• Introduction
• Algebra
• Quadrature
• Common functions
• Mesh
• Finite Elements
  • Fi transform
  • Edges
  • Triangles
  • Quadrilaterals
  • Tetrahedra
  • Pyramids
  • Wedges
  • Hexahedra
• Computations
• Boundary conditions
• Post-processing
• Solved equations
• Time schemes

In this page, we detail the classes related to the transformation F_i. We also detail finite element projectors, that enable to interpolate the solution on predefined sets of points. The transformation F_i transforms a reference element (unit triangle, unit square, unit tetrahedron, symmetric pyramid, unit prism, unit cube) into the real element as shown in the figures below (in 3-D)

The real element can be "straight", i.e. defined with first-order shape functions, or "curved". In that case, we use high-order shape functions to write transformation Fi. When the element is straight, we just need the vertices of the element in order to define the transform, whereas for curved elements, we need intermediary points on the edges and the faces (points placed on the interior are automatically computed). These informations are provided by the mesh.

Usually, we consider three different sets of points which are known on the reference element :

• Nodal points : points used for the expression of transformation Fi when the element is curved
• Quadrature points : points used for integration over the reference element
• Dof (degrees of freedom) points : points used to project a function f to the finite element space. For nodal elements, this will be interpolation points.

Since these points are known, some precomputations are made in finite element classes, so that the computation of transform Fi for these points is fast. The object that effectively stores the images of points by transformation Fi is SetPoints, the jacobian matrices are stored in SetMatrices (see file FiniteElement/PointsReference.hxx), see below an example of use of these classes :

// first you need to define a mesh
Mesh<Dimension2> mesh;
mesh.SetGeometryOrder(4);

// then you read it
mesh.Read("test.mesh");

// you want to know Fi(xi_j) and DF_i(xi_j) for points xi_j
SetPoints<Dimension2> PointsElem;
SetMatrices<Dimension2> MatricesElem;

VectR3 s;
for (int i = 0; i < mesh.GetNbElt(); i++)
  {
    // first step : retrieve the vertices of the element i
    mesh.GetVerticesElement(i, s);

    // we retrieve the finite element associated with the element i
    const ElementGeomReference<Dimension2>& elt = mesh.GetReferenceElement(i);

    // then you ask to compute image of all the points (quadrature, nodal, dofs)
    // if you only need nodal points => FjElemNodal
    // if you only need quadrature points => FjElemQuadrature
    // if you only need dof points => FjElemDof
    elt.FjElem(s, PointsElem, mesh, i);
    
    // then you can loop over nodal points
    for (int j = 0; j < PointsElem.GetNbPointsNodal(); j++)
       cout << PointsElem.GetPointNodal(j) << endl;

    // or quadrature points
    for (int j = 0; j < PointsElem.GetNbPointsQuadrature(); j++)
       cout << PointsElem.GetPointQuadrature(j) << endl;

    // or dof points
    for (int j = 0; j < PointsElem.GetNbPointsDof(); j++)
       cout << PointsElem.GetPointDof(j) << endl;

    // you can require the computation of jacobian matrices
    elt.DFjElemNodal(s, PointsElem, MatricesElem, mesh, i);

    // and display jacobian matrix DFi at each nodal point for instance
    for (int j = 0; j < PointsElem.GetNbPointsNodal(); j++)
       cout << MatricesElem.GetPointNodal(j) << endl;

    // once you have compute FjElemNodal (minimal requirement)
    // you can compute Fi(x) for a single point, and DFi(x) 
    R2 point_local, point global;
    point_local.Init(0.5, 0.4);
    elt.Fj(s, PointsElem, point_local, point_global, mesh, i);
    cout << "The result of Fi(0.5, 0.4) is " << point_global << endl;

    Matrix2_2 dfj;
    elt.DFj(s, PointsElem, point_local, dfj, mesh, i);

    // you can try to invert transformation (using Newton or Minpack solver)
    R2 sol_point;
    FjInverseProblem<Dimension2> invFj(mesh, i); // here nodal points are computed, no need to call FjElemNodal or FjElem
    // inside_elt should be equal to true if the point has been found inside the element 
    bool inside_elt = inverseFj.Solve(point_global, sol_point);

    // you can also compute the image of quadrature points only on the surface
    // num_loc is the local number of the surface in the element
    int num_loc = 2;
    tet.FjSurfaceElem(s, PointsElem, mesh, i, num_loc);

    // and jacobian matrices too
    tet.DFjSurfaceElem(s, PointsElem, MatricesElem, mesh, i, num_loc);

    // then you can display them
    for (int i = 0; i < PointsElem.GetNbPointsQuadratureBoundary(); i++)
      {
         cout << PointsElem.GetPointQuadratureBoundary(i) << endl;
         cout << MatricesElem.GetPointQuadratureBoundary(i) << endl;
      }

    // you can also retrieve normales, and surfacic elements
    for (int i = 0; i < PointsElem.GetNbPointsQuadratureBoundary(); i++)
      {
        // outward normale to the element
        R3 normale = MatricesElem.GetNormaleQuadratureBoundary(i);
        // surfacic element (for integration)
        double ds = MatricesElem.GetDsQuadratureBoundary(i);
      }
  }

Methods of SetPoints

ReallocatePointsDof	allocates the array containing dof points
ReallocatePointsQuadrature	allocates the array containing quadrature points
ReallocatePointsNodal	allocates the array containing nodal points
ReallocatePointsQuadratureBoundary	allocates the array containing quadrature points on a face of the element
ReallocatePointsDofBoundary	allocates the array containing dof points on a face of the element
SetPointDof	sets i-th dof point on the real element
SetPointQuadrature	sets i-th quadrature point on the real element
SetPointNodal	sets i-th nodal point on the real element
SetPointQuadratureBoundary	sets i-th quadrature point on a face of the real element
SetPointDofBoundary	sets i-th dof point on a face of the real element
GetNbPointsDof	returns the number of dof points
GetNbPointsQuadrature	returns the number of quadrature points
GetNbPointsNodal	returns the number of nodal points
GetNbPointsQuadratureBoundary	returns the number of quadrature points on a face of the element
GetNbPointsDofBoundary	returns the number of dof points on a face of the element
GetPointDof	returns i-th dof point on the real element
GetPointQuadrature	returns i-th quadrature point on the real element
GetPointNodal	returns i-th nodal point on the real element
GetPointQuadratureBoundary	returns i-th quadrature point on a face of the real element
GetPointDofBoundary	returns i-th dof point on a face of the real element
CopyNodalToQuadrature	quadrature points are the same as nodal points
CopyNodalToDof	dof points are the same as nodal points
CopyQuadratureToDof	dof points are the same as quadrature points
RotatePoints	applies rotation to points with respect to z-axis
TranslatePoints	applies translation to points

Methods of SetMatrices

ReallocatePointsDof	allocates the array containing matrices on dof points
ReallocatePointsQuadrature	allocates the array containing matrices on quadrature points
ReallocatePointsNodal	allocates the array containing matrices on nodal points
ReallocatePointsQuadratureBoundary	allocates the array containing matrices on quadrature points on a face of the element
ReallocatePointsDofBoundary	allocates the array containing matrices on dof points on a face of the element
GetNbPointsQuadratureBoundary	returns the number of quadrature points on a face of the element
GetNbPointsDofBoundary	returns the number of dof points on a face of the element
SetPointDof	sets the jacobian matrix for the i-th dof point
SetPointQuadrature	sets the jacobian matrix for the i-th quadrature point
SetPointNodal	sets the jacobian matrix for the i-th nodal point
SetPointQuadratureBoundary	sets the jacobian matrix for the i-th quadrature point on a face
SetPointDofBoundary	sets the jacobian matrix for the i-th quadrature point on a face
SetPointNodalBoundary	sets the jacobian matrix for the i-th nodal point on a face
SetInversePointNodalBoundary	sets the inverse of jacobian matrix for the i-th nodal point on a face
SetNormaleQuadratureBoundary	sets the outward normale for the i-th quadrature point on a face
SetDsQuadratureBoundary	sets the element of surface integration for a quadrature point of a face
SetK1QuadratureBoundary	sets the first eigenvalue of curvature tensor for a quadrature point of a face
SetK2QuadratureBoundary	sets the second eigenvalue of curvature tensor for a quadrature point of a face
FillNodal	sets all jacobian matrices for nodal points to a same matrix
FillQuadrature	sets all jacobian matrices for quadrature points to a same matrix
FillDof	sets all jacobian matrices for dof points to a same matrix
ComputeNodalBoundary	computes jacobian matrices for nodal points of a face
GetPointDof	returns the jacobian matrix for the i-th dof point
GetPointQuadrature	returns the jacobian matrix for the i-th quadrature point
GetPointNodal	returns the jacobian matrix for the i-th nodal point
GetPointQuadratureBoundary	returns the jacobian matrix for the i-th quadrature point on a face
GetPointDofBoundary	returns the jacobian matrix for the i-th dof point on a face
GetNormaleQuadratureBoundary	returns the outward normale for the i-th quadrature point on a face
GetPointNodalBoundary	returns the jacobian matrix for nodal points of a face
GetInversePointNodalBoundary	returns the inverse of jacobian matrix for nodal points of a face
GetDsQuadratureBoundary	returns the element of surface integration for a quadrature point of a face
GetK1QuadratureBoundary	returns the first eigenvalue of curvature tensor for a quadrature point of a face
GetK2QuadratureBoundary	returns the second eigenvalue of curvature tensor for a quadrature point of a face
CopyNodalToQuadrature	jacobian matrices for quadrature points are the same as for nodal points
CopyNodalToDof	jacobian matrices for dof points are the same as for nodal points
CopyQuadratureToDof	jacobian matrices for dof points are the same as for quadrature points
RotateNormale	applies rotation to normales with respect to z-axis
InitSetPoints	associates a set of points with the current object
GetSetPoints	returns the associated set of points

Attributes and methods of FjInverseProblem

Below we list the attributes and methods of the class FjInverseProblem, which is used to compute the solution x of the equation F_i(x) = y.

non_linear_solver	which non-linear solver has to be used to solve the equation
nb_max_iterations	maximum number of iterations allowed to find the solution
threshold_newton	stopping criterion used by non-linear solvers
coef_safety_solver	coefficient used to determine if the computed solution is outside the reference element
GetVertices	returns the vertices of the element
GetSetPoints	returns the nodal points of the element
Solve	computes the solution and returns true if the point is outside the reference element

Projection on other sets of points

It is often needed to project a finite element solution on a set of points. The base class FiniteElementProjector does this task from nodal shape functions. The derived classes TensorizedProjector and DenseProjector are implementations of this base class. In the class TensorizedProjector, the projection matrix is stored as a product of sparse matrices because of tensorization of points. In the class DenseProjector, there is no tensorization and the projection matrix is stored as a dense matrix. Finally the class FiniteElementInterpolator stores projectors for the different elements at a given order. For example, in 3-D it will store four projectors (one for tetrahedra, one for pyramids, one for prisms and one for hexahedra). Below, we provide some basic examples of use of these classes

// case of a dense projector (e.g. with triangles)
TriangleGeomReference tri;
tri.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of the projector
DenseProjector<Dimension2> proj;
proj.Init(tri, pts_interp);

// then you can use it to compute the interpolation on the provided points
VectReal_wp u(tri.GetNbPointsNodalElt());
u.FillRand();
VectReal_wp u_interp;
proj.Project(u, u_interp); // u_interp is the value of the field on the interpolated points

// if you want to apply the transpose operator :
VectReal_wp v_interp(pts_interp.GetM());
v_interp.FillRand();
proj.TransposeProject(v_interp, v); // computation of v = P^T v_interp

// if you want a generic writing (for triangles/quads), use a pointer
FiniteElementProjector* p;
p = tri.GetNewNodalInterpolation(); // it will create a DenseProjector or TensorizedProjector

// Init, Project and TransposeProject can be called similarly
p->Init(tri, pts_interp);
p->Project(u, u_interp);

// for mixed meshes, you can use FiniteElementInterpolator
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

// projection for an element :
int num_elem = 12;
liste_proj.Project(u, u_interp, mesh.Element(num_elem).GetHybridType());

Methods of FiniteElementProjector

SetIdentity	inits the projector as a permutation matrix
Project	projects the field on interpolation points
TransposeProject	performs the transpose operation of Project
GetMemorySize	returns the memory used by the object in bytes
Init	computes the projection operator
ProjectScalar	projects a scalar field on interpolation points
TransposeProjectScalar	projects a scalar field on interpolation points

Methods of FiniteElementInterpolator

InitProjection	computes the projection operators
Project	projects the field on interpolation points
TransposeProject	performs the transpose operation of Project
ProjectScalar	projects a scalar field on interpolation points
TransposeProjectScalar	projects a scalar field on interpolation points
ComputeLocalProlongation	computes the prolongation operator between two finite elements
ComputeLocalProlongationLowOrder	computes the prolongation operator between two finite elements by using first-order elements

ReallocatePointsDof

This method reallocates the array containing points associated with degrees of freedom. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsDof(N); // allocating the array
  // filling the array
  pts.SetPointDof(0, R3(0.4, 0.7, 0.2));
  pts.SetPointDof(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsQuadrature

This method reallocates the array containing quadrature points of the element. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsQuadrature(N); // allocating the array
  // filling the array
  pts.SetPointQuadrature(0, R3(0.4, 0.7, 0.2));
  pts.SetPointQuadrature(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsNodal

This method reallocates the array containing nodal points. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsNodal(N); // allocating the array
  // filling the array
  pts.SetPointNodal(0, R3(0.4, 0.7, 0.2));
  pts.SetPointNodal(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsQuadratureBoundary

This method reallocates the array containing the quadrature points of a boundary of the element. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsQuadratureBoundary(N); // allocating the array
  // filling the array
  pts.SetPointQuadratureBoundary(0, R3(0.4, 0.7, 0.2));
  pts.SetPointQuadratureBoundary(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsDofBoundary

This method reallocates the array containing points associated with degrees of freedom of the boundary. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsDofBoundary(N); // allocating the array
  // filling the array
  pts.SetPointDofBoundary(0, R3(0.4, 0.7, 0.2));
  pts.SetPointDofBoundary(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointDof

This method modifies the i-th point associated with degrees of freedom. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsDof(N); // allocating the array
  // filling the array
  pts.SetPointDof(0, R3(0.4, 0.7, 0.2));
  pts.SetPointDof(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointQuadrature

This method sets the i-th quadrature point of the element. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsQuadrature(N); // allocating the array
  // filling the array
  pts.SetPointQuadrature(0, R3(0.4, 0.7, 0.2));
  pts.SetPointQuadrature(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointNodal

This method sets the i-th nodal point of the element. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsNodal(N); // allocating the array
  // filling the array
  pts.SetPointNodal(0, R3(0.4, 0.7, 0.2));
  pts.SetPointNodal(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointQuadratureBoundary

This method sets the i-th quadrature point of a boundary of the element. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsQuadratureBoundary(N); // allocating the array
  // filling the array
  pts.SetPointQuadratureBoundary(0, R3(0.4, 0.7, 0.2));
  pts.SetPointQuadratureBoundary(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointDofBoundary

This method modifies the i-th point associated with degrees of freedom of the boundary. In a regular use, this method should not be called.

Example

  SetPoints<Dimension3> pts;
  int N = 10;
  pts.ReallocatePointsDofBoundary(N); // allocating the array
  // filling the array
  pts.SetPointDofBoundary(0, R3(0.4, 0.7, 0.2));
  pts.SetPointDofBoundary(1, R3(0.8, -0.2, 1.5));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsDof

This method returns the number of points associated with degrees of freedom.

Example

  // finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

  // computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

  // you can retrieve the number of dof points stored in pts
  int nb_pts = pts.GetNbPointsDof();

  // loop over these points
  for (int i = 0; i < nb_pts; i++)
    cout << "Dof point " << i << " = " <<  pts.GetPointDof(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsQuadrature

This method returns the number of quadrature points stored in the object.

Example

  // finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

  // computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

  // you can retrieve the number of quadrature points stored in pts
  int nb_pts = pts.GetNbPointsQuadrature();

  // loop over these points
  for (int i = 0; i < nb_pts; i++)
    cout << "Quadrature point " << i << " = " <<  pts.GetPointQuadrature(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsNodal

This method returns the number of nodal points stored in the object.

Example

  // finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

  // computing transformation Fi for nodal points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemNodal(s, pts, mesh, num_elem);

  // you can retrieve the number of nodal points stored in pts
  int nb_pts = pts.GetNbPointsNodal();

  // loop over these points
  for (int i = 0; i < nb_pts; i++)
    cout << "Nodal point " << i << " = " <<  pts.GetPointNodal(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsQuadratureBoundary

This method returns the number of quadrature points associated with a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);

// you can retrieve the number of boundary quadrature points stored in pts
int nb_pts = pts.GetNbPointsQuadratureBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary quadrature point " << i << " = " <<  pts.GetPointQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsDofBoundary

This method returns the number of points associated with degrees of freedom of the boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// we extract dof points for a given boundary
int num_loc = 2;
wed.FjElemSurfaceDof(s, pts, mesh, num_elem, num_loc);

// you can retrieve the number of boundary dof points stored in pts
int nb_pts = pts.GetNbPointsDofBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary dof point " << i << " = " <<  pts.GetPointDofBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointDof

This method returns the i-th point associated with degrees of freedom.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// you can retrieve the number of dof points stored in pts
int nb_pts = pts.GetNbPointsDof();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Dof point " << i << " = " <<  pts.GetPointDof(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointQuadrature

This method returns the i-th quadrature point stored in the object.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// you can retrieve the number of quadrature points stored in pts
int nb_pts = pts.GetNbPointsQuadrature();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Quadrature point " << i << " = " <<  pts.GetPointQuadrature(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointNodal

This method returns the i-th nodal point stored in the object.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for nodal points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemNodal(s, pts, mesh, num_elem);

// you can retrieve the number of nodal points stored in pts
int nb_pts = pts.GetNbPointsNodal();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Nodal point " << i << " = " <<  pts.GetPointNodal(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointQuadratureBoundary

This method returns the i-th quadrature point associated with a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);

// you can retrieve the number of boundary quadrature points stored in pts
int nb_pts = pts.GetNbPointsQuadratureBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary quadrature point " << i << " = " <<  pts.GetPointQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointDofBoundary

This method returns the i-th point associated with degrees of freedom of the boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// we extract dof points for a given boundary
int num_loc = 2;
wed.FjElemSurfaceDof(s, pts, mesh, num_elem, num_loc);

// you can retrieve the number of boundary dof points stored in pts
int nb_pts = pts.GetNbPointsDofBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary dof point " << i << " = " <<  pts.GetPointDofBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyNodalToQuadrature

This method copies nodal points to quadrature points. On regular use, this method should not be called.

Example

SetPoints<Dimension3> pts;
int N = 10;
pts.ReallocatePointsNodal(N); // allocating the array
// filling the array
pts.SetPointNodal(0, R3(0.4, 0.7, 0.2));
pts.SetPointNodal(1, R3(0.8, -0.2, 1.5));
// etc

// if nodal points coincide with quadrature points
pts.CopyNodalToQuadrature();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyNodalToDof

This method copies nodal points to dof points. On regular use, this method should not be called.

Example

SetPoints<Dimension3> pts;
int N = 10;
pts.ReallocatePointsNodal(N); // allocating the array
// filling the array
pts.SetPointNodal(0, R3(0.4, 0.7, 0.2));
pts.SetPointNodal(1, R3(0.8, -0.2, 1.5));
// etc

// if nodal points coincide with dof points
pts.CopyNodalToDof();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyQuadratureToDof

This method copies quadrature points to dof points. On regular use, this method should not be called.

Example

SetPoints<Dimension3> pts;
int N = 10;
pts.ReallocatePointsQuadrature(N); // allocating the array
// filling the array
pts.SetPointQuadrature(0, R3(0.4, 0.7, 0.2));
pts.SetPointQuadrature(1, R3(0.8, -0.2, 1.5));
// etc

// if quadrature points coincide with dof points
pts.CopyQuadratureToDof();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

RotatePoints

This method rotates points of an angle θ. The axis of rotation if the Ez axis, the new coordinates of the points are given as:

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElem(s, pts, mesh, num_elem);

// you can rotate all points
pts.RotatePoints(pi_wp/3);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

TranslatePoints

This method translates points with a vector u.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElem(s, pts, mesh, num_elem);

// you can translate all points
pts.TranslatePoints(R3(1.0, 0.2, -3.0));

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

ReallocatePointsDof

This method reallocates the array containing jacobian matrices associated with degrees of freedom. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsDof(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointDof(0, A0);
  mat.SetPointDof(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsQuadrature

This method reallocates the array containing jacobian matrices associated with quadrature points. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsQuadrature(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointQuadrature(0, A0);
  mat.SetPointQuadrature(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsNodal

This method reallocates the array containing jacobian matrices associated with nodal points. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsNodal(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointNodal(0, A0);
  mat.SetPointNodal(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ReallocatePointsQuadratureBoundary

This method reallocates the array containing jacobian matrices associated with quadrature points of a face. It also reallocates the array containing normales and surface elements. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointQuadratureBoundary(0, A0);
  mat.SetPointQuadratureBoundary(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

ReallocatePointsDofBoundary

This method reallocates the array containing jacobian matrices associated with dof points of a face. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsDofBoundary(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointDofBoundary(0, A0);
  mat.SetPointDofBoundary(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsQuadratureBoundary

This method returns the number of quadrature points associated with a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// you can retrieve the number of boundary quadrature points stored in mat
int nb_pts = mat.GetNbPointsQuadratureBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary quadrature point " << i << " = " <<  pts.GetPointQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNbPointsDofBoundary

This method returns the number of dof points associated with a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// and jacobian
wed.DFjElemDof(s, pts, mat, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurfaceDof(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurfaceDof(s, pts, mat, mesh, num_elem, num_loc);

// you can retrieve the number of boundary dof points stored in mat
int nb_pts = mat.GetNbPointsDofBoundary();

// loop over these points
for (int i = 0; i < nb_pts; i++)
  cout << "Boundary dof point " << i << " = " <<  pts.GetPointDofBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointDof

This method modifies the i-th jacobian matrix associated with degrees of freedom. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsDof(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointDof(0, A0);
  mat.SetPointDof(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointQuadrature

This method modifies the i-th jacobian matrix associated with quadrature points. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsQuadrature(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointQuadrature(0, A0);
  mat.SetPointQuadrature(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointNodal

This method modifies the i-th jacobian matrix associated with nodal points. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsNodal(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointNodal(0, A0);
  mat.SetPointNodal(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointQuadratureBoundary

This method modifies the i-th jacobian matrix associated with quadrature points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointQuadratureBoundary(0, A0);
  mat.SetPointQuadratureBoundary(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointDofBoundary

This method modifies the i-th jacobian matrix associated with dof points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsDofBoundary(N); // allocating the array
  // filling the array
  Matrix2_2 A0, A1;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
  mat.SetPointDofBoundary(0, A0);
  mat.SetPointDofBoundary(1, A1);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetPointNodalBoundary

This method sets all the jacobian matrices associated with nodal points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  // filling the array of jacobian matrices
  Vector<Matrix2_2> A(2);
  A(0)(0, 0) = 0.4; A(0)(0, 1) = 2.0; A(0)(1, 0) = -1.6; A(0)(1, 1) = 2.5;
  A(1)(0, 0) = 0.8; A(1)(0, 1) = 1.9; A(1)(1, 0) = -2.3; A(1)(1, 1) =
  3.2;
  // all the matrices are provided once
  mat.SetPointNodalBoundary(A);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetInversePointNodalBoundary

This method sets the inverse of all the jacobian matrices associated with nodal points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  // filling the array of jacobian matrices
  Vector<Matrix2_2> A(2);
  A(0)(0, 0) = 0.4; A(0)(0, 1) = 2.0; A(0)(1, 0) = -1.6; A(0)(1, 1) = 2.5;
  A(1)(0, 0) = 0.8; A(1)(0, 1) = 1.9; A(1)(1, 0) = -2.3; A(1)(1, 1) =
  3.2;
  // all the matrices are provided once
  mat.SetPointNodalBoundary(A);

  // you can also compute the inverses
  for (int i = 0; i < A.GetM(); i++)
    GetInverse(A(i));

  // and provide these inverses
  mat.SetInversePointNodalBoundary(A);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetNormaleQuadratureBoundary

This method modifies the i-th outgoing normale associated with quadrature points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension3> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the needed arrays
  // filling the normales
  mat.SetNormaleQuadratureBoundary(0, R3(0.8, 0.5, 0.2));
  mat.SetNormaleQuadratureBoundary(1, R3(0.7, 0.82, -0.3));
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetDsQuadratureBoundary

This method modifies the i-th surface element associated with quadrature points of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension3> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the needed arrays
  // filling the normales and surface elements
  mat.SetNormaleQuadratureBoundary(0, R3(0.8, 0.5, 0.2));
  mat.SetDsQuadratureBoundary(0, 0.23);
  mat.SetNormaleQuadratureBoundary(1, R3(0.7, 0.82, -0.3));
  mat.SetDsQuadratureBoundary(1, 0.35);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetK1QuadratureBoundary

This method modifies the first eigenvalue of the curvature tensor associated with the i-th quadrature point of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension3> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the needed arrays
  // filling the eigenvalues of the curvature tensor 
  mat.SetK1QuadratureBoundary(0, 0.5);
  mat.SetK2QuadratureBoundary(0, 0.23);
  mat.SetK1QuadratureBoundary(1, 0.45);
  mat.SetK2QuadratureBoundary(1, 0.35);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

SetK2QuadratureBoundary

This method modifies the second eigenvalue of the curvature tensor associated with the i-th quadrature point of a boundary. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension3> mat;
  int N = 10;
  mat.ReallocatePointsQuadratureBoundary(N); // allocating the needed arrays
  // filling the eigenvalues of the curvature tensor 
  mat.SetK1QuadratureBoundary(0, 0.5);
  mat.SetK2QuadratureBoundary(0, 0.23);
  mat.SetK1QuadratureBoundary(1, 0.45);
  mat.SetK2QuadratureBoundary(1, 0.35);
  // etc

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

FillNodal

This method sets all the jacobian matrices associated with nodal points with a given matrix. This method is called by finite element classes when the transformation F_i is linear. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsNodal(N); // allocating the needed arrays

  // filling the array of jacobian matrices with a constant matrix
  Matrix2_2 A;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  mat.FillNodal(A);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

FillQuadrature

This method sets all the jacobian matrices associated with quadrature points with a a given matrix. This method is called by finite element classes when the transformation F_i is linear. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsQuadrature(N); // allocating the needed arrays

  // filling the array of jacobian matrices with a constant matrix
  Matrix2_2 A;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  mat.FillQuadrature(A);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

FillDof

This method sets all the jacobian matrices associated with dof points with a a given matrix. This method is called by finite element classes when the transformation F_i is linear. In a regular use, this method should not be called.

Example

  SetMatrices<Dimension2> mat;
  int N = 10;
  mat.ReallocatePointsDof(N); // allocating the needed arrays

  // filling the array of jacobian matrices with a constant matrix
  Matrix2_2 A;
  A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
  mat.FillDof(A);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

ComputeNodalBoundary

This method fills the jacobian matrices associated with nodal points of a boundary. It uses the finite element class to retrieve the numbers of nodal points located on the boundary num_loc.

Example

Mesh<Dimension3> mesh;
MeshNumbering<Dimension3> mesh_num(mesh);

PyramidClassical pyr;
pyr.ConstructFiniteElement(3);
pyr.SetMesh(mesh);

// computing transformation Fi for nodal points
int num_elem = 32;
SetPoints<Dimension3> pts; VectR3 s;
mesh.GetVerticesElement(num_elem, s);
pyr.FjElemNodal(s, pts, mesh, num_elem);

// computing jacobian matrices DFi for nodal points
SetMatrices<Dimension3> mat;
pyr.DFjElemNodal(s, pts, mat, mesh, num_elem);

// DFi on nodal points of boundary
int num_loc = 1;
mat.ComputeNodalBoundary(num_loc, pyr);

// you can retrieve a jacobian matrix on a nodal point of the boundary
int j = 2; // local number of the nodal point on the boundary
Matrix3_3 dfj = mat.GetPointNodalBoundary(j);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointDof

This method returns the i-th jacobian matrix associated with degrees of freedom.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemDof(s, pts, mat, mesh, num_elem);

// you can retrieve the number of dof points stored in pts
int nb_pts = pts.GetNbPointsDof();

// and display jacobian matrices
for (int i = 0; i < nb_pts; i++)
  cout << "Jacobian matrix  " << i << " = " <<  mat.GetPointDof(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointQuadrature

This method returns the jacobian matrix associated with the i-th quadrature point.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// you can retrieve the number of quadrature points stored in pts
int nb_pts = pts.GetNbPointsQuadrature();

// and display jacobian matrices
for (int i = 0; i < nb_pts; i++)
  cout << "Jacobian matrix  " << i << " = " <<  mat.GetPointQuadrature(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointNodal

This method returns the jacobian matrix associated with the i-th nodal point.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for nodal points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemNodal(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemNodal(s, pts, mat, mesh, num_elem);

// you can retrieve the number of nodal points stored in pts
int nb_pts = pts.GetNbPointsNodal();

// and display jacobian matrices
for (int i = 0; i < nb_pts; i++)
  cout << "Jacobian matrix  " << i << " = " <<  mat.GetPointNodal(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointQuadratureBoundary

This method returns the jacobian matrix associated with the i-th quadrature point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// and display jacobian matrices
int nb_pts = mat.GetNbPointsQuadratureBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Jacobian matrix  " << i << " = " <<  mat.GetPointQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointDofBoundary

This method returns the jacobian matrix associated with the i-th dof point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for dof points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemDof(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemDof(s, pts, mat, mesh, num_elem);

// and quadrature points of the boundary
int num_loc = 2;
wed.FjElemSurfaceDof(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurfaceDof(s, pts, mat, mesh, num_elem, num_loc);

// and display jacobian matrices
int nb_pts = mat.GetNbPointsDofBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Jacobian matrix  " << i << " = " <<  mat.GetPointDofBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetNormaleQuadratureBoundary

This method returns the outgoing normale associated with the i-th quadrature point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points/normale of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// and display outgoing normales
int nb_pts = mat.GetNbPointsQuadratureBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Normale  " << i << " = " <<  mat.GetNormaleQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetPointNodalBoundary

This method returns the jacobian matrix for the i-th nodal point of a boundary.

Example

Mesh<Dimension3> mesh;
MeshNumbering<Dimension3> mesh_num(mesh);

PyramidClassical pyr;
pyr.ConstructFiniteElement(3);
pyr.SetMesh(mesh);

// computing transformation Fi for nodal points
int num_elem = 32;
SetPoints<Dimension3> pts; VectR3 s;
mesh.GetVerticesElement(num_elem, s);
pyr.FjElemNodal(s, pts, mesh, num_elem);

// computing jacobian matrices DFi for nodal points
SetMatrices<Dimension3> mat;
pyr.DFjElemNodal(s, pts, mat, mesh, num_elem);

// DFi on nodal points of boundary
int num_loc = 1;
mat.ComputeNodalBoundary(num_loc, pyr);

// you can retrieve a jacobian matrix on a nodal point of the boundary
int j = 2; // local number of the nodal point on the boundary
Matrix3_3 dfj = mat.GetPointNodalBoundary(j);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetInversePointNodalBoundary

This method returns the inverse of jacobian matrix for the i-th nodal point of a boundary.

Example

Mesh<Dimension3> mesh;
MeshNumbering<Dimension3> mesh_num(mesh);

PyramidClassical pyr;
pyr.ConstructFiniteElement(3);
pyr.SetMesh(mesh);

// computing transformation Fi for nodal points
int num_elem = 32;
SetPoints<Dimension3> pts; VectR3 s;
mesh.GetVerticesElement(num_elem, s);
pyr.FjElemNodal(s, pts, mesh, num_elem);

// computing jacobian matrices DFi for nodal points
SetMatrices<Dimension3> mat;
pyr.DFjElemNodal(s, pts, mat, mesh, num_elem);

// DFi on nodal points of boundary
int num_loc = 1;
mat.ComputeNodalBoundary(num_loc, pyr);

// you can retrieve a jacobian matrix on a nodal point of the boundary
int j = 2; // local number of the nodal point on the boundary
Matrix3_3 dfj = mat.GetPointNodalBoundary(j);

// and you can retrieve the inverse as well (inverse has already been computed by ComputeNodalBoundary)
Matrix3_3 inv_dfj = mat.GetInversePointNodalBoundary(j);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetDsQuadratureBoundary

This method returns the surface element associated with the i-th quadrature point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points/normale of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// and display surface elements
int nb_pts = mat.GetNbPointsQuadratureBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Ds  " << i << " = " <<  mat.GetDsQuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetK1QuadratureBoundary

This method returns the first eigenvalue of the curvature tensor associated with the i-th quadrature point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points/normale of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// and display eigenvalues of curvature tensor
int nb_pts = mat.GetNbPointsQuadratureBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Eigenvalues of C on point  " << i << " = " <<  mat.GetK1QuadratureBoundary(i)
       <<  " and " << mat.GetK2QuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

GetK2QuadratureBoundary

This method returns the first eigenvalue of the curvature tensor associated with the i-th quadrature point of a boundary.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi for quadrature points
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElemQuadrature(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElemQuadrature(s, pts, mat, mesh, num_elem);

// and quadrature points/normale of the boundary
int num_loc = 2;
wed.FjElemSurface(s, pts, mesh, num_elem, num_loc);
wed.DFjElemSurface(s, pts, mat, mesh, num_elem, num_loc);

// and display eigenvalues of curvature tensor
int nb_pts = mat.GetNbPointsQuadratureBoundary();
for (int i = 0; i < nb_pts; i++)
  cout << "Eigenvalues of C on point  " << i << " = " <<  mat.GetK1QuadratureBoundary(i)
       <<  " and " << mat.GetK2QuadratureBoundary(i) << endl;

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyNodalToQuadrature

This method copies jacobian matrices computed for nodal points to jacobian matrices associated with quadrature points. On regular use, this method should not be called.

Example

SetMatrices<Dimension2> mat;
int N = 10;
mat.ReallocatePointsNodal(N); // allocating the array
// filling the array
Matrix2_2 A0, A1;
A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
mat.SetPointNodal(0, A0);
mat.SetPointNodal(1, A1);
// etc

// if nodal points coincide with quadrature points, jacobian matrices are equal 
mat.CopyNodalToQuadrature();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyNodalToDof

This method copies jacobian matrices computed for nodal points to jacobian matrices associated with dof points. On regular use, this method should not be called.

Example

SetMatrices<Dimension2> mat;
int N = 10;
mat.ReallocatePointsNodal(N); // allocating the array
// filling the array
Matrix2_2 A0, A1;
A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
mat.SetPointNodal(0, A0);
mat.SetPointNodal(1, A1);
// etc

// if nodal points coincide with dof points, jacobian matrices are equal 
mat.CopyNodalToDof();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

CopyQuadratureToDof

This method copies jacobian matrices computed for quadrature points to jacobian matrices associated with dof points. On regular use, this method should not be called.

Example

SetMatrices<Dimension2> mat;
int N = 10;
mat.ReallocatePointsQuadrature(N); // allocating the array
// filling the array
Matrix2_2 A0, A1;
A0(0, 0) = 0.4; A0(0, 1) = 2.0; A0(1, 0) = -1.6; A0(1, 1) = 2.5;
A1(0, 0) = 0.8; A1(0, 1) = 1.9; A1(1, 0) = -2.3; A1(1, 1) = 3.2;
mat.SetPointQuadrature(0, A0);
mat.SetPointQuadrature(1, A1);
// etc

// if quadrature points coincide with dof points, jacobian matrices are equal 
mat.CopyQuadratureToDof();

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReferenceInline.cxx

RotateNormale

This method rotates normales of an angle θ. The axis of rotation if the Ez axis, the new coordinates of the normales are given as:

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElem(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElem(s, pts, mat, mesh, num_elem);

// computing normales of a boundary
int num_loc = 2;
wed.FjSurfaceElem(s, pts, mesh, num_elem, num_loc);
wed.DFjSurfaceElem(s, pts, mat, mesh, num_elem, num_loc);

// you can rotate all normales
mat.RotateNormale(pi_wp/3);

// and retrieve a rotated normale
R3 n = mat.GetNormaleQuadratureBoundary(1);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

InitSetPoints

This method initializes the pointer storing an object SetPoints in the class, such that GetSetPoints will work correctly. On regular use, you do not need to call this method, because it is already done when DFjElem (and derivates such as DFjElemNodal) is called.

Example

SetPoints<Dimension2> pts;
SetMatrices<Dimension2> mat;

// if you want to store the adress of pts in the object mat :
mat.InitSetPoints(pts);

// then you can retrieve pts from the object mat
const SetPoints<Dimension2> pts2 = mat.GetSetPoints(); // pts and pts2 should share the same address

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

GetSetPoints

This method returns the object SetPoints stored in the current object.

Example

// finite element is constructed
WedgeClassical wed;
wed.ConstructFiniteElement(3);
wed.SetMesh(mesh);

// computing transformation Fi
SetPoints<Dimension3> pts;
VectR3 s; int num_elem = 40;
mesh.GetVerticesElement(num_elem, s);
wed.FjElem(s, pts, mesh, num_elem);

// and jacobian matrices
SetMatrices<Dimension3> mat;
wed.DFjElem(s, pts, mat, mesh, num_elem);

// if you want to access to pts with only mat :
const SetPoints<Dimension2> pts2 = mat.GetSetPoints(); // pts and pts2 are the same variable

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

non_linear_solver

This attribute stores the non-linear solver used to invert the transformation F_i. You can choose among the following non-linear solvers :

• NEWTON_SOLVER : Newton-Raphson solver
• MINPACK_SOLVER : Minpack solver
• LVM_SOLVER : Levenberg-Marquardt solver

If this attribute is not set, the default solver is NEWTON_SOLVER.

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// setting the non-linear solver
FjInverseProblem<Dimension3>::non_linear_solver = FjInverseProblem<Dimension3>::LVM_SOLVER;

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

nb_max_iterations

This attribute stores the maximum number of iterations allowed for the non-linear solver. If this attribute is not set, the default maximum number of iterations is set to 50.

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// setting the maximum number of iterations
FjInverseProblem<Dimension3>::nb_max_iterations = 10;

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

threshold_newton

This attribute stores the stopping criterion used by the non-linear solver. If this attribute is not set, the default value is 4*epsilon_machine where epsilon_machine is the machine precision (2e-16 in double precision).

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// setting the maximum number of iterations
FjInverseProblem<Dimension3>::nb_max_iterations = 10;
// and stopping criterion
FjInverseProblem<Dimension3>::threshold_newton = 1e-14;

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

coef_safety_solver

This attribute stores the safety coefficient used to determine if the solution found is outside the reference element. A larger coefficient leads to admit more solutions that are very close the reference element. A smaller coefficient can lead to exclude some solutions that are computed outside the reference element because of round-off errors.

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// setting the maximum number of iterations
FjInverseProblem<Dimension3>::nb_max_iterations = 10;
// and stopping criterion
FjInverseProblem<Dimension3>::threshold_newton = 1e-14;
// safety coefficient to admit more solutions
FjInverseProblem<Dimension3>::coef_safety_solver = 100;

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

GetVertices

This method gives access to the list of vertices of the element that have been extracted in the constructor.

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// you can retrieve the list of vertices extracted in the constructor :
VectR3& s = fj_inv.GetVertices();

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

GetSetPoints

This method gives access to the list of points of the element that have been computed in the constructor. These points are not computed if the element is affine (in such case, the inversion of transformation F_i is straightforward).

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// you can retrieve the list of points of the element
SetPoints<Dimension3>& pts = fj_inv.GetSetPoints();

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

Solve

This method computes the solution pt_loc of F_i(pt_loc) = pt_glob. It returns true if the element is inside the reference element.

Example

Mesh<Dimension3> mesh;
mesh.Read("test.mesh");  

// initializing the solver
int num_elem = 21; R3 point(4.0, 11.0, -2.0); // global coordinates of point
FjInverseProblem<Dimension3> fj_inv(mesh, num_elem);

// computing pt_loc such that Fi(pt_loc) = point
bool pt_inside = fj_inv.Solve(point, pt_loc);

Location :

FiniteElement/PointsReference.hxx   FiniteElement/PointsReference.cxx

SetIdentity

This method sets the projector as a permutation matrix :

P_{row(i), col(i)} = 1

Example

  DenseProjector<Dimension2> proj;

  // rows and colums such that P(i, j) is equal to 1
  // for other indexes P(k, l) is equal to 0
  IVect row(3), col(3);
  row(0) = 1; row(1) = 2; row(2) = 0;
  col(0) = 0; col(1) = 1; col(2) = 2;

  proj.SetIdentity(col, row, 3);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

Project

This method computes v = P u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points.

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

// random field
VectReal_wp u(quad.GetNbPointsNodalElt()), v;
u.FillRand();

// computation of v = P u to obtain the solution on interpolation points
proj.Project(u, v);

// you can also project vectorial fields :
VectR3 E(quad.GetNbPointsNodalElt()), Einterp;
E(0).Init(0.9, 1.2, -0.5); // etc

proj.Project(E, Einterp);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

TransposeProject

This method computes u = P^T v where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points.

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

// random field
VectReal_wp u(quad.GetNbPointsNodalElt()), v;
u.FillRand();

// computation of v = P u to obtain the solution on interpolation points
proj.Project(u, v);

// application of transpose operator u = P^T v
proj.TransposeProject(v, u);

// you can also transpose project vectorial fields :
VectR3 vE(pts_interp.GetM()), uE;
vE(0).Init(0.9, 1.2, -0.5); // etc

proj.TransposeProject(vE, uE);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

GetMemorySize

This method returns the memory used by the object in bytes.

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

// memory used to store proj ?
size_t mem = proj.GetMemorySize();

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

Init

This method computes the projection operator P usually given as

where φ_j are shape functions and x_i interpolation points. For the class TensorizedProjector, the matrix is not explicitely stored. In this class, sparse matrices are constructed, such that P is a product of sparse matrices. The matrix-vector product with P can be performed by the method Project (method TransposeProject) for the product with the transpose of P). In the second syntax, you can provide the 1-D interpolation points (useful for TensorizedProjector).

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

ProjectScalar

This method computes v = P u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points. This method is virtual such that if you want to define a new class that derives from FiniteElementProjector, this method can be overloaded to define the matrix-vector product with P.

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

// random field
VectReal_wp u(quad.GetNbPointsNodalElt()), v;
u.FillRand();

// computation of v = P u to obtain the solution on interpolation points
proj.ProjectScalar(u, v);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

TransposeProjectScalar

This method computes u = P^T v where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points. This method is virtual such that if you want to define a new class that derives from FiniteElementProjector, this method can be overloaded to define the matrix-vector product with P^T.

Example

// case of a tensorized projector (e.g. with quadrilaterals)
QuadrangleGeomReference quad;
quad.ConstructFiniteElement(4);

// definition of points where we want to interpolate the solution
VectR2 pts_interp(10);
pts_interp(0).Init(0.4, 0.7);
pts_interp(1).Init(0.2, 0.9);
// etc

// computation of operator P
TensorizedProjector<Dimension2> proj;
proj.Init(quad, pts_interp);

// random field
VectReal_wp u(quad.GetNbPointsNodalElt()), v;
u.FillRand();

// computation of v = P u to obtain the solution on interpolation points
proj.ProjectScalar(u, v);

// application of transpose operator u = P^T v
proj.TransposeProjectScalar(v, u);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

InitProjection

This method computes the projection operators P usually given as

where φ_j are shape functions and x_i interpolation points. It computes the operators for the elements present in the first argument. The elements can be a mix of tetrahedra/pyramids/prisms/hexahedra (triangles/quadrilaterals in 2-D). In the second syntax, you can provide the 1-D interpolation points.

Example

// starting with a 3-D mesh
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

ProjectScalar

This method computes v = P u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points.

Example

  // starting with a 3-D mesh
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

// projection for an element of the mesh:
int num_elem = 12;
VectReal_wp u(mesh.GetNbPointsNodalElt(num_elem));
u.FillRand();
VectReal_wp u_interp;
liste_proj.ProjectScalar(u, u_interp, mesh.Element(num_elem).GetHybridType());

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

TransposeProjectScalar

This method computes v = P^T u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points.

Example

  // starting with a 3-D mesh
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

// projection for an element of the mesh:
int num_elem = 12;
VectReal_wp u(mesh.GetNbPointsNodalElt(num_elem));
u.FillRand();
VectReal_wp u_interp;
liste_proj.ProjectScalar(u, u_interp, mesh.Element(num_elem).GetHybridType());

// and transpose operator (u = P^T u_interp)  
liste_proj.TransposeProjectScalar(u_interp, u, mesh.Element(num_elem).GetHybridType());

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

Project

This method computes v = P u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points. The field can be scalar or vectorial.

Example

// starting with a 3-D mesh
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

// projection for an element of the mesh (we take a vectorial field)
int num_elem = 12;
VectR3 u(mesh.GetNbPointsNodalElt(num_elem));
u(0).Init(0.2, 0.5, 1.1); // etc
VectR3 u_interp;
liste_proj.Project(u, u_interp, mesh.Element(num_elem).GetHybridType());

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

TransposeProject

This method computes v = P^T u where P is the projection operator, usually given as

where φ_j are shape functions and x_i interpolation points. The fields u and v can be scalar fields or vectorial fields.

Example

  // starting with a 3-D mesh
Mesh<Dimension3> mesh;
mesh.Read("test.mesh");
Vector<VectR3> pts_3d(4);

// points for the unit tetrahedron
pts_3d(0).Reallocate(5);
pts_3d(0)(0).Init(0.2, 0.2, 0.1); // etc

// for the symmetric pyramid
pts_3d(1).Reallocate(10);
pts_3d(1)(0).Init(-0.3, 0.4, 0.2); // etc

// pts_3d(2) for the prism and pts_3d(3) for the pyramid
// in 2-D, pts(0) for the unit triangle and pts(1) for the unit square

// computing the projection for all elements (tet, pyramids, hex, prisms)
FiniteElementInterpolator liste_proj;
liste_proj.InitProjection(mesh.GetReferenceElement(), pts_3d);

// projection for an element of the mesh (we take a vectorial field)
int num_elem = 12;
VectR3 u(mesh.GetNbPointsNodalElt(num_elem));
u(0).Init(2.1, -1.6, 3.2); // etc
VectR3 u_interp;
liste_proj.Project(u, u_interp, mesh.Element(num_elem).GetHybridType());

// and transpose operator (u = P^T u_interp)  
liste_proj.TransposeProject(u_interp, u, mesh.Element(num_elem).GetHybridType());

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

ComputeLocalProlongation

This method computes the local prolongation operator (as detailed in ComputeLocalProlongation between a coarse finite element and a fine element.

Example

// coarse finite elements
TriangleClassical tri; QuadrangleGauss quad;
tri.ConstructFiniteElement(2); quad.ConstructFiniteElement(2);

// fine finite elements
TriangleClassical tri_f; QuadrangleGauss quad_f;
tri_f.ConstructFiniteElement(4); quad_f.ConstructFiniteElement(4);

// list of elements (usually associated with elements of the mesh)
Vector<const ElementReference<Dimension2, 1>* > elt_fine(6);
Vector<const ElementReference<Dimension2, 1>* > elt_coarse(6);

elt_fine(0) = &tri_f; elt_fine(1) = &quad_f; elt_fine(2) = &quad_f;
elt_fine(3) = &tri_f; elt_fine(4) = &tri_f; elt_fine(5) = &quad_f;

elt_coarse(0) = &tri; elt_coarse(1) = &quad; elt_coarse(2) = &quad;
elt_coarse(3) = &tri; elt_coarse(4) = &tri; elt_coarse(5) = &quad;

// computation of prolongation operator
FiniteElementInterpolator proj;
TinyVector<Matrix<Real_wp>, 4> ProlongationMatrix;
proj.ComputeLocalProlongation(elt_fine, 4, elt_coarse, 2, ProlongationMatrix);

// to prolongate a field on the coarse mesh to the fine mesh => use Project
VectReal_wp u_coarse(tri.GetNbDof()), u_fine(tri_f.GetNbDof());
proj.Project(u_coarse, u_fine, 0); // 0 = triangle, 1 = quad

// for the restriction, use TransposeProject
proj.TransposeProject(u_fine, u_coarse, 0); // 0 = triangle, 1 = quad

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx

ComputeLocalProlongationLowOrder

This method computes the local prolongation operator (as detailed in ComputeLocalProlongationLowOrder between a coarse finite element and a fine element by using first order basis functions.

Example

// coarse finite elements
TriangleClassical tri; QuadrangleGauss quad;
tri.ConstructFiniteElement(2); quad.ConstructFiniteElement(2);

// fine finite elements
TriangleClassical tri_f; QuadrangleGauss quad_f;
tri_f.ConstructFiniteElement(4); quad_f.ConstructFiniteElement(4);

// list of elements (usually associated with elements of the mesh)
Vector<const ElementReference<Dimension2, 1>* > elt_fine(6);
Vector<const ElementReference<Dimension2, 1>* > elt_coarse(6);

elt_fine(0) = &tri_f; elt_fine(1) = &quad_f; elt_fine(2) = &quad_f;
elt_fine(3) = &tri_f; elt_fine(4) = &tri_f; elt_fine(5) = &quad_f;

elt_coarse(0) = &tri; elt_coarse(1) = &quad; elt_coarse(2) = &quad;
elt_coarse(3) = &tri; elt_coarse(4) = &tri; elt_coarse(5) = &quad;

// computation of prolongation operator (with low order basis functions)
FiniteElementInterpolator proj;
TinyVector<Matrix<Real_wp>, 4> ProlongationMatrix;
proj.ComputeLocalProlongationLowOrder(elt_fine, 4, elt_coarse, 2, ProlongationMatrix);

// to prolongate a field on the coarse mesh to the fine mesh => use Project
VectReal_wp u_coarse(tri.GetNbDof()), u_fine(tri_f.GetNbDof());
proj.Project(u_coarse, u_fine, 0); // 0 = triangle, 1 = quad

// for the restriction, use TransposeProject
proj.TransposeProject(u_fine, u_coarse, 0); // 0 = triangle, 1 = quad

Location :

FiniteElement/ProjectionOperator.hxx   FiniteElement/ProjectionOperator.cxx