Computational Mechanics

, Volume 55, Issue 3, pp 543–559

# Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

• N. M. Lafontaine
• R. Rossi
• M. Cervera
• M. Chiumenti
Original Paper

## Abstract

Low-order finite elements face inherent limitations related to their poor convergence properties. Such difficulties typically manifest as mesh-dependent or excessively stiff behaviour when dealing with complex problems. A recent proposal to address such limitations is the adoption of mixed displacement-strain technologies which were shown to satisfactorily address both problems. Unfortunately, although appealing, the use of such element technology puts a large burden on the linear algebra, as the solution of larger linear systems is needed. In this paper, the use of an explicit time integration scheme for the solution of the mixed strain-displacement problem is explored as an alternative. An algorithm is devised to allow the effective time integration of the mixed problem. The developed method retains second order accuracy in time and is competitive in terms of computational cost with the standard irreducible formulation.

## Keywords

Explicit Mixed formulation  Displacement-strain formulation

## Nomenclature

$$(\bullet ):(\bullet )$$

Double contraction of tensor (inner product).

$$(\bullet )\cdot (\bullet )$$

Single contraction of vector and tensor.

$$(\bullet )^T$$

Vector, matrix transpose.

$$(\bullet )_n$$

Variable $$(\bullet )$$ at time $$t_n$$.

$$\varvec{\gamma }$$

Vector of strain weighting function.

$$\varvec{\sigma }$$

Stress tensor.

$$\varvec{\varepsilon }$$

Small strain tensor.

$$\varvec{E}$$

Green-Lagrange strain tensor.

$$\varvec{e}$$

Almansi strain tensor.

$$\varvec{F}$$

$$\varvec{N}$$

Vector of displacement shape functions.

$$\varvec{u}$$

Displacement field.

$$\varvec{w}$$

Vector of displacement weighting functions.

$$\ddot{(\bullet )}$$

Second derivative of $$(\bullet )$$ with respect to time $$t$$.

$$\Delta t$$

Time step.

$$\delta _{ij}$$

Kronecker’s symbol.

$${\mathcal {P}}$$

Projection operator.

$${\mathcal {P}}^{\perp }$$

Orthogonal projection operator.

$$\nabla (\bullet )$$

$$\nabla \cdot (\bullet )$$

Divergence operator.

$$\nabla ^s(\bullet )$$

Symmetric gradient operator $$\nabla ^s(\bullet ) \!=\! \frac{1}{2}(\nabla (\bullet ) \!+\! \nabla (\bullet )^T)$$.

$$\rho$$

Density.

$$\tau _{\varepsilon }$$

Stabilization parameter.

$$\upsilon$$

Poisson’s ratio.

$$\widetilde{\varepsilon }$$

Enhancement strain.

$$E$$

Young’s modulus.

$$h^e$$

Finite element characteristic length.

$$l_{min}^e$$

Minimum element length in the finite element mesh.

## Notes

### Acknowledgments

Nelson Lafontaine thanks to MAEC-AECID scholarships for the financial support given. Funding from the Seventh Framework Programme (FP7/2007–2013) of the ERC under grant agreement n$$^\circ$$ 611636 (NUMEXAS) has helped the development of this project. The authors also wish to thank Mr. Pablo Becker for his help in the revision process.

## References

1. 1.
de Souza NetoEA, Andrade PiresFM, Owen DRJ (2005) F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking. Int J Numer Methods Eng 62(3):353–383
2. 2.
Gil Antonio J, Hean Lee Chun, Bonet Javier, Aguirre Miquel (2014) Stabilised petrov-galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics. Comput Methods Appl Mech Eng 276:659–690
3. 3.
David S Malkus, Hughes Thomas JR (1978) Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81
4. 4.
Simo JC, Taylor RL, Pister KS (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51(1–3):177–208
5. 5.
Taylor Robert L (2000) A mixed-enhanced formulation tetrahedral finite elements. Int J Numer Methods Eng 47(1–3):205–227
6. 6.
Scovazzi G (2012) Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach. J Comput Phys 231(24):8029–8069
7. 7.
Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: part I: formulation. Comput Methods Appl Mech Eng 199(37–40):2559–2570
8. 8.
Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: part II: Strain localization. Comput Methods Appl Mech Eng 199(37–40):2571–2589
9. 9.
Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191(39–40):4295–4321Google Scholar
10. 10.
Donea Jean, Huerta Antonio (2003) Finite element methods for flow problems, 1st edn. Wiley, Chichester
11. 11.
Hughes Thomas JR (1995) Multiscale phenomena: Green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127(1–4):387–401
12. 12.
Oñate E, Valls A, García J (2006) FIC/FEM formulation with matrix stabilizing terms for incompressible flows at low and high reynolds numbers. Comput Mech 38(4–5):440–455
13. 13.
Hughes Thomas J R, Scovazzi Guglielmo, Franca Leopoldo P (2004) Multiscale and stabilized methods. Wiley, New YorkGoogle Scholar
14. 14.
Hughes Thomas JR, Feijóo Gonzalo R, Mazzei Luca, Quincy Jean-Baptiste (1998) The variational multiscale method-a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1—-2):3–24 Advances in Stabilized Methods in Computational Mechanics
15. 15.
Cervera M, Chiumenti M, Codina R (2011) Mesh objective modeling of cracks using continuous linear strain and displacement interpolations. Int J Numer Methods Eng 87(10):962–987
16. 16.
Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New York
17. 17.
Har Jason, Fulton Robert E (2003) A parallel finite element procedure for contact-impact problems. Eng Comput 19:67–84. doi:
18. 18.
Dadvand P, Rossi R, Gil M, Martorell X, Cotela J, Juanpere E, Idelsohn SR, Oñate E (2013) Migration of a generic multi-physics framework to HPC environments. Comput Fluids 80(0):301–309
19. 19.
Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17(3):253–297Google Scholar
20. 20.
GiD (2009) The personal pre and post processorGoogle Scholar
21. 21.
Brezzi F, Fortin M, Marini D (1991) Mixed finite element methods. Springer, New York
22. 22.
Codina R (2000) Stabilization of incompresssibility and convection through orthogonal sub-scales in finite elements methods. Comput Meth Appl Mech Eng 190:1579–1599
23. 23.
Codina R (2008) Analysis of a stabilized finite element approximation of the oseen equations using orthogonal subscales. Appl Numer Math 58(3):264–283Google Scholar
24. 24.
Larese A, Rossi R, Oñate E, Idelsohn SR (2012) A coupled pfem-eulerian approach for the solution of porous FSI problems. Comput Mech 50(6):805–819
25. 25.
Larese A, Rossi R, Oñate E, Toledo M, Morán R, Campos H. Numerical and experimental study of overtopping and failure of rockfill dams. International Journal of Geomechanics, 0(0):04014060, 0Google Scholar
26. 26.
Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamentals, 6th edn. Butterworth-Heinemann, OxfordGoogle Scholar
27. 27.
Zienkiewicz, OC, Taylor, RL, Baynham, JAW (1983) Mixed and irreducible formulations in finite element analysis, in hybrid and mixed finite element methods. In: Atlury SN, Gallagher RH, Zienkiewicz OC (eds.). Wiley, New YorkGoogle Scholar
28. 28.
Arnold Douglas N (1990) Mixed finite element methods for elliptic problems. Comput Methods Appl Mech Eng 82(1—-3):281–300 Proceedings of the Workshop on Reliability in Computational Mechanics
29. 29.
Arnold Douglas N, Winther Ragnar (2003) Mixed finite elements for elasticity in the stress-displacement formulation. Contemp Math 239:33–42 n Z. Chen, R. Glowinski, and K. Li, editors, Current trends in scientific computing (Xi’an, 2002)
30. 30.
Arnold Douglas N, Winther Ragnar (2002) Mixed finite elements for elasticity. Numerische Mathematik 92(3):401–419
31. 31.
Oñate E, Rojek J, Taylor RL, Zienkiewicz OC (2004) Finite calculus formulation for incompressible solids using linear triangles and tetrahedra. Int J Numer Methods Eng 59(11):1473–1500Google Scholar

## Authors and Affiliations

• N. M. Lafontaine
• 1
• 2
Email author
• R. Rossi
• 1
• 2
• M. Cervera
• 1
• 2
• M. Chiumenti
• 1
• 2
1. 1.Universidad Politècnica de Catalunya (UPC)BarcelonaSpain
2. 2.Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)Universidad Politècnica de Catalunya (UPC)BarcelonaSpain