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Two fields mixed formulation to manage incompressibility and nearly-incompressibility

11 February 2011

by S. Fayolle, EDF R&D / AMA

Till now, to deal with incompressibility users had to use the 3 variables mixed finite elements (displacement / pressure / volumetric strain) : see the modelizations 3D_INCO, AXIS_INCO et D_PLAN_INCO.

A new formalism has just been introduced which improves the offer of the previous elements : a 2 variables mixed finite elements (displacement / pressure). They are less rich, but much more efficient for the big sized models required by accurate simulations.

In the context of a J2-plasticity (ex : von Mises, Tresca, …), a one-to-one relation exists between pressure and volumetric strain which permits the elimination of the latter. We therefore obtain an important decrease in the number of DOF and consequently a dramatic decrease of computational time.

As in every mixed formulation, the degree of interpolation of the variables is not arbitrary and must satisfy the so-called LBB condition. A wide range of interpolations families compatible with LBB conditions exist. We have chosen the classic one which consist in a quadratic interpolation of displacements and a linear one for pressures.

In addition to this classical interpolation and in order to reduce the computation times, we implemented a stabilized finite element, called « mini-element ». The geometrical structures compliant with this element are the triangle (2D) and the tetrahedron (3D). Interpolations for displacement and pressure are both linear. To stabilize the element and thus to fulfill the LBB condition, an additional displacement DOF, called « bubble » is added in the center of each element. This DOF is not explicitly represented because it naturally condenses itself in the formulation. Then, we obtain a very costless element.

Users can access this element with the new modelizations : 3D_INCO_UP, AXIS_INCO_UP et D_PLAN_INCO_UP.

In term of performance, for a given mesh :

• P2P1 element (3D_INCO_UP) reduces by about 15% computing time compared to P2P1P1 element (3D_INCO) ;
• P1+P1 element (3D_INCO_UP) reduces by about 85% computing time compared to P2P1P1 element (3D_INCO).

For now, this new formalism is only compatible with small strains (DEFORMATION=’PETIT’). The extension to finite strains is an ongoing work. All the behavior laws derivated from linear elasticity and J2 plasticity are compatible. Except in the case of a linear mesh (mini-element). Indeed, in this case, the formulation is limited yet to ELAS and VMIS_ISOT_XXX laws.

Exemple : Simulation of the crush of an elastic nearly-incompressible material (ν=0,4995):

• Mesh with linear tetrahedrons :
 Standard formalism 2 fields mixed formalism (XXX_INCO_UP)
• Mesh with quadratic tetrahedrons :
 Formalisme standard 3 fields mixed formalism (XXX_INCO) 2 fields mixed formalism (XXX_INCO_UP)