Non-polynomial Shape Functions

[Alex Quinlan]

Hi everyone,

I am looking to modify the FE_Q element to have non-polynomial shape functions.  It looks like FE_Q_Base takes an argument of type ScalarPolynomialBase<dim> (fe_q_base.cc: line 418).

I am looking at a 2D element and instead of 2D polynomial shape functions like this:

        N = (1 - ξ)(1 - η)

I would like to add some additional terms like this:

    N = (1 - ξ)(1 - η) + ζ

where ζ changes at different quadrature points.

It seems that I would need to access the values of ζ before computing fe_values, since the shape functions and their derivatives would be dependent on ζ.

Any thoughts on how feasible this would be to implement?

Thanks for your input,

Alex

[Wolfgang Bangerth]

Alex:
if your shape functions are not polynomial, then this is the wrong approach --
FE_Q_Base is derived from FE_Poly, which is specifically built to create
polynomial bases.

I'm not sure we have a particularly good starting point for a completely
general choice of ζ(ξ,η), but you might want to take a look at what
FE_Enriched is doing.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Alex Quinlan]

Thanks, Wolfgang.  I'll dig into FE_Enriched!

[Alex Quinlan]

Hi Wolfgang,

I've been looking at a few sources regarding FE_Enriched:

(A) https://www.dealii.org/current/doxygen/deal.II/classFE__Enriched.html

(B) https://github.com/dealii/dealii/blob/master/tests/fe/fe_enriched_step-36b.cc

I've been trying to get my problem class to work with a placeholder enrichment function from example (B), but I'm getting tripped up.

I looked at the formatting on Line 146 of fe_enriched.cc:

FE_Enriched<dim, spacedim>::FE_Enriched(
  const FiniteElement<dim, spacedim> &fe_base,
  const FiniteElement<dim, spacedim> &fe_enriched,
  const Function<spacedim> *          enrichment_function)

I have tried to implement this format in the constructor for my problem class.  I am trying to set fe_base as FE_Q, and then fe_enriched as another FE_Q multiplied

by the enrichment function.

template <int dim, int spacedim>
shElast<dim, spacedim>::shElast(
    const std::string &  initial_mesh_filename,
    const std::string &  output_filename)

    : fe( FE_Q<dim>(1)  ,
  FE_Q<dim>(1) ,
  enrichment)
, dof_handler(triangulation)
    , quadrature_formula(fe.degree + 1)
    , initial_mesh_filename(initial_mesh_filename)
    , output_filename(output_filename)
, mapping( FE_Q<dim>(1))

{}

  where I've previously declared:

  Triangulation<dim>             triangulation; 

  const FE_Enriched<dim>         fe;

  DoFHandler<dim>                dof_handler;

  const QGauss<dim>              quadrature_formula;

  const MappingFE<dim>           mapping;

  EnrichmentFunction<dim>        enrichment;

When I try to compile my program, I get this error:

dealii/examples/enrich/enrich9.cc:324:25: error: no matching function for call to ‘dealii::FE_Enriched<3, 3>::FE_Enriched(dealii::FE_Q<3, 3>, dealii::FE_Q<3, 3>, problem_namespace::EnrichmentFunction<3>&)’
  324 |  , mapping( FE_Q<dim>(1))

Then I get the following notes about my arguments and failure to find conversions:

In file included from dealii/examples/enrich/enrich9.cc:53:
fe/fe_enriched.h:263:3: note: candidate: ‘dealii::FE_Enriched<dim, spacedim>::FE_Enriched(const std::vector<const dealii::FiniteElement<dimension_, space_dimension_>*>&, const std::vector<unsigned int>&, const std::vector<std::vector<std::function<const dealii::Function<spacedim>*(const typename dealii::Triangulation<dim, spacedim>::cell_iterator&)> > >&) [with int dim = 3; int spacedim = 3; typename dealii::Triangulation<dim, spacedim>::cell_iterator = dealii::TriaIterator<dealii::CellAccessor<3, 3> >]’
  263 |   FE_Enriched(
      |   ^~~~~~~~~~~

deal.II/fe/fe_enriched.h:264:62: note:   no known conversion for argument 1 from ‘dealii::FE_Q<3, 3>’ to ‘const std::vector<const dealii::FiniteElement<3, 3>*, std::allocator<const dealii::FiniteElement<3, 3>*> >&’
  264 |     const std::vector<const FiniteElement<dim, spacedim> *> &fes,
      |     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~

deal.II/fe/fe_enriched.h:249:3: note: candidate: ‘dealii::FE_Enriched<dim, spacedim>::FE_Enriched(const dealii::FiniteElement<dimension_, space_dimension_>*, const std::vector<const dealii::FiniteElement<dimension_, space_dimension_>*>&, const std::vector<std::vector<std::function<const dealii::Function<spacedim>*(const typename dealii::Triangulation<dim, spacedim>::cell_iterator&)> > >&) [with int dim = 3; int spacedim = 3; typename dealii::Triangulation<dim, spacedim>::cell_iterator = dealii::TriaIterator<dealii::CellAccessor<3, 3> >]’
  249 |   FE_Enriched(
      |   ^~~~~~~~~~~

deal.II/fe/fe_enriched.h:250:62: note:   no known conversion for argument 1 from ‘dealii::FE_Q<3, 3>’ to ‘const dealii::FiniteElement<3, 3>*’
  250 |     const FiniteElement<dim, spacedim> *                     fe_base,
      |     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~~~~~

deal.II/fe/fe_enriched.h:200:3: note: candidate: ‘dealii::FE_Enriched<dim, spacedim>::FE_Enriched(const dealii::FiniteElement<dimension_, space_dimension_>&) [with int dim = 3; int spacedim = 3]’
  200 |   FE_Enriched(const FiniteElement<dim, spacedim> &fe_base);
      |   ^~~~~~~~~~~

deal.II/fe/fe_enriched.h:200:3: note:   candidate expects 1 argument, 3 provided

deal.II/fe/fe_enriched.h:186:3: note: candidate: ‘dealii::FE_Enriched<dim, spacedim>::FE_Enriched(const dealii::FiniteElement<dimension_, space_dimension_>&, const dealii::FiniteElement<dimension_, space_dimension_>&, const dealii::Function<spacedim>*) [with int dim = 3; int spacedim = 3]’
  186 |   FE_Enriched(const FiniteElement<dim, spacedim> &fe_base,
      |   ^~~~~~~~~~~

deal.II/fe/fe_enriched.h:188:51: note:   no known conversion for argument 3 from ‘shell::EnrichmentFunction<3>’ to ‘const dealii::Function<3, double>*’
  188 |               const Function<spacedim> *          enrichment_function);
      |               ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~

It seems that I'm misunderstanding a few things about how FE_Enriched works.  Can you see what I'm doing wrong in constructing the FE_Enriched?  It looks like there's issues with forms I am using for FE_Q and the enrichment function.  Do you have any tips or suggested resources?

Many thanks,

Alex

[Wolfgang Bangerth]

On 6/12/23 14:41, Alex Quinlan wrote:
>
> It seems that I'm misunderstanding a few things about how FE_Enriched works.
> Can you see what I'm doing wrong in constructing the FE_Enriched?  It looks
> like there's issues with forms I am using for FE_Q and the enrichment
> function.  Do you have any tips or suggested resources?

Alex:
The constructor's third argument is a pointer, but you're passing a reference.

[Alex Quinlan]

Hi Wolfgang,

Thanks for the guidance on the pointer.  I have gotten a compiled program using:

const FE_Enriched<dim>   fenr;

PotentialFunction<dim> potential;

shElast<dim, spacedim>::shElast()
:fenr( FE_Q<dim>(2),

                       FE_Q<dim>(1),

                       &potential )

However, I am in need of a 3D vector solution. Previously, I have successfully used FE_System for non-enriched 3D vector problems, i.e.:

const FESystem<dim,spacedim>   fesys;

shElast<dim, spacedim>::shElast()
:fesys( FE_Q<dim>(2), 3)

How can I go about getting a vector solution with FE_Enriched?  I have tried to use vector arguments in FE_Enriched for base elements, enriched elements, and enrichment functions, based on the documentation.  I have also tried to use FE_Enriched as an argument to FE_System:

const FESystem<dim,spacedim>   fesys;

const FE_Enriched<dim>   fenr;

shElast<dim, spacedim>::shElast()
:fesys( fenr( FE_Q<dim>(2),

                              FE_Q<dim>(1),

                              &potential )

                , 3)

Are either of these directions valid?  I can't tell if I'm failing to use the correct syntax or if I'm completely off base with the approach.

Thanks,

Alex

[Wolfgang Bangerth]

On 6/20/23 10:00, Alex Quinlan wrote:
>
> How can I go about getting a vector solution with FE_Enriched?  I have tried
> to use vector arguments in FE_Enriched for base elements, enriched elements,
> and enrichment functions, based on the documentation.  I have also tried to
> use FE_Enriched as an argument to FE_System:
>
> const FESystem<dim,spacedim>   fesys;
> const FE_Enriched<dim>   fenr;
>
> shElast<dim, spacedim>::shElast()

> :fesys(fenr(FE_Q<dim>(2),

>                               FE_Q<dim>(1),
>                               &potential )
>                 , 3)

This is the right idea, but the wrong syntax. In constructors, what comes
after the : is a comma-separated list of the form
member_var_name(arg1,arg2,...),
...

So you need to write this list as

shElast<dim, spacedim>::shElast()
:fenr(FE_Q<dim>(2),

FE_Q<dim>(1),
&potential )

,fesys(fenr, 3)
{...}

[Alex Quinlan]

Thanks, Wolfgang.  Your solution worked for me.

Just for posterity, I got stuck again because I created FE_System and FE_Enriched in the wrong order:

const FESystem<dim,spacedim>   fesys;
const FE_Enriched<dim>         fenr;

my problem was resolved when I switched it around to:

const FE_Enriched<dim>         fenr;

const FESystem<dim,spacedim>   fesys;

Cheers,

Alex