Virtual element method for computational homogenization of composite and heterogeneous materials

2020 ◽  
Vol 232 ◽  
pp. 111523 ◽  
Author(s):  
Marco Lo Cascio ◽  
Alberto Milazzo ◽  
Ivano Benedetti
Keyword(s):  
Heterogeneous Materials ◽  
Computational Homogenization ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

Computational Homogenization of Heterogeneous Materials by a Novel Hybrid Numerical Scheme

2020 ◽  
Vol 11 (04) ◽  
pp. 2050008
Author(s):  
Marco Lo Cascio ◽  
Marco Grifò ◽  
Alberto Milazzo ◽  
Ivano Benedetti
Keyword(s):  
Numerical Technique ◽  
Matrix Material ◽  
Heterogeneous Materials ◽  
Computational Effort ◽  
Computational Homogenization ◽  
Surrounding Matrix ◽  
Unit Cells ◽  
Virtual Element ◽  
Solution Accuracy ◽  
Element Method

The Virtual Element Method (VEM) is a recent numerical technique capable of dealing with very general polygonal and polyhedral mesh elements, including irregular or non-convex ones. Because of this feature, the VEM ensures noticeable simplification in the data preparation stage of the analysis, especially for problems whose analysis domain features complex geometries, as in the case of computational micro-mechanics problems. The Boundary Element Method (BEM) is a well known, extensively used and effective numerical technique for the solution of several classes of problems in science and engineering. Due to its underlying formulation, the BEM allows reducing the dimensionality of the problem, resulting in substantial simplification of the pre-processing stage and in the reduction of the computational effort, without jeopardizing the solution accuracy. In this contribution, we explore the possibility of a coupling VEM and BEM for computational homogenization of heterogeneous materials with complex microstructures. The test morphologies consist of unit cells with irregularly shaped inclusions, representative e.g., of a fiber-reinforced polymer composite. The BEM is used to model the inclusions, while the VEM is used to model the surrounding matrix material. Benchmark finite element solutions are used to validate the analysis results.

Virtual Element Method: Micro-Mechanics Applications

2019 ◽  
Vol 827 ◽  
pp. 128-133
Author(s):  
Marco Lo Cascio ◽  
Alberto Milazzo ◽  
Ivano Benedetti
Keyword(s):  
Material Properties ◽  
Volume Averaging ◽  
Computational Homogenization ◽  
Polycrystalline Materials ◽  
Virtual Element Method ◽  
Structured Materials ◽  
Virtual Element ◽  
Damage Processes ◽  
Computational Micromechanics ◽  
Element Method

In this contribution we present an application of the lowest order Virtual Element Method (VEM) to the problem of material computational homogenization. Material homogenization allows retrieving material properties through suitable volume averaging procedures, starting from a detailed representation of the micro-constituents of the considered material. The representation of such microstructure constitutes a remarkable effort in terms of data/mesh preparation, especially when there is not evident microstructural regularity. For such a reason, computational micromechanics may represent a challenging benchmark for showing the potential of VEM. In this contribution, polycrystalline materials are considered as an application. The proposed technique constitutes a first step towards modelling of damage processes in micro-structured materials

scholarly journals Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions

2018 ◽  
Vol 77 (3) ◽  
pp. 1874-1908 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals The virtual element method for resistive magnetohydrodynamics

2021 ◽  
Vol 381 ◽  
pp. 113815
Author(s):  
S. Naranjo Alvarez ◽  
V. Bokil ◽  
V. Gyrya ◽  
G. Manzini
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

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