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Today's Topics:

   1.  Mixed formulation for incompressible hyperelasticity
      (Bhavesh Shrimali)
   2. Re:  Mixed formulation for incompressible hyperelasticity
      (Bhavesh Shrimali)
   3. Re:  Mixed formulation for incompressible hyperelasticity
      (Patrick Farrell)


----------------------------------------------------------------------

Message: 1
Date: Thu, 24 Jun 2021 22:11:25 -0500
From: Bhavesh Shrimali <bhavesh.shrimali@gmail.com>
To: firedrake@imperial.ac.uk
Subject: [firedrake] Mixed formulation for incompressible
        hyperelasticity
Message-ID:
        <CANw4EA2dOXvzvGtWvphFHBA6azcD9yunrhu_Zad96fdT8rrJGg@mail.gmail.com>
Content-Type: text/plain; charset="utf-8"

Hi everyone,

I was trying to use a mixed-formulation for an incompressible
hyperelasticity problem. See the following MWE which uses a Taylor-Hood
approximation -- P2 for displacement (u) and P1 for pressure (p):

mesh = UnitCubeMesh(5,5,5)
Vh = VectorFunctionSpace(mesh, "CG", 2)
Qh = FunctionSpace(mesh, "CG", 1)
Wh = Vho * Qh
w = Function(Wh)
dw = TrialFunction(Wh)
v = TestFunction(Wh)
u, p = split(w)
mu, lmbda = Constant(1.), Constant(1.e3)
Js = (lmbda + p + sqrt((lmbda + p)**2. + 4.*lmbda*mu ))/(2.*lmbda)
F = Identity(mesh.geometric_dimension()) + grad(u)
J = det(F)
I1 = inner(F, F)
Pi = p * (J - Js) + mu/2 * (I1 - 3) - mu * ln(Js) + lmbda/2 * (Js - 1)**2
Pi *= dx
a = derivative(Pi, w, v)
Jac = derivative(a, w, dw)
bcs = [
DirichletBC(Wh.sub(0).sub(0), Constant(0.), 1),
DirichletBC(Wh.sub(0).sub(1), Constant(0.), 3),
DirichletBC(Wh.sub(0).sub(2), Constant(0.), 5),
DirichletBC(Wh.sub(0).sub(0), Constant(1.), 2)
]
problem = NonlinearVariationalProblem(a, w, bcs, J=Jac)
solver = NonlinearVariationalSolver(problem)
solver.solve()
u, p = w.split()
File("displacement.pvd").write(u)


Everything works fine as expected. Now, I wanted to try another pair of
approximations that is inf-sup stable, namely the one in the celebrated
paper by Crouzeix-Raviart (1973). See (4.36) in
http://archive.numdam.org/article/M2AN_1973__7_3_33_0.pdf.

This amounts to approximating `u` by a "P2 + Bubble" and `p` by a "DG-1"
space. This is also given in (3.2.3) in
https://link.springer.com/content/pdf/10.1007%2F978-3-642-87047-7_5.pdf
and (2.38) in
https://link.springer.com/content/pdf/10.1007%2F978-3-642-61623-5_2.pdf

Now, in 3D as far as I can tell this amounds to approximating `u` with
"P2 + an internal bubble + FacetBubble". Would anyone know how this could
be achieved within firedrake? Because as far as I can see, firedrake
supports `FacetBubble`. However, when I try something like:

B3 = FiniteElement("Bubble", "tetrahedron", 3)
FB3 = FiniteElement("FacetBubble", "tetrahedron", 3)
P2 = FiniteElement("CG", "tetrahedron", 2)
Vh = FunctionSpace(mesh, P2 + B3 + FB3)

I get the following error

  File "/usr/lib/python3.8/functools.py", line 875, in wrapper
    return dispatch(args[0].__class__)(*args, **kw)
  File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
189, in convert_enrichedelement
    elements, deps = zip(*[_create_element(elem, **kwargs)
  File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
189, in <listcomp>
    elements, deps = zip(*[_create_element(elem, **kwargs)
  File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
310, in _create_element
    finat_element, deps = convert(ufl_element, **kwargs)
  File "/usr/lib/python3.8/functools.py", line 875, in wrapper
    return dispatch(args[0].__class__)(*args, **kw)
  File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
177, in convert_finiteelement
    return lmbda(cell, element.degree()), set()
  File "/home/firedrake/firedrake/src/FInAT/finat/fiat_elements.py", line
318, in __init__
    super(Bubble, self).__init__(FIAT.Bubble(cell, degree))
  File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 33, in
__init__
    super(Bubble, self).__init__(ref_el, degree, codim=0)
  File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 24, in
__init__
    raise RuntimeError('Bubble element of degree %d and codimension %d has
no dofs' % (degree, codim))
RuntimeError: Bubble element of degree 3 and codimension 0 has no dofs

Any pointers would be appreciated
--

*Bhavesh Shrimali*
*Ph.D. Student, **Civil and Environmental Engineering,*

*University of Illinois Urbana-Champaign*


*Champaign, IL, United States*

*Email : bshrima2@illinois.edu <bshrima2@illinois.edu>*
*Web   : **https://bhaveshshrimali.github.io/
<https://bhaveshshrimali.github.io/>*
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Message: 2
Date: Thu, 24 Jun 2021 22:31:48 -0500
From: Bhavesh Shrimali <bhavesh.shrimali@gmail.com>
To: firedrake@imperial.ac.uk
Subject: Re: [firedrake] Mixed formulation for incompressible
        hyperelasticity
Message-ID:
        <CANw4EA010sANtvZXdYrX8bVKZ_4UfStFy3Xqot1JycP+vNQdpw@mail.gmail.com>
Content-Type: text/plain; charset="utf-8"

For the 3D Crouzeix-Raviart element see also Example 8.7.2 in the textbook
by Brezzi :
https://link.springer.com/content/pdf/10.1007%2F978-3-642-36519-5.pdf

"We therefore start from a quadratic approximation and enrich it by face
bubbles to control the fluxes on the faces and by internal bubbles to
control linear pressure. This yields "

[image: image.png]

Please let me know what you think! Note that this is different from the P1
non-conforming element that is denoted by "CR" and often simply called the
"Crouzeix-Raviart" element.

Thanks again for your help

On Thu, 24 Jun 2021 at 22:11, Bhavesh Shrimali <bhavesh.shrimali@gmail.com>
wrote:

> Hi everyone,
>
> I was trying to use a mixed-formulation for an incompressible
> hyperelasticity problem. See the following MWE which uses a Taylor-Hood
> approximation -- P2 for displacement (u) and P1 for pressure (p):
>
> mesh = UnitCubeMesh(5,5,5)
> Vh = VectorFunctionSpace(mesh, "CG", 2)
> Qh = FunctionSpace(mesh, "CG", 1)
> Wh = Vho * Qh
> w = Function(Wh)
> dw = TrialFunction(Wh)
> v = TestFunction(Wh)
> u, p = split(w)
> mu, lmbda = Constant(1.), Constant(1.e3)
> Js = (lmbda + p + sqrt((lmbda + p)**2. + 4.*lmbda*mu ))/(2.*lmbda)
> F = Identity(mesh.geometric_dimension()) + grad(u)
> J = det(F)
> I1 = inner(F, F)
> Pi = p * (J - Js) + mu/2 * (I1 - 3) - mu * ln(Js) + lmbda/2 * (Js - 1)**2
> Pi *= dx
> a = derivative(Pi, w, v)
> Jac = derivative(a, w, dw)
> bcs = [
> DirichletBC(Wh.sub(0).sub(0), Constant(0.), 1),
> DirichletBC(Wh.sub(0).sub(1), Constant(0.), 3),
> DirichletBC(Wh.sub(0).sub(2), Constant(0.), 5),
> DirichletBC(Wh.sub(0).sub(0), Constant(1.), 2)
> ]
> problem = NonlinearVariationalProblem(a, w, bcs, J=Jac)
> solver = NonlinearVariationalSolver(problem)
> solver.solve()
> u, p = w.split()
> File("displacement.pvd").write(u)
>
>
> Everything works fine as expected. Now, I wanted to try another pair of
> approximations that is inf-sup stable, namely the one in the celebrated
> paper by Crouzeix-Raviart (1973). See (4.36) in
> http://archive.numdam.org/article/M2AN_1973__7_3_33_0.pdf.
>
> This amounts to approximating `u` by a "P2 + Bubble" and `p` by a "DG-1"
> space. This is also given in (3.2.3) in
> https://link.springer.com/content/pdf/10.1007%2F978-3-642-87047-7_5.pdf
> and (2.38) in
> https://link.springer.com/content/pdf/10.1007%2F978-3-642-61623-5_2.pdf
>
> Now, in 3D as far as I can tell this amounds to approximating `u` with
> "P2 + an internal bubble + FacetBubble". Would anyone know how this could
> be achieved within firedrake? Because as far as I can see, firedrake
> supports `FacetBubble`. However, when I try something like:
>
> B3 = FiniteElement("Bubble", "tetrahedron", 3)
> FB3 = FiniteElement("FacetBubble", "tetrahedron", 3)
> P2 = FiniteElement("CG", "tetrahedron", 2)
> Vh = FunctionSpace(mesh, P2 + B3 + FB3)
>
> I get the following error
>
>   File "/usr/lib/python3.8/functools.py", line 875, in wrapper
>     return dispatch(args[0].__class__)(*args, **kw)
>   File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
> 189, in convert_enrichedelement
>     elements, deps = zip(*[_create_element(elem, **kwargs)
>   File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
> 189, in <listcomp>
>     elements, deps = zip(*[_create_element(elem, **kwargs)
>   File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
> 310, in _create_element
>     finat_element, deps = convert(ufl_element, **kwargs)
>   File "/usr/lib/python3.8/functools.py", line 875, in wrapper
>     return dispatch(args[0].__class__)(*args, **kw)
>   File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line
> 177, in convert_finiteelement
>     return lmbda(cell, element.degree()), set()
>   File "/home/firedrake/firedrake/src/FInAT/finat/fiat_elements.py", line
> 318, in __init__
>     super(Bubble, self).__init__(FIAT.Bubble(cell, degree))
>   File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 33, in
> __init__
>     super(Bubble, self).__init__(ref_el, degree, codim=0)
>   File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 24, in
> __init__
>     raise RuntimeError('Bubble element of degree %d and codimension %d has
> no dofs' % (degree, codim))
> RuntimeError: Bubble element of degree 3 and codimension 0 has no dofs
>
> Any pointers would be appreciated
> --
>
> *Bhavesh Shrimali*
> *Ph.D. Student, **Civil and Environmental Engineering,*
>
> *University of Illinois Urbana-Champaign*
>
>
> *Champaign, IL, United States*
>
> *Email : bshrima2@illinois.edu <bshrima2@illinois.edu>*
> *Web   : **https://bhaveshshrimali.github.io/
> <https://bhaveshshrimali.github.io/>*
>


--

*Bhavesh Shrimali*
*Ph.D. Student, **Civil and Environmental Engineering,*

*University of Illinois Urbana-Champaign*


*Champaign, IL, United States*

*Email : bshrima2@illinois.edu <bshrima2@illinois.edu>*
*Web   : **https://bhaveshshrimali.github.io/
<https://bhaveshshrimali.github.io/>*
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Message: 3
Date: Fri, 25 Jun 2021 08:48:50 +0100
From: Patrick Farrell <patrick.farrell@maths.ox.ac.uk>
To: firedrake@imperial.ac.uk
Subject: Re: [firedrake] Mixed formulation for incompressible
        hyperelasticity
Message-ID: <e94f233d-f114-cdca-5b8b-2ca8b1724cfb@maths.ox.ac.uk>
Content-Type: text/plain; charset=utf-8; format=flowed

Dear Bhavesh,

I'm not an expert on the topic you raise, but gazing into a crystal ball:
does it work if you try

B4 = FiniteElement("Bubble", "tetrahedron", 4)

instead of the code you had

B3 = FiniteElement("Bubble", "tetrahedron", 3)

?

Example 8.7.2 in the book you cite uses degree 4 cell bubbles.

Patrick



On 25/06/2021 04:31, Bhavesh Shrimali wrote:
> For the 3D Crouzeix-Raviart element see also Example 8.7.2 in the textbook by Brezzi : https://link.springer.com/content/pdf/10.1007%2F978-3-642-36519-5.pdf <https://link.springer.com/content/pdf/10.1007%2F978-3-642-36519-5.pdf>
>
> "We therefore start from a quadratic approximation and enrich it by face bubbles to control the fluxes on the faces and by internal bubbles to control linear pressure. This yields "
>
> image.png
>
> Please let me know what you think! Note that this is different from the P1 non-conforming element that is denoted by "CR" and often simply called the "Crouzeix-Raviart" element.
>
> Thanks again for your help
>
> On Thu, 24 Jun 2021 at 22:11, Bhavesh Shrimali <bhavesh.shrimali@gmail.com <mailto:bhavesh.shrimali@gmail.com>> wrote:
>
>     Hi everyone,
>
>     I was trying to use a mixed-formulation for an incompressible hyperelasticity problem. See the following MWE which uses a Taylor-Hood approximation -- P2 for displacement (u) and P1 for pressure (p):
>
>     mesh?=?UnitCubeMesh(5,5,5)
>     Vh?=?VectorFunctionSpace(mesh,?"CG",?2)
>     Qh?=?FunctionSpace(mesh,?"CG",?1)
>     Wh?=?Vho?*?Qh
>     w?=?Function(Wh)
>     dw?=?TrialFunction(Wh)
>     v?=?TestFunction(Wh)
>     u,?p?=?split(w)
>     mu,?lmbda?=?Constant(1.),?Constant(1.e3)
>     Js?=?(lmbda?+?p?+?sqrt((lmbda?+?p)**2.?+?4.*lmbda*mu?))/(2.*lmbda)
>     F?=?Identity(mesh.geometric_dimension())?+?grad(u)
>     J?=?det(F)
>     I1?=?inner(F,?F)
>     Pi?=?p?*?(J?-?Js)?+?mu/2?*?(I1?-?3)?-?mu?*?ln(Js)?+?lmbda/2?*?(Js?-?1)**2
>     Pi?*=?dx
>     a?=?derivative(Pi,?w,?v)
>     Jac?=?derivative(a,?w,?dw)
>     bcs?=?[
>     DirichletBC(Wh.sub(0).sub(0),?Constant(0.),?1),
>     DirichletBC(Wh.sub(0).sub(1),?Constant(0.),?3),
>     DirichletBC(Wh.sub(0).sub(2),?Constant(0.),?5),
>     DirichletBC(Wh.sub(0).sub(0),?Constant(1.),?2)
>     ]
>     problem?=?NonlinearVariationalProblem(a,?w,?bcs,?J=Jac)
>     solver?=?NonlinearVariationalSolver(problem)
>     solver.solve()
>     u,?p?=?w.split()
>     File("displacement.pvd").write(u)
>
>
>     Everything works fine as expected. Now, I wanted to try another pair of approximations that?is?inf-sup stable, namely the one in the celebrated paper by Crouzeix-Raviart (1973). See (4.36) in http://archive.numdam.org/article/M2AN_1973__7_3_33_0.pdf <http://archive.numdam.org/article/M2AN_1973__7_3_33_0.pdf>.
>
>     This amounts to approximating `u` by a "P2?+ Bubble" and `p` by a "DG-1" space. This is also given in (3.2.3) in https://link.springer.com/content/pdf/10.1007%2F978-3-642-87047-7_5.pdf <https://link.springer.com/content/pdf/10.1007%2F978-3-642-87047-7_5.pdf>
>     and (2.38) in https://link.springer.com/content/pdf/10.1007%2F978-3-642-61623-5_2.pdf <https://link.springer.com/content/pdf/10.1007%2F978-3-642-61623-5_2.pdf>
>
>     Now, in 3D as far as I can tell this amounds to approximating `u` with "P2?+ an internal bubble?+ FacetBubble". Would anyone know how this could be achieved within firedrake? Because as far as I can see, firedrake supports `FacetBubble`. However, when I try something like:
>
>     B3 = FiniteElement("Bubble", "tetrahedron", 3)
>     FB3 = FiniteElement("FacetBubble", "tetrahedron", 3)
>     P2 = FiniteElement("CG", "tetrahedron", 2)
>     Vh = FunctionSpace(mesh, P2?+ B3?+ FB3)
>
>     I get the following error
>
>      ? File "/usr/lib/python3.8/functools.py", line 875, in wrapper
>      ? ? return dispatch(args[0].__class__)(*args, **kw)
>      ? File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line 189, in convert_enrichedelement
>      ? ? elements, deps = zip(*[_create_element(elem, **kwargs)
>      ? File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line 189, in <listcomp>
>      ? ? elements, deps = zip(*[_create_element(elem, **kwargs)
>      ? File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line 310, in _create_element
>      ? ? finat_element, deps = convert(ufl_element, **kwargs)
>      ? File "/usr/lib/python3.8/functools.py", line 875, in wrapper
>      ? ? return dispatch(args[0].__class__)(*args, **kw)
>      ? File "/home/firedrake/firedrake/src/tsfc/tsfc/finatinterface.py", line 177, in convert_finiteelement
>      ? ? return lmbda(cell, element.degree()), set()
>      ? File "/home/firedrake/firedrake/src/FInAT/finat/fiat_elements.py", line 318, in __init__
>      ? ? super(Bubble, self).__init__(FIAT.Bubble(cell, degree))
>      ? File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 33, in __init__
>      ? ? super(Bubble, self).__init__(ref_el, degree, codim=0)
>      ? File "/home/firedrake/firedrake/src/fiat/FIAT/bubble.py", line 24, in __init__
>      ? ? raise RuntimeError('Bubble element of degree %d and codimension %d has no dofs' % (degree, codim))
>     RuntimeError: Bubble element of degree 3 and codimension 0 has no dofs
>
>     Any pointers would be appreciated
>     --
>     *Bhavesh Shrimali
>     *
>     *Ph.D. Student, **Civil and Environmental Engineering,*
>     *University of Illinois Urbana-Champaign
>     *
>     *Champaign, IL, United States
>
>     *
>     *Email : bshrima2@illinois.edu <mailto:bshrima2@illinois.edu>
>     *
>     *Web ? : **https://bhaveshshrimali.github.io/ <https://bhaveshshrimali.github.io/>*
>
>
>
> --
> *Bhavesh Shrimali
> *
> *Ph.D. Student, **Civil and Environmental Engineering,*
> *University of Illinois Urbana-Champaign
> *
> *Champaign, IL, United States
>
> *
> *Email : bshrima2@illinois.edu <mailto:bshrima2@illinois.edu>
> *
> *Web ? : **https://bhaveshshrimali.github.io/ <https://bhaveshshrimali.github.io/>*
>
> _______________________________________________
> firedrake mailing list
> firedrake@imperial.ac.uk
> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
>




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