Transformations and extrusions#

Transformations on meshes#

XLiFE++ allows you to apply geometrical transformations on Mesh.

Then, if you want to apply a transformation and modify the input object, you can use one of the following functions:

  • translate to apply a translation

  • rotate2d to apply a 2d rotation

  • rotate3d to apply a 3d rotation

  • homothetize to apply a homothety

  • pointReflect to apply a point reflection

  • reflect2d to apply a 2d reflection

  • reflect3d to apply a 3d reflection

For instance:

Segment s(_v1=Point(0.,0.,0.), _v2=Point(0.,0.,0.), _nnodes=10, _domain_name="S1");
s.translate(_direction={0.,0.,1.});

Please see Geometrical transformations for definition and use of transformations routines.

However, if you want now to create a new object by applying a transformation on a geometry, you should use one of the related external functions instead.

For instance:

Segment s(_v1=Point(0.,0.,0.), _v2=Point(0.,0.,0.), _nnodes=10, _domain_name="S1");
Segment s2=translate(s, _direction={0.,0.,1.});

Important

When transforming a Mesh, domain names are changed. Indeed, the transformation adds a suffix “_prime”.

Mesh extrusion#

Instead of extruding a geometry, it is possible to extrude a 1d or 2d mesh (section) by applying a geometric transformation to it. To do this, use the function extrude.

Mesh extrude(s_mesh, tranformation, [[key-value1,[key-value2]], ...])

where

  • the mesh of the section (s_mesh) may be either

    • a 1d mesh made of segments, the extruded mesh is then a 2d mesh made of quadrangles

    • a 2d mesh made of triangles or quadrangles, the extruded mesh is then a 3d mesh made of prisms or hexahedra.

  • the transformation may be either

    • a Transformation object handling a tranlsation, a rotation or a combination of basic transformations; the same transformation being applied to each section of the extrusion

    • a collection of Transformation objects, each transformation being applied to one section of the extrusion. The number of transformations has to be equal to the number of layers of the extrusion.

    • a c++ function describing the extrusion path, with the following form:

      Vector<real_t> path(const Point&, Parameters&, DiffOpType);

Key values (up to 5) can be specified to control the extrusion process:

  • _layers to give the number of layers (mandatory)

  • _center to give the center of rotation when using a curve parameterization (computed by default as the barycenter of the input section)

  • _init_transformation to specify a tranformation to apply to the reference section before extruding it (none by default)

  • _naming_domain to control the naming of domains (noName, globalName, indexedName)

  • _naming_section to control the naming of sections(noName, globalName, indexedName)

  • _naming_side to control the naming of boundaries (noName, globalName, indexedName)

The meaning of the naming arguments is

  • noName: no domains, no section domains, noboundaries domains are created. The full domain is always created and named “#Omega”.

  • globalName: only some global extruded domains, a domain including end sections or some global boundary domains included all boundary domains are created (domain names are postfixed _e).

  • indexedName: for each layer, one extruded domain, section domain or boundary domains are created (domain names are postfixed _0, _1,… or _e0, _e1, …).

extrusion_domain_naming
extrusion_domain_naming_table

The following figure illustrates the naming rules of domains when meshing a tubular domain using crown mesh extrusion:

Disk dext(_center=Point(0.,0.), _radius=1., _nnodes=20, _domain_name="Omega", _edge_names="Sigma");
Disk dint(_center=Point(0.,0.), _radius=0.5, _nnodes=10, _edge_names="Gamma");
Mesh crown(dext-dint, _generator=gmsh);
Mesh tube(crown, Translation(0,0,.1), _layers=10, _naming_domain=1, _naming_section=1, _naming_side=1);
crown_extrude

Fig. 113 Tube prismatic mesh from extrusion of a crown mesh#

Example 1: extrusion of a disk using a rotation leading to a torus

Here a 3d rotation of angle \(\frac{\pi}{20}\) is applied to each section and _layers key-value specifies the number of layers (40). Note that in that case, a full turn is done and the last section coincides with the first section, the nodes being not duplicated. ..code:

Mesh mdi(Disk(_center=Point(0.,0.), _radius=1., _nnodes=8), _shape=_triangle);
Mesh m=extrude(mdi, Rotation3d(_center=Point(3.,0.,0.), _axis={0.,1.,0.}, _angle=2*pi_/40), _layers=40);
mesh_extrusion_torus

Fig. 114 Extrusion using a rotation leading to a torus#

Example 2: extrusion of a rectangle using a combination of transformations

The transformation may be a combination of transformations (see Geometrical transformations for more details):

Mesh mr(Rectangle(_xmin=0, _xmax=1, _ymin=1, _ymax=3, _nnodes=Numbers(6, 10)), _shape=_triangle, _order=1, _generator=_structured);
Mesh m=extrude(mr, Translation(0.,0.1,0.)*Rotation3d(Point(3.,0.,0.), Reals(0.,1.,0.), 2.5*pi_/100), _layers=100);
mesh_extrusion_helix

Fig. 115 Extrusion using a combination of transformations#

Example 3 : extrusion of a rectangle using a list of transformations

Instead of specifying a transformation, it is possible to specify the list of transformations to apply to the sections:

Mesh mr(Rectangle(_xmin=0, _xmax=1, _ymin=1, _ymax=3, _nnodes=Numbers(6, 10)), _shape=_triangle, _order=1, _generator=_structured);
Translation tran(0.2,0.,0.);
Rotation3d rot1(Point(3.,0.,0.),Reals(0.,1.,0.),pi_/40);
Rotation3d rot2(Point(5.,0.,6.),Reals(0.,1.,0.),-pi_/40);
std::vector<Transformation*> trs(50);
for (number_t i=0; i<20; i++)  { trs[i]=&rot1; trs[i+30]=&rot2; }
for (number_t i=0; i<10; i++)  { trs[i+20]=&tran; }
Mesh m=extrude(mesh2dP1, trs, _naming_domain=1);
mesh_extrusion_S

Fig. 116 Extrusion using a list of Transformations#

Example 4 : extrusion of a disk using a path parametrization

To deal with more general extrusions, a C++ function giving the parametrization of the extrusion path: \(t\in[0,1]\rightarrow \gamma(t)\in \mathbb R^d\) can be given. Be cautious, the parametrization has to statisfy \(\gamma(0)=0\) and \(\gamma'(0)=n_s\) where \(n_s\) is the normal vector to the section to be extruded. The extrusion works as following:

  • the nodes \(M_0\) of section \(S_0\) are translated: \(M_k=M_0+\gamma(\frac{k}{n})\)

  • then, a rotation defined from the center \(C_k=C_0+\gamma(\frac{k}{n})\) and the two vectors \(n_0,\ \gamma'(\frac{k}{n})\) is applied to the nodes \(M_k\)

By default, the rotation center \(C_0\) is the barycenter of all nodes of the section \(S_0\) but it can be changed using the key-value _center. Note that this process guarantees that all sections are orthogonal to the extrusion curve.

Reals gamma(const Point& P, Parameters& par, DiffOpType dop)
{
  real_t t=P(1);
  if (dop==_dt) return Reals{16*pi_*std::sin(4*pi_*t), 0., 40.};
  return Reals{2*(1-std::cos(4*pi_*t)), 0., 40*t};6
}

Mesh mdi(Disk(_center=Point(0.,0.), _radius=1., _nnodes=8), _shape=_triangle, _generator=subdiv);
Mesh m=extrude(mdi, gamma, _layers=100);
mesh_extrusion_snake

Fig. 117 Extrusion using a curve parametrization#

Tip

The orientation of normal vector to the section to be extruded is not predictable. But using the routine setNormalOrientation it can be set:

Mesh mdi(Disk(_center=Point(0.,0.), _radius=1., _nnodes=8), _shape=_triangle);
mdi.domain(0).setNormalOrientation(_towardsInfinite,Point(0.,0.,-1.));

where the given point (here (0.,0.,-1.)) specifies an ‘interior’ point.

Danger

If curvature radius is too small compared to the section size, the extrusion process may lead to incoherent domain!