Getfem++ User Guide

• Introduction
• How to install
• Build a mesh
• • Add an element to a mesh
  • Remove an element from a mesh
  • Simple structured meshes
  • Mesh regions
  • Methods of the getfem::mesh object
  • Using dal::bit_vector
  • Face numbering
  • Save and load meshes
  • • From getfem file format
    • Import a mesh
• Selecting finite element methods
• • Examples
  • Other methods of the mesh_fem object
  • Obtaining generic mesh_fems
• Selecting integration methods
• • Methods of the mesh_im object
• Mesh refinement
• Linear algebra procedures
• Standard assembly procedures
• • Laplacian (Poisson) problem
  • Linear Elasticity problem
  • Stokes Problem with mixed finite element method
  • Assembling a mass matrix
• Compute arbitrary elementary matrices - generic assembly procedures
• Incorporate new finite element methods in getfem++ 
• Incorporate new approximated integration methods in getfem++ 
• Level-sets, Xfem, fictitious domains
• • Representation of level-sets
  • Mesh cut by level-sets
  • Adapted integration methods
  • Discontinuous field across some level-sets
  • Fictitious domain approach with Xfem
• Support for Xfem methods
• Interpolation of a finite element method on non-matching meshes
• • mixed methods with different meshes
  • mortar methods
• Compute L² and H¹ norms
• Compute derivatives
• Export and view a solution
• • Saving mesh and mesh_fem objects for the Matlab interface
  • Producing mesh slices
  • Exporting mesh, mesh_fem or slices to VTK
  • Exporting mesh, mesh_fem or slices to OpenDX
• Interpolation on different meshes
• The model bricks
• • The model state variable
  • Basic properties of a brick
  • Brick parameters
  • generic elliptic brick
  • Source term brick
  • Constraint brick
  • Dirichlet condition brick
  • Example of a complete Poisson problem
  • Predefined solvers
  • Isotropic linearized elasticity brick
  • Qu term brick
  • Helmholtz brick
  • linear incompressibility (or nearly incompressibility) brick
  • Small displacement plasticity brick
  • Contact and friction conditions brick
  • Linearized plate brick
  • Large strain elasticity brick
  • Large strain incompressibility brick
• Parallelization of getfem++ 
• Catch errors
• Example : Laplacian program
• Appendix A. Finite element method list
• • Dof graphical codification
  • Classical P_K Lagrange elements on simplices
  • Classical Lagrange elements on other geometries
  • Elements with hierarchical basis
  • • Hiercarchical elements with respect to the degree
    • Composite elements
    • Hierarchical composite elements
  • Classical vectorial elements
  • • Raviart-Thomas of lowest order elements
    • Nedelec (or Whitney) edge elements
  • Specific elements in dimension 1
  • • GaussLobatto element
    • Hermite element
    • Lagrange element with an additional bubble function
  • Specific elements in dimension 2
  • • Elements with additional bubble functions
    • Non-conforming P₁ element
    • Hermite element
    • Morley element
    • Argyris element
    • Hsieh-Clough-Tocher element
    • A composite C¹ element on quadrilaterals
  • Specific elements in dimension 3
  • • Elements with additional bubble functions
    • Hermite element
• Appendix B. Cubature method list
• • Exact Integration methods
  • Newton cotes Integration methods
  • Gauss Integration methods on dimension 1
  • Gauss Integration methods on dimension 2
  • Gauss Integration methods on dimension 3
  • Direct product of integration methods
  • Composite integration methods
• References
• Index

Appendix A. Finite element method list

Let us recall that all finite element methods defined in getfem++ are declared in the file getfem_fem.h and that a descriptor on a finite element method is obtained thanks to the function

getfem::pfem pf = getfem::fem_descriptor("name of method");

where "name of method" is a string to be choosen among the existing methods.

Dof graphical codification

Classical P_K Lagrange elements on simplices

It is possible to define a classical P_K Lagrange element of arbitrary dimension and arbitrary degree. Each degree of freedom of such an element corresponds to the value of the function on a corresponding node. The grid of node is the so-called Lagrange grid. Figures *, * and * show examples in dimension 1, 2 and 3.

 

The number of degree of freedom for a classical P_K Lagrange element of dimension P and degree K is ((P+K)!)/(P!K!). For instance, in dimension 2 (P = 2), this value is ((P+1) (P+2))/(2) and in dimension 3 (P = 3), it is ((P+1) (P+2) (P+3))/(6).

 

The particular way used in getfem++ to numerate the nodes are also shown in figures *, * and *. Using another numeration, let

be somme indices such that Then, the coordinate of a node can be computed as where S₀, S₁, ... S_N are the vertices of the simplex (for K = 0 the particular choice a_{0, 0, ... 0} = ∑_{n = 0}^P (1)/(P+1)S_n has been chosen). Then each base function, corresponding of each node a_{i0, i1, ... iP} is defined by where λ_n are the barycentric coordinates, i.e. the polynomials of degree 1 whose value is 1 on the vertex S_n and whose value is 0 on other vertices. On the reference element, one has When between two elements of the same degrees (even with different dimensions), the d.o.f. of a common face are linked, the element is of class C⁰. This means that the global polynomial is continuous. If you try to link elements of different degrees, you will get some trouble with the unlinked d.o.f. This is not automatically supported by getfem++ , so you will have to support it (add constraints on these d.o.f.).
For some applications (computation of a gradient for instance) one may not want the d.o.f. of a common face to be linked. This is why there are two versions of the classical P_K Lagrange element.
 
 
 
 
Even thought Lagrange elements are defined for arbitrary degrees, to choose a hight degree can be problematic for a large number of applications due to the "noisy" caracteristic of the lagrange basis. Those element are recommended for the basic interpolation but for p.d.e. applications elements with hierarchical basis are preferable (see the corresponding section).

Classical Lagrange elements on other geometries

Classical Lagrange elements on parallelepipeds or prisms are obtained as tensor product of Lagrange elements on simplices. When two element are defined, one on a dimension P¹ and the other in dimension P², one obtains the base functions of the tensorial product (on the reference element) as

where φ'¹_i and φ'²_i are respectively the base functions of the first and second element.  

The Q_K element on a parallelepiped of dimension P is obtained as the tensorial product of P classical P_K element on the segment. Examples in dimension 2 are shown in figure * and in dimension 3 in figure *.
A prism in dimension P > 1 is the direct product of a simplex of dimension P-1 with a segment. The P_K ⊗P_K element on this prism is the tensorial product of the classical P_K element on a simplex of dimension P-1 with the classical P_K element on a segment. For P=2 this coincide with a parallelepiped. Examples in dimension 3 are shown in figure *. This is also possible not to have the same degree on each dimension. An example is shown on figure *.

     
 
 
 
 
 
 
 

Elements with hierarchical basis

The idea behind hierarchical basis is the description of the solution at different level : a rought level, a more refined level ... In the same discretisation some degrees of freedom represent the rought description, some other the more rafined and so on. This correspond to imbricated spaces of discretisation. The hierarchical basis contains a basis of each of these spaces (this is not the case in classical Lagrange elements when the mesh is refined).
Among the advantages, the condition number of rigidity matrices can be greatly improved, it allows local raffinement and a resolution with a multigrid approach.

Hiercarchical elements with respect to the degree

   
 
 
 
 
 
some particular choices : P₄ will be build with the basis of the P₁, the additional basis of the P₂ then the additionnal basis of the P₄.P₆ will be build with the basis of the P₁, the additional basis of the P₂ then the additionnal basis of the P₆ (not with the basis of the P₁,

the additional basis of the P₃ then the additionnal basis of the P₆, this is possible to build the latter with "FEM_GEN_HIERARCHICAL(a,b))

Composite elements

The principal interest of the composite elements is to build hierarchical elements. But this tool can also be used to build piecewise polynomial elements.

   
 
It is important to use a corresponding composite integration method.

Hierarchical composite elements

   
 
 
 
Other constructions are possible thanks to "FEM_GEN_HIERARCHICAL(FEM1, FEM2)" and
"FEM_STRUCTURED_COMPOSITE(FEM1, S)"

It is important to use a corresponding composite integration method.

Classical vectorial elements

Raviart-Thomas of lowest order elements

   
 
 
 

Nedelec (or Whitney) edge elements

   
 

Specific elements in dimension 1

GaussLobatto element

The 1D GaussLobatto P_K element is similar to the classical P_K fem on the segment, but the nodes are given by the Gauss-Lobatto-Legendre quadrature rule of order 2K-1. This FEM is known to lead to better conditioned linear systems, and can be used with the correspounding quadrature to perform mass-lumping (on segments or parallelepipeds).

The polynomials coefficients have been pre-computed with Maple (they require the inversion of an ill-conditionned system), hence they are only available for the following values
of K: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 24, 32. Note that for K=1 and K=2, this is the classical P1 and P2 fem.

 
 

Hermite element

 

Base functions on the reference element

This element is close to be τ-equivalent but it is not. On the real element the value of the gradient on vertices will be multiplied by the gradient of the geometric transformation. The matrix M is not equal to identity but is still diagonal. 
 

Lagrange element with an additional bubble function

   
 

Specific elements in dimension 2

Elements with additional bubble functions

   
 
   
 
   
 
   
 

Non-conforming P₁ element

   
 

Hermite element

 

Base functions on the reference element:

This element is not τ-equivalent (The matrix M is not equal to identity). On the real element linear combinaisons of φ'₄ and φ'₇ are used to match the gradient on the corresponding vertex. Idem for the two couples (φ'₅, φ'₈) and (φ'₆, φ'₉) for the two other vertices.  
 

Morley element

 

This element is not τ-equivalent (The matrix M is not equal to identity). In particular, it can be used for non-conforming discretization of fourth order problems, despite the fact that it is not C⁰.

 
 

Argyris element

 

The base functions on the reference element are:

This element is not τ-equivalent (The matrix M is not equal to identity). On the real element linear combinaisons of the transformed base functions φ'_i are used to match the gradient, the second derivatives and the normal derivatives on the faces. Note that the use of the matrix M allows to define Argyris element even with nonlinear geometric transformations (for instance to treat curved boundaries). 
 

Hsieh-Clough-Tocher element

 

This element is not τ-equivalent. This is a composite element. Polynomial of degree 3 on each of the three sub-triangles (see figure * and [1]). It is strongly advised to use a IM_HCT_COMPOSITE integration method with this finite element. The numeration of the dof is the following : 0, 3 and 6 for the lagrange dof on the first second and third vertex respectively; 1, 4, 7 for the derivative with respects to the first variable; 2, 5, 8 for the derivative with respects to the second variable and 9, 10, 11 for the normal derivatives on face 0, 1, 2 respectively.
 
 

 

This element exists also in its reduced form, where the normal derivatives is assumed to be polynomial of degree one on each edge (see figure *)

 
 

A composite C¹ element on quadrilaterals

 

This element is not τ-equivalent. This is a composite element. Polynomial of degree 3 on each of the four sub-triangles (see figure *). At least on the reference element it correponds to the Fraeijs de Veubeke-Sander element (see [1]). It is strongly advised to use a IM_QUADC1_COMPOSITE integration method with this finite element.
 
 

 

This element exists also in its reduced form, where the normal derivatives is assumed to be polynomial of degree one on each edge (see figure *)

 
 

Specific elements in dimension 3

Elements with additional bubble functions

   
 
   
 

Hermite element

 

Base functions on the reference element:

This element is not τ-equivalent (The matrix M is not equal to identity). On the real element linear combinaisons of φ'₈, φ'₁₂ and φ'₁₆ are used to match the gradient on the corresponding vertex. Idem on the orther vertices.  
 

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