed to analyse angle‐ply laminates under static loading. The minimum potential energy principle has been used for the development of the mixed FE model, where the transverse stress components (τ_xz, τ_yz and σ_z, where z is the thickness direction) have been incorporated as the nodal degrees of freedom, in addition to the three displacement fields. Further, continuity of transverse stress fields through the thickness of the plate and layerwise continuity of displacement fields have been enforced in the formulation. Because all the constitutive and the compatibility conditions have been ensured within the continuum, the present formulation is unique amongst the family of mixed FE models. Results have been obtained for various angle‐ply laminates and compared with analytical and finite‐element solutions, which have been found to be in good agreement with them. Some new results on angle‐ply with clamped–clamped support condition have also been presented to serve as benchmark results. Copyright © 2003 John Wiley & Sons, Ltd.     

三维复合材料层合板渐进损伤非线性分析模型

为有效反映复合材料层合板层间相互作用和材料损伤非线性,建立了中等尺度的三维复合材料层合板渐进损伤分析模型。非线性渐进损伤分析过程包括应力求解、材料损伤失效判据及材料性能退化方案3个方面。讨论了损伤材料性能退化方案,引入与材料损伤模式相对应的损伤变量表征材料点的损伤状态,材料的刚度矩阵按损伤变量退化。基于该模型可成功预测复合材料层合板损伤起始、扩展直至最终失效的整个过程和极限强度。经文献试验数据验证,12种不同铺层顺序层合板的计算强度与试验数据均吻合较好,表明该模型在复合材料层合板极限强度预测上的有效性。     

Analysis of fracture and delamination in laminates using 3D numerical modelling

The static failure behaviour of the fibre-metal laminate GLARE is examined using 3D finite element simulations. The configuration analysed is a centre-cracked tensile specimen composed of two aluminium layers sandwiching a cross-plied, fibre-epoxy layer. The crack and delamination growths are simulated by means of interface elements equipped with a mixed-mode damage model. The mode-mixity is derived from an energy criterion typically used in linear elastic fracture mechanics studies. The damage kinetic law is rate-dependent, in order to simulate rate effects during interfacial delamination and to avoid numerical convergence problems due to crack bifurcations. The numerical implementation of the interface damage model is based on a backward Euler approach. In the boundary value problem studied, the failure responses of GLARE specimens containing elastic aluminium layers and elasto-plastic aluminium layers are compared. The development of plastic deformations in the aluminium layers stabilizes the effective failure response, and increases the residual strength of the laminate. For a ‘quasi-brittle’ GLARE specimen with elastic aluminium layers, the residual strength is governed by the toughness for interfacial delamination, and is in close correspondence with the residual strength obtained from a closed-form expression derived from energy considerations. Conversely, for a ‘ductile’ GLARE specimen with elasto-plastic aluminium layers, the residual strength is also determined by the relation between the fracture strength and the yield strength of the aluminium. The amount of constraint by the horizontal displacements at the vertical specimen edges has a moderate to small influence on the residual strength. Furthermore, the ultimate laminate strength is lower for a larger initial crack length, and shows to be in good correspondence with experimental values.     

On mixed variational formulations in finite elasticity

Summary In extension of a recent variational formulation of the problem of infinitesimal elasticity, with arbitrary variations of displacements andsome stresses [6], we use a recent consideration of variational formulations of finite elasticity, with arbitrary variations of translational and rotational displacements and of all stresses, or with arbitrary variations of these displacements and of three reactive stress measures [5], for the purpose of obtaining generalizations of the results in [6] which are valid for the equations of finite elasticity in terms of what we have called [5] distinguished generalized Piola stress components.Based upon work supported by National Science Foundation Grant No. CEE-8213256.     

Different finite element formulations of 3D magnetostatic fields

Several finite-element formulations of three-dimensional magnetostatic fields are reviewed. Both nodal and edge elements are considered. The aim is to suggest remedies to some shortcomings of widely used methods. Various formulations are compared based on results for Problem No. 13 of the TEAM Workshops, a nonlinear magnetostatic problem involving thin iron plates     

Fracture simulation of CFRP laminates in mixed mode bending

This paper analyses the progressive mixed mode delamination failure in unidirectional and multidirectional composite laminates using fracture experiments, finite element (FE) simulations and an analytical solution. The numerical model of the laminate is described as an assembly of damageable layers and bilinear interface elements subjected to mixed mode bending. The analytical approach is used to estimate the total mixed mode and decomposed fracture energies for laminates with different stacking sequences, which is also validated through experiments. It is concluded that the interlaminar fracture toughness of multidirectional laminates is considerably higher than that of the unidirectional ones. The effect of initial interfacial stiffness and element size is studied and it is also shown that their value must not exceed a definite limit for the numerical simulations to converge. The model can also be further extended to simulate the mixed mode fracture in hybrid fiber metal laminates.     

考虑三维应力的复合材料层压板疲劳寿命分析

由于层间应力的存在,受面内载荷作用的复合材料层压板实际处于多轴应力状态。构建了由刚性元、弹簧元和二维板元构成的准三维有限元模型,结合单向板在典型应力状态下的疲劳试验结果和疲劳损伤模型,发展了一种考虑三维应力的、预测任意铺层多向层压板疲劳寿命的分析方法,包括应力分析、静力和疲劳累积损伤失效分析及材料性能退化3个主要部分,能够模拟面内和层间损伤产生、发展直至层压板整体破坏的完整过程,并得到疲劳寿命。对2种T300/QY8911多向铺层板进行了实际计算,寿命预测结果与试验结果吻合较好。      

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 John Wiley & Sons, Ltd.     

Investigation of 3D crack propagation problems via fast BEM formulations

Abstact The simulation of 3D fatigue crack growth is investigated and in combination with suitable experiments it is aimed to identify a reliable crack growth criterion. To perform the crack growth simulation as effectively as possible the boundary element method (BEM) in terms of the 3D Dual BEM is utilized. The relevant boundary integral equations are evaluated in the framework of a collocation procedure. To compress the system matrix and to speed-up the solution procedure the adaptive cross approximation (ACA) method is applied. It is an algebraic technique acting purely on the system matrix itself. The efficiency of this procedure with respect to crack problems will be shown on both a standard fracture specimen and an industrial application.     

High gradient magnetic particle separation in viscous flows by 3D BEM

The boundary element method was applied to study the motion of magnetic particles in fluid flow under the action of external nonuniform magnetic field. The derived formulation combines the velocity-vorticity resolved Navier–Stokes equations with the Lagrange based particle tracking model, where the one-way coupling with fluid phase was considered. The derived algorithm was used to test a possible design of high gradient magnetic separation in a narrow channel by computing particles trajectories in channel flow under the influence of hydrodynamic and magnetic forces. Magnetic field gradient was obtained by magnetization wires placed outside of the channel. Simulations with varying external magnetic field and flow rate were preformed in order to asses the collection efficiency of the proposed device. We found that the collection efficiency decreases linearly with increasing flow rate. Also, the collection efficiency was found to increase with magnetic field strength only up a saturation point. Furthermore, we found that high collection efficiently is not feasible at high flow velocity and/or at weak magnetic field. Recommendation for optimal choice of external magnetic field and flow rate is discussed.     

2D/3D growth of GaN by molecular beam epitaxy: towards GaN quantum dots

The first stages of the growth of strained GaN on AlN were studied using reflection high energy electron diffraction, atomic force microscopy and high resolution electron microscopy. It was shown that GaN grows in the Stranski–Krastanov mode, with three-dimensional islanding occuring after deposition of two monolayers. This 2D/3D transition was found to depend on the growth temperature. At low growth temperature, coalescence of 3D islands rapidly leads to a smooth surface. At high temperature, no smoothing process is observed. It is shown that the size of the 3D islands is controlable and that it is small enough to expect quantum effects.     

A shell/3D modeling technique for the analysis of delaminated composite laminates

A shell/3D modeling technique was developed for which a local three-dimensional solid finite element model is used only in the immediate vicinity of the delamination front. The goal was to combine the accuracy of the full three-dimensional solution with the computational efficiency of a plate or shell finite element model. Multi-point constraints provided a kinematically compatible interface between the local three-dimensional model and the global structural model which has been meshed with plate or shell finite elements. Double cantilever beam (DCB), end notched flexure (ENF), and single leg bending (SLB) specimens were modeled using the shell/3D technique to study the feasibility for pure mode I (DCB), mode II (ENF) and mixed mode I/II (SLB) cases. Mixed mode strain energy release rate distributions were computed across the width of the specimens using the virtual crack closure technique. Specimens with a unidirectional layup and with a multidirectional layup where the delamination is located between two non-zero degree plies were simulated. For a local three-dimensional model, extending to a minimum of about three specimen thicknesses on either side of the delamination front, the results were in good agreement with mixed mode strain energy release rates obtained from computations where the entire specimen had been modeled with solid elements. For large built-up composite structures modeled with plate elements, the shell/3D modeling technique offers a great potential for reducing the model size, since only a relatively small section in the vicinity of the delamination front needs to be modeled with solid elements.     

缝合复合材料层板三维纤维弯曲模型及压缩强度预报

提出了三维纤维弯曲模型, 基于有限元法和周期性边界条件建立了缝合层板压缩强度分析方法, 采用桥联模型和最大应力判据分析损伤扩展并获得压缩强度, 预报结果与试验吻合较好。详细探讨了缝合参数对层合板压缩强度的影响规律, 结果表明: 缝合方向与表面纤维方向一致时, 较小的缝合针距和行距、较大的缝合线半径对压缩强度较为有利; 缝合方向与表面纤维方向垂直时, 较小的缝合针距和缝合线半径、较大的缝合行距对压缩强度较为有利。      

Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

Procedures based on group representation theory, allowing the exploitation of geometrical symmetry in symmetric Galerkin BEM formulations, are investigated. In particular, this investigation is based on the weaker assumption of partial geometrical symmetry, where the boundary has two disconnected components, one of which is symmetric; e.g. this can be very useful for defect identification problems. The main development is expounded in the context of 3D Neumann elastostatic problems, considered as model problems; and then extended to SGBIE formulations for Dirichlet and/or scalar problems. Both Abelian and non‐Abelian finite symmetry groups are considered. The effectiveness of the present approach is demonstrated through numerical examples, where both partial and complete symmetry are considered, in connection with both Abelian and non‐Abelian symmetry groups. Copyright © 2003 John Wiley & Sons, Ltd.     

Development of a shock-fitting field-panel method for 3D transonic flows

The paper presents the development of a shock-fitting field-panel method for three-dimension (3D) transonic flows. In this method, the full-potential equation, written in the form of the Poisson's equation, is solved by integral equation field-panel method. The solution consists of a wing surface source panel integral term, a field-volume panel integral term of compressibility over a small limited domain, and a shock panel integral term. Due to the non-linearity of flows, solutions are obtained through an iterative procedure. Instead of using a field-panel refinement procedure, a shock-fitting technique is used to fit the shock. Finally, numerical examples are provided to demonstrate the accuracy of the method.This work is supported by NASA-Langley Research Center under the Grant No. NAG-1-1170. Dr. E. Yates, Jr. and Mr. W. Silva were the technical monitors