Transformations for Piola-mapped elements

Francis R. A. Aznaran; Patrick E. Farrell; Robert C. Kirby

doi:10.5802/smai-jcm.91

Francis R. A. Aznaran ¹; Patrick E. Farrell ¹; Robert C. Kirby ²

¹ Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
² Department of Mathematics, Baylor University; Sid Richardson Science Building; 1410 S. 4th St.; Waco, TX 76706.

The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 399-437.

Abstract

The Arnold–Winther element successfully discretizes the Hellinger–Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior calculus. Despite its great utility, it is not available in standard finite element software, because its degrees of freedom are not preserved under the standard Piola push-forward. In this work we apply the novel transformation theory recently developed by Kirby [SMAI J. Comput. Math., 4:197–224, 2018] to devise the correct map for transforming the basis on a reference cell to a generic physical triangle. This enables the use of the Arnold–Winther elements, both conforming and nonconforming, in the widely-used Firedrake finite element software, composing with its advanced symbolic code generation and geometric multigrid functionality. Similar results also enable the correct transformation of the Mardal–Tai–Winther element for incompressible fluid flow. We present numerical results for both elements, verifying the correctness of our theory.

Published online: 2023-07-31

DOI: 10.5802/smai-jcm.91

Classification: 65N30, 65F08
Keywords: Finite element method, Piola transform, pullback, linear elasticity, Stokes, reference basis, Firedrake

Author's affiliations:

Francis R. A. Aznaran ¹; Patrick E. Farrell ¹; Robert C. Kirby ²

¹ Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
² Department of Mathematics, Baylor University; Sid Richardson Science Building; 1410 S. 4th St.; Waco, TX 76706.

License:

CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{SMAI-JCM_2022__8__399_0,
     author = {Francis R.~A. Aznaran and Patrick E. Farrell and Robert C. Kirby},
     title = {Transformations for {Piola-mapped} elements},
     journal = {The SMAI Journal of computational mathematics},
     pages = {399--437},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {8},
     year = {2022},
     doi = {10.5802/smai-jcm.91},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.91/}
}

TY  - JOUR
AU  - Francis R. A. Aznaran
AU  - Patrick E. Farrell
AU  - Robert C. Kirby
TI  - Transformations for Piola-mapped elements
JO  - The SMAI Journal of computational mathematics
PY  - 2022
SP  - 399
EP  - 437
VL  - 8
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.91/
DO  - 10.5802/smai-jcm.91
LA  - en
ID  - SMAI-JCM_2022__8__399_0
ER  -

%0 Journal Article
%A Francis R. A. Aznaran
%A Patrick E. Farrell
%A Robert C. Kirby
%T Transformations for Piola-mapped elements
%J The SMAI Journal of computational mathematics
%D 2022
%P 399-437
%V 8
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.91/
%R 10.5802/smai-jcm.91
%G en
%F SMAI-JCM_2022__8__399_0

Francis R. A. Aznaran; Patrick E. Farrell; Robert C. Kirby. Transformations for Piola-mapped elements. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 399-437. doi : 10.5802/smai-jcm.91. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.91/

References
Cited by

[1] M. S. Alnæs; A. Logg; K. B. Ølgaard; M. E. Rognes; G. N. Wells Unified Form Language: a Domain-Specific Language for weak formulations of partial differential equations, ACM Trans. Math. Softw., Volume 40 (2014) no. 2, pp. 1-37 | DOI | MR | Zbl

[2] P. R. Amestoy; I. S. Duff; J.-Y. L’Excellent; J. Koster A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., Volume 23 (2001) no. 1, pp. 15-41 | DOI | MR | Zbl

[3] D. N. Arnold Finite element exterior calculus, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 2018 no. 93 | DOI

[4] D. N. Arnold; G. Awanou; R. Winther Finite elements for symmetric tensors in three dimensions, Math. Comput., Volume 77 (2008) no. 263, pp. 1229-1251 | DOI | MR | Zbl

[5] D. N. Arnold; G. Awanou; R. Winther Nonconforming tetrahedral mixed finite elements for elasticity, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 04, pp. 783-796 | DOI | MR | Zbl

[6] D. N. Arnold; F. Brezzi; J. Douglas PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math., Volume 1 (1984) no. 2, pp. 347-367 | DOI | MR | Zbl

[7] D. N. Arnold; R. S. Falk; R. Winther Multigrid in $H (div)$ and $H (curl)$ , Numer. Math., Volume 85 (2000) no. 2, pp. 197-217 | DOI | MR | Zbl

[8] D. N. Arnold; R. S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155 | DOI | MR | Zbl

[9] D. N. Arnold; R. S. Falk; R. Winther Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comput., Volume 76 (2007) no. 260, pp. 1699-1723 | DOI | MR | Zbl

[10] D. N. Arnold; R. S. Falk; R. Winther Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Eng., Volume 198 (2009) no. 21-26, pp. 1660-1672 | DOI | MR | Zbl

[11] D. N. Arnold; R. Winther Mixed finite elements for elasticity, Numer. Math., Volume 92 (2002) no. 3, pp. 401-419 | DOI | MR | Zbl

[12] D. N. Arnold; R. Winther Nonconforming mixed elements for elasticity, Math. Models Methods Appl. Sci., Volume 13 (2003) no. 03, pp. 295-307 | DOI | MR | Zbl

[13] S. Balay; S. Abhyankar; M. F. Adams; J. Brown; P. Brune; K. Buschelman; L. Dalcin; V. Eijkhout; W. D. Gropp; D. Karpeyev; D. Kaushik; M. G. Knepley; D. May; L. C. McInnes; R. T. Mills; T. Munson; K. Rupp; P. Sanan; B. F. Smith; S. Zampini; H. Zhang; H. Zhang PETSc Users Manual (2019) (ANL-95/11 - Revision 3.11) (Technical report)

[14] M. Benzi; G. H. Golub; J. Liesen Numerical solution of saddle point problems, Acta Numer., Volume 14 (2005), pp. 1-137 | DOI | MR | Zbl

[15] M. Benzi; M. A. Olshanskii An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., Volume 28 (2006) no. 6, pp. 2095-2113 | DOI | MR | Zbl

[16] J. Bleyer Numerical Tours of Computational Mechanics with FEniCS (2018) (https://comet-fenics.readthedocs.io) | DOI

[17] D. Boffi; F. Brezzi; M. Fortin Mixed finite element methods and applications, Springer Series in Computational Mathematics, 44, Springer, 2013 | DOI

[18] J. Bramwell; L. Demkowicz; J. Gopalakrishnan; W. Qiu A locking-free $h p$ DPG method for linear elasticity with symmetric stresses, Numer. Math., Volume 122 (2012) no. 4, pp. 671-707 | DOI | MR | Zbl

[19] F. Brezzi; J. Douglas; M. Fortin; L. D. Marini Efficient rectangular mixed finite elements in two and three space variables, ESAIM, Math. Model. Numer. Anal., Volume 21 (1987) no. 4, pp. 581-604 | DOI | Numdam | MR | Zbl

[20] F. Brezzi; J. Douglas; L. D. Marini Two families of mixed finite elements for second order elliptic problems, Numer. Math., Volume 47 (1985) no. 2, pp. 217-235 | DOI | MR | Zbl

[21] E. Burman; R. Puppi Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow, J. Numer. Math., Volume 30 (2022) no. 2, pp. 141-162 | DOI | MR | Zbl

[22] Z. Cai; G. Starke Least-squares methods for linear elasticity, SIAM J. Numer. Anal., Volume 42 (2004) no. 2, pp. 826-842 | DOI | MR | Zbl

[23] C. Carstensen; M. Eigel; J. Gedicke Computational competition of symmetric mixed FEM in linear elasticity, Comput. Methods Appl. Mech. Eng., Volume 200 (2011) no. 41-44, pp. 2903-2915 | DOI | MR | Zbl

[24] C. Carstensen; D. Gallistl; M. Schedensack $L^{2}$ best approximation of the elastic stress in the Arnold–Winther FEM, IMA J. Numer. Anal., Volume 36 (2016) no. 3, pp. 1096-1119 | DOI | MR | Zbl

[25] C. Carstensen; D. Günther; J. Reininghaus; J. Thiele The Arnold–Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification, Comput. Methods Appl. Mech. Eng., Volume 197 (2008) no. 33-40, pp. 3014-3023 | DOI | Zbl

[26] L. Chen; X. Huang Geometric decompositions of div-conforming finite element tensors (2021) (https://arxiv.org/abs/2112.14351)

[27] S. H. Christiansen; J. Hu; K. Hu Nodal finite element de Rham complexes, Numer. Math., Volume 139 (2018) no. 2, pp. 411-446 | DOI | MR | Zbl

[28] G. Danisch; G. Starke First-order system least-squares for Darcy–Stokes flow, SIAM J. Numer. Anal., Volume 45 (2007) no. 2, pp. 731-745 | DOI | MR | Zbl

[29] P. E. Farrell; L. F. Gatica; B. P. Lamichhane; R. Oyarzúa; R. Ruiz-Baier Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity, Comput. Methods Appl. Mech. Eng., Volume 374 (2021), p. 113562 | DOI | MR | Zbl

[30] P. E. Farrell; P. A. Gazca-Orozco An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow, SIAM J. Sci. Comput., Volume 42 (2020) no. 6, p. B1329-B1349 | DOI | MR | Zbl

[31] P. E. Farrell; P. A. Gazca-Orozco; E. Süli Finite element approximation and preconditioning for anisothermal flow of implicitly-constituted non-Newtonian flow, Math. Comput., Volume 91 (2021) no. 334, pp. 659-697 | DOI | Zbl

[32] P. E. Farrell; M. G. Knepley; F. Wechsung; L. Mitchell PCPATCH: software for the topological construction of multigrid relaxation methods, ACM Trans. Math. Softw., Volume 47 (2021) no. 3, pp. 1-22 | DOI | MR | Zbl

[33] P. E. Farrell; L. Mitchell; L. R. Scott; F. Wechsung A Reynolds-robust preconditioner for the Scott–Vogelius discretization of the stationary incompressible Navier–Stokes equations, SMAI J. Comput. Math., Volume 7 (2021), pp. 75-96 | DOI | MR | Zbl

[34] P. E. Farrell; L. Mitchell; L. R. Scott; F. Wechsung Robust multigrid methods for nearly incompressible elasticity using macro elements, IMA J. Numer. Anal. (2022) | DOI | MR | Zbl

[35] P. E. Farrell; L. Mitchell; F. Wechsung An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier–Stokes equations at high Reynolds number, SIAM J. Sci. Comput., Volume 41 (2019) no. 5, p. A3073-A3096 | DOI | MR | Zbl

[36] T. Frankel The geometry of physics: An introduction, Cambridge University Press, 2003 | DOI

[37] J. Gedicke; A. Khan Arnold–Winther mixed finite elements for Stokes eigenvalue problems, SIAM J. Sci. Comput., Volume 40 (2018) no. 5, p. A3449-A3469 | DOI

[38] J. S. Hale; M. Brunetti; S. P. A. Bordas; C. Maurini Simple and extensible plate and shell finite element models through automatic code generation tools, Computers & Structures, Volume 209 (2018), pp. 163-181 | DOI

[39] K. Hellan Analysis of elastic plates in flexure by a simplified finite element method, Acta polytechn. Scand., Civil Engin. Building Construct. Ser., Volume 46 (1967), pp. 1-29 (Finnish Academy of Technology, Tekniikentie 12, FIN-02150 Espoo, Finland) | Zbl

[40] L. R. Herrmann Finite-element bending analysis for plates, J. Eng. Mech. Div., Volume 93 (1967) no. 5, pp. 13-26 | DOI

[41] M. Homolya; R. C. Kirby; D. A. Ham Exposing and exploiting structure: Optimal code generation for high-order finite element methods (2017) (https://arxiv.org/abs/1711.02473)

[42] M. Homolya; L. Mitchell; F. Luporini; D. A. Ham TSFC: A structure-preserving form compiler, SIAM J. Sci. Comput., Volume 40 (2018) no. 3, p. C401-C428 | DOI | MR | Zbl

[43] Q. Hong; J. Hu; L. Ma; J. Xu An extended Galerkin analysis for linear elasticity with strongly symmetric stress tensor (2020) (https://arxiv.org/abs/2002.11664)

[44] Q. Hong; J. Kraus Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem, SIAM J. Numer. Anal., Volume 54 (2016) no. 5, pp. 2750-2774 | DOI | MR | Zbl

[45] Q. Hong; J. Kraus; J. Xu; L. Zikatanov A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations, Numer. Math., Volume 132 (2015), pp. 23-49 | DOI | MR

[46] J. Hu Finite element approximations of symmetric tensors on simplicial grids in $ℝ^{n}$ : The higher order case, J. Comput. Math., Volume 33 (2015) no. 3, pp. 283-296 | DOI | MR

[47] J. Hu A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation, SIAM J. Numer. Anal., Volume 53 (2015) no. 3, pp. 1438-1463 | DOI | MR | Zbl

[48] J. Hu; S. Zhang A family of conforming mixed finite elements for linear elasticity on triangular grids (2015) (https://arxiv.org/abs/1406.7457)

[49] J. Hu; S. Zhang Finite element approximations of symmetric tensors on simplicial grids in $ℝ^{n}$ : The lower order case, Math. Models Methods Appl. Sci., Volume 26 (2016) no. 09, pp. 1649-1669 | DOI | MR

[50] V. John; A. Linke; C. Merdon; M. Neilan; L. G. Rebholz On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., Volume 59 (2017) no. 3, pp. 492-544 | DOI | MR | Zbl

[51] C. Johnson On the convergence of a mixed finite-element method for plate bending problems, Numer. Math., Volume 21 (1973) no. 1, pp. 43-62 | DOI | MR

[52] T. Karper; K.-A. Mardal; R. Winther Unified finite element discretizations of coupled Darcy–Stokes flow, Numerical Methods for Partial Differential Equations: An International Journal, Volume 25 (2009) no. 2, pp. 311-326 | DOI | MR | Zbl

[53] R. C. Kirby Algorithm 839: FIAT, A New Paradigm for Computing Finite Element Basis Functions, ACM Trans. Math. Softw., Volume 30 (2004) no. 4, pp. 502-516 | DOI | MR | Zbl

[54] R. C. Kirby A general approach to transforming finite elements, SMAI J. Comput. Math., Volume 4 (2018), pp. 197-224 | DOI | Numdam | MR | Zbl

[55] R. C. Kirby; L. Mitchell Solver composition across the PDE/linear algebra barrier, SIAM J. Sci. Comput., Volume 40 (2018) no. 1, p. C76-C98 | DOI | MR | Zbl

[56] R. C. Kirby; L. Mitchell Code generation for generally mapped finite elements, ACM Trans. Math. Softw., Volume 45 (2019) no. 4, pp. 1-23 | DOI | MR | Zbl

[57] J. Könnö; D. Schötzau; R. Stenberg Mixed finite element methods for problems with Robin boundary conditions, SIAM J. Numer. Anal., Volume 49 (2011) no. 1, pp. 285-308 | DOI | MR | Zbl

[58] M. Kuchta; K.-A. Mardal; M. Mortensen On the singular Neumann problem in linear elasticity, Numer. Linear Algebra Appl., Volume 26 (2019) no. 1, e2212 | DOI | MR | Zbl

[59] F. Laakmann; P. E. Farrell; L. Mitchell An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers, SIAM J. Sci. Comput., Volume 44 (2022) no. 4, p. B1018-B1044 | DOI | MR

[60] H. P. Langtangen; A. Tveito Advanced topics in computational partial differential equations: Numerical methods and Diffpack programming, Lecture Notes in Computational Science and Engineering, 33, Springer, 2012

[61] J. J. Lee Robust three-field finite element methods for Biot’s consolidation model in poroelasticity, BIT Numer. Math., Volume 58 (2018) no. 2, pp. 347-372 | DOI | MR | Zbl

[62] L. Li Regge finite elements with applications in solid mechanics and relativity, Ph. D. Thesis, University of Minnesota (2018) (http://www-users.math.umn.edu/~arnold/papers/LiThesis.pdf)

[63] Automated solution of differential equations by the finite element method: The FEniCS book (A. Logg; K.-A. Mardal; G. N. Wells, eds.), Lecture Notes in Computational Science and Engineering, 84, Springer, 2012 | DOI | Zbl

[64] K.-A. Mardal; X.-C. Tai; R. Winther A robust finite element method for Darcy–Stokes flow, SIAM J. Numer. Anal., Volume 40 (2002) no. 5, pp. 1605-1631 | DOI | MR

[65] K.-A. Mardal; R. Winther An observation on Korn’s inequality for nonconforming finite element methods, Math. Comput., Volume 75 (2006) no. 253, pp. 1-6 | DOI | MR | Zbl

[66] L. A. Mihai; A. Goriely Numerical simulation of shear and the Poynting effects by the finite element method: An application of the generalised empirical inequalities in non-linear elasticity, Int. J. Non-Linear Mech., Volume 49 (2013), pp. 1-14 | DOI

[67] P. Monk Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003 | DOI

[68] B. Müller; G. Starke; A. Schwarz; J. Schröder A first-order system least squares method for hyperelasticity, SIAM J. Sci. Comput., Volume 36 (2014) no. 5, p. B795-B816 | DOI | MR | Zbl

[69] S. Nicaise; K. Witowski; B. Wohlmuth An a posteriori error estimator for the Lamé equation based on $H (div)$ -conforming stress approximations, IMA J. Numer. Anal., Volume 28 (2008), pp. 331-353

[70] T. Nilssen; X.-C. Tai; R. Winther A robust nonconforming $H^{2}$ -element, Math. Comput., Volume 70 (2001) no. 234, pp. 489-505 | DOI | MR | Zbl

[71] J. Nitsche Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, Volume 36, Springer (1971), pp. 9-15 | DOI | Zbl

[72] S. Olver; Y. Xu Orthogonal structure on a wedge and on the boundary of a square, Found. Comput. Math., Volume 19 (2019) no. 3, pp. 561-589 | DOI | MR

[73] J. E. Pasciak; Y. Wang A multigrid preconditioner for the mixed formulation of linear plane elasticity, SIAM J. Numer. Anal., Volume 44 (2006) no. 2, pp. 478-493 | DOI | MR | Zbl

[74] D. Pauly; W. Zulehner The divDiv-complex and applications to biharmonic equations, Appl. Anal., Volume 99 (2020) no. 9, pp. 1579-1630 | DOI | MR | Zbl

[75] R. Rannacher; S. Turek Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differ. Equations, Volume 8 (1992) no. 2, pp. 97-111 | DOI | MR | Zbl

[76] F. Rathgeber; D. A. Ham; L. Mitchell; M. Lange; F. Luporini; A. T. T. McRae; G.-T. Bercea; G. R. Markall; P. H. J. Kelly Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Softw., Volume 43 (2016) no. 3, pp. 1-27 | DOI | MR | Zbl

[77] P.-A. Raviart; J.-M. Thomas A mixed finite element method for 2 $^{nd}$ order elliptic problems, Mathematical Aspects of Finite Element Methods, Springer, 1977, pp. 292-315 | DOI | Zbl

[78] R. Riedlbeck; D. A. Di Pietro; A. Ern; S. Granet; K. Kazymyrenko Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis, Comput. Math. Appl., Volume 73 (2017) no. 7, pp. 1593-1610 | DOI | MR | Zbl

[79] M. E. Rognes; R. C. Kirby; A. Logg Efficient assembly of $H (div)$ and $H (curl)$ conforming finite elements, SIAM J. Sci. Comput., Volume 31 (2009) no. 6, pp. 4130-4151 | DOI | MR | Zbl

[80] J. Schöberl Robust multigrid methods for parameter dependent problems, Ph. D. Thesis, Johannes Kepler University (1999) (https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/diss.pdf)

[81] J. Schöberl; A. S. Sinwel Tangential-Displacement and Normal-Normal-Stress continuous mixed finite Elements for elasticity (2007) (https://ricamwww.ricam.oeaw.ac.at/files/reports/07/rep07-10.pdf) (Technical report)

[82] D. Shapero Nitsche’s method, https://shapero.xyz/posts/nitsches-method/, 2019 (Accessed 2020.11.16)

[83] D. Silvester; A. Wathen Fast iterative solution of stabilised Stokes systems Part II: Using general block preconditioners, SIAM J. Numer. Anal., Volume 31 (1994) no. 5, pp. 1352-1367 | DOI | Zbl

[84] X.-C. Tai; R. Winther A discrete de Rham complex with enhanced smoothness, Calcolo, Volume 43 (2006) no. 4, pp. 287-306 | DOI | MR | Zbl

[85] Y. Wang Preconditioning for the mixed formulation of linear plane elasticity, Ph. D. Thesis, Texas A&M University (2004) (https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/2781/etd-tamu-2004B-MATH-Wang.pdf)

[86] S. Wu; J. Xu Nonconforming finite element spaces for $2 m$ th order partial differential equations on $ℝ^{n}$ simplicial grids when $m = n + 1$ , Math. Comput., Volume 88 (2019) no. 316, pp. 531-551 | DOI

[87] J. Xia; P. E. Farrell; F. Wechsung Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals, BIT Numer. Math. (2021), pp. 1-38 | DOI | MR | Zbl

[88] J. Xu Iterative methods by space decomposition and subspace correction, SIAM Rev., Volume 34 (1992) no. 4, pp. 581-613 | DOI | MR | Zbl

[89] S-Y Yi Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions, Calcolo, Volume 42 (2005) no. 2, pp. 115-133 | DOI | MR | Zbl

[90] Software used in ‘Transformations for Piola-mapped elements’, 2021 (https://doi.org/10.5281/zenodo.5596313) | DOI

Cited by Sources: