Extended finite element methods

Significant improvements in modeling have been achieved with the use of extended finite element methods (x-FEM) coupled with more accurate constitutive properties and continuum damage mechanics [15,16]. In this and similar approaches, improved failure criteria such as the LaRC 03 are combined with accurate stress—strain curves... 

The extended finite element method (XFEM) for treating fracture in composite materials is proposed by Huynh and Belytschko [191]. This methods work with meshes that are independent of matrix/inclusion interfaces and the discontinuities and neartip enrichments were modeled. In order to describe the geometry of the interfaces and cracks in this method, level sets were employed, so that there is no need for explicit representation of either the cracks or the material interfaces. The other researchers such as Du et al. [192] and Ying et al. [193] used XFEM to model material interfaces in particulate composites with more in detail. 

Loh WK, Crocombe AD et al (2003) J Adhes 79 1135 MacNeal RH (1994) Finite elements their design and performance. Marcel Dekker, New York Mohammadi S (2008) Extended finite element method for fracture analysis of structures. Blackwell, Oxford Mubashar A, Ashcroft lA et al (2009) J Adhes 85 711 Rybicki EE, Kanninen MF (1977) Eng Eract Mech 9 931... 

The finite-element method (FEM) is based on shape functions which are defined in each grid cell. The imknown fimction O is locally expanded in a basis of shape fimctions, which are usually polynomials. The expansion coefficients are determined by a Ritz-Galerkin variational principle [80], which means that the solution corresponds to the minimization of a functional form depending on the degrees of freedom of the system. Hence the FEM has certain optimality properties, but is not necessarily a conservative method. The FEM is ideally suited for complex grid geometries, and the approximation order can easily be increased, for example by extending the set of shape fimctions. 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

For intermediate viscosities ratios, the resolution of the Navier—Stokes equations is more difficult because of the coupled flows inside and outside the fluid sphere. In this case there are only few works. Abdel-Alim and Hamielec [19] used a finite-difference method to calculate the steady motion for Re < 50 and viscosity ratio k< 1.4. This work was extended to higher Reynolds number (up to 200) by Rivkind and Ryskin [20] and Rivkind et al. [21]. Oliver and Chung [22] used a different method (series truncation method with a cubic finite element method) for moderate Reynolds numbers Re < 50. Feng and Michaelides [7], Saboni and Alexandrova [23], Saboni et al. [15] used a finite-difference method to calculate the flow field inside and outside the fluid sphere. The results provide information on the two-flow field and values for drag coefficients of viscous sphere over the entire range of the viscosity ratio. 

A few solutions exist for 3-D PZT bodies. Most well-known solutions for finite PZT plates were obtained from approximated two-dimensional (2-D) equations of extended Mindlin s solutions (Herrmann 1974). But, these solutions are not directly applicable to the analysis of AE sensors commercially available. In order to clarify the frequency response of AE sensor (function W(f) in eq. 3.5) and to optimize the design of PZT elements, resonance characteristics of PZT element were analyzed by using the finite element method (FEM) (Ohtsu Ono 1983). 

Equations [18], [54], and [55] constitute a system of three equations with three unknowns, and this system is solved numerically on one-, two-or three-dimensional domains. For the sake of simplicity, we will discuss the one-dimensional case (the equations are easily extended to three-dimensional). Although finite element methods have been used extensively for the solution of Eqs. [18], and [55] in solid state electronics, flux-based approaches for the simulation of ion channels rely primarily on finite difference schemes. 

Fig. 4.8. The dependence of load transfer length near the free edge of a film with a remote biaxial mismatch stress (Tm on the modulus ratio of the two materials. The load transfer length is defined as the distance from the free edge of the film at which the internal force resultant has increased to a value equal to 90% of its remote asymptotic value of (Tm/tf, and this length was determined by means of the numerical finite element method. Results are shown for the case when the substrate extends indefinitely far beyond the free film edge and when the substrate edge coincides with the film edge.

Adams et al. [6] detailed axisymmetric analysis of a butt-tension joint and considered the influence of the detailed geometry at the edges of the joint on the stress distribution. The single lap joint was further analysed by Crocombe and Adams [7] using the finite element method and the effect of the spew fillet was included which was seen to significantly redistribute, and decrease, the stresses at the ends of the adhesive layer. In complementary work, Crocombe and Adams [8,9] analysed the mechanics of behaviour of the peel test and included the effects of non-linear deformations and also plasticity in their work. Harris and Adams [10] extended this work and accounted for the non-linear behaviour of the single lap joint. Crocombe et al. [11] quantified the influence of this non-linearity, both material and geometric. 

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