question about a lage deformation model based on the updated Lagrangian foemolation

[ha.ba...@gmail.com]

Dear all

I am working on a solid material model with large deformation using a updated Lagrangian formulation. In each time step and Newton iteration the mesh is moved. The mesh is highly deformed and and the deformation localized in a narrow band(due to the nature of the material behavior). So some element is distorted and the following error occurs:

An error occurred in line <2587> of file </home/hojat/deal/dealii-8.4.1/source/fe/mapping_q_generic.cc> in function

    dealii::CellSimilarity::Similarity dealii::MappingQGeneric<dim, spacedim>::fill_fe_values(const typename dealii::Triangulation<dim, spacedim>::cell_iterator&, dealii::CellSimilarity::Similarity, const dealii::Quadrature<dim>&, const typename dealii::Mapping<dim, spacedim>::InternalDataBase&, dealii::internal::FEValues::MappingRelatedData<dim, spacedim>&) const [with int dim = 2; int spacedim = 2; typename dealii::Triangulation<dim, spacedim>::cell_iterator = dealii::TriaIterator<dealii::CellAccessor<2, 2> >; typename dealii::Mapping<dim, spacedim>::InternalDataBase = dealii::Mapping<2, 2>::InternalDataBase]

The violated condition was: 

    det > 1e-12*Utilities::fixed_power<dim>(cell->diameter()/ std::sqrt(double(dim)))

The name and call sequence of the exception was:

    (typename Mapping<dim,spacedim>::ExcDistortedMappedCell(cell->center(), det, point))

Additional Information: 

The image of the mapping applied to cell with center [0.295666 0.850622] is distorted. The cell geometry or the mapping are invalid, giving a non-positive volume fraction of -0.000119973 in quadrature point 0.

Stacktrace:

-----------

#0  /home/hojat/deal/install_8.4.1/lib/libdeal_II.g.so.8.4.1: dealii::MappingQGeneric<2, 2>::fill_fe_values(dealii::TriaIterator<dealii::CellAccessor<2, 2> > const&, dealii::CellSimilarity::Similarity, dealii::Quadrature<2> const&, dealii::Mapping<2, 2>::InternalDataBase const&, dealii::internal::FEValues::MappingRelatedData<2, 2>&) const

#1  /home/hojat/deal/install_8.4.1/lib/libdeal_II.g.so.8.4.1: dealii::FEValues<2, 2>::do_reinit()

#2  /home/hojat/deal/install_8.4.1/lib/libdeal_II.g.so.8.4.1: void dealii::FEValues<2, 2>::reinit<dealii::DoFHandler, false>(dealii::TriaIterator<dealii::DoFCellAccessor<dealii::DoFHandler<2, 2>, false> > const&)

#3  /home/hojat/deal/install_8.4.1/examples/HB/: FEM<2>::assemble_large_deformation()

The attached figures show the deformed and initial mesh. How to solve this problem (without adaptive mesh refinement)? Can I use the update Lagrangian formulation in Deal.II without moving the mesh? Do you think that the total Lagrangian formulation is a solution?

Best Regards,

H.B

[Jean-Paul Pelteret]

Dear H.B.

For this mesh and deformation field, you can expect to run into this problem regardless of whether you use the total or updated Lagrangian formulation. As you can see, with your updated formulation you run into an issue when the mapping Jacobian becomes singular (due to element skewness/inversion). For the total Lagrangian approach, the same problem would manifest itself in that the deformation gradient would be non-invertible i.e. det(F) <= 0. You only really have two "easy" options here:

1. Perform some mesh adaptivity

2. Change your initial mesh, keeping in mind the expected deformation.

A third option would be to use an ALE solid formulation, but I have no experience with this so I have no idea how hard it is to do.

What I suggest might help is if you make some sort of c-grid that focusses on the crack tip. That way the mesh opens up as the crack does, rather than thins out like you have here.

I hope this helps,

J-P

[Bruno Turcksin]

H.B.,

On Wednesday, January 25, 2017 at 3:02:01 PM UTC-5, ha.ba...@gmail.com wrote:

The image of the mapping applied to cell with center [0.295666 0.850622] is distorted. The cell geometry or the mapping are invalid, giving a non-positive volume fraction of -0.000119973 in quadrature point 0.

The attached figures show the deformed and initial mesh. How to solve this problem (without adaptive mesh refinement)? Can I use the update Lagrangian formulation in Deal.II without moving the mesh? Do you think that the total Lagrangian formulation is a solution?

This is common problem with the Lagrangian formulation. What some people do is to periodically remesh the domain. So your restart with a good mesh. Of course this means that you need to project all the quantities from the old mesh to the new one.

Best,

Bruno

[ha.ba...@gmail.com]

Hi,

I tried to solve the issue using adaptive mesh refinement (with 3 refinement levels limit for each cell). However, the problem occurs again since the crack tip opening process continues during next loading steps and cells around the crack tip highly deform again. The attached figure demonstrates the mesh with adaptive h-refinement. 

[Jean-Paul Pelteret]

Hi H.B,

I suppose its not entirely surprising, because performing local mesh refinement doesn't really address the core issue that you're facing here (but the resulting picture does look great!). I think that perhaps I should have said "remeshing" as opposed to "mesh adaptivity" - that would have made this part of my suggestion more clear. If it were me trying to solve this problem, I would change the initial meshing strategy itself. Considering the type of deformation that you're modelling, a Cartesian-aligned grid is clearly not a good approach. If you use an external preprocessor like Cubit then you could produce a more complex and suitable mesh quite easily.

Best,

Jean-Paul

[Jean-Paul Pelteret]

e.g. Perhaps the mesh strategy I've sketched out might be more appropriate?

[ha.ba...@gmail.com]

Thanks so much for your help J-P. Generally, the direction of the crack propagation is unknown. The complete remeshing is interesting for me. However it can be difficult specially for three-dimensional analyses.