## E. Boundary Conditions

Boundary conditions may be described using free format. Forall the data inputs, node number is referenced as N, a degree-of-freedom is referencedas D, with the following correspondence: 1 for Ux (or P), 2 for Uy, 3 for Uz, 4 for qx (or F or f), 5 for qy, and 6 for qz. Moreover, the lines that areparallel to a global coordinate axis and the planes that are normal to these axes are referenced as P. The correspondence is:
1 for a plane normal to the Ox axis, 4 for a line parallel to the Ox axis,
2 for a plane normal to the Oy axis, 5 for a line parallel to the Oy axis,
3 for a plane normal to the Oz axis, 6 for a line parallel to the Oz axis.

Note that the loading definitions availablein previous versions are superseded by LOADS or EXCITATIONS entries.

Boundary conditions enable the user to force clamped, hingedor simply supported conditions, or identity between degrees-of-freedom. Theseconditions are generally associated with symmetry planes or axes, electrodes orpressure-release surfaces. The easiest way to understand how to provideboundary conditions is to refer to the following rules:

- Boundary conditions acting on one node are defined by the nodenumber N and no line or plane P (or P = 0);
- Boundary conditions acting a group of nodes on the same line orplane are defined by a negative value of N and a corresponding line or plane Pnot equal to 0;
- Boundary conditions deleting a degree-of-freedom are defined by apositive value of the line or plane indicator P;
- Boundary conditions making an identity between degrees-of-freedomof a group of nodes are defined by a negative value of the line or planeindicator P;
- A node number N = 0 means the whole structure.

### 1. BoundaryConditions in Global Axes

The degree-of-freedom in the D direction is deleted for all the structure nodes:

0 d 0

The degree-of-freedom is deleted in the D direction at node N:

n d 0

The degree-of-freedom is deleted in the D direction for all the nodes that belong to the P plane or line containing the node N:

-n d p

For fluid-structure problems, this condition can berestricted to the Solid part of a mesh using the fourthfield of the format:

-n d p S

It can also be restricted to the Fluid part:

-n d p F

The degree-of-freedom in the D direction at node N is identical to thedegree-of-freedom in the D direction at node M:

n d M

Degrees-of-freedom in the D direction for all the nodes that belong to the plane or line P containing N are identical:

-n d -p

When using piezoelectric materials, excitation electrodesare defined by setting identical electric potential degree-of-freedom for thecorresponding nodes; voltage reference electrodes are defined by deletingidentical electric potential degree-of-freedom for the corresponding nodes. Thepotential on excitation electrodes may be prescribed using the EXCITATIONS entry.

When doing the analysis of a periodic elastic or piezoelectric structure two kinds of surfaces limit the mesh.

Surfaces limiting the mesh that are orthogonal to theperiodicity directions are constant X planes (in fact, lines) for 1-Dperiodicity, and constant X and Y planes for 2-D periodicity. For surfacesfacing each other, nodes must be identical after elementary translation.

Surfaces between the mesh and semi-infinite fluid domainsare constant Y planes (in fact, lines) for 1-D periodicity, and constant Zplanes for 2-D periodicity. On these surfaces, the pressure is linked to aplane-wave series expansion.

In the case of a structure with fluid on the front and backsurfaces, the user must provide the extreme nodes of the elementary cell:

-N1 1 1 X1
-N2 1 1 X2

In the case of a structure with fluid on the front side only(free back face), the condition becomes:

-N1 1 1 X1
-N2 1 1 XS

To get a clamped back face for a 1D periodicity, theadditional condition:

-N2 2 2

must be introduced. For a 2D periodicity, the condition is:

-N2 3 3

When the periodic structure is piezoelectric, an additionalcondition prescribing the kind of analysis must be introduced after the blank line closing the boundary condition set. For a scatteringproblem, the condition is:

-1 N

where N is the node number of the electrode. For a radiationproblem, the condition is:

-2 N

where N is the node number of the hot electrode. In thiscase, the applied voltage of this electrode has to be prescribed using the EXCITATIONS command.

### 2. Boundary Conditions in Local Axes

Boundary conditions in terms of local axes can be specifiedin several cases. First, a displacement to be constrained at a node can be ina direction that is not parallel to a global axis (Fig. 1). Second, a boundarycondition can be applied on a line that is not parallel to a global axis or ona plane that is not perpendicular to a global axis (Fig. 2). These twocases are generally merged (Fig. 3), which constitutes case 3.

| Case 1 | Case 2 | Case 3

Case 1: To define local axes at agiven node N, the user must issue the following lines:

n 10 0
A11 A12 A13 A21 A22 A23

where A11, A12, A13 (respectively A21, A22, A23 ) are the direction cosines ofthe first (second) local axis expressed in the global coordinate system. Then, degrees-of-freedom at node N are automaticallydefined by the code in the new local axes and they can be constrained using thesame data lines as for the first group. If the same global axes have to bedefined for all the nodes that belong to the same line or plane P containing the node N, thepreceding data set has to be modified by simply substituting for the first linethe following one:

-n 10 P

Case 2: To constrain nodes belongingto a plane that is not perpendicular to a global axis (1, 2 or 3), thecorresponding line of the first group has to be modified by substituting 90001 for P and adding immediately following this line asecond line that contains the direction cosines A1, A2 and A3 of the normal to this newplane, expressed in the global coordinate system. Thus, if thedegrees-of-freedom of type D of all the nodes thatbelong to this plane have to be deleted, the two data lines are:

-n d 90001
A1 A2 A3

To constrain nodes belonging to a line that is not parallelto a global axis, (3, 4, or 5), 90004 must be substituted for P and following a second line must be added that contains the direction cosines A1, A2 and A3 ofthis new line, in the global coordinate system. One obtains:

-n d 90004
A1 A2 A3

Case 3: Finally, the two precedingcases can be merged simply by mixing the corresponding data lines. Thus, ifthe preceding local axes have to be defined for all the nodes belonging to theplane described in case 2:

-n 10 90001
A11 A12 A13 A21 A22 A23
A1 A2 A3

For a line that is different from the global axes system,the data are written in the same order, substituting 90004 for 90001, with A1, A2 and A3 becoming the direction cosines of this line.

Notes

- A blank line closes this data line set.
- Constraints on different degrees-of-freedom can be merged. Thusif the same constraint has to be applied to Ux and Uz, D = 13.
- When D = 0, all degrees-of-freedom are clamped.
- The null flux condition is the natural condition of a fluiddomain (pressure gradient flux) or of a piezoelectric domain (electrical fieldflux) or of a magnetic domain (reduced magnetic field flux).

Examples:

+ -1 1 1 * yOz is a SYMMETRY PLANE(node 1 is on O)
+ -1 12 3 * xOy is an ANTISYMMETRY PLANE
+ -45 4 1 * Electrode at v=0
+ -68 4 1 * Electrode at v=0
+ -66 4 -1 * Excitation electrode
+

Note: For a modal analysis, the electrical short-circuitconditions (resonance) would have to be written:

+ -45 4 1 * Electrode at v=0
+ -68 4 1 * Electrode at v=0
+ -66 4 1 * Excitation electrode at V=0
+

and the open-circuitconditions (antiresonance) would have to be written:

+ -45 4 1 * Electrode at v=0
+ -68 4 1 * Electrode at v=0
+ -66 4 -1 * Floating voltage electrode
+

Example of a non standard symmetry plane (xOz plane rotated15° around Oz.

+ -1 10 90001 * Define local axes on the plane
+ 0.965926 0.258819 0.0 -0.258819 0.965926 0.0
+ -0.258819 0.965926 0.0
+ -1 2 90001 * Freeze local Uy displacements on that plane
+ -0.258819 0.965926 0.0
+