Higher order Bernstein-Bézier and Nédélec finite elements for the relaxed micromorphic model. (English) Zbl 07756773
Summary: The relaxed micromorphic model is a generalized continuum model that is well-posed in the space \(X = [H^1]^3 \times [H (\operatorname{curl})]^3\). Consequently, finite element formulations of the model rely on \(H^1\)-conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein-Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein-Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein-Bézier basis. We introduce the Bernstein-Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein-Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model.
MSC:
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
Keywords:
Nédélec elements; Bernstein-Bézier elements; relaxed micromorphic model; dual numbers; automatic differentiation; hp-FEMReferences:
| [1] | Neff, P.; Ghiba, I.-D.; Madeo, A.; Placidi, L.; Rosi, G., A unifying perspective: the relaxed linear micromorphic continuum, Contin. Mech. Thermodyn., 26, 5, 639-681 (2014) · Zbl 1341.74135 |
| [2] | Forest, S., Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media, Phil. Trans. R. Soc. A, 378, 20190169 (2020) · Zbl 1462.74013 |
| [3] | Eringen, A., Microcontinuum Field Theories. I. Foundations and Solids (1999), Springer-Verlag New York · Zbl 0953.74002 |
| [4] | Mindlin, R., Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302 |
| [5] | Madeo, A.; Neff, P.; Ghiba, I.-D.; Rosi, G., 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 (2016) · Zbl 1482.74103 |
| [6] | Madeo, A.; Barbagallo, G.; Collet, M.; d’Agostino, M. V.; Miniaci, M.; Neff, P., 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 (2018) · Zbl 1430.74028 |
| [7] | d’Agostino, M. V.; Barbagallo, G.; Ghiba, I.-D.; Eidel, B.; Neff, P.; Madeo, A., Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model, J. Elasticity, 139, 2, 299-329 (2020) · Zbl 1433.74009 |
| [8] | Barbagallo, G.; Tallarico, D.; D’Agostino, M. V.; Aivaliotis, A.; Neff, P.; Madeo, A., Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures, Int. J. Solids Struct., 162, 148-163 (2019) |
| [9] | Demore, F.; Rizzi, G.; Collet, M.; Neff, P.; Madeo, A., Unfolding engineering metamaterials design: Relaxed micromorphic modeling of large-scale acoustic meta-structures, J. Mech. Phys. Solids, 168, Article 104995 pp. (2022) |
| [10] | Rizzi, G.; Neff, P.; Madeo, A., Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach, Phil. Trans. R. Soc. A, 380, 2231 (2022) |
| [11] | Perez-Ramirez, L. A.; Rizzi, G.; Madeo, A., Multi-element metamaterial’s design through the relaxed micromorphic model (2022), arXiv:2210.14697 |
| [12] | Rizzi, G.; d’Agostino, M. V.; Neff, P.; Madeo, A., Boundary and interface conditions in the relaxed micromorphic model: Exploring finite-size metastructures for elastic wave control, Math. Mech. Solids, Article 10812865211048923 pp. (2021) |
| [13] | Alberdi, R.; Robbins, J.; Walsh, T.; Dingreville, R., Exploring wave propagation in heterogeneous metastructures using the relaxed micromorphic model, J. Mech. Phys. Solids, 155, Article 104540 pp. (2021) |
| [14] | Rizzi, G.; Hütter, G.; Madeo, A.; Neff, P., Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua, Contin. Mech. Thermodyn., 33, 4, 1505-1539 (2021) · Zbl 1537.74245 |
| [15] | Rizzi, G.; Hütter, G.; Khan, H.; Ghiba, I.-D.; Madeo, A.; Neff, P., Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations), Math. Mech. Solids, Article 10812865211023530 pp. (2021) |
| [16] | Rizzi, G.; Hütter, G.; Madeo, A.; Neff, P., Analytical solutions of the simple shear problem for micromorphic models and other generalized continua, Arch. Appl. Mech., 91, 5, 2237-2254 (2021) |
| [17] | Rizzi, G.; Khan, H.; Ghiba, I.-D.; Madeo, A.; Neff, P., Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations), Arch. Appl. Mech. (2021) |
| [18] | Ghiba, I.-D.; Neff, P.; Madeo, A.; Placidi, L.; Rosi, G., The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics, Math. Mech. Solids, 20, 10, 1171-1197 (2015) · Zbl 1338.74007 |
| [19] | Neff, P.; Ghiba, I. D.; Lazar, M.; Madeo, A., The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations, Quart. J. Mech. Appl. Math., 68, 1, 53-84 (2015) · Zbl 1310.74037 |
| [20] | Knees, D.; Owczarek, S.; Neff, P., A local regularity result for the relaxed micromorphic model based on inner variations, J. Math. Anal. Appl., 519, 2, Article 126806 pp. (2023) · Zbl 1503.35219 |
| [21] | Owczarek, S.; Ghiba, I.-D.; Neff, P., A note on local higher regularity in the dynamic linear relaxed micromorphic model, Math. Methods Appl. Sci., 44, 18, 13855-13865 (2021) · Zbl 1479.35173 |
| [22] | Sky, A.; Neunteufel, M.; Muench, I.; Schöberl, J.; Neff, P., Primal and mixed finite element formulations for the relaxed micromorphic model, Comput. Methods Appl. Mech. Engrg., 399, Article 115298 pp. (2022) · Zbl 1507.74048 |
| [23] | Nédélec, J. C., A new family of mixed finite elements in \(\mathbb{R}^3\), Numer. Math., 50, 1, 57-81 (1986) · Zbl 0625.65107 |
| [24] | Nedelec, J. C., Mixed finite elements in \(\mathbb{R}^3\), Numer. Math., 35, 3, 315-341 (1980) · Zbl 0419.65069 |
| [25] | Bergot, M.; Lacoste, P., Generation of higher-order polynomial basis of Nédélec H(curl) finite elements for Maxwell’s equations, J. Comput. Appl. Math., 234, 6, 1937-1944 (2010), Eighth International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007) · Zbl 1191.78055 |
| [26] | Sky, A.; Muench, I., Polytopal templates for the formulation of semi-continuous vectorial finite elements of arbitrary order (2022), arXiv:2210.03525 |
| [27] | Lai, M.-J.; Schumaker, L. L., Spline Functions on Triangulations (2007), Cambridge University Press · Zbl 1185.41001 |
| [28] | Ainsworth, M.; Andriamaro, G.; Davydov, O., Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures, SIAM J. Sci. Comput., 33, 6, 3087-3109 (2011) · Zbl 1237.65120 |
| [29] | El-Amrani, M.; Kacimi, A. E.; Khouya, B.; Seaid, M., A Bernstein-Bézier Lagrange-Galerkin method for three-dimensional advection-dominated problems, Comput. Methods Appl. Mech. Engrg., 403, Article 115758 pp. (2023) · Zbl 1536.76061 |
| [30] | El-Amrani, M.; El-Kacimi, A.; Khouya, B.; Seaid, M., Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems, J. Sci. Comput., 92, 2, 58 (2022) · Zbl 1502.65117 |
| [31] | Fike, J. A.; Alonso, J. J., Automatic differentiation through the use of hyper-dual numbers for second derivatives, (Forth, S.; Hovland, P.; Phipps, E.; Utke, J.; Walther, A., Recent Advances in Algorithmic Differentiation (2012), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 163-173 · Zbl 1251.65026 |
| [32] | Margossian, C. C., A review of automatic differentiation and its efficient implementation, WIREs Data Min. Knowl. Discov., 9, 4, Article e1305 pp. (2019) |
| [33] | Buffa, A.; Costabel, M.; Dauge, M., Algebraic convergence for anisotropic edge elements in polyhedral domains, Numer. Math., 101, 1, 29-65 (2005) · Zbl 1116.78020 |
| [34] | Ciarlet, P.; Zou, J., Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math., 82, 2, 193-219 (1999) · Zbl 1126.78310 |
| [35] | Ciarlet, P.; Wu, H.; Zou, J., Edge element methods for maxwell’s equations with strong convergence for Gauss’ laws, SIAM J. Numer. Anal., 52, 2, 779-807 (2014) · Zbl 1304.35668 |
| [36] | Neunteufel, M.; Pechstein, A. S.; Schöberl, J., Three-field mixed finite element methods for nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 382, Article 113857 pp. (2021) · Zbl 1506.74053 |
| [37] | Neunteufel, M.; Schöberl, J., The Hellan-Herrmann-Johnson and TDNNS method for linear and nonlinear shells (2023), arXiv:2304.13806 |
| [38] | Pechstein, A. S.; Schöberl, J., An analysis of the TDNNS method using natural norms, Numer. Math., 139, 1, 93-120 (2018) · Zbl 1412.65224 |
| [39] | Pechstein, A.; Schöberl, J., Anisotropic mixed finite elements for elasticity: Anisotropic mixed finite elements for elasticity, Internat. J. Numer. Methods Engrg., 90, 2, 196-217 (2012) · Zbl 1242.74148 |
| [40] | Faghih Shojaei, M.; Yavari, A., Compatible-strain mixed finite element methods for 3D compressible and incompressible nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 357, Article 112610 pp. (2019) · Zbl 1442.74034 |
| [41] | Ding, Q.; Long, X.; Mao, S., Error estimate of a fully discrete finite element method for incompressible vector potential magnetohydrodynamic system, J. Sci. Comput., 88, 3, 71 (2021) · Zbl 1493.65148 |
| [42] | Zaglmayr, S., High Order Finite Element Methods for Electromagnetic Field Computation (2006), Johannes Kepler Universität Linz, URL https://www.numerik.math.tugraz.at/ zaglmayr/pub/szthesis.pdf |
| [43] | Schöberl, J.; Zaglmayr, S., High order Nédélec elements with local complete sequence properties, COMPEL - Int. J. Comput. Math. Electr. Electron. Eng., 24, 2, 374-384 (2005) · Zbl 1135.78337 |
| [44] | Solin, P.; Segeth, K.; Dolezel, I., Higher-Order Finite Element Methods (1st Ed.) (2003), Chapman and Hall/CRC |
| [45] | Anjam, I.; Valdman, J., Fast \(\text{MATLAB}\) assembly of \(\text{FEM}\) matrices in 2D and 3D: Edge elements, Appl. Math. Comput., 267, 252-263 (2015) · Zbl 1410.65441 |
| [46] | Sky, A.; Muench, I.; Neff, P., On \([ \mathit{H}^1 ]^{3 \times 3}, [ \mathit{H} ( \operatorname{curl} ) ]^3\) and \(\mathit{H} ( \operatorname{sym} \operatorname{Curl} )\) finite elements for matrix-valued Curl problems, J. Eng. Math., 136, 1, 5 (2022) · Zbl 1524.65859 |
| [47] | Sky, A.; Neunteufel, M.; Münch, I.; Schöberl, J.; Neff, P., A hybrid \(\mathit{H}^1 \times \mathit{H} ( \operatorname{curl} )\) finite element formulation for a relaxed micromorphic continuum model of antiplane shear, Comput. Mech., 68, 1, 1-24 (2021) · Zbl 1480.74291 |
| [48] | Schröder, J.; Sarhil, M.; Scheunemann, L.; Neff, P., Lagrange and \(\mathit{H} ( \operatorname{curl} , \mathcal{B} )\) based finite element formulations for the relaxed micromorphic model, Comput. Mech. (2022) · Zbl 1509.74048 |
| [49] | Sarhil, M.; Scheunemann, L.; Schröder, J.; Neff, P., Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model (2022), arXiv:2210.17117 |
| [50] | Voss, J.; Baaser, H.; Martin, R. J.; Neff, P., More on anti-plane shear, J. Optim. Theory Appl., 184, 1, 226-249 (2020) · Zbl 1431.74026 |
| [51] | d’Agostino, M. V.; Rizzi, G.; Khan, H.; Lewintan, P.; Madeo, A.; Neff, P., The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters, Contin. Mech. Thermodyn. (2022) · Zbl 1514.74068 |
| [52] | Lewintan, P.; Müller, S.; Neff, P., Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. Partial Differential Equations, 60, 4, 150 (2021) · Zbl 1471.35009 |
| [53] | Lewintan, P.; Neff, P., Nečas-Lions lemma revisited: An \(\mathit{L}^p\)-version of the generalized Korn inequality for incompatible tensor fields, Math. Methods Appl. Sci., 44, 14, 11392-11403 (2021) · Zbl 1473.35015 |
| [54] | Lewintan, P.; Neff, P., \( \mathit{L}^p\)-Versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with \(p\)-integrable exterior derivative, C. R. Math., 359, 6, 749-755 (2021) · Zbl 1472.35015 |
| [55] | Neff, P.; Pauly, D.; Witsch, K.-J., Maxwell meets Korn: A new coercive inequality for tensor fields with square-integrable exterior derivative, Math. Methods Appl. Sci., 35, 1, 65-71 (2012) · Zbl 1255.35220 |
| [56] | Neff, P.; Eidel, B.; d’Agostino, M. V.; Madeo, A., Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization, J. Elasticity, 139, 2, 269-298 (2020) · Zbl 1433.74014 |
| [57] | Aivaliotis, A.; Tallarico, D.; d’Agostino, M.-V.; Daouadji, A.; Neff, P.; Madeo, A., Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model, Arch. Appl. Mech., 90, 5, 1073-1096 (2020) |
| [58] | Neff, P.; Forest, S., A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results, J. Elasticity, 87, 2, 239-276 (2007) · Zbl 1206.74019 |
| [59] | Barbagallo, G.; Madeo, A.; d’Agostino, M. V.; Abreu, R.; Ghiba, I.-D.; Neff, P., Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics, Int. J. Solids Struct., 120, 7-30 (2017) |
| [60] | Monk, P., (Finite Element Methods for Maxwell’s Equations. Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation (2003), Oxford University Press: Oxford University Press New York), xiv+450 · Zbl 1024.78009 |
| [61] | Neidinger, R. D., Introduction to automatic differentiation and MATLAB object-oriented programming, SIAM Rev., 52, 3, 545-563 (2010) · Zbl 1196.65048 |
| [62] | Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A.; Siskind, J. M., Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res., 18, 1-43 (2018) · Zbl 06982909 |
| [63] | Eisenträger, S.; Saputra, A.; Gravenkamp, H., High order transition elements: The xNy-element concept - Part I: Statics, Comput. Methods Appl. Mech. Engrg., 362, Article 112833 pp. (2020) · Zbl 1439.74408 |
| [64] | Ainsworth, M.; Fu, G., Bernstein-Bézier bases for tetrahedral finite elements, Comput. Methods Appl. Mech. Engrg., 340, 178-201 (2018) · Zbl 1440.65168 |
| [65] | Dunavant, D. A., High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg., 21, 6, 1129-1148 (1985) · Zbl 0589.65021 |
| [66] | Xiao, H.; Gimbutas, Z., A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions, Comput. Math. Appl., 59, 2, 663-676 (2010) · Zbl 1189.65047 |
| [67] | Witherden, F.; Vincent, P., On the identification of symmetric quadrature rules for finite element methods, Comput. Math. Appl., 69, 10, 1232-1241 (2015) · Zbl 1443.65378 |
| [68] | Papanicolopulos, S.-A., Efficient computation of cubature rules with application to new asymmetric rules on the triangle, J. Comput. Appl. Math., 304, 73-83 (2016) · Zbl 1337.65023 |
| [69] | Jaśkowiec, J.; Sukumar, N., High-order cubature rules for tetrahedra, Internat. J. Numer. Methods Engrg., 121, 11, 2418-2436 (2020) · Zbl 07841925 |
| [70] | Demkowicz, L.; Monk, P.; Vardapetyan, L.; Rachowicz, W., De Rham diagram for hp-finite element spaces, Comput. Math. Appl., 39, 7, 29-38 (2000) · Zbl 0955.65084 |
| [71] | Johnen, A.; Remacle, J.-F.; Geuzaine, C., Geometrical validity of high-order triangular finite elements, Eng. Comput., 30, 3, 375-382 (2014) |
| [72] | Johnen, A.; Remacle, J.-F.; Geuzaine, C., Geometrical validity of curvilinear finite elements, J. Comput. Phys., 233, 359-372 (2013) |
| [73] | Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal., 23, 3, 562-580 (1986) · Zbl 0605.65071 |
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