1. Discretization#

1.1. Degrees of freedom#

DX, DY refer to degrees of freedom of movement.

PRE1 and PRE2 refer to two degrees of freedom of pressure, whose precise meaning depends on the laws of behavior used. TEMP refers to temperature.

Modeling

Degrees of Freedom (at each vertex node)

D_ PLAN_HM AXIS_HM D_ PLAN_HMD AXIS_HMD D_ PLAN_HMS AXIS_HMS

DX, DY, PRE1

D_ PLAN_HHM AXIS_HHM D_ PLAN_HHMD AXIS_HHMD D_ PLAN_HHMS AXIS_HHMS

DX, DY, PRE1, PRE2

D_ PLAN_THM AXIS_THM D_ PLAN_THMD AXIS_THMD D_ PLAN_THMS AXIS_THMS

DX, DY, PRE1, TEMP

D_ PLAN_THHD AXIS_THHD D_ PLAN_THHS AXIS_THHS

PRE1, PRE2, TEMP

D_ PLAN_THHMD AXIS_THHMD D_ PLAN_THHMS AXIS_THHMS

DX, DY, PRE1,, PRE2, TEMP

D_ PLAN_HM_DIL D_ PLAN_HMS_DIL

DX, DY, PRES,, PRE1, GONF

D_ PLAN_THMS_DIL

DX, DY, PRES,, PRE1, TEMP, GONF

D_ PLAN_HHM_DIL

DX, DY, PRES,, PRE1, PRE2, GONF

1.2. Support mesh for stiffness matrices#

The finite element support meshes can be tetrahedra, pyramids, prisms or hexahedra. The elements are iso-parametric. The notations (_, S, D) concern the type of integration which can be classical, lumped (“D”) or selective (“S”).

Modeling

Mesh

Interpolation

Remarks

D_ PLAN_HM (_, S, D) AXIS_HM (_, S, D)

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure

The pressure of a middle node is the average of the node vertices of the segment

D_ PLAN_HHM (_, S, D) AXIS_HHM (_, S, D)

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure

The pressures of a middle node are the averages of the vertices of the segment nodes

D_ PLAN_THM (_, S, D) AXIS_THM (_, S, D)

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure and temperature

The pressure and the temperature of a middle node are the average of the node vertices of the segment

D_ PLAN_THH (S, D) AXIS_THH (S, D)

QUAD8

Bi-linear over 4 knots in pressure and temperature

The pressures and temperature of a middle node are the average of the peak nodes of the segment

D_ PLAN_THHM (S, D) AXIS_THHM (S, D)

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure and temperature

The pressures and the temperature of a middle node are the average of the node vertices of the segment

D_ PLAN_HM_DIL

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure, volume deformation and Lagrange multiplier

The pressure of a middle node is the average of the node vertices of the segment

D_ PLAN_THM_DIL

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure and temperature, volume deformation and Lagrange multiplier

The pressures and the temperature of a middle node are the average of the node vertices of the segment

D_ PLAN_HHM_DIL

QUAD8

Serendip 8 knots in bi-linear motion over 4 knots in pressure, volume deformation and Lagrange multiplier

The pressures of a middle node are the averages of the node vertices of the segment

D_ PLAN_HM (_, S, D) AXIS_HM (_, S, D)

TRIA6

Quadratic in linear displacement under pressure

The pressure of a middle node is the average of the vertex nodes of the segment

D_ PLAN_HHM (_, S, D) AXIS_HHM (_, S, D)

TRIA6

Quadratic in linear displacement under pressure

The pressures of a middle node are the averages of the vertices of the segment nodes

D_ PLAN_THM (_, S, D) AXIS_THM (_, S, D)

TRIA6

Quadratic in linear displacement in pressure and temperature

The pressure and temperature of a middle node are the average of the vertices of the segment

D_ PLAN_THH (S, D) AXIS_THH (S, D)

TRIA6

Linear

The pressures and temperature of a middle node are the average of the top nodes in the segment

D_ PLAN_THHM (S, D) AXIS_THHM (S, D)

TRIA6

Quadratic in linear displacement in pressure and temperature

The pressures and temperature of a middle node are the average of the vertices of the segment

D_ PLAN_HM_DIL

TRIA6

Quadratic in linear displacement in pressure, volume deformation, and Lagrange multiplier

The pressure of a middle node is the average of the node vertices of the segment

D_ PLAN_THM_DIL

TRIA6

Quadratic in linear displacement in pressure and temperature, volume deformation, and Lagrange multiplier

The pressure and temperature of a middle node are the average of the node vertices of the segment

D_ PLAN_HHM_DIL

TRIA6

Quadratic in linear displacement in pressure, volume deformation, and Lagrange multiplier

The pressure of a middle node is the average of the node vertices of the segment

1.3. Cargo support mesh#

Modeling

Mesh

Interpolation

Remarks

D_ PLAN_HM (_, S, D) AXIS_HM (_, S, D)

SEG3

Quadratic in displacement, linear in pressure

The middle node pressure is the average of the vertex nodes in the segment

D_ PLAN_HHM (_, S, D) AXIS_HHM (_, S, D)

SEG3

Quadratic in displacement, linear in pressure

Middle node pressures are the averages of the segment’s vertex nodes

D_ PLAN_THM (_, S, D) AXIS_THM (_, S, D)

SEG3

Quadratic in motion, linear in pressure and temperature

The pressure and temperature of the middle node are the average of the vertices of the segment

D_ PLAN_THH (S, D) AXIS_THH (S, D)

SEG3

Linear

Middle node pressures and temperature are the average of the peak nodes in the segment

D_ PLAN_THHM (S, D) AXIS_THHM (S, D)

SEG3

Quadratic in displacement, linear in pressure and temperature

The pressures and temperature of the middle node are the average of the vertices of the segment

The load support cells for the _ DIL models are identical to those of the corresponding HM, THM or HHM models.