Ellipsoids & balls

Ellipsoid

To define an ellipsoidal volume, do the same way as for an ellipse or a disk (See Ellipse or Disk), namely using 4 points \(c\), \(v_1\), \(v_2\), \(v_6\), defined as in the following figure:

There is a parameter for each of them: _center, _v1, _v2, and _v6 taking points or a single value (in this case, it is like a 1D point).

For ellipsoidal volumes where main axes are parallel to x-axis, y-axis and z-axis, the ellipsoid can be defined with the center and 3 axis lengths. For this purpose, use _xlength, _ylength and _zlength, taking one single positive value. Semi-axis lengths can be also defined by using _xradius, _yradius and _zradius parameters.

_nnodes can take one single value, an explicit list of 3 or 12 values or a Numbers object, one for each quarter of ellipse. _hsteps can take one real value, an explicit list of 6 real values or a Reals object. If required, give names of main domain and side domains as explained in Geometry definition.

As for Parallelepiped, Cuboid and Cube objects, the number of octants to deal with (parameter _nboctants) may be selected. Let us explain this with the following figure:

Considering the center of the ellipsoid, and the associated trihedron, symbolized by black dashed arrows, the ellipsoid can be split into 8 parts, corresponding to one octant. Octants having a numbering convention, when giving the number of octants, for instance 5, to build intersection of the ellipsoid with octants 1 to 5. The default value is 8, so that the whole ellipsoid is considered. This is a way to define some specific geometries, such as:

• the Fichera ellipsoid (7 octants):

• the ellipsoidal L-shape (6 octants):

• the ellipsoid with 5 octants:

• the half ellipsoid (4 octants):

• the half ellipsoidal L-shape (3 octants):

• the quarter of ellipsoid (2 octants):

• the eighth of ellipsoid(1 octants):

Examples:

Point o(0.,0.,0.), a(3.,0.,0.), b(0.,2.,0.), c(0.,0.,1.);
Ellipsoid e1(_center=o, _v1=a, _v2=b, _v6=c, _nnodes={35, 30, 25},
             _domain_name="Omega1", _side_names="Gamma");
Ellipsoid e2(_center=o, _v1=a, _v2=b, _v6=c,
             _nnodes={35, 35, 35, 35, 30, 30, 30, 30, 25, 25, 25, 25},
             _domain_name="Omega1", _side_names="Gamma");
Ellipsoid e3(_center=o,_xlength=6, _ylength=4, _zlength=2,
             _nnodes={35, 30, 25}, _domain_name="Omega1", _side_names="Gamma");

These are 3 definitions of the same Ellipsoid object. The difference between \(e_1\) and \(e_2\) explains the ability to give 3 or 12 values for _nnodes.

Let’s summarize information about geometrical keys on ellipsoids:

key(s)	authorized types	examples
_center, _v1, _v2, _v6	Point	_center=Point(0.,0.,0.)
_xlength, _ylength, _zlength, _xradius, _yradius, _zradius	single unsigned integer or real positive value	_xlength=1, _zradius=2.5
_nboctants	single unsigned integer value between 1 and 8	_nboctants=3

Ball

To define a ball, do the same way as for an ellipsoid (See Ellipsoid, namely using 4 points \(c\), \(p_1\), \(p_2\), \(p_6\), defined as in the following figure:

There is a parameter for each of them: _center, _v1, _v2, and _v6 taking points or a single value (in this case, it is like a 1D point). For balls where main axes are parallel to x-axis, y-axis and z-axis, the ball can be defined with the center and the radius. For this purpose, use _radius, taking one single positive value.

_nnodes can take one single value, an explicit list of 3 or 12 values or a Numbers object, one for each quarter of ellipse. _hsteps can take one real value, an explicit list of 6 real values or a Reals object.

Give an additional argument: the number of octants to deal with (parameter _nboctants). See Ellipsoid for details.

If required give names of main domain and side domains as explained in Geometry definition.

Point o(0.,0.,0.), a(3.,0.,0.), b(0.,3.,0.), c(0.,0.,3.);
Ball b1(_center=o, _v1=a, _v2=b, _v6=c, _nnodes={35, 30, 25},
        _domain_name="Omega1", _side_names="Gamma");
Ball b2(_center=o, _v1=a, _v2=b, _v6=c, _domain_name="Omega1", _side_names="Gamma",
        _nnodes={35, 35, 35, 35, 30, 30, 30, 30, 25, 25, 25, 25});
Ball b3(_center=o, _radius=3, _nnodes={35, 30, 25}, _domain_name="Omega1",
        _side_names="Gamma");

These are 3 definitions of the same Ball object. The difference between b1 and b2 explains the ability to give 3 values for _nnodes.

Important

The Ball object has another name Sphere

Let’s summarize information about geometrical keys on balls:

key(s)	authorized types	examples
_center, _v1, _v2, _v6	Point	_center=Point(0.,0.,0.)
_radius	single unsigned integer or real positive value	_radius=1, _radius=2.5
_nboctants	single unsigned integer value between 0 and 8	_nboctants=3