On the use of mixed formulations for computational polyconvexity and multi-variable convexity

Antonio J. Gil (University of Swansea, UK)

08-Mar-2021, 14:40-16:10 (3 years ago)

Abstract: The computational modelling of large strain multi-physics problems is nowadays an area of intensive research by numerous scientists. Among problems of this kind, two pressing applications underpin the motivation behind this presentation. On the one hand, the in-silico design of highly flexible dielectric elastomers for soft robotics applications and, on the other hand, the realistic simulation of the electromechanical interactions of the cardiac muscle from the physiological and pathological standpoints. In both applications, the requirement for extremely accurate and stable results at, possibly, large values of the field variables (i.e. strains and electric/cardiac potential) proves very challenging from the computational point of view. The presentation starts by revisiting the now well-accepted requirement of polyconvexity (a sufficient condition for rank-one convexity) in the case of reversible hyperelasticity, where the strain energy functional is described by a convex combination of the minors of the deformation gradient tensor. We then extend this concept to the case of energies which depend on non-strain based variables, for instance, other physical measures such as the electric displacement, leading to the introduction of a new computational paradigm, Multi-Variable Convexity. Crucially, this new definition of the internal energy enables ellipticity to be extended to the entire range of deformations and electric fields and, thus, spurious numerical instabilities can then effectively be removed from the model whilst maintaining real physical instabilities. The polyconvexity of the energy functional with respect to its set of arguments guarantees the definition of well-posed counterpart work conjugates, where a one to one (invertible) relationship can be established. This ensures the existence of alternative energy potentials via the use of partial or total Legendre transforms and opens up the possibility for new extended Hu–Washizu-type variational principles in terms of using various interpolation spaces for the different unknown fields, leading to enhanced type of formulations. A series of examples will be presented demonstrating the potential of this new paradigm when coupled with the use of Finite Element type discretisation techniques.

MathematicsPhysics

Audience: researchers in the topic

( video )


Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor).

Organizers: Miloslav Feistauer, Petr Knobloch, Martin Kružík*, Šárka Nečasová*
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