# Level-sets, Xfem, fictitious domains, Cut-fem¶

Since v2.0, *GetFEM* offers a certain number of facilities to support Xfem
and fictitious domain methods with a cut-fem strategy. Most of these
tools have been initially mainly developed by Julien Pommier for the
study published in [LA-PO-RE-SA2005].

The implementation is a fairly large generality, based on the use of level-sets, as suggested in [SU-CH-MO-BE2001] and allows simultaneous use of a large number of level-sets which can cross.

The Xfem implementation for the discretization of the jump follows the strategy of [HA-HA2004] although we had no knowledge of this work during implementation. This means that there is no degree of freedom representing the jump across the level-set. Instead, the degrees of freedom represent the displacement of each side of the level-set. This is essential in any way in the presence of level-set that intersect each other because it may exist more than two different zones of continuity inside a single element.

The cut fem strategy for fictitious domain method has been used for
the first time with *GetFEM* for the study published in [HA-RE2009] where
a quite simple stabilization strategy is proposed. Here also, before
knowing the existence of the Work of
E. Burman and P. Hanbo [bu-ha2010] on that topic.

The tools for Xfem have been then enriched by the PhD works of J. Larsy (see for instance [LA-RE-SA2010]) the one of E. Chahine (see for instance [CH-LA-RE2011], [NI-RE-CH2011]), of S. Amdouni (see for instance [AM-MO-RE2014], [AM-MO-RE2014b]) and of M. Fabre (see for instance [Fa-Po-Re2015]).

Important

All the tools listed below needs the package qhull installed on your system. This package is widely available. It computes convex hull and Delaunay triangulations in arbitrary dimension.

The programs `tests/crack.cc`

, `interface/tests/matlab/crack.m`

and `interface/tests/python/crack.py`

are some good examples of use of these tools.

## Representation of level-sets¶

Some structure are defined to manipulate level-set functions defined by
piecewise polynomial function on a mesh. In the file
`getfem/getfem_levelset.h`

a level-set is represented by a
function defined on a Lagrange fem of a certain degree on a mesh.
The constructor to define a new `getfem::level_set`

is the following:

```
getfem::level_set ls(mesh, degree = 1, with_secondary = false);
```

where `mesh`

is a valid mesh of type `getfem::mesh`

, `degree`

is the degree of the
polynomials (1 is the default value), and `with_secondary`

is a boolean whose
default value is false. The secondary level-set is used to represent
fractures (if \(p(x)\) is the primary level-set function and
\(s(x)\) is the secondary level-set function, the crack is defined
by \(p(x) = 0\) and \(s(x) \leq 0\): the role of the secondary
is to delimit the crack).

Each level-set function is defined by a mesh_fem `mf`

and the dof values
over this mesh_fem, in a vector. The object `getfem::level_set`

contains a mesh_fem and the
vectors of dof for the corresponding function(s). The method
`ls.value(0)`

returns the vector of dof for the primary level-set
function, so that these values can be set. The method `ls.value(1)`

returns the dof vector for the secondary level-set function if any.
The method `ls.get_mesh_fem()`

returns a reference on the `getfem::mesh_fem`

object.

Note that, in applications, the level-set function often evolves thanks to an Hamilton-Jacobi equation (for its re-initialization for instance). See the A pure convection method which can be used in the approximation of a Hamilton-Jacobi equation.

## Mesh cut by level-sets¶

In order to compute adapted integration methods and finite element methods to
represent a field which is discontinuous across one or several level-sets,
a certain number of pre-computations have to be done at the mesh level. In
`getfem/getfem_mesh_level_set.h`

is defined the object `getfem::mesh_level_set`

which
handles these pre-computations. The constructor of this object is the
following:

```
getfem::mesh_level_set mls(mesh);
```

where `mesh`

is a valid mesh of type `getfem::mesh`

. In order to indicate that
the mesh is cut by a level-set, one has to call the method
`mls.add_level_set(ls)`

, where `ls`

is an object of type `getfem::level_set`

.
An arbitrary number of level-sets can be added. To initialize the object
or to actualize it when the value of the level-set function is modified,
one has to call the method `mls.adapt()`

.

In particular a subdivision of each element cut by the level-set is made with simplices. Note that the whole cut-mesh is generally not conformal.

The cut-mesh can be obtained for instance for post-treatment thanks to `mls.global_cut_mesh(m)`

which fill `m`

with the cut-mesh.

## Adapted integration methods¶

For fields which are discontinuous across a level-set, integration methods have
to be adapted. The object `getfem::mesh_im_level_set`

defined in the file
`getfem/getfem_mesh_im_level_set.h`

defines a composite integration method
for the elements cut by the level-set. The constructor of this object is the
following:

```
getfem::mesh_im_level_set mim(mls, where, regular_im = 0, singular_im = 0);
```

where `mls`

is an object of type `getfem::mesh_level_set`

, `where`

is an enum for which
possible values are

`getfem::mesh_im_level_set::INTEGRATE_INSIDE`

(integrate over \(p(x)<0\)),`getfem::mesh_im_level_set::INTEGRATE_OUTSIDE`

(integrate over \(p(x)>0\)),`getfem::mesh_im_level_set::INTEGRATE_ALL`

,`getfem::mesh_im_level_set::INTEGRATE_BOUNDARY`

(integrate over \(p(x)=0\) and \(s(x)\leq 0\))

The argument `regular_im`

should be of type `pintegration_method`

, and will be
the integration method applied on each sub-simplex of the composite integration
for elements cut by the level-set. The optional `singular_im`

should be also of
type `pintegration_method`

and is used for crack singular functions: it is
applied to sub-simplices which share a vertex with the crack tip (the specific
integration method `IM_QUASI_POLAR(..)`

is well suited for this purpose).

The object `getfem::mesh_im_level_set`

can be used as a classical `getfem::mesh_im`

object (for instance the
method `mim.set_integration_method(...)`

allows to set the integration methods
for the elements which are not cut by the level-set).

To initialize the object or to actualize it when the value of the level-set
function is modified, one has to call the method `mim.adapt()`

.

When more than one level-set is declared on the `getfem::mesh_level_set`

object, it is possible to set more precisely the integration domain using the method:

```
mim.set_level_set_boolean_operations("desc");
```

where “desc” is a string containing the description of the boolean operation which defines the integration domain. The syntax is simple, for example if there are 3 different level-set,

“a*b*c” is the intersection of the domains defined by each level-set (this is the default behavior if this function is not called).

“a+b+c” is the union of their domains.

“c-(a+b)” is the domain of the third level-set minus the union of the domains of the two others.

“!a” is the complementary of the domain of a (i.e. it is the domain where a(x)>0)

The first level-set is always referred to with “a”, the second with “b”, and so on.

## Cut-fem¶

The implementation of a cut finite element method such as described in [bu-ha2010], i.e. a finite element on a fictitious domain restricted to a smaller real domain, is possible just using the previous tools and mainly the adapted integration method. Several examples are available on *GetFEM* test programs. See for instance `interface/tests/python/demo_fictitious_domain.py`

or `interface/tests/matlab/demo_fictitious_domain.m`

.

In this context, one often needs to restrict the unknown finite element field to the degrees of freedom whose corresponding shape function supports have an intersection with the real domain. This can be done using the `partial_mesh_fem`

object. See for instance `interface/tests/matlab/demo_structural_optimization.m`

.

Note that often, a stabilization technique have to be considered in order to treat eventual locking phenomena due to element with very small intersection with the real domain for example when applying a Dirichlet condition. See for instance [bu-ha2010], [HA-RE2009] and [Fa-Po-Re2015].

## Discontinuous field across some level-sets¶

The object `getfem::mesh_fem_level_set`

is defined in the file
`getfem/getfem_mesh_fem_level_set.h`

. It is derived from `getfem::mesh_fem`

object
and can be used in the same way. It defines a finite element method with
discontinuity across the level-sets (it can deal with an arbitrary number of
level-sets). The constructor is the following:

```
getfem::mesh_fem_level_set mfls(mls, mf);
```

where `mls`

is a valid mesh of type `getfem::mesh_level_set`

and `mf`

is the an object of type
`getfem::mesh_fem`

which defines the finite element method used for elements which are not
cut by the level-sets.

To initialize the object or to actualize it when the value of the level-set
function is modified, one has to call the method `mfls.adapt()`

.

To represent discontinuous fields, the finite element method is enriched
with discontinuous functions which are the product of some Heaviside functions
by the shape functions of the finite element method represented by `mf`

(see [HA-HA2004] and [Xfem] for more details).

## Xfem¶

The Xfem (see [Xfem]) consists not only in the enrichment with some Heaviside functions (which is done by the object `getfem::mesh_fem_level_set`

) but also the enrichment with asymptotic displacement at the crack tip. There is several manner to enrich with an asymptotic displacement: enrichment only on the element containing the crack tip as in [Xfem], enrichment in a fixed size zone as in [LA-PO-RE-SA2005] or [Be-Mi-Mo-Bu2005], enrichment with a cut-off function as in [CH-LA-RE2008] or [NI-RE-CH2011] or with an integral matching condition between the enriched and non-enriched zones as in [CH-LA-RE2011]. The choice in Getfem fell on maximum flexibility to easily implement all possibilities. As it is mainly a transformation of the finite element method itself, two tools have been defined to produce some enriched finite elements:

```
getfem::mesh_fem_product mf_asympt(mf_part_unity, mf_sing)
getfem::mesh_fem_sum mf_sum(mf1, mf2)
```

where `mf_sing`

should be a global ‘finite element method’, in fact just a collection of global functions (with or without a cut-off function) defined thanks to the object `getfem::mesh_fem_global_function`

(see the file `src/getfem/getfem_mesh_fem_global_function.h`

) and `mf_part_unity`

a basic scalar finite element method. The resulting `` getfem::mesh_fem_product`` is the linear combination of all the product of the shape function of the two given finite element methods, possibly restricted to a sub-set of degrees of freedom of the first finite element method given by the method `mf_asympt.set_enrichment(enriched_dofs)`

.

Once the asymptotic enrichment is defined, the object `getfem::mesh_fem_sum`

allows to produce the direct sum of two finite element methods. For instance of the one enriched by the Heaviside functions (`getfem::mesh_fem_level_set`

object) and the asymptotic enrichment.

See `interface/tests/matlab/demo_crack.m`

, `interface/tests/python/demo_crack.py`

or `tests/crack.cc`

for some examples of use of these tools.

Additionally, GWFL, the generic weak form language, defines the two commands `Xfem_plus`

and `Xfem_minus`

allowing to take into account the jump of any field or derivative of any field across a level-set (see Xfem discontinuity evaluation (with mesh_fem_level_set)). This a priori allows to write any interface law easily.

Note also that some procedures are available in the file `src/getfem/getfem_crack_sif.h`

to compute the stress intensity factors in 2D (restricted to homogeneous isotropic linearized elasticity).

## Post treatment¶

Several tools are available to represent the solution only on a side of a levels-set or on both taking into account the discontinuity (for Xfem approximation).

When a cut-mesh `mls`

is used (i.e. a `getfem::mesh_level_set`

object), is is possible to obtain the set of all sub-elements with the command:

```
mls.global_cut_mesh(mcut);
```

where `mcut`

has to be an empty mesh which will be fill by the sub-elements. Note that the resulting mesh is a non-regular one in the sense that the sub-mesh of all elements are not conformal at the element edges/faces. It is however possible to interolate on a Lagrange fem on this mesh and make a post-treatment with it to correctly represent a discontinuous field.

Another mean to represent only the interesting part of the solution when a fictitious domain method is used is to use the mesh slices defined by an isovalue level-set (see Producing mesh slices).

see for instance files `interface/tests/matlab/demo_crack.m`

, `interface/tests/python/demo_fictitious_domain.py`

and `interface/tests/matlab/demo_structural_optimization.m`

.