SymFEL.jl Documentation

Testing

This package contains several functions usefull for the implementation of the finite elements method (FEM). We use the package SymPy for the computation of the basis finite element functions for both Lagrange and Hermite finite elements. We also compute the elementary matrices in 1d and for quadragular elements in 2d. For the two dimensional meshes we employ Gmsh. The Gmsh SDK should be installed in your path following the instruction on the Gmsh site.

The utilisation of the package is illustrated by several examples in 1d and 2d.

A documentation website is available at http://lmbp.uca.fr/~cindea/software/SymFEL.jl/.

Here are some features to develop in the next versions:

3d finite elements on quadragular meshes
2d triangular finite elements
3d tetrahedral finite elements
combine the symbolic integration to numerical integration to speed-up the execution.

Contents

SymFEl.jl

SymFEL.x — Constant

x = SymPy.symbols("x")

Symbolic variable x.

This variable is exported.

SymFEL.y — Constant

y = SymPy.symbols("y")

Symbolic variable y.

This variable is exported.

SymFEL.z — Constant

z = SymPy.symbols("z")

Symbolic variable z.

This variable is exported.

SymFEL.h — Constant

h = SymPy.symbols("h")

Symbolic variable h.

This variable is exported.

SymFEL.hx — Constant

hx = SymPy.symbols("hx")

Symbolic variable h.

This variable is exported.

SymFEL.hy — Constant

hy = SymPy.symbols("hy")

Symbolic variable hy.

This variable is exported.

SymFEL.hz — Constant

hz = SymPy.symbols("hz")

Symbolic variable hz.

This variable is exported.

SymFEL.get_em — Function

get_em(deg1=1, deg2=1, der1=0, der2=0; fe1="Lagrange", fe2="Lagrange")

Gives 1d elementary matrices for Lagrange and Hermite elements.

The elementary matrices are computed for elements of length h.

SymFEL.get_square_em — Method

get_square_em(Mx::Array{SymPy.Sym, 2},
              My::Array{SymPy.Sym, 2},
              nc::Tuple{Array{Int64,1},Array{Int64,1}},
              nr::Tuple{Array{Int64,1},Array{Int64,1}})

Get elementary matrices for a squared or a rectangular element.

Arguments

Mx : elementary matrix for the x variable.
My : elementary matrix for the y variable
nc : order of nodes for column
nr : order of nodes for row

SymFEL.get_cube_em — Method

get_cube_em(Mx::Array{SymPy.Sym, 2},
            My::Array{SymPy.Sym, 2},
            Mz::Array{SymPy.Sym, 2},
            nc::Tuple{Array{Int64,1},Array{Int64,1},Array{Int64,1}},
            nr::Tuple{Array{Int64,1},Array{Int64,1}},Array{Int64,1})

Get elementary matrices for a cube or a rectangular hexahedron element.

Arguments

Mx : elementary matrix for the x variable.
My : elementary matrix for the y variable
Mz : elementary matrix for the z variable
nc : order of nodes for column
nr : order of nodes for row

Lagrange finite elements

SymFEL.get_lagrange_basis — Function

get_lagrange_basis(n = 1, varcoeff = false)

Get Lagrange basis function of order n.

SymFEL.get_lagrange_em — Function

get_lagrange_em(p = 1, m = 0, n = 0)

Get Lagrange finite elements elementary matrices.

Arguments

p: degree of polynomials in the basis.
m: number of derivatives on the first function.
n: number of derivatives on the second function.

SymFEL.get_lagrange_em_varcoeff — Function

get_lagrange_em_varcoeff(p = 1, m = 0, n = 0, f = 1)

Get Hermite finite elements elementary matrices for variable coefficients.

Arguments

p: degree of polynomials in the basis.
m: number of derivatives on the first function.
n: number of derivatives on the second function.
f: the variable coefficient.

SymFEL.get_square_lagrange_em — Function

get_square_lagrange_em((px, py) = (1, 1), (mx, my) = (0, 0), (nx, ny) = (0, 0))

Get Lagrange finite elements elementary matrices for a squared element.

Arguments

(px, py) : degree of polynomials in the basis.
(mx, my) : number of derivatives on the first function wrt x and y
(nx, ny) : number of derivatives on the second function wrt x and y

Remarks

The current version works only for px, py in {1, 2}

Hermite finite elements

SymFEL.get_hermite_basis — Function

get_hermite_basis(n = 3, varcoeff = false)

Get Hermite basis function of order n.

SymFEL.get_hermite_em — Function

get_hermite_em(p = 3, m = 0, n = 0)

Get Hermite finite elements elementary matrices.

Arguments

p: degree of polynomials in the basis.
m: number of derivatives on the first function.
n: number of derivatives on the second function.

SymFEL.get_hermite_em_varcoeff — Function

get_hermite_em_varcoeff(p = 3, m = 0, n = 0, f = 1)
Get Hermite finite elements elementary matrices for variable coefficients.

Arguments

p: degree of polynomials in the basis.
m: number of derivatives on the first function.
n: number of derivatives on the second function.
f: the variable coefficient.

SymFEL.get_square_hermite_em((px, py) = (3, 3), (mx, my) = (0, 0), (nx, ny) = (0, 0))

SymFEL.interpolate — Method

interpolate(fd::Matrix{Float64}, t::Vector{Float64}, ti::Vector{Float64})

Interpolates fd from t to ti.

Arguments:

fd: values and derivatives of function to interpolate.
t: values in which fd is known.
ti: values in which fd is interpolated.

Assembling functions – 1d

SymFEL.Mesh1d — Type

Mesh1d

A type describing a mesh for a one dimensional domain.

SymFEL.assemble_1d_FE_matrix — Method

assemble_1d_FE_matrix(elem::Array{Float64, 2}, nbNodes::Int64;
  intNodes = 0, dof1 = 1, dof2 = 1)

Assemble a finite elements matrix corresponding to a 1 dimensional uniform mesh.

Arguments

elem : elementary finite elements matrix
nbNodes : number of nodes in the mesh
intNodes1 : number of interior nodes in the interior of each element for lhs
intNodes2 : number of interior nodes in the interior of each element for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

SymFEL.assemble_1d_nu_FE_matrix — Method

assemble_1d_nu_FE_matrix(elem::Matrix{SymPy.Sym}, nodes::Array{Float64, 1};
  intNodes = 0, dof1 = 1, dof2 = 1)

Assemble a finite elements matrix corresponding to a 1 dimensional non-uniform mesh.

Arguments

elem : elementary finite elements matrix (with elements of type SymPy.Sym)
nodes : vector of nodes composing the non uniform-mesh
intNodes1 : number of interior nodes in the interior of each element for lhs
intNodes2 : number of interior nodes in the interior of each element for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

Assembling functions – 2d

SymFEL.assemble_squaremesh_FE_matrix — Method

assemble_squaremesh_FE_matrix(el_mat::Array{Float64, 2},
                              elements::Array{Int64, 2};
                              order1 = 1, order2 = 1,
                              dof1 = 1, dof2 = 1)

Assemble a finite elements matrix corresponding to a 2 dimensional square mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

SymFEL.assemble_squaremesh_FE_matrix — Method

assemble_squaremesh_FE_matrix(el_mat::Array{Float64, 2},
                              elements::Array{Int64, 2},
                              el_labels::Array{Int64, 1};
                              order1 = 1, 
                              order2 = 1,
                              dof1 = 1, 
                              dof2 = 1)

Assemble a finite elements matrix corresponding to a 2 dimensional square mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
el_labels: list of element numbers on which we integrate
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

SymFEL.assemble1d_squaremesh_FE_matrix — Method

assemble1d_squaremesh_FE_matrix(el_mat::Array{Float64, 2},
                                elements::Array{Int64, 2},
                                el_labels::Array{Int64, 1};
                                order1 = 1, 
                                order2 = 1,
                                dof1 = 1, 
                                dof2 = 1)

Assemble a finite elements matrix corresponding to a 2 dimensional square mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
elements1s: list of 1d elements
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

Assembling functions – 3d

SymFEL.assemble_cubemesh_FE_matrix — Method

assemble_cubemesh_FE_matrix(el_mat::Array{Float64, 2},
                            elements::Array{UInt64, 2};
                            order1 = 1, order2 = 1,
                            dof1 = 1, dof2 = 1)

Assemble a finite elements matrix corresponding to a 3 dimensional cube mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

SymFEL.assemble_cubemesh_FE_matrix — Method

assemble_cubemesh_FE_matrix(el_mat::Array{Float64, 2},
                              elements::Array{UInt64, 2},
                              el_labels::Array{UInt64, 1};
                              order1 = 1, 
                              order2 = 1,
                              dof1 = 1, 
                              dof2 = 1)

Assemble a finite elements matrix corresponding to a 3 dimensional cube mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
el_labels: list of element numbers on which we integrate
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs

SymFEL.assemble2d_cubemesh_FE_matrix — Method

assemble2d_cubemesh_FE_matrix(el_mat::Array{Float64, 2},
                              elements::Array{UInt64, 2},
                              el_labels::Array{UInt64, 1};
                              order1 = 1, 
                              order2 = 1,
                              dof1 = 1, 
                              dof2 = 1)

Assemble a finite elements matrix corresponding to a 3 dimensional cube mesh.

Arguments

el_mat : elementary finite elements matrix
elements : list of elements
elements1s: list of 1d elements
order1 : order for lhs
order2 : order for rhs
dof1 : number of degrees of freedom for each node for lhs
dof2 : number of degrees of freedom for each node for rhs