## C. Element Description

The ATILA element library isdescribed on the following pages. The information in brackets following theelement name refers to the list of Section I.B and specifies one or several references concerning the element design.

### 1. SPRI02AM

#### a) Description

The SPRI02AM element is a two-node spring element used tomodel the memory effect of a shape memory material.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz (3 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond to a shape memory alloy with a memorybehavior (Section III.D).

GEOMETRY entry parameters

S

where S is the cross section area of the rod.

#### e) Remarks

- For this element, the performed analysis is restricted to acombination of loading from 0 to the requested static forces and unloading fromthat maximum back to 0
- In three-dimensional models, the section S must be divided by thenumber of symmetry planes + 1. In axisymmetrical models, the section S must bedivided by 2π.
- Care should be taken about lateral displacements, as there is nostiffness associated to them. The common rule is either to block them or tomake them equal to their neighboring elements.

#### f) Example

GEOMETRY
1 = 0.0005 * Section 1
2 = 0.0015 * Section 2
…
ELEMENTS
SPRI02AM SMA1 2 = 4 5 * The rod
SPRI02AM SMA1 1 = 5 1 / 5 6 / 5 8 * The “umbrella”
QUAD08E STEEL
1 3 8 10 2 6 7 9
…
END
4 2 6 * Same Oy displacement as that of node 6
5 2 6 * Idem

### 2. SPRI02AS

#### a) Description

The SPRI02AS element is a two-node spring element used tomodel the superelastic effect in a shape memory material.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz (3 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond to a shape memory alloy with a superelasticbehavior (Section III.D).

GEOMETRY entry parameters

S

where S is the cross section area of the rod.

#### e) Remarks

- For this element, the performed analysis is restricted to acombination of loading from 0 to the requested static forces and unloading fromthat maximum back to 0
- In three-dimensional models, the section S must be divided by thenumber of symmetry planes + 1. In axisymmetrical models, the section S must bedivided by 2π.
- Care should be taken about lateral displacements, as there is nostiffness associated to them. The common rule is either to block them or tomake them equal to their neighboring elements.

#### f) Example

GEOMETRY
1 = 0.0005 * Section 1
2 = 0.0015 * Section 2
…
ELEMENTS
SPRI02AS SMA2 2 = 4 5 * The rod
SPRI02AS SMA2 1 = 5 1 / 5 6 / 5 8 * The “umbrella”
QUAD08E STEEL
1 3 8 10 2 6 7 9
…
END
4 2 6 * Same Oy displacement as that of node 6
5 2 6 * Idem

### 3. HEXA20C (27)

#### a) Description

The HEXA20C element is a twenty-node isoparametrichexahedron used to model multi-layer composite materials. The formulation ofthis element relies on the Halpin and Tsai model. The element must be a partof a thick plate or of a thick cylindrical shell having a circularcross-section. It must be simply curved with a constant thickness and aconstant curvature in the element Oxy plane. No curvature or thicknessvariation is allowed in the element Oyz plane.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz, (3 translations).

#### c) Topology

(fiber skew angle PHI1 shown here)

#### d) Parameters

MATERIALS entry parameters

Fiber and resin characteristics. See the detaileddescription of the MATERIALS entry, (Section III.D).

GEOMETRY entry parameters

NLAY PHI1 PHI2 ... PHIN

where NLAY is the number of layers, PH1 to PHIN are, foreach layer, the values of the fiber skew angle referenced to the element Ozaxis. All layers have the same thickness, deduced from element size and NLAY.

#### e) Remarks

- Modeling of shells of various thicknesses and curvatures ispossible by assembling several elements so as to discretize this variation.ATILA does not detect non allowed element curvature or thickness variations.
- Shells with double curvature can be modeled with the facetassumption.

#### f) Example

GEOMETRY
2 = 4 30. -30. 30. -30.
…
…
ELEMENTS
HEXA20C PLC3 2
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 4. SHEL08C, SHEL06C (9)

#### a) Description

These eight-node or six-node curved shell elements are usedto model multi-layer composite materials. The formulation of this elementrelies on the Halpin and Tsai model.

#### b) Active degrees-of-freedom

Ux, Uy, Uz, qx, and qy (3 translations, 2 rotations).

#### c) Topology

(Fiber skew angle PHI1 shown here)

#### d) Parameters

MATERIALS entry parameters

Fiber and matrix characteristics. See the detaileddescription of the MATERIALS entry, (Section III.D).

GEOMETRY entry parameters

NLAY PHI1 PHI2 ... PHIN T

where NLAY is the number of layers, PH1 to PHIN are, foreach layer, the values of the fiber skew angle referenced to the element Oxaxis, T is the shell thickness.

#### e) Remark

Degrees-of-freedom qx and qy areexpressed in local axis. For each node, if is the vector normal to the element, , and are the directions of the global axis, qx is the rotation around the direction and qy the rotation around thedirection . If and are parallel, qx is the rotationaround the direction and qy the rotation around the direction .

#### f) Example

GEOMETRY
2 = 4 30. -30. 30. -30. 0.01
…
…
ELEMENTS
SHEL08C PLC3 2
6 1 8 3 4 7 2 5

### 5. QUAD08C (27)

#### a) Description

The QUAD08C element is an eight-node isoparametric elementused to model composite materials in axisymmetrical analysis. The formulationof this element relies on the Halpin and Tsai model. Ox is the axis ofsymmetry. The shell can be doubly curved.

#### b) Active degrees-of-freedom

Ux, and Uy (2 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Fiber and matrix characteristics. See the detaileddescription of the MATERIALS entry, (Section III.D).

GEOMETRY entry parameters

NLAY PHI1 PHI2 ... PHIN

where NLAY is the number of layers, PH1 to PHIN are, foreach layer, the values of the fiber skew angle referenced to the element Ozaxis.

#### e) Remark

Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

GEOMETRY
2 = 4 30. -30. 30. -30.
…
…
ELEMENTS
QUAD08C PLC3 2
6 1 8 3 4 7 2 5

### 6. HEXA20E, PRIS15E, PYRA13E, TETR10E (1)

#### a) Description

These isoparametric elements are used to model isotropicelastic materials, with or without material losses.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz, (3 translations).

#### c) Topologies

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

Not required.

#### e) Remark

Local axes (directions 1-2, 1-3, and 1-5 for HEXA20E andPYRA13E, directions 1-2, 1-3 and 1-4 for PRIS15E and TETR10E) should form adirect system. ATILA detects inverted elements and renumbers themautomatically.

#### f) Example

ELEMENTS
HEXA20E AU4G
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 7. PLAT08E, PLAT06Ex (4)

#### a) Description

These flat plate elements are used to model isotropicelastic materials with or without material losses. They rely on the classicalLove-Kirchhoff hypotheses and must be flat.

#### b) Active degrees-of-freedom

Uz, qx, and qy (1 translation, 2 rotations).

#### c) Topologies

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

T

where T is the plate thickness

#### e) Remarks

- The longitudinal and lateral dimensions of the plate must be atleast ten times the thickness.
- These elements can be superimposed on the QUAD08E or TRIA06Eelements (plane-stress condition) to generate a plane element that enables thefacet modeling of shells (see FACE08E).
- Local axes must be defined (section III.E.2) so that the x and y directions for the degrees of freedom of each node of the element are the local x and y directions of the element.

#### f) Example

GEOMETRY = 10 = 0.065 * Thickness
…
…
ELEMENTS
PLAT08E AU4G 10
6 1 8 3 4 7 2 5

### 8. FACE08E (4)

#### a) Description

The FACE08E element is an eight-node quadrilateral thinshell element, used for homogeneous isotropic elastic materials, with orwithout material losses. It is plane and enables the facet modeling ofshells. It relies on the classical Love-Kirchhoff hypotheses and must be flat (facet).

#### b) Active degrees-of-freedom

Ux, Uy, Ux, qx, and qy (3 translations, 2 rotations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

T

where T is the shell thickness.

#### e) Remarks

- The longitudinal and lateral dimensions of the shell must be atleast ten times the thickness.
- This element is equivalent to the superposition of the QUAD08Eelement (plane-stress condition) and the PLAT08E element.
- Local axes must be defined (section III.E.2) so that the x and y directions for the degrees of freedom of each node of the element are the local x and y directions of the element.

#### f) Example

GEOMETRY = 10 = 0.065 * Thickness
…
…
ELEMENTS
FACE08E AU4G 10
6 1 8 3 4 7 2 5

### 9. SPRI02E (1)

#### a) Description

The SPRI02E element is a two-node linear spring element,which is only able to transmit to its two limiting nodes a restoring force dueto a change of its length. The spring constitutive material may or may notinclude material losses. Its classical use is the modeling of prestress rods.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz (3 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

S

where S is the cross section area of the rod.

#### e) Remarks

- In three-dimensional models, the section S must be divided by thenumber of symmetry planes + 1. In axisymmetrical models, the section S must bedivided by 2π.
- This element is nothing more than a classical spring with mass. Use of this element can lead to spurious but real modes in a modal or harmonicanalysis. These “rod” modes can be shifted without effect on the otherresults, by artificially dividing the rod density by 100, for example, tomultiply their resonance frequencies by 10.
- Care should be taken about lateral displacements, as there is nostiffness associated to them. The common rule is either to block them or tomake them equal to their neighboring elements.

#### f) Example

GEOMETRY
1 = 0.0005 * Section 1
2 = 0.0015 * Section 2
…
ELEMENTS
SPRI02E STEEL 2 = 4 5 * The rod
SPRI02E STEEL 1 = 5 1 / 5 6 / 5 8 * The “umbrella”
QUAD08E STEEL
1 3 8 10 2 6 7 9
…
END
4 2 6 * Same Oy displacement as that of node 6
5 2 6 * Idem

### 10. QUAD08E, TRIA06E (1)

#### a) Description

These isotropic elements are used to model isotropic elasticmaterials with or without material losses. They can be used for plane-stress,plane-strain or axisymmetrical analyses. For axisymmetrical problems, theglobal Ox axis is the axis of symmetry.

#### b) Active degrees-of-freedom

Ux, and Uy (2 translations).

#### c) Topologies

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

T

where T is the thickness of the element. Not required andignored for plane-strain or axisymmetrical analyses.

#### e) Remark

Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

CLASS PLSTRESS
GEOMETRY = 10 = 0.065 * Thickness
…
…
ELEMENTS
QUAD08E AU4G 10
6 1 8 3 4 7 2 5

### 11. SHEL03E (4)

#### a) Description

The SHEL03E element is a three-node axisymmetrical thinshell element, with or without material losses, the formulation of which takesaccount of its double curvature. It relies on the classical Love-Kirchhoffhypotheses. The global Ox axis is the symmetry axis.

#### b) Active degrees-of-freedom

Ux, Uy, and qz (2 translations, 1 rotation).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicelastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

T R

where T is the thickness of the element and R its radius ofcurvature.

#### e) Remarks

- If R is null, the element is assumed to be straight.
- The y ordinate of a shell point must be positive.
- The numbering of the element nodes must be generated in ascendingorder of x:
  X(N1) .LE. X(N2) .LE. X(N3)
- The longitudinal and radial dimensions of the shell must be atleast ten times the thickness.

#### f) Example

GEOMETRY
4 = 0.065 0.7 * Thickness + radius
5 = 0.065 0.9 * Thickness + radius
…
…
ELEMENTS
SHEL03E AU4G 5
25 23 22
SHEL03E AU4G 4
16 19 25

### 12. HEXA20ES, PRIS15ES, PYRA13ES, TETR10ES

#### a) Description

These isoparametric elements are used to modelelectrostrictive materials.

#### b) Active degrees-of-freedom

Ux, Uy, Uz , and F (3 translations, 1electrical potential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, electrostrictive, dielectric tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL]. Note that electrostrictive materials are often isotropic and that in this case the data of the GEOMETRYPOLARIZATION do not affect the computation.
- Local axes (directions 1-2, 1-3, and 1-5 for HEXA20ES andPYRA13ES, directions 1-2, 1-3 and 1-4 for PRIS15ES and TETR10ES) should form adirect system. ATILA detects inverted elements and renumbers themautomatically.
- The STRESS command is required when using these elements.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 0. 0. 0. *
…
ELEMENTS
HEXA20ES PMN 1
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 13. QUAD08ES, TRIA06ES

#### a) Description

These isoparametric elements are designed to model anyelectrostrictive material. They can be used for plane-strain or axisymmetricalanalyses.

#### b) Active degrees-of-freedom

Ux, Uy, and F (2 translations, 1 electrical potential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, electrostrictive, dielectric tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to an isotropic elastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL]. Note that electrostrictivematerials are often isotropic and that in this case the data of the GEOMETRYPOLARIZATION do not affect the computation.
- Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.
- The STRESS command is required when using these elements.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 0. 0. 0.
…
ELEMENTS
QUAD08P PMN 1
6 1 8 3 4 7 2 5

### 14. HEXA20F, PRIS15F, PYRA13F, TETR10F (1)

#### a) Description

These isoparametric elements are used to model homogeneousfluid media, with or without losses.

#### b) Active degrees-of-freedom

The pressure P.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to a fluid, withor without material losses (Section III.D).

GEOMETRY entry parameters

Not required.

#### e) Remark

Local axes (directions 1-2, 1-3, and 1-5 for HEXA20F andPYRA13F, directions 1-2, 1-3 and 1-4 for PRIS15F and TETR10F) should form adirect system. ATILA detects inverted elements and renumbers them automatically.

#### f) Example

ELEMENTS
HEXA20F WATER
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 15. QUAD08F, TRIA06F (1)

#### a) Description

These isoparametric elements are used to model homogeneousfluid media, with or without material losses. These elements can be used forplane-strain or axisymmetrical analyses.

#### b) Active degrees-of-freedom

The pressure P.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to a fluid, withor without material losses (Section III.D).

GEOMETRY entry parameters

Not required.

#### e) Remark

Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

ELEMENTS
QUAD08F WATER
6 1 8 3 4 7 2 5

### 16. HEXA20G, PRIS15G, PYRA13G, TETR10G (6)

#### a) Description

These isoparametric elements are used to model isotropicmagnetic media.

#### b) Active degrees-of-freedom

The magnetic potential f.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicmagnetic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

Not required.

#### e) Remarks

- The elements may be safely superimposed to elastic and fluidelements, allowing the extent of the magnetic domain inside purely elasticparts of the structure.
- Local axes (directions 1-2, 1-3, and 1-5 for HEXA20G and PYRA13G,directions 1-2, 1-3 and 1-4 for PRIS15G and TETR10G) should form a directsystem. ATILA detects inverted elements and renumbers them automatically.

#### f) Example

ELEMENTS
HEXA20G VACUUM
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 17. QUAD08G, TRIA06G (6)

#### a) Description

These isoparametric elements are designed to model anyisotropic magnetic media. These elements can be used for plane-strain oraxisymmetrical analyses.

#### b) Active degrees-of-freedom

The magnetic potential f.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond, depending upon the case, to an isotropicmagnetic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

Not required.

#### e) Remarks

- The elements may be safely superimposed to elastic and fluidelements, allowing the extent of the magnetic domain inside purely elasticparts of the structure.
- Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

ELEMENTS
QUAD08G VACUUM
6 1 8 3 4 7 2 5

### 18. QUAD16I, TRIA12I (1)

#### a) Description

These isoparametric elements are used to ensure the matchingbetween solid and fluid meshes along their interface. An element includes 8solid nodes and 8 fluid nodes that have the same coordinates as the solidnodes. It has no thickness.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz for each solid node (3 translations), and P (the pressure)for each fluid node.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Not required.

GEOMETRY entry parameters

Not required.

#### e) Remarks

- Nodes 1, 2, 3, 4, 9, 10, 11, 12 are fluid nodes.
- Nodes 5, 6, 7, 8, 13, 14, 15, 16 are solid nodes.
- The local Oz axis, which is computed from local Ox (direction1-2) and Oy (direction 1-3) axes, must be oriented from the fluid domain intothe solid domain. ATILA does not detect inverted elements.

#### f) Example

ELEMENTS
QUAD16I
9 14 11 16 1 6 3 8 12 10 15 13 4 2 7 5

### 19. LINE06I (1)

#### a) Description

The LINE06I element is a six-node isoparametric element usedfor plane-strain (two-dimensional) or axisymmetrical analyses. It ensures thematching between solid and fluid meshes along their interface. It includes 3solid nodes and 3 fluid nodes that have the same coordinates as the solidnodes. It has no thickness.

#### b) Active degrees-of-freedom

Ux, and Uy for each solid node (2 translations), and P (the pressure) for each fluid node.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Not required.

GEOMETRY entry parameters

Not required.

#### e) Remarks

- Nodes 1, 2, 5 are fluid nodes.
- Nodes 2, 3, 6 are solid nodes.
- The local Oy axis, which is computed from the local Ox (direction1-2) axis and the local Oz axis of other 2D elements, must be oriented from thefluid domain into the solid domain. ATILA does not detect inverted elements.

#### f) Example

ELEMENTS
QUAD08E AU4G
1 3 6 8 2 4 5 7
LINE06I
9 14 3 8 12 5
QUAD08F WATER
9 11 14 16 10 12 13 15

### 20. HEXA20M, PRIS15M, PYRA13M, TETR10M (6)

#### a) Description

These isoparametric elements are used to modelmagnetostrictive materials, with or without material losses.

#### b) Active degrees-of-freedom

Ux, Uy, Uz and f (3 translations, 1 magneticpotential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, piezomagnetic, magnetic tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to a magnetostrictive material, with or without material losses (Section III.D).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL].
- Local axes (directions 1-2, 1-3, and 1-5 for HEXA20M and PYRA13M,directions 1-2, 1-3 and 1-4 for PRIS15M and TETR10M) should form a directsystem. ATILA detects inverted elements and renumbers them automatically.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 0. 90. 0. * polarization along global Oz
…
ELEMENTS
HEXA20M TERFENOL 1
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 21. QUAD16M

#### a) Description

The QUAD16M element is a sixteen-node isoparametric elementused to model magnetostrictive thin films, with or without material losses.Because the thickness is small compared to other dimensions, a HEXA20M elementis not suitable. This element is made of two node layers having the samedisplacements but different magnetic potential degrees-of-freedom.

#### b) Active degrees-of-freedom

Ux, Uy, Uz and f (3 translations, 1 magneticpotential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, piezomagnetic, magnetic tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to a magnetostrictive material, with or without material losses (Section III.D).

GEOMETRY entry parameters

ALPHA BETA GAMMA T

These are the three Euler angles that define the naturalaxes of the material with respect to the global coordinate system, followed bythe element thickness (see the detailed description of the GEOMETRYPOLARIZATION CARTESIAN entry, Section III.D).

#### e) Remarks

- The material’s polarization must be CARTESIAN.
- The longitudinal and lateral dimensions of the element must be atleast ten times the thickness.
- On the film boundaries, the user must provide appropriateboundary conditions: the magnetic degrees of freedom of the two node layersmust be set equal.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
6 = 0. 0. 0. 0.0001 * polarization along global Ox
…
ELEMENTS
QUAD16M TERFENOL 6
9 14 11 16 1 6 3 8 12 10 15 13 4 2 7 5
...
END
14 4 6 * Same magnetic potential
12 4 4 * on the magnetostrictive film edges
9 4 1

### 22. QUAD08M, TRIA06M (6)

#### a) Description

These isoparametric elements are designed to model anymagnetostrictive material, with or without material losses. These elements canbe used for plane-strain or axisymmetrical analyses.

#### b) Active degrees-of-freedom

Ux, Uy, and f (2 translations, 1 magnetic potential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, piezomagnetic, magnetic tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to a magnetostrictive material, with or without material losses (Section III.D).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL].
- Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 90. 0. 0. * polarization along global Oy
…
ELEMENTS
QUAD08M TERFENOL 1
6 1 8 3 4 7 2 5

### 23. HEXA20P, PRIS15P, PYRA13P, TETR10P (1, 5)

#### a) Description

These isoparametric elements are used to model piezoelectricmaterials with or without material losses.

#### b) Active degrees-of-freedom

Ux, Uy, Uz , and F (3 translations, 1electrical potential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, piezoelectric, dielectric tensors, and the density(see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to an isotropic elastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL].
- Local axes (directions 1-2, 1-3, and 1-5 for HEXA20P and PYRA13P,directions 1-2, 1-3 and 1-4 for PRIS15P and TETR10P) should form a directsystem. ATILA detects inverted elements and renumbers them automatically.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 0. 90. 0. * polarization along global Oz
…
ELEMENTS
HEXA20P X51 1
1 3 6 8 13 15 18 20 2 4 5 7 9 10 11 12 14 16 17 19

### 24. TRIL08P, TRIL06P (5)

#### a) Description

The TRIL08P element is an eight-node quadrilateral used tomodel piezoelectric trilaminars. These trilaminars are composed of a metalliccore sandwiched between two plates of a piezoelectric material. The elementformulation relies, for the mechanical part, on the classical Love-Kirchhoffhypotheses. The polarization and the electrical field are takenperpendicular to the plane and uniform on the piezoelectric plates.

#### b) Active degrees-of-freedom

Uz, qx, qy, and F (1 translation, 2 rotations, 1 electrical potential). Inthis case, the LCPPDC command must be used to reorder the degrees-of-freedom.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Young’s modulus, Poisson’s ratio, and density for the metalliclayer, elastic, piezoelectric, dielectric tensors, and density for thepiezoelectric part. See the detailed description of the materials entry,Section III.D.

GEOMETRY entry parameters

TM TP

where TM is the thickness of the metallic layer and TP thepiezoelectric plate thickness.

#### e) Remarks

- The longitudinal and lateral dimensions of the whole plate mustbe at least ten times the thickness.
- The MATERIALS entry has a specific form because of the existenceof two distinct materials (see Section III.D).

#### f) Example

GEOMETRY = 4 = 0.065 0.003 * Thicknesses
…
…
ELEMENTS
TRIL08P AG5X31 4
6 1 8 3 4 7 2 5

### 25. QUAD08P, TRIA06P (1, 5)

#### a) Description

These isoparametric elements are designed to model anypiezoelectric material, with or without material losses. These elements can beused for plane-strain or axisymmetrical analyses.

#### b) Active degrees-of-freedom

Ux, Uy, and F (2 translations, 1 electrical potential).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

The elastic, piezoelectric, dielectric tensors, and thedensity (see the detailed description of the MATERIALS entry in Chapter III). These correspond, depending upon the case, to an isotropic elastic material, with or without material losses (Section III.D).

GEOMETRY entry parameters

These are the coordinates of the O'x'y'z' system origin andthe Euler angles that define the natural axes of the material with respect tothe global coordinate system (see the detailed description of the GEOMETRYPOLARIZATION entry, Section III.D).

#### e) Remarks

- The GEOMETRY entry must be given in the particular form GEOMETRYPOLARIZATION [CARTESIAN, CYLINDRICAL, SPHERICAL].
- Local axes (directions 1-2, 1-3, and global Oz) of all 2Delements must form either a uniquely direct or a uniquely inverted system.ATILA does not detect inconsistent axes.

#### f) Example

GEOMETRY POLARIZATION CARTESIAN
1 = 90. 0. 0. * polarization along global Oy
…
ELEMENTS
QUAD08P X51 1
6 1 8 3 4 7 2 5

### 26. QUAD08R, TRIA06R (1, 2)

#### a) Description

These isoparametric elements are used to prescribe amonopolar or dipolar radiation condition. They are attached to the outsidesurface of a three-dimensional fluid mesh.

#### b) Active degrees-of-freedom

The pressure P.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond to a fluid, withoutmaterial losses (Section III.D).

GEOMETRY entry parameters

R

where R is the element radius of curvature.

#### e) Remarks

- The local Oz axis, which is computed from local 0x (direction1-2) and Oy (direction 1-3) axes, must point away from the fluid mesh. ATILAdoes not detect inverted elements.
- All radiating elements must lie on a unique spherical surface,whose center is consistent with the structure’s center of radiation. Thissurface cannot reside in the near-field, except when the RADIATION DIPOLARentry is provided.
- ATILA does not check the consistency between the radius providedby the geometry entry and the element curvature. It does not check the presenceof multiple spherical surfaces.

#### f) Example

GEOMETRY = 3 = 7.5 * Radius
…
…
ELEMENTS
QUAD08R WATER 3
6 1 8 3 4 7 2 5

### 27. LINE03R (1, 2)

#### a) Description

The LINE03R element is a three-node isoparametric linearelement used to prescribe a monopolar or dipolar radiation condition. It mustbe circular and attached on the outer surface of a two-dimensional fluiddomain. It can be used for plane-strain and axisymmetrical analyses.

#### b) Active degrees-of-freedom

The pressure P.

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

These correspond to a fluid, withoutmaterial losses (Section III.D).

GEOMETRY entry parameters

R

where R is the element radius of curvature.

#### e) Remarks

- The local Oy axis, which is computed from the local Ox (direction1-2) axis and the local Oz axis of other 2D elements, must be oriented awayfrom the fluid domain. ATILA does not detect inverted elements.
- All radiating elements must lie on a unique spherical orcylindrical surface, whose center is consistent with the structure’s center ofradiation. This surface cannot reside in the near-field, except when the RADIATION DIPOLAR entry is provided.
- ATILA does not check the consistency between the radius provided bythe geometry entry and the element curvature. It does not check the presence ofmultiple spherical spheres or cylinders.

#### f) Example

GEOMETRY = 4 = 0.75 * Radius
…
…
ELEMENTS
LINE03R WATER 4
16 18 17

### 28. QUAD08Z, TRIA06Z

#### a) Description

These isoparametric elements are used to couple an ATILA finite-element model with a CHIEF-type or EQI-type boundary-element model, or to load a solidstructure with a frequency-dependent local mechanical impedance. In the firstuse (BEM coupling), it is superimposed on the radiating surfaces of the mesh ofthe solid structure. These elements must be ordered in the file in the same wayas the mutual impedance matrix described in the JOB.ZRAD file. In the seconduse (local impedance), the local mechanical impedance is provided by means of afluid material’s constants and a geometry number corresponding to thecoefficients of a ratio of two frequency dependent polynomials.

#### b) Active degrees-of-freedom

Ux, Uy, and Uz (3 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Not required for BEM coupling. Not used but required forlocal impedance.

GEOMETRY entry parameters

Not required for BEM coupling.

N D CN0R CN0I CN1R CN1I ... CN(N-1)R CN(N-1)I&
CD0R CD0I CD1R CD1I ... CD(D-1)R CD(D-1)I

where N and D are the number of terms in the numerator and denominator (polynomial orders minus one), CNiR + j CNiIis the complex constant of the ith power of pulsation w in the numerator and CDiR + j CDiIis the complex constant of the ith power of pulsation w in the denominator.

The local impedance Z(w) is :

#### e) Remark

- The local Oz axis, which is computed from local 0x (direction1-2) and Oy (direction 1-3) axes, must point away from the ATILA mesh.
- The local impedance is the default element behavior. The EQI orCHIEF commands (Section III.D) set the BEM coupling behavior.
- If N £ 0, Z(w) is set to 0. If D £0, the denominator of Z(w) is set to 1.
- The geometry numbers are designed to deal with a maximum numberof ten values. If more values are needed, the user must provide geometrynumbers increasing by a value of more than 1, in order to leave space for extrageometry values. For example, if geometry number 3 needs 28 values, then thenext used geometry number must be greater or equal to 6, that is, 3 +next_integer(28/10).

#### f) Example 1

CHIEF
ELEMENTS
QUAD08Z
6 1 8 3 4 7 2 5

#### >g) Example2

GEOMETRY
2 = 2 2 & * This geometry to set Z() = c jka/(1 + jka)
0. 0. 0. 100. & * that is, the radiation impedance ofa pulsating sphere
1. 0. 0. 6.7156E-05 * of radius = 0.1 in water.
ELEMENTS
QUAD08Z DUMMY 2
6 1 8 3 4 7 2 5

### 29. LINE03Z

#### a) Description

The LINE03Z element is a three-node isoparametric elementused either to couple an ATILA finite-element modelwith a CHIEF-type or EQI-type boundary element model, or to load a solid structure with a frequency-dependentlocal mechanical impedance. In the first use (BEM coupling), it is superimposedon the radiating surfaces of the mesh of the solid structure. These elementsmust be ordered in the file in the same way as the mutual impedance matrixdescribed in the JOB.ZRAD file. In the second use (local impedance), the localmechanical impedance is provided by means of a fluid materials constants and ageometry number corresponding to the coefficients of a ratio of two frequencydependent polynomials.

#### b) Active degrees-of-freedom

Ux, and Uy (2 translations).

#### c) Topology

#### d) Parameters

MATERIALS entry parameters

Not required for BEM coupling. Not used but required forlocal impedance.

GEOMETRY entry parameters

Not required for BEM coupling.

N D CN0R CN0I CN1R CN1I ... CN(N-1)R CN(N-1)I&
CD0R CD0I CD1R CD1I ... CD(D-1)R CD(D-1)I

where N and D are the number of terms in the numerator anddenominator (polynomial orders minus one), CNiR + j CNiIis the complex constant of the ith power of pulsation w in the numerator and CDiR + j CDiIis the complex constant of the ith power of pulsation w in the denominator.

The local impedance Z(w) is :

#### e) Remarks

- The local Oy axis, which is computed from the local Ox (direction1-2) axis and the local Oz axis of other 2D elements, must be oriented awayfrom the fluid domain. ATILA does not detect inverted elements.
- The local impedance is the default element behavior. The EQI orCHIEF commands (Section III.D) set the BEM coupling behavior.
- If N £ 0, Z(w) is set to 0. If D £0, the denominator of Z(w) is set to 1.
- The geometry numbers are designed to deal with a maximum numberof ten values. If more values are needed, the user must provide geometrynumbers increasing by a value of more than 1, in order to leave space for extrageometry values. For example, if geometry number 3 needs 28 values, then thenext used geometry number must be greater or equal to 6, that is, 3 +next_integer(28/10).

#### f) Example 1

CHIEF
ELEMENTS
QUAD08E AU4G
9 11 4 6 10 7 8 5
LINE03Z
4 6 5

#### g)Example 2

GEOMETRY
2 = 2 2 & * This geometry to set Z() = c jka/(1 + jka)
0. 0. 0. 100. & * that is, the radiation impedance ofa pulsating sphere
1. 0. 0. 6.7156E-05 * of radius = 0.1 in water.
ELEMENTS
LINE03Z DUMMY 2
4 6 5