The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture (2) dislocations (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

SUMMARYThe introduction of discontinuous/non-differentiable functions in the eXtended Finite-Element Method allows to model discontinuities independent of the mesh structure. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity line is commonly adopted.In the paper, it is shown how standard Gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation. The technique is illustrated and developed in one, two and three dimensions for crack and material discontinuity problems.

SUMMARYA new vector level set method for modelling propagating cracks in the element-free Galerkin (EFG) method is presented. With this approach only nodal data are used to describe the crack; no geometrical entity is introduced for the crack trajectory, and no partial di erential equations need to be solved to update the level sets. The nodal description is updated as the crack propagates by geometric equations. The advantages of this approach, here introduced and analysed for the two-dimensional case, are particularly promising in three-dimensional applications, where the geometrical description and evolution of an arbitrary crack surface in a complex solid is very awkward. In addition, new methods for crack approximations in EFG are introduced, using a jump function accounting for the displacement discontinuity along the crack faces and the Westergard's solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack and are naturally coupled to the level set crack representation.

SUMMARYTwo issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self-equilibrating and blending of the enrichment. The integration is based on transforming the domain integrals in the weak form into equivalent contour integrals. It is shown that the contour form is computationally more efficient than the domain form, especially when the enrichment function is singular and/or discontinuous. A method for alleviating the errors in the blending elements is also studied. In this method, the enrichment function is pre-multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains. A method for blending step function enrichment with singular enrichments is described. It is also shown that if the enrichment is not shifted properly, the weighted enrichment is equivalent to the standard enrichment. An edge dislocation and a crack problem are used to benchmark the technique; the influence of the variables that parameterize the weight function is analyzed. The resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing XFEM codes.

For survivors of severe COVID-19 disease, having defeated the virus is just the beginning of an uncharted recovery path. What follows after the acute phase of SARS-CoV-2 infection depends on the extension and severity of viral attacks in different cell types and organs. Despite the ridiculously large number of papers that have flooded scientific journals and preprinthosting websites, a clear clinical picture of COVID-19 aftermath is vague at best. Without larger prospective observational studies that are only now being started, clinicians can retrieve information just from case reports and or small studies. This is the time to understand how COVID-19 goes forward and what consequences survivors may expect to experience. To this aim, a multidisciplinary post-acute care service involving several specialists has been established at the Fondazione Policlinico Universitario A. Gemelli IRCSS (Rome, Italy). Although COVID-19 is an infectious disease primarily affecting the lung, its multi-organ involvement requires an interdisciplinary approach encompassing virtually all branches of internal medicine and geriatrics. In particular, during the post-acute phase, the geriatrician may serve as the case manager of a multidisciplinary team. The aim of this article is to describe the importance of the interdisciplinary approach-coordinated by geriatrician-to cope the potential post-acute care needs of recovered COVID-19 patients.

SUMMARYA new level set method is developed for describing surfaces that are frozen behind a moving front, such as cracks. In this formulation, the level set is described in two dimensions by a three-tuple: the sign of the level set function and the components of the closest point projection to the surface. The update of the level set is constructed by geometric formulas, which are easily implemented. Results are given for growth of lines in two dimensions that show the method is very accurate. The method combines very naturally with the extended ÿnite element method (XFEM) where the discontinuous enrichment for cracks is best described in terms of level set functions. Examples of crack growth simulations obtained by combining this level set method with the extended ÿnite element method are given.

SUMMARYA new technique for the modelling of multiple dislocations based on introducing interior discontinuities is presented. In contrast to existing methods, the superposition of infinite domain solutions is avoided; interior discontinuities are specified on the dislocation slip surfaces and the resulting boundary value problem is solved by a finite element method. The accuracy of the proposed method is verified and its efficiency for multi-dislocation problems is illustrated. Bounded core energies are incorporated into the method through regularization of the discontinuities at their edges. Though the method is applied to edge dislocations here, its extension to other types of dislocations is straightforward.

SUMMARYThis work focuses on the modelling through the extended finite element method of structural problems characterized by discontinuous displacement. As a model problem, an elastic isotropic domain characterized by a displacement discontinuity across a surface is studied. A regularization of the displacement field is introduced depending on a scalar parameter. The regularized solution is defined in a layer. The emerging strain and stress fields are independently modelled using specific constitutive assumptions. In particular, it is shown that the mechanical work spent within the regularization layer can be interpreted as an interface work provided that a spring-like constitutive law is adopted. The accuracy of the integration procedures adopted for the stiffness matrix is assessed, as highly non-linear terms appear. Standard Gauss quadrature is compared with adaptive quadrature and with a new technique, based on an equivalent polynomial approach. One-and two-dimensional results are reported for varying discretization size, regularization parameter, and constitutive parameters.