PSI - Issue 23

Michal Jedlička et al. / Procedia Structural Integrity 23 (2019) 445– 450 Michal Jelička, Václav Rek, Ji ří Vala / Structural Integrity Procedia 00 (2019) 000 – 000

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gust 2018. The application of the standard finite element method (FEM) to structural defects Jirásek (2002) is a geometrically complicated and computationally expensive task, which stimulates the development of various modi fied or alternative approaches, based on the partition of unity method (PUM) Babuška & Melenk (199 7), namely the extended (or similar generalized) extended finite element method (XFEM) Jirásek & Belytschko (2002), Khoei (2015). In the original PUM the standard space of trial functions, representing a primary field of variables, is en riched with knowledge of the analytical solution for a particular type of discontinuity. Unlike such PUM, the XFEM works only at the level at the level of the affected elements, not on the entire physical domain of interest. This paper refers to the effective implementation of XFEM into the existing robust FEM solver NE- XX Kolář & Němec (1986) from the FEM consulting company (see http://www.fem.cz/?lang=en), in the scope of the well known structural mechanics analysis software RFEM from the Dlubal Software (see https://www.dlubal.com/en US). To handle the above mentioned discontinuities, a single type of enrichment, represented by certain signed func tion in an extrinsic formulation, will be implemented here. Assessment of enriched trial functions in primary fields of variables leads to the increasing number of variables in nodal equations, thus such functions have to be integrated properly and effectively during their assembly process. The direct utilization of standard integration formulae like the Gaussian quadrature is impossible because of the presence of discontinuities. This can be avoided using an available mesh generator to form the tetrahedral mesh within an element, considering the position and shape of the respective type of discontinuity. Such internal mesh does not force substantial increase of number of degrees of freedom, working with selected values of particular element volumes for the numerical quantification of certain energy functional (where integrands represent density of some energy). It can also provide possible positions of integration nodes for quantification of stresses for both linear and nonlinear analysis. However, the application of such approach to XFEM in the environment of spatial finite elements used and general crack shapes, represented by some surfaces in the 3-dimensional Euclidean space, is not trivial and needs still new ideas for the development of both numerical and computational formulae and implementation procedures. The benefit of XFEM is the increase of precision of numerical approximations of engineering problems with cracks for a wide class of utilizes functions. Another advantage of this method is the almost independence of a mesh on discontinuities. From the introduction of this technique Belytschko & Black (1999), expanding conventional FEM, new types of algorithms Melenk & Babuška (1996), Babuška & Melenk (1998), allow special enrichments of approximation functions. Two basic variants of XFEM can be distinguished: the singularity-based approach and the phantom node method. Let us consider the displacement field u ( x ), related to certain initial configuration, in its rather general form, which can be seen as a slight modification of Huynh et al. (2019), u ( x ) = N i ( x ) u i + N j ( x ) H j ( f ( x )) a j + N k ( x ) F l ( x ) b kl (1) where i  I , j  J , k  K , l  L are the Einstein summation indices from the set of all nodes of the mesh I , the subset of nodes enriched by some Heaviside function J , the subset of nodes enriched by some branch function K and the set of all branch functions L , accounting for the crack tip singularities, N i ( x ) are the standard shape functions, N j ( x ) and N k ( x ) those corresponding to particular enrichments, H j (.) are the Heaviside functions with values 1 or – 1, corre sponding to the location of sampling points of particular crack sides, working with the crack surface function f ( x ), F l ( x ) are the branch functions extracted from asymptotic fields and, finally, u i , a j and b kl contain the a priori un known real parameters. Frequently in dynamical or even quasi-static calculations such parameters are considered as dependent on the time t , unlike all remaining functions, to generate a time-dependent u ( x , t ). Clearly (1) has to be assembled over all finite elements including some nodes i  I , j  J , k  K . All parameters u i , a j and b kl then create a global vector of unknowns d , whose evaluation will come from the weak form of the Cauchy equilibrium condi tions. The singularity-based approach Sih (2009) pays attention to the last additive term of (1), whereas the phan tom node method Song et al. (2006) tries to handle cracking using the second one only. However, the notation by (1), compatible with Pike and Oskay (2005) and Kaliske et al. (2012), does not cover all possible approaches, in particular the intrinsic method Fries and Belytschko (2006). 2. Physical, mathematical and numerical background