PSI - Issue 43

Vladislav Kozák et al. / Procedia Structural Integrity 43 (2023) 47–52 V. Koza´k & J. Vala / Structural Integrity Procedia 00 (2023) 000–000

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proximation space by all needed types if (especially locally) discontinuous functions, and similar approaches, derived from the Partition of Unity Method, namely the Partition of Unity Finite Element Method (Babusˇka et al., 1997), the Generalized Finite Element Method (Duarte et al., 2001), or the Discontinuous Galerkin XFEM (Aduloju et al., 2019). Especially XFEM adds some degrees of freedom in relevant regions during the computation, typically along all curves and surfaces of discontinuities and in singular points, exploiting the Moving Least Squares technique: the usual extrinsic XFEM works with additional variables and functions, whereas the intrinsic XFEM (Fries et al., 2006) tries to avoid them, only with one additional shape function in each relevant node. However, although no singularity exist at the tip of cohesive crack, the stresses obtained by discretization of the displacement are not accurate, and cannot be used to predict accurately the growth of the (Li at al., 2017) and (Li at al., 2018). In general, in case of time dependent problems coming from the conservation principle of classical mechanics, namely of mass, (linear and angular) momentum and energy, one can come to the variational, or weak, etc. formu lation of an initial and boundary value problem for (in general nonlinear) hyperbolic system of partial di ff erential equations where at least piecewise knowledge of some analytical solutions can be taken into account just for very special cases, much di ff erent from those needed by engineering practice. The design of computational algorithms for construction of approximate solutions comes then either i) from the method of discretization in time, leading, on base on the construction of several types of Rothe sequences, to the repeated solution elliptic systems of partial di ff erential equations in so-called space variables (unlike a time one), or ii) from the method of lines where, thanks to the mul tiplicative Fourier decomposition of a required solution and to the subsequent implementation of XFEM, generating large sparse systems of ordinary di ff erential equations in a time variable.

2. A model problem

Computational homogenization of macroscopic material e.g. (Vala et al., 2015) is based on semi-analytical mixing formulas for special fibre shapes (acceptable especially for their low volume shares), two-scale homogenization of periodic structures or alternative results from asymptotic analysis (G-convergence, H-convergence, Γ -convergence, etc.) to very general (deterministic and stochastic) results for σ -convergence on homogenizations structures with numerous open problems, see (Necˇas at al., 1991) and (Zhang at. al., 2010). A unified approach is desirable from a macroscopic and microscopic scale, covering elastic and plastic behaviour along with degradation and fracture.

Fig. 1. X-ray image of a concrete sample with an edge of 150 mm, axonometric view of the surface and inside of the sample.

From the point of view of finite element modelling, the essential problem of determining the representative volume of RVE (Representative Volume Element) for correct calculation is solved, see Fig. 1. As with the simplified model example, based on the considerations detailed in (Vala and Koza´k, to be published), we can start from an abstract (generally nonlinear) quasi-static problem for the purpose of computational modelling G ( ˙ u ) , v + A ( u ) , v = F , v , (1) where parentheses refer to certain dual assignments (in the simplest cases scalar products) for reflexive and separable Banach spaces V , v ∈ V indicates the required virtual quantity, for example, o ff sets relative to the reference configura-