# How to apply quadratic element type

**URL:** <https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719>\
**Category:** Uncategorized\
**Created:** [March 25, 2020, 6:16pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719 "2020-03-25T18:16:46Z")\
**Posts on this page:** 9\
**Page:** 1

<div class="post-metadata">

**Author:** ![BilleBili](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@BilleBili](https://fenicsproject.discourse.group/u/BilleBili)\
**Post date:** [March 25, 2020, 6:16pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/1 "2020-03-25T18:16:46Z")

</div>

Hello,

I recently found a code for the cantilever beam with uniform distributed load problem. It works well with CG element type. I want it run with quadratic element type. So I tried to edit some codes inside.  
The code is here:

> from **future** import print\_function  
> from dolfin import \*  
> from fenics import \*  
> import matplotlib.pyplot as plt  
> import math
> 
> L = 25.  
> H = 1.  
> Nx = 250  
> Ny = 10  
> mesh = RectangleMesh(Point(0., 0.), Point(L, H), Nx, Ny, “crossed”)
> 
> def eps(v):  
> return sym(grad(v))
> 
> E = Constant(1e5)  
> nu = Constant(0.3)
> 
> lmbda = E_nu/(1+nu)/(1-2_nu)  
> mu = E/2/(1+nu)  
> lmbda = 2_mu_lmbda/(lmbda+2\*mu)
> 
> def sigma(v):  
> return lmbda_tr(eps(v))Identity(2) + 2.0mu_eps(v)
> 
> rho\_g = 1e-3  
> f = Constant((0, -rho\_g))
> 
> fe = VectorElement(“Quadrature”, cell=mesh.ufl\_cell(), degree = 1, quad\_scheme = “default”)  
> V = VectorFunctionSpace(mesh, fe, degree = 1)  
> du = TrialFunction(V)  
> u\_ = TestFunction(V)  
> a = inner(sigma(du), eps(u\_))\*dx  
> l = inner(f, u\_)\*dx
> 
> def left(x, on\_boundary):  
> return near(x[0],0.)
> 
> bc = DirichletBC(V, Constant((0.,0.)), left)
> 
> u = Function(V, name=“Displacement”)  
> solve(a == l, u, bc)
> 
> plot(1e3\*u, mode=“displacement”)  
> plt.title(“2D elastic bending model”)  
> plt.xlabel(“length”)  
> plt.ylabel(“bending”)  
> plt.show()
> 
> print(“Maximal deflection:”, -u(L,H/2.)[1])  
> print(“Beam theory deflection:”, float(3_rho\_g_L **4/2/E/H** 3))
> 
> Vsig = TensorFunctionSpace(mesh, “DG”, degree=0)  
> sig = Function(Vsig, name=“Stress”)  
> sig.assign(project(sigma(u), Vsig))  
> print(“Stress at (0,H):”, sig(0, H))
> 
> error = abs(-u(L,H/2.)[1] - float(3_rho\_g_L **4/2/E/H** 3))  
> print(error)

But the an err occurred:

> runcell(0, ‘/Users/pigdtt12345/Desktop/Bill/FEniCS/Training/2D\_elasticity\_simple.py’)  
> Calling FFC just-in-time (JIT) compiler, this may take some time.  
> Calling FFC just-in-time (JIT) compiler, this may take some time.  
> Calling FFC just-in-time (JIT) compiler, this may take some time.  
> Symmetric part of tensor with rank != 2 is undefined.  
> Traceback (most recent call last):
> 
> File “/Users/pigdtt12345/Desktop/Bill/FEniCS/Training/2D\_elasticity\_simple.py”, line 34, in   
> a = inner(sigma(du), eps(u\_))\*dx
> 
> File “/Users/pigdtt12345/Desktop/Bill/FEniCS/Training/2D\_elasticity\_simple.py”, line 25, in sigma  
> return lmbda_tr(eps(v))Identity(2) + 2.0mu_eps(v)
> 
> File “/Users/pigdtt12345/Desktop/Bill/FEniCS/Training/2D\_elasticity\_simple.py”, line 14, in eps  
> return sym(grad(v))
> 
> File “/Users/pigdtt12345/opt/anaconda3/envs/fenicsproject/lib/python3.8/site-packages/ufl/operators.py”, line 318, in sym  
> return Sym(A)
> 
> File “/Users/pigdtt12345/opt/anaconda3/envs/fenicsproject/lib/python3.8/site-packages/ufl/tensoralgebra.py”, line 457, in **new**  
> error(“Symmetric part of tensor with rank != 2 is undefined.”)
> 
> File “/Users/pigdtt12345/opt/anaconda3/envs/fenicsproject/lib/python3.8/site-packages/ufl/log.py”, line 172, in error  
> raise self.\_exception\_type(self.\_format\_raw(\*message))
> 
> UFLException: Symmetric part of tensor with rank != 2 is undefined.

Could some please help me with it? I just want it work with “Quadrature” element type. I am sorry if I make some format error here. I am a new member with this qa forum.  
Thanks a lot in advance.

---

<div class="post-metadata">

**Author:** ![JoeKW](https://avatars.discourse-cdn.com/v4/letter/j/c4cdca/32.png) [@JoeKW](https://fenicsproject.discourse.group/u/JoeKW)\
**Post date:** [March 25, 2020, 6:37pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/2 "2020-03-25T18:37:27Z")

</div>

If what you want is to use quadratic shape functions, I think you just need to set degree=2.

---

<div class="post-metadata">

**Author:** ![BilleBili](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@BilleBili](https://fenicsproject.discourse.group/u/BilleBili)\
**Post date:** [March 25, 2020, 8:26pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/4 "2020-03-25T20:26:02Z")

</div>

Hi,  
Thanks for your help. I have tried it before.  
I directly tried

> V = VectorFunctionSpace(mesh, “Quadrature”, degree = 2)

but the error comes out too. The error said: Missing quad\_scheme in quadrature element. So I checked online to solve how to fix it, and I found this page that has same issue.

[https://fenicsproject.org/qa/9635/quadrature-element/](https://fenicsproject.org/qa/9635/quadrature-element/)

But when I try it, it has the exception as the initial problem as “Symmetric part of tensor with rank != 2 is undefined.”

So here I am totally lost what I can do next to fix this bug. Sorry that I am beginner and this is a bit out of my knowledge.  
Thanks in advance for your help.

---

<div class="post-metadata">

**Author:** ![bhaveshshrimali](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/bhaveshshrimali/32/124_2.png) [@bhaveshshrimali](https://fenicsproject.discourse.group/u/bhaveshshrimali)\
**Post date:** [March 25, 2020, 9:14pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/6 "2020-03-25T21:14:15Z")

</div>

Additionally for using quadrature elements, you might have to change the representation for the form compiler. The default is `uflacs`. This can be done using something like:

```auto
parameters["form_compiler"]["representation"] = 'quadrature'
# And to suppress `depecration` warnings 
import warnings
from ffc.quadrature.deprecation import QuadratureRepresentationDeprecationWarning
warnings.simplefilter("once", QuadratureRepresentationDeprecationWarning)

```

see [here](https://comet-fenics.readthedocs.io/en/latest/demo/2D_plasticity/vonMises_plasticity.py.html)

---

<div class="post-metadata">

**Author:** ![bleyerj](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/bleyerj/32/6020_2.png) [@bleyerj](https://fenicsproject.discourse.group/u/bleyerj)\
**Post date:** [March 25, 2020, 9:38pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/7 "2020-03-25T21:38:46Z")

</div>

Hello,  
you are mixing “quadratic” interpolation (implictly Lagrange P2 elements) with “quadrature” elements which correspond to fields evaluated at quadrature points only and from which one cannot compute gradients and so on.  
For quadratic elements, just use ` V = VectorFunctionSpace(mesh, "CG", 2)`

---

<div class="post-metadata">

**Author:** ![BilleBili](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@BilleBili](https://fenicsproject.discourse.group/u/BilleBili)\
**Post date:** [March 25, 2020, 10:25pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/8 "2020-03-25T22:25:28Z")

</div>

Ah! I see! Thank you so much!

---

<div class="post-metadata">

**Author:** ![BilleBili](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@BilleBili](https://fenicsproject.discourse.group/u/BilleBili)\
**Post date:** [March 25, 2020, 10:28pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/9 "2020-03-25T22:28:18Z")

</div>

Hello,

I put it at the top and the error occurs again. It said, “Symmetric part of tensor with rank != 2 is undefined.”. But I think I mixed the quadratic interpolation with quadrature elements. But thank you for your help very much too! Sorry for taking up your time.

---

<div class="post-metadata">

**Author:** ![bhaveshshrimali](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/bhaveshshrimali/32/124_2.png) [@bhaveshshrimali](https://fenicsproject.discourse.group/u/bhaveshshrimali)\
**Post date:** [March 25, 2020, 10:50pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/11 "2020-03-25T22:50:43Z")

</div>

Yeah. I just figured that you wanted to use `quadratic` and not `quadrature` elements. Both

```auto
deg = 2
V = VectorFunctionSpace(mesh, "CG", deg)

```

and

```auto
W = VectorElement("CG", mesh.ufl_cell(), deg)
V = FunctionSpace(mesh, W)

```

are equivalent!!

---

<div class="post-metadata">

**Author:** ![BilleBili](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@BilleBili](https://fenicsproject.discourse.group/u/BilleBili)\
**Post date:** [March 25, 2020, 11:04pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-quadratic-element-type/2719/12 "2020-03-25T23:04:32Z")

</div>

Yeah, because actually I have just got familiar with FEniCS and FEM. I am sorry I did not make my question clear. Thank you for your added up information. This also helps me a lot.