2. Choice a priory#

2.1. The mesh#

The elements can be either:

  • triangular elements or quadrangles in 2D,

  • tetrahedra or hexahedra in 3D.

In fact, contrary to the often widespread idea, elements of the triangle or tetrahedron type give good results, even in terms of plasticity, provided of course that they do not use a mesh that is too coarse. It is also possible to use the software HOMARD which adapts 2D/3D meshes for triangular, quadrangular, tetrahedral or hexahedral finite elements by refinement and deraffination. It is thus possible to obtain the optimum mesh as a function of an error indicator (cf. [R4.10.01], [R4.10.02], [R4.10.03], or the test case TPLL01j [V4.02.01] for a demonstration) by calling the command MACR_ADAP_MAIL in the command file*Code_Aster*.

On the other hand, it is recommended to use:

  • linear thermal elements for chained calculations and fast transient thermal calculations. For other cases, we can also opt for quadratic elements,

  • quadratic elements in mechanics.

This choice is all the more important when performing thermal and then mechanical chained calculations. It is then necessary to use two different meshes for thermal and mechanical purposes. Two strategies are then possible:

  • or mesh the structure independently for thermal calculation and for mechanical calculation

  • or create a mesh with linear elements and then transform it into a quadratic mesh using the command CREA_MAILLAGE, keyword factor LINE_QUAD.

Regardless of the method chosen, each mesh can be optimized separately with*Lobster* thanks to the thermal and mechanical error indicators available in Aster (cf. test case forma05b [V6.03.120]).

Note:

We recall here that all stress or deformation quantities are calculated at Gauss points, and that any passage through the nodes leads to a bias. This is all the more true when we then try to calculate standards; we thus noticed that tetrahedra were more sensitive than hexahedra to the method of calculating equivalent stresses for example. It is therefore necessary to have an even more critical eye on the results calculated at the nodes.

2.2. Modeling#

Whether for the resolution of thermal or mechanical problems, several models are available in Code_Aster. These different models can be distinguished by the number or type of degrees of freedom, the number of integration points, particular treatments… Depending on the calculation carried out, some are of course more suitable than others.

2.2.1. In thermal#

To do a thermal calculation with Code_Aster, two types of models are available ([U3.23.01], [U3.24.01], [R3.06.02], [R3.06.07]):

  • classical finite elements: 3D modeling, AXIS or PLAN

  • lumped or diagonalized finite elements: 3D modeling DIAG, AXIS_DIAG or PLAN_DIAG

As a choice by default, we offer:

modelling XXXX_DIAG with linear elements

Justice

In thermal, the time step \(\Delta t\) cannot be any one, it must satisfy a condition \(\Delta {t}_{\mathrm{min}}<\Delta t<\Delta {t}_{\mathrm{max}}\), \(\Delta {t}_{\mathrm{min}}\) and \(\Delta {t}_{\mathrm{max}}\) depending on material properties, the size of the finite elements and the time integration parameters (cf. [R3.06.07]).

In the case of rapid transient thermal problems, it may be necessary to use a time step that is too small. One can then observe oscillations of the solution and non-physical temperatures due to the violation of the principle of the maximum (temperature higher than the initial temperature of a room that is being cooled). Modeling DIAG, which consists in diagonalizing the mass matrix, makes it possible to get rid of the condition on \(\Delta {t}_{\mathit{min}}\) and to avoid the associated problems.

However, it should be noted that this diagonalization is not sufficient to eliminate oscillations in all configurations (cf. [R3.06.07]). It does not guarantee non-oscillation with quadratic elements for example. This is why linear elements are recommended.

2.2.2. In mechanics#

Four types of models are available to solve non-linear mechanical problems using « classical » laws of behavior (of the elasto-plasticity type):

  • classical isoparametric finite elements: 3D, D_ PLAN, C_, C_ PLAN, AXIS ([U3.14.01], [U3.13.01]),

  • sub-integrated elements: 3D_SI, D_ PLAN_SI, C_, C_ PLAN_SI, AXIS_SI ([U3.14.01], [U3.13.05]),

  • elements based on a nearly incompressible formulation with 3 fields (displacement, swelling, pressure): 3D_ INCO_UPG, D_ PLAN_INCO_UPG, AXIS_INCO_UPG ([U3.14.06], [U3.13.07], [], [R3.06.08]),

  • elements based on an incompressible formulation with 2 fields (displacement, pressure): 3D_ INCO_UP, D_ PLAN_INCO_UP and AXIS_INCO_UP for small deformations and large deformations (GDEF_LOG).

As a prima facie choice, we suggest using:

quadratic elements

As for the choice of modeling, it depends on the type of elements and the need to treat the incompressibility condition. These considerations are summarized in the table below.

normal

almost incompressible (high plasticity or \(\nu >0.45\) )

triangles/tetraheders

standard

INCO

quadrilaterals/hexaheders

SI

SI or INCO

Justifications and precautions:

  • If the material is almost incompressible (\(\nu >\mathrm{0,45}\)), it is preferable to use one of the INCO formulations, because the standard formulation on the move does not give good results.

The plastic flow is at a constant volume. This condition of incompressibility can cause difficulties with classical modeling, namely too rigid behavior and especially the appearance of oscillations at the stress level. Sub-integration makes it possible to improve these problems, because the incompressibility condition is then verified in fewer Gauss points. However,**only the elements QUAD8 and HEXA20 are really under-integrated, * for the other meshes, the classic integration is maintained. Consequently, when oscillatory phenomena are observed for a mesh composed of triangles or tetrahedra, it is preferable to use one of the formulations INCO. This significantly improves the results but the calculations will take longer.

  • In the general case, sub-integrated modeling gives as good results as classical finite elements, and this for a faster calculation time since fewer Gauss points are used. In addition, in the case of thermo-mechanical calculations, this makes it possible to limit the difficulties during the transition from thermal deformation to mechanical calculation when the refinements of the thermal and mechanical meshes differ. However, under-integration can sometimes lead to the appearance of parasitic modes. If, at the end of the calculation, the deformation presents this kind of non-physical deformation modes, it is better to do the calculation with classical or almost incompressible modeling if the levels of plasticity are very high.