Computational Mechanics

, Volume 55, Issue 3, pp 543–559 | Cite as

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

  • N. M. LafontaineEmail author
  • R. Rossi
  • M. Cervera
  • M. Chiumenti
Original Paper


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.


Explicit Mixed formulation  Displacement-strain formulation 


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

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.


Green-Lagrange strain tensor.


Almansi strain tensor.


Gradient deformation tensor.


Vector of displacement shape functions.


Displacement field.


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.

\(\nabla (\bullet )\)

Gradient operator.

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

Divergence operator.

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

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

\(\rho \)


\(\tau _{\varepsilon }\)

Stabilization parameter.

\(\upsilon \)

Poisson’s ratio.

\(\widetilde{\varepsilon }\)

Enhancement strain.


Young’s modulus.


Finite element characteristic length.


Minimum element length in the finite element mesh.



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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

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

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