[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A discrete elasticity complex on three-dimensional Alfeld splits

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alfeld, P.: A trivariate Clough-Tocher scheme for tetrahedral data. Computer Aided Geometric Design 1(2), 169–181 (1984)

    Article  Google Scholar 

  2. Amstutz, S., Van Goethem, N.: The incompatibility operator: from Riemann’s intrinsic view of geometry to a new model of elasto-plasticity. In: Hintermuller, M., Rodrigues, J. (eds.) Topics in Applied Analysis and Optimisation, pp. 33–70. Springer, Berlin (2019)

    Chapter  Google Scholar 

  3. Angoshtari, A., Yavari, A.: Differential complexes in continuum mechanics. Arch. Ration. Mech. Anal. 216(1), 193–220 (2015)

    Article  MathSciNet  Google Scholar 

  4. Arnold, D., Awanou, G., Winther, R.: Finite elements for symmetric tensors in three dimensions. Math. Comput. 77(263), 1229–1251 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  5. Arnold, D., Falk, R., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76(260), 1699–1723 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  6. Arnold, D.N.: Finite Element Exterior Calculus. SIAM, New Delhi (2018)

    Book  Google Scholar 

  7. Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods II: The elasticity complex. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds.) Compatible Spatial Discretizations, pp. 47–67. Springer, Berlin (2006)

    Chapter  Google Scholar 

  8. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  9. Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76(260), 1699–1723 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Arnold, D. N., Hu, K.: Complexes from complexes. arXiv preprint arXiv:2005.12437, (2020)

  11. Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92(3), 401–419 (2002)

    Article  MathSciNet  Google Scholar 

  12. Calabi, E.: On compact, Riemannian manifolds with constant curvature I. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1961, pages 155–180, (1961)

  13. Čap, A., Slovák, J., Souček, V.: Bernstein-Gelfand-Gelfand sequences. Ann. Math. 154(1), 97–113 (2001)

    Article  MathSciNet  Google Scholar 

  14. Christiansen, S.H.: On the linearization of Regge calculus. Numer. Math. 119(4), 613–640 (2011)

    Article  MathSciNet  Google Scholar 

  15. Christiansen, S.H., Hu, J., Hu, K.: Nodal finite element de Rham complexes. Numer. Math. 139(2), 411–446 (2018)

    Article  MathSciNet  Google Scholar 

  16. Christiansen, S. H., Hu, K.: Finite element systems for vector bundles: elasticity and curvature. arXiv preprint arXiv:1906.09128, (2019)

  17. Ciarlet, P.G., Gratie, L., Mardare, C.: Intrinsic methods in elasticity: a mathematical survey. Discrete Contin. Dyn. Syst. A 23(1 &2), 133 (2009)

    MathSciNet  Google Scholar 

  18. Costabel, M., McIntosh, A.: On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265(2), 297–320 (2010)

    Article  MathSciNet  Google Scholar 

  19. Eastwood, M.: A complex from linear elasticity. In: Proceedings of the 19th Winter School" Geometry and Physics", pp. 23–29. Circolo Matematico di Palermo, (2000)

  20. Farin, G.: Bézier polynomials over triangles. Technical Report TR/91, Department of Mathematics, Brunel University, Uxbridge, UK, (1980)

  21. Farrell, P. E., Mitchell, L., Scott, L. R., Wechsung, F.: A Reynolds-robust preconditioner for the Reynolds-robust Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations. arXiv preprint arXiv:2004.09398, (2020)

  22. Fu, G., Guzman, J., Neilan, M.: Exact smooth piecewise polynomial sequences on Alfeld splits. Mathematics of Computation, (2020)

  23. Goethem, N.V.: The non-Riemannian dislocated crystal: a tribute to Ekkehart Kröner (1919–2000). J. Geom. Mech. 2, 303 (2010)

    Article  MathSciNet  Google Scholar 

  24. Gopalakrishnan, J.: Oberwolfach Reports: MFO Workshop 2225. MFO, (2022). Workshop on “Hilbert Complexes: Analysis, Applications, and Discretizations” held 19 Jun – 25 Jun 2022. https://doi.org/10.14760/OWR-2022-29

  25. Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32(1), 352–372 (2012)

    Article  MathSciNet  Google Scholar 

  26. Guzmán, J., Neilan, M.: Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions. SIAM J. Numer. Anal. 56(5), 2826–2844 (2018)

    Article  MathSciNet  Google Scholar 

  27. Hauret, P., Kuhl, E., Ortiz, M.: Diamond elements: a finite element/discrete-mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity. Int. J. Numer. Meth. Eng. 72(3), 253–294 (2007)

    Article  MathSciNet  Google Scholar 

  28. Hu, J., Zhang, S.: A family of conforming mixed finite elements for linear elasticity on triangular grids. arXiv preprint arXiv:1406.7457, (2014)

  29. Hu, J., Zhang, S.: A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. 58(2), 297–307 (2015)

    Article  MathSciNet  Google Scholar 

  30. Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math. 30(1), 103–116 (1978)

    Article  MathSciNet  Google Scholar 

  31. Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rational Mech. Anal. 4(273–334), 1960 (1960)

    MathSciNet  Google Scholar 

  32. Lee, Y.-J., Wu, J., Xu, J., Zikatanov, L.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937–1963 (2007)

    Article  MathSciNet  Google Scholar 

  33. Li, L.: Regge finite elements with applications in solid mechanics and relativity. PhD thesis, University of Minnesota, (2018)

  34. Nédélec, J.-C.: A new family of mixed finite elements in \({ R}^3\). Numer. Math. 50(1), 57–81 (1986)

    Article  MathSciNet  Google Scholar 

  35. Neunteufel, M., Schöberl, J.: Avoiding Membrane Locking with Regge Interpolation. Preprint: arXiv:1907.06232, (2020)

  36. Regge, T.: General relativity without coordinates. Il Nuovo Cimento (1955-1965) 19(3), 558–571 (1961)

    Article  MathSciNet  Google Scholar 

  37. Schöberl, J.: Robust multigrid methods for parameter dependent problems. PhD thesis, Johannes Kepler Universität Linz, (1999)

  38. Stenberg, R.: A nonstandard mixed finite element family. Numer. Math. 115(1), 131–139 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Snorre H. Christiansen

  2. Portland State University (MTH), PO Box 751, Portland, OR, 97207, USA

    Jay Gopalakrishnan

  3. Division of Applied Mathematics, Brown University, Box F, 182 George Street, Providence, RI, 02912, USA

    Johnny Guzmán

  4. School of Mathematics, the University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, UK

    Kaibo Hu

Authors
  1. Snorre H. Christiansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jay Gopalakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Johnny Guzmán
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kaibo Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johnny Guzmán.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by the National Science Foundation (USA) under grants 1913083, 2245077, and 1912779, and a Royal Society University Research Fellowship (URF\(\backslash \)R1\(\backslash \)221398). The first and last two authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Geometry, compatibility and structure preservation in computational differential equations (EPSRC grant number EP/R014604/1).

Appendix A: Supersmoothness

Appendix A: Supersmoothness

Consider a tetrahedron T and its Alfeld split \(\{ T_i\}\) as in the rest of the paper. Proposition 2.1’s items (1) and (2) are a consequence of the following fact proved in [1]: if \(v \in C^1(T)\) and \(v|_{T_i}\) is in \(C^\infty (T_i)\), then v is \(C^2\) at the vertices of T. Such serendipitous “supersmoothness” at some points was observed on triangles earlier [20]. In Theorem A.1 below, we establish a supersmoothness result in the same spirit for 1-forms. In fact, the earlier result of Alfeld follows from the theorem, as noted in Corollary A.2. Items (3) and (4) of Proposition 2.1 follow from the arguments below. (The proof will show that the assumption that \(v|_{T_i}\) is infinitely smooth can be relaxed, but this generalization is not important for our purposes.)

Theorem A.1

Suppose v is in \(C^0(T)^3\), \(v_i = v|_{T_i} \in C^\infty (T_i)^{{3}}\), and \({\mathop {\textrm{curl}\,}}v\) is \(C^0\) at the vertices \(x_i\) of T. Then v is \(C^1\) at \(x_i\).

Proof

Let \(F_{ij} = \partial T_i \cap \partial T_j\) and let \(T F_{ij}\) denote the tangent plane of \(F_{ij}\). Let \(c \in {{\mathbb {V}}}\) and \(\tau \in TF_{ij}\). The first observation needed for this proof is that

$$\begin{aligned} c \cdot ({\mathop {\textrm{ grad}\,}}v_i) \tau = c \cdot ({\mathop {\textrm{ grad}\,}}v_j) \tau \quad \text { on } F_{ij}. \end{aligned}$$
(A.1)

This is because the continuity of v requires \((v _i - v_j)\cdot c\) to vanish on \(F_{ij}\) for any \(c \in {{\mathbb {V}}}\), so its tangential derivatives also vanish on \(F_{ij}\).

We claim that at a vertex of T on \(F_{ij}\), we also have

$$\begin{aligned} \tau \cdot ({\mathop {\textrm{ grad}\,}}v_i) c = \tau \cdot ({\mathop {\textrm{ grad}\,}}v_j) c. \end{aligned}$$
(A.2)

To show this, consider \(x_1\), a common vertex of T and \(F_{23}\). Then, since the scalar \(\tau \cdot ({\mathop {\textrm{ grad}\,}}v_i)(x_1) \, c\) equals its transpose, we have

$$\begin{aligned} \tau \cdot ({\mathop {\textrm{ grad}\,}}v_2)\, c&= c \cdot ({\mathop {\textrm{ grad}\,}}v_2)' \tau{} & {} \text { at } x_1, \\&= c \cdot \left( ({\mathop {\textrm{ grad}\,}}v_2)' - ({\mathop {\textrm{ grad}\,}}v_2) \right) \tau + c \cdot ({\mathop {\textrm{ grad}\,}}v_2) \tau \\&= c \cdot \left( ({\mathop {\textrm{ grad}\,}}v_2)' - ({\mathop {\textrm{ grad}\,}}v_2) \right) \tau + c \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau{} & {} \text { by }(\text {A.1}), \\&= c \cdot \left( ({\mathop {\textrm{ grad}\,}}v_3)' - ({\mathop {\textrm{ grad}\,}}v_3) \right) \tau + c \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau{} & {} \text { as }{\mathop {\textrm{curl}\,}}v\text { is }C^0\text { at }x_1 \\&= \tau \cdot ({\mathop {\textrm{ grad}\,}}v_3)\, c. \end{aligned}$$

This argument can be repeated at other vertices to finish the proof of (A.2).

Now we are ready to show that v is \(C^1\) at \(x_i\). Let \(\tau _i = (x_i - z) / \Vert x_i - z\Vert \) and \(\tau _{ij} = (x_i - x_j) / \Vert x_i - x_j \Vert \). Without loss of generality, we focus on one vertex, say \(x_1\). At \(x_1\),

$$\begin{aligned} c \cdot ({\mathop {\textrm{ grad}\,}}v_2) \tau _0&= c \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau _0&\tau _0 ({\mathop {\textrm{ grad}\,}}v_2) c&= \tau _0 ({\mathop {\textrm{ grad}\,}}v_3) c \end{aligned}$$
(A.3a)
$$\begin{aligned} c \cdot ({\mathop {\textrm{ grad}\,}}v_2) \tau _{10}&= c \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau _{10}&\tau _{10} ({\mathop {\textrm{ grad}\,}}v_2) c&= \tau _{10} ({\mathop {\textrm{ grad}\,}}v_3) c. \end{aligned}$$
(A.3b)

The left equalities follow from (A.1) and the right ones from (A.2). Furthermore, at \(x_1\) we have

$$\begin{aligned} \tau _{12} \cdot ({\mathop {\textrm{ grad}\,}}v_2 ) \tau _{13}&= \tau _{12} \cdot ({\mathop {\textrm{ grad}\,}}v_0) \tau _{13}{} & {} \text {by } (\text {A.1})\text { applied to }F_{20} \\&= \tau _{12} \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau _{13}{} & {} \text {by }(\text {A.2}) \text {applied to }F_{03}. \end{aligned}$$

Therefore, \(\tau _{12} \cdot ({\mathop {\textrm{ grad}\,}}v_2 )\tau _{13} = \tau _{12} \cdot ({\mathop {\textrm{ grad}\,}}v_3 )\tau _{13}\) at \(x_1\). Writing \(\tau _{12}\) as a linear combination of \(\tau _0, \tau _{10}\) and \(\tau _{13}\), and using the equalities of (A.3) in the right panel, we conclude that

$$\begin{aligned} \tau _{13} \cdot ({\mathop {\textrm{ grad}\,}}v_2) \tau _{13} = \tau _{13} \cdot ({\mathop {\textrm{ grad}\,}}v_3) \tau _{13} \qquad \text { at } x_1. \end{aligned}$$
(A.4)

The identities of (A.4) and (A.3) together yield the equality of \({\mathop {\textrm{ grad}\,}}v_2\) and \({\mathop {\textrm{ grad}\,}}v_3\) at \(x_1\). Repeating this argument for every pair of \(v_i\) meeting at a vertex, the proof is finished. \(\square \)

Corollary A.2

If \(w \in C^1(T)\) and \(w|_{T_i}\) is in \(C^\infty (T_i)\), then w is \(C^2\) at the vertices of T.

Proof

This follows by applying Theorem A.1 with \(v = {\mathop {\textrm{ grad}\,}}w\). \(\square \)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Christiansen, S.H., Gopalakrishnan, J., Guzmán, J. et al. A discrete elasticity complex on three-dimensional Alfeld splits. Numer. Math. 156, 159–204 (2024). https://doi.org/10.1007/s00211-023-01381-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-023-01381-9

Mathematics Subject Classification

Search

Navigation