A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

Abdulle A (2006) Analysis of a heterogeneous multiscale FEM for problems in elasticity. Math Models Methods Appl Sci 16(04):615–635 Aivaliotis A, Daouadji A, Barbagallo G, Tallarico D, Neff P, Madeo A (2018) Low-and high-frequency stoneley waves, reflection and transmission at a \(\text{C}\)auchy/relaxed micromorphic interface. arXiv:1810.12578 Aivaliotis A, Daouadji A, Barbagallo G, Tallarico D, Neff P, Madeo A (2019) Microstructure-related stoneley waves and their effect on the scattering properties of a 2\(\text{ D }\)\(\text{ C }\)auchy/relaxed-micromorphic interface. Wave Motion 90:99–120 Anjam I, Valdman J (2015) Fast \(\text{ MATLAB }\) assembly of \(\text{ FEM }\) matrices in 2d and 3d: edge elements. Appl Math Comput 267:252–263 Askes H, Aifantis E (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48:1962–1990 Braess D (2013) Finite Elemente - Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, 5th edn. Springer-Verlag, Berlin d’Agostino MV, Barbagallo G, Ghiba ID, Eidel B, Neff P, Madeo A (2020) Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. J Elast 139(2):299–329 Demkowicz L (2006) Computing with hp-adaptive finite elements. One- and two-dimensional elliptic and maxwell problems, vol 1. Chapman and Hall/CRC, London Demkowicz L, Kurtz J, Pardo D, Paszynski M, Rachowicz W, Zdunek A (2007) Computing with hp-adaptive finite elements. Frontiers: three-dimensional elliptic and maxwell problems with applications, vol 2. Chapman and Hall/CRC, London Efendiev Y, Hou T (2009) Multiscale finite element methods. Springer-Verlag, New York Eidel B, Fischer A (2018) The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the \(\text{ FE}^2\) method. Comput Methods App Mech Eng 329:332–368 Eringen A (1999) Microcontinuum field theories. I. Foundations and solids. Springer-Verlag, New York Forest S, Sievert R (2006) Nonlinear microstrain theories. Int J Solids Struct 43(24):7224–7245 Ghiba ID, Neff P, Owczarek S (2021) Existence results for non-homogeneous boundary conditions in the relaxed micromorphic model. Math Methods Appl Sci 44(2):2040–2049 Hütter G (2016) Application of a microstrain continuum to size effects in bending and torsion of foams. Int J Eng Sci 101:81–91 Jeong J, Neff P (2010) Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math Mech Solids 15(1):78–95 Kirchner N, Steinmann P (2006) Mechanics of extended continua: modeling and simulation of elastic microstretch materials. Comput Mech 40(4):651 Lehrenfeld C, Schöberl J (2016) High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput Methods Appl Mech Eng 307:339–361 Madeo A, Barbagallo G, Collet M, dAgostino MV, Miniaci M, Neff P (2018) Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: new perspectives towards metastructural design. Math Mech Solids 23(12):1485–1506 Madeo A, Neff P, Ghiba ID, Rosi G (2016) Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J Mech Phys Solids 95:441–479 Meunier G (2010) The finite element method for electromagnetic modeling. Wiley-ISTE, United States Mindlin R (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78 Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124 Monk P (2003) Finite element methods for Maxwell’s Equations. Numerical mathematics and scientific computation. Oxford University Press, New York Münch I, Neff P, Madeo A, Ghiba ID (2017) The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. Zeitschrift für Angewandte Mathematik und Mechanik 97(12):1524–1554 Münch I, Neff P, Wagner W (2011) Transversely isotropic material: nonlinear Cosserat versus classical approach. Contin Mech Thermodyn 23(1):27–34 Nedelec JC (1980) Mixed finite elements in \(\mathbb{R}^3\). Numerische Mathematik 35(3):315–341 Nédélec JC (1986) A new family of mixed finite elements in \(\mathbb{R}^3\). Numerische Mathematik 50(1):57–81 Neff P, Eidel B, d’Agostino MV, Madeo A (2020) Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization. J Elast 139(2):269–298 Neff P, Forest S (2007) A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J Elast 87(2):239–276 Neff P, Ghiba ID, Madeo A, Placidi L, Rosi G (2014) A unifying perspective: the relaxed linear micromorphic continuum. Contin Mech Thermodyn 26(5):639–681 Neff P, Jeong J, Ramézani H (2009) Subgrid interaction and micro-randomness - novel invariance requirements in infinitesimal gradient elasticity. Int J Solids Struct 46(25):4261–4276 Neff P, Madeo A, Barbagallo G, D’Agostino MV, Abreu R, Ghiba ID (2017) Real wave propagation in the isotropic-relaxed micromorphic model. Proceed R Soc A Math Phys Eng Sci 473:2197 Owczarek S, Ghiba ID, Neff P (2020) A note on local higher regularity in the dynamic linear relaxed micromorphic model. submitted. arXiv:2006.05448 Rizzi G, Hütter G, Madeo A, Neff P (2021) Analytical solutions of the simple shear problem for micromorphic models and other generalized continua. Arch Appl Mech. https://doi.org/10.1007/s00419-021-01881-w Rizzi G, Htter G, Khan H, Madeo A, Neff P. Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations) (in preparation) Rizzi G, Htter G, Madeo A, Neff P (2021) Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations). to appear in continuum mechanics and thermodynamics. arXiv:2012.10391 Romeo M (2020) A microstretch continuum approach to model dielectric elastomers. Zeitschrift für angewandte Mathematik und Physik 71(2):44 Schöberl J (1997) NETGEN an advancing front 2D/3D-mesh generator based on abstract rules. Comput Vis Sci 1(1):41–52 Schöberl J (2014) C++ 11 implementation of finite elements in NGSolve. Institute for analysis and scientific computing, Vienna University of Technology. URL https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/ngs-cpp11.pdf Schöberl J, Zaglmayr S (2005) High order Nédélec elements with local complete sequence properties. COMPEL - Int J Comput Math Electr Electron Eng 24(2):374–384 Steigmann DJ (2012) Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int J Non-Linear Mech 47(7):734–742 Voss J, Baaser H, Martin RJ, Neff P (2020) More on anti-plane shear. J Optim Theory Appl 184(1):226–249 Zaglmayr S (2006) High order finite element methods for electromagnetic field computation. Ph.D. thesis, Johannes Kepler Universität Linz. URL https://www.numerik.math.tugraz.at/~zaglmayr/pub/szthesis.pdf