# Developers

## File formats#

### Mesh file data structure#

The mesh data structure, output of a mesh generation algorithm, refers to the geometric data structure and in some case to another mesh data structure.

In this case, the fields are

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 MeshVersionFormatted 0 Dimension [DIM](int) Vertices [Number of vertices](int) X_1(double) Y_1(double) (Z_1(double)) Ref_1(int) ... X_nv(double) Y_nv(double) (Z_nv(double)) Ref_nv(int) Edges [Number of edges](int) Vertex1_1(int) Vertex2_1(int) Ref_1(int) ... Vertex1_ne(int) Vertex2_ne(int) Ref_ne(int) Triangles [Number of triangles](int) Vertex1_1(int) Vertex2_1(int) Vertex3_1(int) Ref_1(int) ... Vertex1_nt(int) Vertex2_nt(int) Vertex3_nt(int) Ref_nt(int) Quadrilaterals [Number of Quadrilaterals](int) Vertex1_1(int) Vertex2_1(int) Vertex3_1(int) Vertex4_1(int) Ref_1(int) ... Vertex1_nq(int) Vertex2_nq(int) Vertex3_nq(int) Vertex4_nq(int) Ref_nq(int) Geometry [File name of geometric support](char*) VertexOnGeometricVertex [Number of vertex on geometric vertex](int) Vertex_1(int) VertexGeometry_1(int) ... Vertex_nvg(int) VertexGeometry_nvg(int) EdgeOnGeometricEdge [Number of geometric edge](int) Edge_1(int) EdgeGeometry_1(int) ... Edge_neg(int) EdgeGeometry_neg(int) CrackedEdges [Number of cracked edges](int) Edge1_1(int) Edge2_1(int) ... Edge1_nce(int) Edge2_nce(int) 

When the current mesh refers to a previous mesh, we have in addition

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 MeshSupportOfVertices [File name of mesh support](char*) VertexOnSupportVertex [Number of vertex on support vertex](int) Vertex_1(int) VertexSupport_1(int) ... Vertex_nvsv(int) VertexSupport_nvsv(int) VertexOnSupportEdge [Number of vertex on support edge](int) Vertex_1(int) EdgeSupport_1(int) USupport_1(double) ... Vertex_nvse(int) EdgeSupport_nvse(int) USupport_nvse(double) VertexOnSupportTriangle [Number of vertex on support triangle](int) Vertex_1(int) TriangleSupport_1(int) USupport_1(double) VSupport_1(double) ... Vertex_nvst(int) TriangleSupport_nvst(int) USupport_nvst(double) VSupport_nvst(double) VertexOnSupportQuadrilaterals [Number of vertex on support quadrilaterals] Vertex_1(int) TriangleSupport_1(int) USupport_1(double) VSupport_1(double) ... Vertex_nvsq(int) TriangleSupport_nvsq(int) USupport_nvsq(double) VSupport_nvsq(double) 
• nv means the number of vertices
• ne means the number of edges
• nt means the number of triangles
• nq means the number of quadrilaterals
• nvg means the number of vertex on geometric vertex
• neg means the number of edges on geometric edge
• nce means the number of cracked edges

### bb file type to Store Solutions#

The file is formatted such that:

 1 2 3 4 5 2 [Number of solutions](int) [Number of vertices](int) 2 U_1_1(double) ... U_ns_1(double) ... U_1_nv(double) ... U_ns_nv(double) 
• ns means the number of solutions
• nv means the number of vertices
• U_i_j is the solution component i at the vertex j on the associated mesh.

### BB file type to store solutions#

The file is formatted such that:

  1 2 3 4 5 6 7 8 9 10 11 2 [Number of solutions](int) [Type 1](int) ... [Type ns](int) [Number of vertices](int) 2 U_1_1_1(double) ... U_(type_k)_1_1(double) ... U_1_1_1(double) ... U_(type_k)_nbv_1(double) ... U_1_1_ns(double) ... U_(type_k)_1_ns(double) ... U_1_nbv_ns(double) ... U_(type_k)_nbv_ns(double) 
• ns means the number of solutions
• type_k mean the type of solution k:
• 1: the solution is scalar (1 value per vertex)
• 2: the solution is vectorial (2 values per vertex)
• 3: the solution is a $2\times 2$ symmetric matrix (3 values per vertex)
• 4: the solution is a $2\times 2$ matrix (4 values per vertex)
• nbv means the number of vertices
• U_i_j_k is the value of the component iof the solution k at vertex j on the associated mesh

### Metric file#

A metric file can be of two types, isotropic or anisotropic.

The isotropic file is such that

 1 2 3 4 [Number of vertices](int) 1 h_0(double) ... h_nv(double) 
• nv is the number of vertices
• h_i is the wanted mesh size near the vertex i on associated mesh.

The metric is $\mathcal{M}_i = h_i^{-2}I$ where $I$ is the identity matrix.

The anisotropic file is such that

 1 2 3 4 [Number of vertices](int) 3 a11_0(double) a21_0(double) a22_0(double) ... a11_nv(double) a21_nv(double) a22_nv(double) 
• nv is the number of vertices
• a11_i, a21_i and a22_i represent metric $\mathcal{M}_i = \left(\begin{array}{cc}a_{11,i} & a_{12,i}\\a{12}_i & a_{22,i}\end{array}\right)$ which define the wanted size in a vicinity of the vertex i such that $h$ in direction $u \in \R^2$ is equal to $|u|/\sqrt{u\cdot\mathcal{M}_i\, u}$, where $\cdot$ is the dot product in $\R^2$, and $|\cdot|$ is the classical norm.

### List of AM_FMT, AMDBA Meshes#

The mesh is only composed of triangles and can be defined with the help of the following two integers and four arrays:

• nbt the number of triangles
• nbv the number of vertices
• nu(1:3, 1:nbt) an integer array giving the three vertex numbers counterclockwise for each triangle
• c(1:2, 1:nbv) a real array giving tje two coordinates of each vertex
• refs(1:nbv) an integer array giving the reference numbers of the vertices
• reft(1:nbt) an integer array giving the reference numbers of the triangles

AM_FMT Files

In Fortran the am_fmt files are read as follows:

 1 2 3 4 5 6 7 open (1, file='xxx.am_fmt', form='formatted', status='old') read (1, *) nbv, nbt read (1, *) ((nu(i, j), i=1, 3), j=1, nbt) read (1, *) ((c(i, j), i=1, 2), j=1, nbv) read (1, *) ( reft(i), i=1, nbt) read (1, *) ( refs(i), i=1, nbv) close(1) 

AM Files

In Fortran the am files are read as follows:

 1 2 3 4 5 6 7 open (1, file='xxx.am', form='unformatted', status='old') read (1, *) nbv, nbt read (1) ((nu(i, j), i=1, 3), j=1, nbt), & ((c(i, j), i=1, 2), j=1, nbv), & (reft(i), i=1, nbt), & (refs(i), i=1, nbv) close(1) 

AMDBA Files

In Fortran the amdba files are read as follows:

 1 2 3 4 5 open (1, file='xxx.amdba', form='formatted', status='old') read (1, *) nbv, nbt read (1, *) (k, (c(i, k), i=1, 2), refs(k), j=1, nbv) read (1, *) (k, (nu(i, k), i=1, 3), reft(k), j=1, nbt) close(1) 

msh Files

First, we add the notions of boundary edges

• nbbe the number of boundary edge
• nube(1:2, 1:nbbe) an integer array giving the two vertex numbers of boundary edges
• refbe(1:nbbe) an integer array giving the reference numbers of boundary edges

In Fortran the msh files are read as follows:

 1 2 3 4 5 6 open (1, file='xxx.msh', form='formatted', status='old') read (1, *) nbv, nbt, nbbe read (1, *) ((c(i, k), i=1, 2), refs(k), j=1, nbv) read (1, *) ((nu(i, k), i=1, 3), reft(k), j=1, nbt) read (1, *) ((ne(i, k), i=1, 2), refbe(k), j=1, nbbe) close(1) 

ftq Files

In Fortran the ftq files are read as follows:

 1 2 3 4 5 open(1,file='xxx.ftq',form='formatted',status='old') read (1,*) nbv,nbe,nbt,nbq read (1,*) (k(j),(nu(i,j),i=1,k(j)),reft(j),j=1,nbe) read (1,*) ((c(i,k),i=1,2),refs(k),j=1,nbv) close(1) 

where if k(j) = 3 when the element j is a triangle and k(j) = 4 when the the element j is a quadrilateral.

### sol and solb files#

With the keyword savesol, we can store a scalar functions, a scalar finite element functions, a vector fields, a vector finite element fields, a symmetric tensor and a symmetric finite element tensor.

Such format is used in medit.

Extension file .sol

The first two lines of the file are :

• MeshVersionFormatted 0

• Dimension [DIM](int)

The following fields begin with one of the following keyword: SolAtVertices, SolAtEdges, SolAtTriangles, SolAtQuadrilaterals, SolAtTetrahedra, SolAtPentahedra, SolAtHexahedra.

In each field, we give then in the next line the number of elements in the solutions (SolAtVertices: number of vertices, SolAtTriangles: number of triangles, ...). In other lines, we give the number of solutions, the type of solution (1: scalar, 2: vector, 3: symmetric tensor). And finally, we give the values of the solutions on the elements.

The file must be ended with the keyword End.

The real element of symmetric tensor :

\begin{eqnarray} \label{savesol.def.symtensor} ST^{3d}=\left( \begin{array}{ccc} ST_{xx}^{3d} & ST_{xy}^{3d} & ST_{xz}^{3d}\\ ST_{yx}^{3d} & ST_{yy}^{3d} & ST_{yz}^{3d} \\ ST_{zx}^{3d} & ST_{zy}^{3d} & ST_{zz}^{3d} \end{array} \right) \qquad ST^{2d}= \left( \begin{array}{cc} ST_{xx}^{2d} & ST_{xy}^{2d} \\ ST_{yx}^{2d} & ST_{yy}^{2d} \end{array} \right) \end{eqnarray}

stored in the extension .sol are respectively $ST_{xx}^{3d}, ST_{yx}^{3d}, ST_{yy}^{3d}, ST_{zx}^{3d}, ST_{zy}^{3d}, ST_{zz}^{3d}$ and $ST_{xx}^{2d}, ST_{yx}^{2d}, ST_{yy}^{2d}$

An example of field with the keyword SolAtTetrahedra:

  1 2 3 4 5 6 7 8 9 10 11 12 13 SolAtTetrahedra [Number of tetrahedra](int) [Number of solutions](int) [Type of solution 1](int) ... [Type of soution nt](int) U_1_1_1(double) ... U_nrs_1_1(double) ... U_1_ns_1(double) ... U_(nrs_k)_ns_1(double) ... U_1_1_nt(double) ... U_nrs_1_nt(double) ... U_1_ns_nt(double) ... U_(nrs_k)_ns_nt(double) 
• ns is the number of solutions
• typesol_k, type of the solution number k
• typesol_k = 1 the solution k is scalar
• typesol_k = 2 the solution k is vectorial
• typesol_k = 3 the solution k is a symmetric tensor or symmetric matrix
• nrs_k is the number of real to describe solution k
• nrs_k = 1 if the solution k is scalar
• nrs_k = dim if the solution k is vectorial (dim is the dimension of the solution)
• nrs_k = dim*(dim+1)/2 if the solution k is a symmetric tensor or symmetric matrix
• U_i_j_^k is a real equal to the value of the component i of the solution k at tetrahedron j on the associated mesh

The format .solb is the same as format .sol but in binary (read/write is faster, storage is less).

A real scalar functions $f1$, a vector fields $\mathbf{\Phi} = [\Phi1, \Phi2, \Phi3]$ and a symmetric tensor $ST^{3d}$ \eqref{savesol.def.symtensor} at the vertices of the three dimensional mesh Th3 is stored in the file f1PhiTh3.sol using :

 1 savesol("f1PhiST3dTh3.sol", Th3, $f1$, [Phi(1), Phi(2), Phi(3)], VV3, order=1); 

where $VV3 = [ST_{xx}^{3d}, ST_{yx}^{3d}, ST_{yy}^{3d}, ST_{zx}^{3d}, ST_{zy}^{3d}, ST_{zz}^{3d}]$.

For a two dimensional mesh Th, A real scalar functions $f2$, a vector fields $\mathbf{\Psi} = [\Psi1, \Psi2]$ and a symmetric tensor $ST^{2d}$ \eqref{savesol.def.symtensor} at triangles is stored in the file f2PsiST2dTh3.solb using :

 1 savesol("f2PsiST2dTh3.solb", Th, f2, [Psi(1), Psi(2)], VV2, order=0); 

where $VV2 = [ST_{xx}^{2d}, ST_{yx}^{2d}, ST_{yy}^{2d}]$

The arguments of savesol functions are the name of a file, a mesh and solutions. These arguments must be given in this order.

The parameters of this keyword are :

• order = 0 is the solution is given at the center of gravity of elements. 1 is the solution is given at the vertices of elements.

In the file, solutions are stored in this order : scalar solutions, vector solutions and finally symmetric tensor solutions.

## Adding a new finite element#

### Some notations#

For a function $\boldsymbol{f}$ taking value in $\R^{N},\, N=1,2,\cdots$, we define the finite element approximation $\Pi_h\boldsymbol{f}$ of $\boldsymbol{f}$.

Let us denote the number of the degrees of freedom of the finite element by $NbDoF$. Then the $i$-th base $\boldsymbol{\omega}^{K}_{i}$ ($i=0,\cdots,NbDoF-1$) of the finite element space has the $j$-th component $\mathbf{\omega}^{K}_{ij}$ for $j=0,\cdots,N-1$.

The operator $\Pi_{h}$ is called the interpolator of the finite element.

We have the identity $\boldsymbol{\omega}^{K}_{i} = \Pi_{h} \boldsymbol{\omega}^{K}_{i}$.

Formally, the interpolator $\Pi_{h}$ is constructed by the following formula: where $P_{p}$ is a set of $npPi$ points,

In the formula \eqref{eq-interpo}, the list $p_{k},\, j_{k},\, i_{k}$ depend just on the type of finite element (not on the element), but the coefficient $\alpha_{k}$ can be depending on the element.

Classical scalar Lagrange finite element

With the classical scalar Lagrange finite element, we have $\mathtt{kPi}=\mathtt{npPi}=\mathtt{NbOfNode}$ and

• $P_{p}$ is the point of the nodal points
• the $\alpha_k=1$, because we take the value of the function at the point $P_{k}$
• $p_{k}=k$ , $j_{k}=k$ because we have one node per function.
• $j_{k}=0$ because $N=1$

The Raviart-Thomas finite element

$$RT0_{h} = \{ \mathbf{v} \in H(div) / \forall K \in \mathcal{T}_{h} \quad \mathbf{v}_{|K}(x,y) = \vecttwo{\alpha_{K}}{\beta_{K}} + \gamma_{K}\vecttwo{x}{y} \} \label{eq:RT0-fe}$$

The degrees of freedom are the flux through an edge $e$ of the mesh, where the flux of the function $\mathbf{f} : \R^2 \longrightarrow \R^2$ is $\int_{e} \mathbf{f}.n_{e}$, $n_{e}$ is the unit normal of edge $e$ (this implies a orientation of all the edges of the mesh, for example we can use the global numbering of the edge vertices and we just go to small to large number).

To compute this flux, we use a quadrature formula with one point, the middle point of the edge. Consider a triangle $T$ with three vertices $(\mathbf{a},\mathbf{b},\mathbf{c})$.

Let denote the vertices numbers by $i_{a},i_{b},i_{c}$, and define the three edge vectors $\mathbf{e}^{0},\mathbf{e}^{1},\mathbf{e}^{2}$ by $sgn(i_{b}-i_{c})(\mathbf{b}-\mathbf{c})$, $sgn(i_{c}-i_{a})(\mathbf{c}-\mathbf{a})$, $sgn(i_{a}-i_{b})(\mathbf{a}-\mathbf{b})$.

The three basis functions are: where $|T|$ is the area of the triangle $T$.

So we have $N=2$, $\mathtt{kPi}=6; \mathtt{npPi}=3;$ and:

• $P_{p} = \left\{\frac{\mathbf{b}+\mathbf{c}}{2}, \frac{\mathbf{a}+\mathbf{c}}{2}, \frac{\mathbf{b}+\mathbf{a}}{2} \right\}$

• $\alpha_{0}= - \mathbf{e}^{0}_{2}, \alpha_{1}= \mathbf{e}^{0}_{1}$, $\alpha_{2}= - \mathbf{e}^{1}_{2}, \alpha_{3}= \mathbf{e}^{1}_{1}$, $\alpha_{4}= - \mathbf{e}^{2}_{2}, \alpha_{5}= \mathbf{e}^{2}_{1}$ (effectively, the vector $(-\mathbf{e}^{m}_{2}, \mathbf{e}^{m}_{1})$ is orthogonal to the edge $\mathbf{e}^{m}= (e^m_{1},e^m_{2})$ with a length equal to the side of the edge or equal to $\int_{e^m} 1$).

• $i_{k}=\{0,0,1,1,2,2\}$,
• $p_{k}=\{0,0,1,1,2,2\}$ , $j_{k}=\{0,1,0,1,0,1,0,1\}$.

Add file FE_ADD.cpp in directory FreeFem-sources/src/femlib for example first to initialize :

 1 2 3 4 5 6 7 8 9 #include "error.hpp" #include "rgraph.hpp" using namespace std; #include "RNM.hpp" #include "fem.hpp" #include "FESpace.hpp" #include "AddNewFE.h" namespace Fem2D { ... } 

Then add a class which derive for public TypeOfFE like:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 class TypeOfFE_RTortho : public TypeOfFE { public: static int Data[]; //some numbers TypeOfFE_RTortho(): TypeOfFE( 0+3+0, //nb degree of freedom on element 2, //dimension N of vectorial FE (1 if scalar FE) Data, //the array data 1, //nb of subdivision for plotting 1, //nb of sub finite element (generaly 1) 6, //number kPi of coef to build the interpolator 3, //number npPi of integration point to build interpolator 0 //an array to store the coef \alpha_k to build interpolator //here this array is no constant so we have //to rebuilt for each element ) { const R2 Pt[] = {R2(0.5, 0.5), R2(0.0, 0.5), R2(0.5, 0.0) }; // the set of Point in hat{K} for (int p = 0, kk = 0; p < 3; p++){ P_Pi_h[p] = Pt[p]; for (int j = 0; j < 2; j++) pij_alpha[kk++] = IPJ(p, p, j); } } //definition of i_k, p_k, j_k in interpolator void FB(const bool *watdd, const Mesh &Th, const Triangle &K, const R2 &PHat, RNMK_ &val) const; void Pi_h_alpha(const baseFElement &K, KN_ &v) const; } ; 

where the array data is formed with the concatenation of five array of size NbDoF and one array of size N.

This array is:

 1 2 3 4 5 6 7 8 9 int TypeOfFE_RTortho::Data[] = { //for each df 0, 1, 3: 3, 4, 5, //the support of the node of the df 0, 0, 0, //the number of the df on the node 0, 1, 2, //the node of the df 0, 0, 0, //the df come from which FE (generally 0) 0, 1, 2, //which are the df on sub FE 0, 0 }; //for each component j=0, N-1 it give the sub FE associated 

where the support is a number $0,1,2$ for vertex support, $3,4,5$ for edge support, and finally $6$ for element support.

The function to defined the function $\boldsymbol{\omega}^{K}_{i}$, this function return the value of all the basics function or this derivatives in array val, computed at point Phat on the reference triangle corresponding to point R2 P=K(Phat); on the current triangle K.

The index $i,j,k$ of the array $val(i,j,k)$ correspond to:

• $i$ is the basic function number on finite element $i \in [0,NoF[$
• $j$ is the value of component $j \in [0,N[$
• $k$ is the type of computed value $f(P),dx(f)(P), dy(f)(P), ...\ i \in [0,\mathtt{last\_operatortype}[$.

Note

For optimization, this value is computed only if whatd[k] is true, and the numbering is defined with

  1 2 3 4 5 6 7 8 9 10 11 enum operatortype { op_id = 0, op_dx = 1, op_dy = 2, op_dxx = 3,op_dyy = 4, op_dyx = 5,op_dxy = 5, op_dz = 6, op_dzz = 7, op_dzx = 8, op_dxz = 8, op_dzy = 9, op_dyz = 9 }; const int last_operatortype = 10; 

The shape function :

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 void TypeOfFE_RTortho::FB(const bool *whatd, const Mesh &Th, const Triangle & K, const R2 &PHat,RNMK_ &val) const { R2 P(K(PHat)); R2 A(K[0]), B(K[1]), C(K[2]); R l0 = 1 - P.x-P.y; R l1 = P.x, l2 = P.y; assert(val.N() >= 3); assert(val.M() == 2); val = 0; R a = 1./(2*K.area); R a0 = K.EdgeOrientation(0) * a; R a1 = K.EdgeOrientation(1) * a; R a2 = K.EdgeOrientation(2) * a; if (whatd[op_id]){ //value of the function assert(val.K() > op_id); RN_ f0(val('.', 0,0)); //value first component RN_ f1(val('.', 1,0)); //value second component f1[0] = (P.x - A.x)*a0; f0[0] = -(P.y - A.y)*a0; f1[1] = (P.x - B.x)*a1; f0[1] = -(P.y - B.y)*a1; f1[2] = (P.x - C.x)*a2; f0[2] = -(P.y - C.y)*a2; } if (whatd[op_dx]){ //value of the dx of function assert(val.K() > op_dx); val(0,1,op_dx) = a0; val(1,1,op_dx) = a1; val(2,1,op_dx) = a2; } if (whatd[op_dy]){ assert(val.K() > op_dy); val(0,0,op_dy) = -a0; val(1,0,op_dy) = -a1; val(2,0,op_dy) = -a2; } for (int i = op_dy; i < last_operatortype; i++) if (whatd[op_dx]) assert(op_dy); } 

The function to defined the coefficient $\alpha_{k}$:

  1 2 3 4 5 6 7 8 9 10 11 void TypeOfFE_RT::Pi_h_alpha(const baseFElement &K, KN_ &v) const { const Triangle &T(K.T); for (int i = 0, k = 0; i < 3; i++){ R2 E(T.Edge(i)); R signe = T.EdgeOrientation(i) ; v[k++] = signe*E.y; v[k++] = -signe*E.x; } } 

Now , we just need to add a new key work in FreeFem++.

Two way, with static or dynamic link so at the end of the file, we add:

With dynamic link it is very simple (see section Dynamical link), just add before the end of FEM2d namespace:

 1 2 3  static TypeOfFE_RTortho The_TypeOfFE_RTortho; static AddNewFE("RT0Ortho", The_TypeOfFE_RTortho); } //FEM2d namespace 

Try with ./load.link command in examples++-load/ and see BernardiRaugel.cpp or Morley.cpp new finite element examples.

  1 2 3 4 5 6 7 8 9 10 11 //let the 2 globals variables static TypeOfFE_RTortho The_TypeOfFE_RTortho; //the name in freefem static ListOfTFE typefemRTOrtho("RT0Ortho", &The_TypeOfFE_RTortho); //link with FreeFem++ do not work with static library .a //so add a extern name to call in init_static_FE //(see end of FESpace.cpp) void init_FE_ADD() { }; //end } //FEM2d namespace 

To inforce in loading of this new finite element, we have to add the two new lines close to the end of files src/femlib/FESpace.cpp like:

 1 2 3 4 5 6 7 8 //correct problem of static library link with new make file void init_static_FE() { //list of other FE file.o extern void init_FE_P2h() ; init_FE_P2h() ; extern void init_FE_ADD(); //new line 1 init_FE_ADD(); //new line 2 } 

and now you have to change the makefile.

First, create a file FE_ADD.cpp contening all this code, like in file src/femlib/Element_P2h.cpp, after modify the Makefile.am by adding the name of your file to the variable EXTRA_DIST like:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # Makefile using Automake + Autoconf # ---------------------------------- # Id # This is not compiled as a separate library because its # interconnections with other libraries have not been solved. EXTRA_DIST=BamgFreeFem.cpp BamgFreeFem.hpp CGNL.hpp CheckPtr.cpp \ ConjuguedGradrientNL.cpp DOperator.hpp Drawing.cpp Element_P2h.cpp \ Element_P3.cpp Element_RT.cpp fem3.hpp fem.cpp fem.hpp FESpace.cpp \ FESpace.hpp FESpace-v0.cpp FQuadTree.cpp FQuadTree.hpp gibbs.cpp \ glutdraw.cpp gmres.hpp MatriceCreuse.hpp MatriceCreuse_tpl.hpp \ MeshPoint.hpp mortar.cpp mshptg.cpp QuadratureFormular.cpp \ QuadratureFormular.hpp RefCounter.hpp RNM.hpp RNM_opc.hpp RNM_op.hpp \ RNM_tpl.hpp FE_ADD.cpp 

and do in the FreeFem++ root directory

 1 2 3 autoreconf ./reconfigure make 

For codewarrior compilation add the file in the project an remove the flag in panal PPC linker FreeFm++ Setting Dead-strip Static Initializition Code Flag.

Now, it's possible to add built-in functionnalites in FreeFem++ under the three environnents Linux, Windows and MacOS X 10.3 or newer.

It is agood idea to first try the example load.edp in directory example++-load.

You will need to install a compiler (generally g++/gcc compiler) to compile your function.

• Windows Install the cygwin environnent or the mingw one
• MacOs Install the developer tools Xcode on the apple DVD
• Linux/Unix Install the correct compiler (gcc for instance)

Now, assume that you are in a shell window (a cygwin window under Windows) in the directory example++-load.

Note

In the sub directory include, they are all the FreeFem++ include file to make the link with FreeFem++.

Note

If you try to load dynamically a file with command load "xxx" * Under Unix (Linux or MacOs), the file xxx.so will be loaded so it must be either in the search directory of routine dlopen (see the environment variable $LD_LIBRARY_PATH. or in the current directory, and the suffix ".so" or the prefix "./" is automatically added. • Under Windows, the file xxx.dll will be loaded so it must be in the loadLibary search directory which includes the directory of the application, Compilation of your module: The script ff-c++ compiles and makes the link with FreeFem++, but be careful, the script has no way to known if you try to compile for a pure Windows environment or for a cygwin environment so to build the load module under cygwin you must add the -cygwin parameter. ### A first example myfunction.cpp# The following defines a new function call myfunction with no parameter, but using the $x,y$ current value.   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 #include #include using namespace std; #include "error.hpp" #include "AFunction.hpp" #include "rgraph.hpp" #include "RNM.hpp" #include "fem.hpp" #include "FESpace.hpp" #include "MeshPoint.hpp" using namespace Fem2D; double myfunction(Stack stack){ //to get FreeFem++ data MeshPoint &mp = *MeshPointStack(stack); //the struct to get x, y, normal, value double x = mp.P.x; //get the current x value double y = mp.P.y; //get the current y value //cout << "x = " << x << " y=" << y << endl; return sin(x)*cos(y); }  Now the Problem is to build the link with FreeFem++, to do that we need two classes, one to call the function myfunction. All FreeFem++ evaluable expression must be a C++ struct/class which derivate from E_F0. By default this expression does not depend of the mesh position, but if they derivate from E_F0mps the expression depends of the mesh position, and for more details see HECHT2002.   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 //A class build the link with FreeFem++ //generaly this class are already in AFunction.hpp //but unfortunatly, I have no simple function with no parameter //in FreeFem++ depending of the mesh template class OneOperator0s : public OneOperator { //the class to define and evaluate a new function //It must devive from E_F0 if it is mesh independent //or from E_F0mps if it is mesh dependent class E_F0_F :public E_F0mps { public: typedef R (*func)(Stack stack); func f; //the pointeur to the fnction myfunction E_F0_F(func ff) : f(ff) {} //the operator evaluation in FreeFem++ AnyType operator()(Stack stack) const {return SetAny(f(stack));} }; typedef R (*func)(Stack); func f; public: //the function which build the FreeFem++ byte code E_F0 *code(const basicAC_F0 &) const { return new E_F0_F(f); } //the constructor to say ff is a function without parameter //and returning a R OneOperator0s(func ff) : OneOperator(map_type[typeid(R).name()]),f(ff){} };  To finish we must add this new function in FreeFem++ table, to do that include :   1 2 3 4 5 6 7 8 9 10 11 12 13 14 void init(){ Global.Add("myfunction", "(", new OneOperator0s(myfunction)); } LOADFUNC(init); cpp It will be called automatically at load module time. To compile and link, use the ff-c++ script : cpp ff-c++ myfunction.cpp g++ -c -g -Iinclude myfunction.cpp g++ -bundle -undefined dynamic_lookup -g myfunction.o -o ./myfunction.dylib  To try the simple example under Linux or MacOS, do FreeFem++-nw load.edp The output must be:   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -- FreeFem++ v *.****** (date *** ** *** ****, **:**:** (UTC+0*00)) Load: lg_fem lg_mesh lg_mesh3 eigenvalue 1 : // Example of dynamic function load 2 : // -------------------------------- 3 : //$Id\$ 4 : 5 : load "myfunction" 6 : // dumptable(cout); 7 : mesh Th=square(5,5); 8 : fespace Vh(Th,P1); 9 : Vh uh= myfunction(); // warning do not forget () 10 : cout << uh[].min << " " << uh[].max << endl; 11 : cout << " test io ( " << endl; 12 : testio(); 13 : cout << " ) end test io .. " << endl; sizestack + 1024 =1416 ( 392 ) -- Square mesh : nb vertices =36 , nb triangles = 50 , nb boundary edges 20 0 0.841471 test io ( test cout 3.14159 test cout 512 test cerr 3.14159 test cerr 512 ) end test io .. times: compile 0.012854s, execution 0.000313s, mpirank:0 CodeAlloc : nb ptr 2715, size :371104 mpirank: 0 Ok: Normal End 

Under Windows, launch FreeFem++ with the mouse (or ctrl O) on the example.

### Example: Discrete Fast Fourier Transform#

This will add FFT to FreeFem++, taken from FFTW. To download and install under download/include just go in download/fftw and try make.

The 1D dfft (fast discret fourier transform) for a simple array $f$ of size $n$ is defined by the following formula

Note

The value $n$ is given by $size(f)/m$, and the numbering is row-major order.

So the classical discrete DFFT is $\hat{f}=\mathtt{dfft}(f,-1)/\sqrt{n}$ and the reverse dFFT $f=\mathtt{dfft}(\hat{f},1)/\sqrt{n}$

Note

The 2D Laplace operator is and we have So where $\tilde{k} = k$ if $k \leq n/2$ else $\tilde{k} = k-n$ and $\tilde{l} = l$ if $l \leq m/2$ else $\tilde{l} = l-m$.

And to have a real function we need all modes to be symmetric around zero, so $n$ and $m$ must be odd.

Compile to build a new library

 1 2 3 4 ff-c++ dfft.cpp ../download/install/lib/libfftw3.a -I../download/install/include export MACOSX_DEPLOYMENT_TARGET=10.3 g++ -c -Iinclude -I../download/install/include dfft.cpp g++ -bundle -undefined dynamic_lookup dfft.o -o ./dfft.dylib ../download/install/lib/libfftw3.a 

To test, try FFT example.

### Load Module for Dervieux' P0-P1 Finite Volume Method#

The associed edp file is examples++-load/convect_dervieux.edp.

### More on Adding a new finite element#

First read the Adding a new finite element section, we add two new finite elements examples in the directory examples++-load.

#### The Bernardi-Raugel Element#

The Bernardi-Raugel finite element is meant to solve the Navier Stokes equations in $u,p$ formulation; the velocity space $P^{br}_K$ is minimal to prove the inf-sup condition with piecewise constant pressure by triangle.

The finite element space $V_h$ is where with notation $4=1, 5=2$ and where $\lambda^K_i$ are the barycentric coordinates of the triangle $K$, $(e_k)_{k=1,2}$ the canonical basis of $\R^2$ and $n^K_k$ the outer normal of triangle $K$ opposite to vertex $k$.

A way to check the finite element

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 load "BernardiRaugel" // Macro //a macro the compute numerical derivative macro DD(f, hx, hy) ( (f(x1+hx, y1+hy) - f(x1-hx, y1-hy))/(2*(hx+hy)) ) // // Mesh mesh Th = square(1, 1, [10*(x+y/3), 10*(y-x/3)]); // Parameters real x1 = 0.7, y1 = 0.9, h = 1e-7; int it1 = Th(x1, y1).nuTriangle; // Fespace fespace Vh(Th, P2BR); Vh [a1, a2], [b1, b2], [c1, c2]; for (int i = 0; i < Vh.ndofK; ++i) cout << i << " " << Vh(0,i) << endl; for (int i = 0; i < Vh.ndofK; ++i) { a1[] = 0; int j = Vh(it1, i); a1[][j] = 1; plot([a1, a2], wait=1); [b1, b2] = [a1, a2]; //do the interpolation c1[] = a1[] - b1[]; cout << " ---------" << i << " " << c1[].max << " " << c1[].min << endl; cout << " a = " << a1[] <

A real example using this finite element, just a small modification of the Navier-Stokes P2-P1 example, just the begenning is change to

 1 2 3 4 5 6 7 8 load "BernardiRaugel" real s0 = clock(); mesh Th = square(10, 10); fespace Vh2(Th, P2BR); fespace Vh(Th, P0); Vh2 [u1, u2], [up1, up2]; Vh2 [v1, v2]; 

And the plot instruction is also changed because the pressure is constant, and we cannot plot isovalues of peacewise constant functions.

#### The Morley Element#

See the example bilapMorley.edp`.

## References#

[HECHT2002] HECHT, Frédéric. C++ Tools to construct our user-level language. ESAIM: Mathematical Modelling and Numerical Analysis, 2002, vol. 36, no 5, p. 809-836.