ValueError: Found Argument in <Cos id=6262508512>, this is an invalid expression

General
1

Hello, I am trying to use FEniCsX to solve a time-dependent PDE with a non-linear coefficient. However, I am getting an error stating: “ValueError: Found Argument in Cos id=6262508512, this is an invalid expression.” …when I formulate my variational problem (a,L)=system(F). I am wondering if this is because system() sets up a linear system to solve with a linear solver? Also can NonLinearSolver() be used to solve time-dependent PDEs? My PDE is essentially a Fokker-plank equation with a non-linear drift (or velocity) term. Please find my code below. Any help/clarification would be greatly appreciated. Thanks in advance!


import numpy as np
import matplotlib

import matplotlib.pyplot as plt

from dolfinx import mesh, fem, io, plot, la
import pyvista
import ufl
import numpy as np
from dolfinx import *
import dolfinx
import numpy, sys


from mpi4py import MPI
from petsc4py import PETSc
from ufl import (VectorElement, FiniteElement,
                 SpatialCoordinate, TrialFunction, TestFunction,
                 as_vector, cos, sin, inner, div, grad, dx, pi, system)

from dolfinx import mesh, fem, io, nls, log
from dolfinx.fem.petsc import NonlinearProblem
from dolfinx.nls.petsc import NewtonSolver


##cretae a rectangular mesh with quadrilateral elements
##note: here theta=x

from mpi4py import MPI
length, height = 10, 3
Nx, Ny = 80, 60
extent = [[0., 0.], [length, height]]
domain = mesh.create_rectangle(
    MPI.COMM_WORLD, extent, [Nx, Ny], mesh.CellType.quadrilateral)



##define the non-linear coeff
omega=(np.pi/2)
##define the non-linear coefficient
def vel(x):
    return omega + cos(x)

##setting up the variational problem
from ufl import (TestFunction, SpatialCoordinate, TrialFunction,
                 as_vector, dx, grad, inner, system, equation, Constant)
V = fem.FunctionSpace(domain, ("Lagrange", 1))
u = TrialFunction(V) ##time-dep c at n+1
v = TestFunction(V)
un = fem.Function(V) ##time dep C at n
D= fem.Constant(domain, 1.0) ##diffusion constant
dt = fem.Constant(domain, 0.05)

##define the variational problem
F= (un*v*ufl.dx) - (v*ufl.dx) ##note: i think it should be  L=un*v -v*dx
F-= u*dx - v*dx - vel(u)*u*v*dt*dx  - D*dt*inner(grad(u), grad(v))*dx + D*dt*v*dx
#variational problem stated as a=L
(a,L)=system(F)```
2

Hi

This is a non-linear problem, so considering u=TrialFunction(V) is wrong for system(F) unless you linearize and intend to implement manually a fixed-point iteration or a Newton iteration. Please see how @dokken implemented a Non-linear Poisson equation in the Dolfinx tutorial. One of the most important things is that you need to redefine u as u=fem.Function(V). Also, be aware that using the class NonlinearProblem will give you an ArityMismatch error because you have non-linear terms involving TrialFunctions like cos(u)*u*v*dt*dx.

I’m unfamiliar with the type of PDE that your are solving, but the following runs with no error. I’m not sure about the expression u*dx (maybe is related to your previous post ?)

from ufl import (TestFunction, SpatialCoordinate, TrialFunction,
                 as_vector, dx, grad, inner, system, equation, Constant)
import numpy as np
import matplotlib

import matplotlib.pyplot as plt

from dolfinx import mesh, fem, io, plot, la
# import pyvista
import ufl
import numpy as np
from dolfinx import *
import dolfinx
import numpy
import sys


from mpi4py import MPI
from petsc4py import PETSc
from ufl import (VectorElement, FiniteElement,
                 SpatialCoordinate, TrialFunction, TestFunction,
                 as_vector, cos, sin, inner, div, grad, dx, pi, system)

from dolfinx import mesh, fem, io, nls, log
from dolfinx.fem.petsc import NonlinearProblem
from dolfinx.nls.petsc import NewtonSolver


# cretae a rectangular mesh with quadrilateral elements
# note: here theta=x

from mpi4py import MPI
length, height = 10, 3
Nx, Ny = 80, 60
extent = [[0., 0.], [length, height]]
domain = mesh.create_rectangle(
    MPI.COMM_WORLD, extent, [Nx, Ny], mesh.CellType.quadrilateral)


# define the non-linear coeff
omega = (np.pi/2)
# define the non-linear coefficient


def vel(x):
    return omega + cos(x)


# setting up the variational problem
V = fem.FunctionSpace(domain, ("Lagrange", 1))
u = fem.Function(V)  # time-dep c at n+1
v = TestFunction(V)
un = fem.Function(V)  # time dep C at n
D = fem.Constant(domain, 1.0)  # diffusion constant
dt = fem.Constant(domain, 0.05)

# define the variational problem
# note: i think it should be  L=un*v -v*dx
F = (un*v*ufl.dx) - (v*ufl.dx)
F -= u*dx - fem.Constant(domain, 1.0)*v*dx - cos(u)*u*v*dt*dx - D*dt * \
    inner(grad(u), grad(v))*dx + D*dt*v*dx
# variational problem stated as a=L
(a, L) = system(F)

# problem = NonlinearProblem(F, u, bcs=[]) # works if taking u*v*dx instead of u*dx

Hope this helps.

Cheers.

2 Likes
3

Hi! Thanks so much for your reply! Would a backward Euler scheme be appropriate for solving the time-dependent problem? This is a time-dependent PDE (which is different from the non-linear poisson example). Would you suggest applying a Newton Solver for the spatial discretization in combination with a backward Euler scheme for the time-dependent part? Also can you refer me to where I might find documentation on how to use NonlinearVariationalSolver() with examples? Would I want to use this function? Thank you so much!
Also, would I still want to linearize the non-linear term?

I found the page below, but it did not help clarify how the NonLinearVariationalSolver works. https://fenicsproject.org/olddocs/dolfin/1.4.0/python/programmers-reference/fem/solving/NonlinearVariationalSolver.html

4

Yes, backward Euler could work. It all depends on the stability of your scheme. You could also use the Newton solver, or implement a fixed-point iteration.

In the forum you could find some post on non-linear time-dependent problems:

https://fenicsproject.discourse.group/t/nonlinear-heat-equation-performance-dolfin-vs-dolfinx/11961

https://fenicsproject.discourse.group/t/how-to-check-if-the-nonlinearproblem-is-setup-properly/10218

Also, in this dolfinx-tutorial by @bleyerj you can find a thermo-elastic evolution problem.

Cheers.

1 Like