3. Thermal#

For the resolution of chained thermomechanical problems, we must use thermal shell finite elements [R3.11.01] for thermal calculation. These elements are plate elements, or linear elements in the case of structures of revolution or structures that are invariant along an axis. The curvature of the structure is not taken into account by itself. The tangential plane metric for each element is calculated assuming that all vertices are coplanar. These elements assume a priory parabolic distribution of temperature in the thickness, which results from an asymptotic development in linear thermics for a low shell thickness, when the temperature variations are not too significant. It should be noted that a model based on a richer temperature field development in thickness sees its terms of order greater than two converge to zero when the shell is thin. It is therefore not possible to treat thermal shock problems with a large variation in the temperature profile in thickness with these shells. The terms of use of these elements are presented in [U1.22.01].

3.1. Problem definition#

3.1.1. Spatial discretization and modeling assignment: operator AFFE_MODELE#

3.1.1.1. Degrees of freedom#

The degrees of freedom are temperatures TEMP_MIL (temperature on the average surface of the shell), TEMP_INF (temperature on the lower surface of the shell), and TEMP_SUP (temperature on the upper surface of the shell).

3.1.1.2. Stiffness matrix support meshes#

Modeling	Mesh	Nature of the mesh	Finished element	Remarks
COQUE	QUAD9 QUAD8 QUAD4 TRIA7 TRIA6 TRIA3	plane Plane Plane Plane Plane Plane	THCOQU9 THCOQU8 THCOQU4 THCOTR7 THCOTR6 THCOTR3	nodes with 3 coordinates \(x,y,z\)
COQUE_PLAN	SEG3	not assumed to be plane	THCPSE3	nodes with 2 coordinates \(x,y\)
COQUE_AXIS	SEG3	not assumed to be plane	THCASE3	nodes with 2 coordinates \(x,y\)

For THCOTRi, only the three vertices are used to define the local geometry (tangential plane, normal). For THCOQUi, we consider that the element is plane and its tangential plane is defined by default by 3 of the 4 vertices of the element.

3.1.1.3. Load support meshes#

Modeling	Mesh	Finished Element	Remarks
COQUE	SEG2	THCOSE2	with TRIA3 and QUAD4
COQUE	SEG3	THCOSE3	with TRIA6, 7 and QUAD8 ,9

All the loads applicable to the facets of the shell elements are treated by direct discretization on the support mesh of the element in temperature formulation. No loading mesh is therefore necessary for the faces of the shell elements.

For loads applicable to the edges of the shell elements, a support mesh of type SEG2 (element THCOSE2) or SEG3 (element THCOSE3) must be used.

For the temperatures imposed, the support meshes are meshes reduced to one point.

3.1.1.4. Model: AFFE_MODELE#

The modeling assignment is done through the operator AFFE_MODELE [U4.41.01].

AFFE_MODELE		Remarks
AFFE
PHENOMENE:	“THERMIQUE”
MODELISATION	“COQUE”
	“COQUE_PLAN”
	“COQUE_AXIS”

3.1.2. Basic characteristics: AFFE_CARA_ELEM#

In this part, the operands characteristic of thermal plate and shell elements are described. The documentation for using the AFFE_CARA_ELEM operator is [U4.42.01].

AFFE_CARA_ELEM	COQUE	COQUE_PLAN	COQUE_AXIS	Remarks
COQUE
EPAIS

The characteristics assigned to the materials are the same as for a mechanical calculation. It should be noted that it is not useful to define a particular frame of reference for the exploitation of the results of thermal calculation because they are limited to temperature fields, scalar quantities, independent of the reference frame used.

3.1.3. Materials: DEFI_MATERIAU#

DEFI_MATERIAU	COQUE	COQUE_PLAN	COQUE_AXIS	Remarks
THER
THER_FO

The materials used with plate or shell elements in thermal can have linear thermal characteristics that are constant or dependent on the loading increment.

3.1.4. Loads and limit conditions: AFFE_CHAR_THER and AFFE_CHAR_THER_F#

The assignment of loads and boundary conditions on a thermal model is performed using the operator AFFE_CHAR_THER, if the loads and mechanical boundary conditions on a system are real values that do not depend on any parameter, or AFFE_CHAR_THER_F, if these values are a function of the position or the load increment.

The documentation for using AFFE_CHAR_THER and AFFE_CHAR_THER_F is [U4.44.02].

3.1.4.1. List of keywords AFFE_CHAR_THER factor#

The values of the loads affected are real and do not depend on any parameters.

AFFE_CHAR_THER general	COQUE		COQUE_PLAN	COQUE_AXIS	Remarks
TEMP_IMPO

TEMP_IMPO: Keyword factor that can be used to impose a temperature on nodes or groups of nodes.
FLUX_REP: Keyword factor that can be used to apply normal flows to a thermal shell face defined by one or more cells or groups of cells such as triangle or quadrangle cells.
ECHANGE: Keyword factor that can be used to apply exchange conditions with an external temperature to a shell face, defined by one or more cells or groups of cells such as triangle or quadrangle cells.

3.1.4.2. List of keywords AFFE_CHAR_THER_F factor#

The values of the loads affected can be a function of global coordinates and time, or of the temperature in non-linear thermal terms (except in shells).

AFFE_CHAR_THER_F general	COQUE		COQUE_PLAN	COQUE_AXIS	Remarks
TEMP_IMPO

3.2. Resolution#

3.2.1. Transient calculations: operator THER_LINEAIRE#

Transitional calculation option	COQUE		COQUE_PLAN	COQUE_AXIS	Remarks
CHAR_THER_EVOL

This is the treatment of evolutionary thermal problems.

3.3. Additional calculations and post-treatments#

3.3.1. Post-processing calculations#

Post-treatment options for plate and shell elements are presented below.

OPTIONS elemental	COQUE	COQUE_PLAN	COQUE_AXIS	Remarks
“FLUX_ELNO”
“FLUX_ELGA”

FLUX_ELNO: This option calculates heat flow at the nodes based on temperature.
FLUX_ELGA: This option calculates the heat flow at the integration points from the temperature.

3.4. Examples#

One gives here the list of test cases available for thermal shells. These are stationary thermal test cases. The results are correct for all of these test cases, regardless of the element used.

Name	Model	Element	Notes
tplp301a	COQUE	THCOTR3		Title: Plate with imposed temperature distributed sinusoidally on one side. Documentation: [V4.05.301]
tplp302a	COQUE	THCOTR6		Title: Rectangular plate with temperature imposed on the sides. Documentation: [V4.05.302]
TPLS100A tpls100b	COQUE COQUE_PLAN	THCOTR6/THCOTR3 THCPSE3	Title: Infinite plate subjected to a pair of antisymmetric heat flows that are stationary on its two half-faces. Documentation: [V4.03.100] Conduction is linear, homogeneous, and isotropic.
TPLS101A TPLS101B TPLS101C TPLS101D tpls101e	COQUE	THCOTR6/THCOSE3 THCOQU4/THCOSE2 THCOQU8/THCOSE3 THCOQU9/THCOSE3 THCOTR7/THCOSE3	Title: Infinite plate subjected to a couple of thermal conditions with the outside, symmetric with respect to the middle sheet. Documentation: [V4.03.101] Conduction is linear, homogeneous, and isotropic.
TPLS302A TPLS302B TPLS302C tpls302d	COQUE	THCOQU8/THCOSE3 THCOQU4/THCOSE2 THCOQU9/THCOSE3 THCOTR7/THCOSE3	Title: Rectangular plate with convection and imposed temperature Documentation: [V4.03.302]