How to use sympy in firedrake

[zhyxue0]

Hello, everyone! I am trying to use sympy in firerake to make expressions general.

Here is my idea, but it seems does not work.

import sympy

x, y, t = sympy.symbols('x y t')
u = 3*x*y + x**2 + 2*y**2 + t**3

Do you have any good suggestions?

[wence-]

What would you like to do with the expression? It may be that you can get away with using UFL for your symbolics (and the things will be easier).

[zhyxue0]

For example, given a exact solution expression u. I try to solve the heat conduction equation u_t = a(u_xx + u_yy) + f, and obtain the corresponding expression of the source term f. f depends on the expression of the exact solution u (actually depends on x,y,t). I want a way to get a unified expression of f, so that f can be updated in time iteration.

[wence-]

OK, so you want to symbolically differentiate to find f = u_t - a(u_xx + u_yy). You can do this with ufl if you use UFL objects and variable and diff (see https://fenics.readthedocs.io/projects/ufl/en/latest/manual/form_language.html#variable-derivatives)

So, suppose you have x y and z as being spatial coordinates:

x, y = SpatialCoordinate(mesh)
t = Constant(...) # current time

# define objects that we can differentiate with respect to
xv = variable(x)
yv = variable(y)
tv = variable(t)

u = 3*xv*yv + xv**2 + 2*yv**2 + tv**3
# a is defined previous as a normal python function
f = diff(u, tv) - a(diff(diff(u, xv), xv) + diff(diff(u, yv), yv))

# Now you can just use `f` as a normal UFL expression

[rckirby]

This is also done several places in Irksome for time-dependent problems. For example, check the demo: https://github.com/firedrakeproject/Irksome/blob/master/demos/heat/demo_heat.py.rst There are other demos and tests in similar vein.

…

On Thu, Oct 26, 2023 at 12:09 PM Lawrence Mitchell ***@***.***> wrote: OK, so you want to symbolically differentiate to find f = u_t - a(u_xx + u_yy). You can do this with ufl if you use UFL objects and variable and diff (see https://fenics.readthedocs.io/projects/ufl/en/latest/manual/form_language.html#variable-derivatives ) So, suppose you have x y and z as being spatial coordinates: x, y = SpatialCoordinate(mesh)t = Constant(...) # current time # define objects that we can differentiate with respect toxv = variable(x)yv = variable(y)tv = variable(t) u = 3*xv*yv + xv**2 + 2*yv**2 + tv**3# a is defined previous as a normal python functionf = diff(u, tv) - a(diff(diff(u, xv), xv) + diff(diff(u, yv), yv)) # Now you can just use `f` as a normal UFL expression — Reply to this email directly, view it on GitHub <#3187 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AA4XULY54YZDFE3S7OZMV4TYBKKMZAVCNFSM6AAAAAA6RNWGRSVHI2DSMVQWIX3LMV43SRDJONRXK43TNFXW4Q3PNVWWK3TUHM3TGOJVGM3DK> . You are receiving this because you are subscribed to this thread.Message ID: <firedrakeproject/firedrake/repo-discussions/3187/comments/7395365@ github.com>

[zhyxue0]

Thanks a lot. But I have some questions. I note

uexact = B * atan(t)*(pi / 2.0 - atan(S * (R - t)))
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))

f I want to solve for f by this equation, $u=(u_0, u_1)$ is a vector,a vector, therefore f is also a vector,
Does expand_derivatives(diff(uexact, t)) - div(grad(uexact)) also work in vector form?

And this equation is
$f = u_{t} - \Delta u - |u|^{p} u$, where $|u| =（u_0^2 + u_1^2)^p$.
How to express f if I code

def u_exact(x, y, t):
    u = as_vector(sin(x)*t, sin(y)*t)
return u


# How to code it?
def f(u):

[rckirby]

It should respect the rules of vector algebra/calculus. You can stick UFL expressions into arrays by `as_vector` (it takes a list?) and then you need to compute the vector Laplacian that's right for you. You should try it out and post specific questions when something breaks.

…

On Thu, Oct 26, 2023 at 9:41 PM Sean ***@***.***> wrote: Thanks a lot. But I have some questions. I note uexact = B * atan(t)*(pi / 2.0 - atan(S * (R - t)))rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact)) f I want to solve for f by this equation, $u=(u_0, u_1)$ is a vector,a vector, therefore f is also a vector, Does expand_derivatives(diff(uexact, t)) - div(grad(uexact)) also work in vector form? And this equation is $f = u_{t} - \Delta u - |u|^{p} u$, where $|u| =（u_0^2 + u_1^2)^p$. How to express f if I code def u_exact(x, y, t): u = as_vector(sin(x)*t, sin(y)*t)return u # How to code it?def f(u): — Reply to this email directly, view it on GitHub <#3187 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AA4XULZBI34JKPOUL57LT6DYBMNMVAVCNFSM6AAAAAA6RNWGRSVHI2DSMVQWIX3LMV43SRDJONRXK43TNFXW4Q3PNVWWK3TUHM3TGOJYHEZDC> . You are receiving this because you commented.Message ID: ***@***.*** com>