Computing shape function derivatives for each cell

General
1

Hi FEniCS family,

I am using lagrange linear element mesh for my analysis of periodic homogenization. I wanted to compute shape function derivatives for each cell.
The mesh data is taken from this source.
Periodic homogenization of linear elastic materials — Numerical tours of continuum mechanics using FEniCS master documentation (comet-fenics.readthedocs.io)

Ve = VectorElement("CG", mesh.ufl_cell(), 1)
Re = VectorElement("R", mesh.ufl_cell(), 0)
W = FunctionSpace(mesh, MixedElement([Ve, Re]), constrained_domain=PeriodicBoundary(vertices, tolerance=1e-10))
V = FunctionSpace(mesh, Ve)

v_,lamb_ = TestFunctions(W)
dv, dlamb = TrialFunctions(W)
w = Function(W)
dx = Measure('dx')(subdomain_data=subdomains)

Eps = Constant(((1, 0), (0, 0)))
F = sum([inner(sigma(dv, i, Eps), eps(v_))*dx(i) for i in range(nphases)])
a, L = lhs(F), rhs(F)
a += dot(lamb_,dv)*dx + dot(dlamb,v_)*dx 
solve(a == L, w, [])

I wanted shape function derivatives w.r.t spatial coordinates (x,y) at each cell (triangle) of the mesh.
This means dN1/dx, dN2/dx; dN1/dy, dN2/dy data for each cell in the mesh.

Can I get a list of shape function derivatives of each cell ? /
is there any way to use function w for computing shape function derivatives like project function ?.

Thanking You

Any help is greatly appreciated.

2

I would suggest having a look at Automated shape differentiation in the Unified Form Language | SpringerLink

Note that shape derivatives are not computer per cell, but for each mesh node (usually in fenics a vector function space of first order Lagrange elements).

See for instance: Shape Derivatives in UFL - #2 by dokken as well

3

Thanks @dokken for your response and sharing the reference. I have doubts regarding your statement for above case.

“Note that shape derivatives are not computer per cell, but for each mesh node (usually in fenics a vector function space of first order Lagrange elements).”

  1. Does this mean that, at each node we get dN1/dx, dN2/dx, dN3/dx; dn1/dy, dN2/dy , dN3/dy (6X1) evaluated at each node
  2. For entire mesh, would it be
    mesh.num_vertices() X 6 scalar matrix ?
  3. would shape function derivative be a scalar of 6X1 dimension for each node (after using w.compute_vertex_values )?

Thanks for your help.

4

I would suggest you look at the references I have provided. The first one:

goes into detail about the mathematics, and the expected input/output, while the second reference goes more into the usage of this derivative for optimization problems

5

Hi @dokken, thanks for your response.

The resources you suggested shows Automatic differentiation algorithm to find change in functional w.r.t. some variable and looks more of time dependent application. I wanted to compute the derivatives of shape functions w.r.t x, y (for an element, evaluated through guass points). I don’t find way I can interpret the shape functions as it is formed through argument (test and trial functions in backend) and therefore, couldn’t find way to minimize the basis or shape functions.

I wanted to use the shape function derivatives w.r.t spatial coordinates for each element like
dN1/dx_element= (Summation dN1/dx_i * weight_i) , i=1… no. of guass points.

I wanted the direction related to my application. Kindly help me.

Thanking You

6

I dont understand what you mean by this.
A shape derivative for me is the change of an integral when you perturb the integration domain
This can be achieved in FEniCS by.

u = TrialFunction(V)
v = TestFunction(V)
a = inner(u,v)*dx

x= SpatialCoordinate(mesh)
dadx =derivative(a,x)
7

Based on the requirement, shape derivative means derivative of lagrangian basis functions used in general finite element formulation.
\:u=\:\sum _{i=1}^Nu_i\:\phi i\: , N is the number of nodes.
\phi -lagrangian shape/basis function.

This u is used as Function (Coefficient) u=Function(W) and
\phi as argument in UFL language. dv= TrialFunction(W)

Here, \phi is the lagrangian shape functions and the derivative which I want is \frac{d}{dx}\left(\phi \right)

Can you tell me how this derivative is elated to perturbation of integral which you described? I apologize for any mistakes I committed during explanation.

8

I think you just want: Bitbucket