How to difine UserExpression that takes the solution u

1

Hi all,
I have a question regarding the implementaiton of non-linear coefficient such as:

def q(u, alpha):
return 1 + alpha*u**2

I want to evaluate q in integrals using the values of u at the nodes. To achieve that (Treatment of non-linear coefficient - #2 by kamensky), we can use Expression:

expr = dl.Expression(“alpha * u * u”,u=u, alpha = alpha, degree=1)

My question is how can we write the expression using the UserExpression. My code below does not work (i.e., computation never finishes somehow):

import dolfin as dl
import matplotlib.pyplot as plt

N = 100
X = 1.0
mesh = dl.IntervalMesh(N, 0, X)
degree = 1 # linear basis functions
Vh = dl.FunctionSpace(mesh, “Lagrange”, degree)
u = dl.interpolate(dl.Expression(‘x[0]’, degree = 1), Vh)
alpha = 2.0

class user_expr(dl.UserExpression):
def set_alpha(self, u, alpha): # in the user_expr class
self.u = u
self.alpha = alpha

def eval(self, value, x): # in the user_expr class
“set value[0] to value at point x”
value[0] = self.alpha * self.u* self.u

user_expr2 = user_expr(degree = 0)
user_expr2.set_alpha(u, alpha)

user_expr2.compute_vertex_values(mesh) # not working

I appreciate any help!

2

In not sure what you mean by this. Do you want the quadrature rule for your integral to be the set of vertices in your mesh? Why would you want that?
Such a quadrature rule would underintegrate the polynomial.

3

Thank you for your comment. The reason is I wanted to test the method used in existing papers.

4

Consider the following:

import dolfin as dl

N = 100
X = 1.0
mesh = dl.IntervalMesh(N, 0, X)
degree = 1 # linear basis functions
Vh = dl.FunctionSpace(mesh, "Lagrange", degree)
u = dl.interpolate(dl.Expression("x[0]", degree = 1), Vh)
alpha = 2.0

class user_expr(dl.UserExpression):
    def set_alpha(self, u, alpha): # in the user_expr class
        self.u = u
        self.alpha = alpha

    def eval(self, value, x): # in the user_expr class
        "set value[0] to value at point x"
        value[0] = self.alpha * self.u(x)* self.u(x)

user_expr2 = user_expr(degree = 0)
user_expr2.set_alpha(u, alpha)

q = dl.interpolate(user_expr2, dl.FunctionSpace(mesh, "Lagrange", 1))
d_to_v = dl.dof_to_vertex_map(q.function_space())
print(q.vector()[:][d_to_v])
5

Thank you! I worked very well. I appreciate your help.