An hp‐adaptive finite element method for electromagnetics. Part 3: A three‐dimensional infinite element for Maxwell's equations

This paper is a continuation of Reference [26] (Cecot, Demkowicz and Rachowicz, Computer Methods in Applied Mechanics and Engineering 2000; 188 : 625–643) and describes an implementation of the infinite element for three‐dimensional, time harmonic Maxwell's equations, proposed in Reference [15] (Demkowicz and Pal, Computer Methods in Applied Mechanics and Engineering 1998; 164 : 77–94). The element is compatible with the hp finite element discretizations for Maxwell's equations in bounded domains reported in References [16–18] (Computer Methods in Applied Mechanics and Engineering 1998; 152 : 103–124, 1999; 169 : 331–344, 2000; 187 : 307–337). Copyright © 2003 John Wiley & Sons, Ltd.     

On the approximation in the smoothed finite element method (SFEM)

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295. DOI: 10.1002/nme.2460 ) and answered by (Int. J. Numer. Meth. Engng 2009; DOI: 10.1002/nme.2587 ) by proposing a systematic approximation scheme based on non‐mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ; Commun. Numer. Meth. Engng 2009; 25 (1):19–34. DOI: 10.1002/cnm.1098 ; Int. J. Numer. Meth. Engng 2007; 71 (8):902–930; Comput. Meth. Appl. Mech. Engng 2008; 198 (2):165–177. DOI: 10.1016/j.cma.2008.05.029 ; Comput. Meth. Appl. Mech. Engng 2007; submitted; Int. J. Numer. Meth. Engng 2008; 74 (2):175–208. DOI: 10.1002/nme.2146 ; Comput. Meth. Appl. Mech. Engng 2008; 197 (13–16):1184–1203. DOI: 10.1016/j.cma.2007.10.008 ) and resolve the existence, linearity and positivity deficiencies pointed out in (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295). We show that Wachspress interpolants (A Rational Basis for Function Approximation. Academic Press, Inc.: New York, 1975) computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results that are almost identical to those of the SFEM initially proposed in (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ). These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs. Copyright © 2009 John Wiley & Sons, Ltd.     

The boundary contour method for piezoelectric media with linear boundary elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158 : 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley & Sons, Ltd.     

A partition of unity‐based multiscale approach for modelling fracture in piezoelectric ceramics

The development of models for a priori assessment of the reliability of micro electromechanical systems is of crucial importance for the further development of such devices. In this contribution a partition of unity‐based cohesive zone finite element model is employed to mimic crack nucleation and propagation in a piezoelectric continuum. A multiscale framework to appropriately represent the influence of the microstructure on the response of a miniaturized component is proposed. It is illustrated that using the proposed multiscale method a representative volume element exists. Numerical simulations are performed to demonstrate the constitutive homogenization framework. Copyright © 2009 John Wiley & Sons, Ltd.     

The determination of the mode II adhesive fracture resistance, GIIC, of structural adhesive joints: an effective crack length approach

This paper reports results from the mode II testing of adhesively-bonded carbon-fibre-reinforced composite substrates using the end-loaded split (ELS) method. Two toughened, structural epoxy adhesives were employed (a general purpose grade epoxy-paste adhesive, and an aerospace grade epoxy-film adhesive). Linear Elastic Fracture Mechanics was employed to determine values of the mode II adhesive fracture energy, G_IIC for the joints via various forms of corrected beam theory. The concept of an effective crack length is invoked and this is then used to calculate values of G_IIC. The corrected beam theory analyses worked consistently for the joints bonded with the epoxy-paste adhesive, but discrepancies were encountered when analysing the results of joints bonded with the epoxy-film adhesive. During these experiments, a microcracked region ahead of the main crack was observed, which led to difficulties in defining the true crack length. The effective crack length approach provides an insight into the likely errors encountered when attempting to measure mode II crack growth experimentally.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

Computational multiscale modelling of heterogeneous material layers

A computational homogenization procedure for a material layer that possesses an underlying heterogeneous microstructure is introduced within the framework of finite deformations. The macroscopic material properties of the material layer are obtained from multiscale considerations. At the macro level, the layer is resolved as a cohesive interface situated within a continuum, and its underlying microstructure along the interface is treated as a continuous representative volume element of given height. The scales are linked via homogenization with customized hybrid boundary conditions on this representative volume element, which account for the deformation modes along the interface. A nested numerical solution scheme is adopted to link the macro and micro scales. Numerical examples successfully display the capability of the proposed approach to solve macroscopic boundary value problems with an evaluation of the constitutive properties of the material layer based on its micro-constitution.     

A multiscale parametric study of mode I fracture in metal-to-metal low-toughness adhesive joints

The failure of adhesively-bonded joints, consisting of metallic adherends and epoxy-based structural adhesive with a relatively low toughness ~200 J/m², has been studied. The failure was via quasi-static mode I, steady-state crack propagation and has been modelled numerically. The model implements a ‘top-down approach’ to fracture using a dedicated steady-state, finite-element formulation. The damage mechanisms responsible for fracture are condensed onto a row of cohesive zone elements with zero thickness, and the responses of the bulk adhesive and of the adherends are represented by continuum elements spanning the full geometry of the joint. The material parameters employed in the model are first quantitatively identified for the particular epoxy adhesive of interest, and their validity is verified by comparison with experimental results. The model is then used to conduct a detailed study on the effects of (a) large variations in the geometrical configuration of the different types of specimens and (b) the adherend stiffness on the predicted value of the adhesive fracture energy, G _a . These numerical modelling results reveal that the adhesive fracture energy is a strong nonlinear function of the thickness of the adhesive layer, the other variables being of secondary importance in influencing the value of G _a providing the adhesive does not contribute significantly to the bending stiffness of the joint. These results which fully agree with experimental observations are explained in detail by identifying, and quantifying, the different sources of energy dissipation in the bulk adhesive contributing to the value of G _a . These sources are the locked-in elastic energy, crack tip plasticity, reverse plastic loading and plastic shear deformation at the adhesive/adherend interface. Further, the magnitudes of these sources of energy dissipation are correlated to the degree of constraint at the crack tip, which is quantified by considering the opening angle of the cohesive zone at the crack tip.     

The toughening of cyanate-ester polymers Part I Physical modification using particles, fibres and woven-mats

A fracture mechanics approach has been used to investigate the effects of the addition of physical modifiers on the fracture energy, G _c, of brittle cyanate-ester polymers. Tests were performed using adhesive joint specimens at –55, 21 and 150°C, with all the specimens exhibiting cohesive failure in the cyanate-ester adhesive layer. The fracture energies of systems modified using a range of inorganic and thermoplastic particles, fibres and woven-mats have been measured, and scanning electron microscopy has been used to determine the toughening micromechanisms involved. Firstly, it is shown that the addition of 10% by weight of particulate modifiers can increase the fracture energy of the cyanate-ester polymer by 100%, due to a combination of toughening micromechanisms such as crack deflection, pinning and matrix cavitation around the second-phase particles. These experimental data have been compared to predictions from an analytical model. Secondly, it is demonstrated that the use of long fibres or woven-mats can give an a major increase in the value of the fracture energy, G _c, at initiation, and a further increase with increasing crack length, i.e. a significant R-curve effect is observed. At relatively long crack lengths, the measured fracture energy may be six times greater than that of the unmodified polymer value, due to fibres debonding and bridging across the fracture surfaces. Finally, it is shown that several of the physically-modified polymers developed in the present work have fracture energies that are greater than a typical commercially-available cyanate-ester film adhesive.     

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd.     

A multiscale framework for localizing microstructures towards the onset of macroscopic discontinuity

This paper presents a multiscale computational homogenization model for the post localization behavior of a macroscale domain crossed by a cohesive discontinuity emanating from microstructural damage. The stress–strain and the cohesive macroscopic responses are obtained incorporating the underlying microstructure, in which the damage evolution results in the formation of a strain localization band. The macro structural kinematics entails a discontinuous displacement field and a non-uniform deformation field across the discontinuity. Novel scale transitions are formulated to provide a consistent coupling to the continuous microscale kinematics. From the solution of the micromechanical boundary value problem, the macroscale stress responses at both sides of the discontinuity are recovered, providing automatically the cohesive tractions at the interface. The effective displacement jump and deformation field discontinuity are derived from the same microscale analysis. This contribution focusses on scale transition relations and on the solution procedure at the microlevel; the highlights of the approach are demonstrated on microscale numerical examples. Coupled two-scale solution strategy will be presented in a subsequent paper.     

Numerical differentiation for non‐trivial consistent tangent matrices: an application to the MRS‐Lade model

In a companion paper Pérez‐Foguet, A., Rodríguez‐Ferran, A. and Huerta, A. ‘Numerical differentiation for local and global tangent operators in computational plasticity’. Computer Methods in Applied Mechanics and Engineering, 2000, in press, the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS‐Lade model. This plastic model has a cone‐cap yield surface and exhibits strong coupling between the flow vector and the hardening moduli. Because of this, differentiating these quantities with respect to stresses and internal variables—the crucial step in obtaining consistent tangent matrices—is rather involved. Numerical differentiation is used here to approximate these derivatives. The approximated derivatives are then used to (1) compute consistent tangent matrices (global problem) and (2) integrate the constitutive equation at each Gauss point (local problem) with the Newton–Raphson method. The choice of the stepsize (i.e. the perturbation in the approximation schemes), based on the concept of relative stepsize, poses no difficulties. In contrast to previous approaches for the MRS‐Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational efficiency (CPU time) and robustness of the proposed approach is illustrated by means of several numerical examples, where the major relevant topics are discussed in detail. Copyright © 2000 John Wiley & Sons, Ltd.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

Hysteresis heating based induction bonding of thermoplastic composites

The bonding of polymer matrix composites using magnetic particulate susceptor materials for hysteresis induction heating is investigated in this study. Hysteresis heating is tailored through careful design of the microstructure of magnetic particulate polymer films. The bond strength of hysteresis-welded materials is comparable to that of autoclave-welded materials while offering an order of magnitude reduction in cycle time. The relative contribution of the intimate contact and healing mechanisms to fusion bonding process indicates that it is intimate contact controlled. The macroscopic failure modes of hysteresis bonded specimens include adhesive composite/film, cohesive film and cohesive composite. Inspection of the microscopic failure at the nickel particle/polymer interface in the film indicates quasi-brittle failure mode. The XPS peaks confirm nickel oxide in the form of NiO on the failure surface indicating adhesive failure at the particle/polymer interface. The area fraction of adhesive failure is found to increase with decreasing particle size and increasing particle volume fraction.     

Non-uniform breaking of molecular bonds,peripheral morphology and releasable adhesion by elastic anisotropy in bio-adhesive contacts

Biological adhesive contacts are usually of hierarchical structures, such as the clustering of hundreds of sub-micrometre spatulae on keratinous hairs of gecko feet, or the clustering of molecular bonds into focal contacts in cell adhesion. When separating these interfaces, releasable adhesion can be accomplished by asymmetric alignment of the lowest scale discrete bonds (such as the inclined spatula that leads to different peeling force when loading in different directions) or by elastic anisotropy. However, only two-dimensional contact has been analysed for the latter method (Chen & Gao 2007 J. Mech. Phys. Solids 55, 1001–1015 (doi:10.1016/j.jmps.2006.10.008)). Important questions such as the three-dimensional contact morphology, the maximum to minimum pull-off force ratio and the tunability of releasable adhesion cannot be answered. In this work, we developed a three-dimensional cohesive interface model with fictitious viscosity that is capable of simulating the de-adhesion instability and the peripheral morphology before and after the onset of instability. The two-dimensional prediction is found to significantly overestimate the maximum to minimum pull-off force ratio. Based on an interface fracture mechanics analysis, we conclude that (i) the maximum and minimum pull-off forces correspond to the largest and smallest contact stiffness, i.e. ‘stiff-adhere and compliant-release’, (ii) the fracture toughness is sensitive to the crack morphology and the initial contact shape can be designed to attain a significantly higher maximum-to-minimum pull-off force ratio than a circular contact, and (iii) since the adhesion is accomplished by clustering of discrete bonds or called bridged crack in terms of fracture mechanics terminology, the above conclusions can only be achieved when the bridging zone is significantly smaller than the contact size. This adhesion-fracture analogy study leads to mechanistic predictions that can be readily used to design biomimetics and releasable adhesives.     

Multiscale design of three-dimensional nonlinear composites using an interface-enriched generalized finite element method

A computational framework is developed to model and optimize the nonlinear multiscale response of three-dimensional particulate composites using an interface-enriched generalized finite element method. The material nonlinearities are associated with interfacial debonding of inclusions from a surrounding matrix which is modeled using C⁻¹ continuous enrichment functions and a cohesive failure model. Analytic material and shape sensitivities of the homogenized constitutive response are derived and used to drive a nonlinear inverse homogenization problem using gradient-based optimization methods. Spherical and ellipsoidal particulate microstructures are designed to match a component of the homogenized stress-strain response to a desired constructed macroscopic stress-strain behavior.