Volume element

In this section, Metafor volume element are described. A FieldApplicator interaction is associated to them.

Volume2DElement and Volume3DElement are basic elements in Metafor. Volume2DElement is a quadrangle with 4 nodes in 2D, while Volume3DElement is a hexaedron with 8 nodes in 3D. Each element has respectively 8 and 24 degrees of freedom.

By default, if $n$ is the degree of mechanical interpolation and $d$ the element dimension, the number of integration points in the deviatoric part of the stress field is $(n+1)^{d}$, unless stated otherwise with the definition of NIPDKSI, NIPDETA or NIPDZETA (see Parameters).
For example, the interpolation of Volume2DElement is of the first degree ($n=1$), and it is a 2D element ($d=2$), leading to 4 integration points.

The number of integration points in the volume part of the stress field (pressure) depend on the chosen method to integrate stresses and cannot be modified.

• The angular velocity OMEGA is used when loading comes from centrifugal forces (whatever the integration scheme).
• Both start and end points of the rotation axis can be moved (see fixation and loading).

• When MDE_IQSI=1 and MDE_NDYN=2, centrifugal forces are computed during a quasi-static equilibrium phase. Then, they are replaced by the real structural rotation.
• If MDE_IQSI=2 and MDE_NDYN=2: The quasi-static equilibrium phase is applied, but the frame of the blade is still used.

* MDE_IQSI=0 :

ti.setUseInitialRotationBalancing(False) 

* MDE_IQSI=1 :

ti.setUseInitialRotationBalancing(True) 
ti.setShiftToFixedFrame(True) 

* MDE_IQSI=2 :

ti.setUseInitialRotationBalancing(True) 
ti.setShiftToFixedFrame(False) 

* MDR_DTMR :

ti.setInitialRotationFactorIncrement(_initialRotationFactorIncrement)

ti is a reference towards the current dynamic time integration scheme.

• In axisymmetric modeling, Selective Reduced Integration with Pressure Report VES_CMVIM_SRIPR must be used.
• EAS Formulation is not valid in axisymmetric modeling, some EAS modes are missing along $\theta$.

Thermomechanical formulation of Volume[2][3]DElement.

For the mechanical part, linear elements are still used, so a 4-nodes quadrangle in 2D and a 8-nodes hexaedron in 3D, leading to 8 and 24 degrees of freedom.

For the thermal part, first (TmVolume[2|3]DElement) or second (Tm2Volume[2|3]DElement) order thermal nodes are added.
TmVolume[2|3]DElement has therefore 4/8 degrees of fredom in 2/3D, when Tm2Volume[2|3]DElement has 8/20 dofs.

By default, the number of integration points in the deviatoric and volume part of the stress field is equal to those of Volume[2|3]DElement.

Likewise, the default number of integration points is given by $(m+1)^d$, where $m$ is the thermal degree of interpolation (different from the mechanical one for Tm2Volume[2|3]DElement). This number can be changed by the definition of NIPTKSI, NIPTETA or NIPTZETA.

Volume mechanical element with the shape of a pentahedron (a triangular prism to be more accurate). This pentahedron has 6 nodes, so 18 dofs.

By default, stresses are integrate over 2 integration points in the deviatoric part, unless stated otherwise by the definition of NIPDKSI, NIPDETA or NIPDZETA. The number of integration points in the volume part of the stress field (pressure) depends on the chosen method to integrate stresses are cannot be modified.

3-nodes triangular element in 2D, 6 dofs.

By default, stresses are integrate over 1 integration points in the deviatoric part.

4-nodes tetrahedron in 3D, 12 dofs.

By default, stresses are integrate over 1 integration points in the deviatoric part.

QuadVolume2DElement and QuadVolume3DElement are quadratic elements. QuadVolume2DElement is a 8-nodes quadratic quadrangle, QuadVolume2DElement a 20 quadratic hexaedron. They are purely mechanical, with respectively 16 and 60 dofs.

By default, stresses are integrate over 4/27 integration points in the deviatoric part in 2/3D, with the formula already stated for Volume[2|3]DElement. The standard formulation must be used (CAUCHYMECHVOLINTMETH = VES_CMVIM_STD), which is not the default one.

Remark: To use these higher order elements, the mesh must be also defined as second or third degree.

Quadratic, 6-nodes triangular element in 2D, 12 dofs. The standard formulation must be used (CAUCHYMECHVOLINTMETH = VES_CMVIM_STD).

By default, stresses are integrated over 3 integration points in the deviatoric part.

Remark: To use these higher order elements, the mesh must be also defined as second or third degree.

Quadratic, 10-nodes tetrahedron in 3D, 30 dofs. As with quadratic elements in 2D, the standard formulation must be used (CAUCHYMECHVOLINTMETH = VES_CMVIM_STD).

By default, stresses are integrated over 4 integration points in the deviatoric part.

Remark: To use these higher order elements, the mesh must be also defined as second or third degree.

The AEJ element (“Average Elemental Jacobian”) is a first degree triangle/tetrahedron, used to avoid locking which appears with standard formulation in incompressible configurations (Poisson ratio close to 0.5, Von Mises plasticity).

Careful : This elements must be used together with the integration scheme CAUCHYMECHVOLINTMETH = VES_CMVIM_AEJ.

The idea is to replace the kinematic Jacobian $J_e$ of the element by a new value called $J_{e,aej}$. This modified Jacobian is obtained by:

1. defining nodal Jacobians $J_I$, equal to the rate of volume change around this node (which requires a sum over neighboring triangles/tetrahedra).
2. averaging the $J_I$ over the element.

This corresponds to setting the incompressibility constraints to the nodes, and no longer to the integration points. The degrees of freedom of the AEJ element are then extended, corresponding to these of the standard triangle/tetrahedron augmented by the dofs of neighboring nodes.