# How to apply symmetry boundary conditions on symmetry planes in thermoelastic calculations?
**URL:** https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406
**Category:** dolfinx
**Created:** [April 8, 2025, 10:58am UTC](https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406 "2025-04-08T10:58:51Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@French\_fries](https://fenicsproject.discourse.group/u/French_fries)
#### Post date: [April 8, 2025, 10:58am UTC](https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406/1 "2025-04-08T10:58:51Z")
Assume I am solving the thermal expansion of a cylinder at 100°C (with the axial direction as the z-axis and the height being 10mm), where the mechanical boundary conditions restrict the z-direction displacement at the top and bottom faces to be 0. Clearly, this is a problem that is symmetric about z = 5mm. If I only model half of the cylinder (with a height of 5mm) and apply a displacement restriction of 0 in the z-direction at one of the bottom faces, how can I apply symmetric boundary conditions on the symmetry plane to obtain the same results as the original model?
Here is part of my code:
```auto
from mpi4py import MPI
from petsc4py import PETSc
from dolfinx.io import gmshio
from dolfinx import default_scalar_type
from ufl import (TestFunction, TrialFunction, dx, grad, tr, Identity, inner, lhs, rhs)
from dolfinx.fem import (functionspace, Function, dirichletbc, locate_dofs_topological, form, Constant)
from dolfinx.fem.petsc import (assemble_matrix, assemble_vector, apply_lifting, set_bc)
import nullspace1
# mesh
mesh, cell_markers, facet_markers = gmshio.read_from_msh("sym.msh", MPI.COMM_WORLD, gdim=3)
#parameters
E = 2e5
nu = 0.3
mu = E/2/(1+nu)
lmbda = E*nu/(1+nu)/(1-2*nu)
alpha = 1e-5
Delta_T = 100
def eps(v):
return 0.5 * (grad(v) + grad(v).T)
def sigma(v, dT):
return (lmbda*tr(eps(v))-alpha*(3*lmbda+2*mu)*dT)*Identity(3)+2.0*mu*eps(v)
VU = functionspace(mesh, ("CG", 1, (mesh.geometry.dim,)))
#The z-direction displacement at one bottom face is 0.
U_D = Constant(mesh, default_scalar_type(0))
fdim = mesh.topology.dim - 1
facets_bottom = facet_markers.find(4)
dofs_bottom_z = locate_dofs_topological(VU.sub(2), fdim, facets_bottom)
bcU1 = dirichletbc(U_D, dofs_bottom_z, VU.sub(2))
bcU = [bcU1]
#symmetry boundary condition:
facets_sym = facet_markers.find(5) #The symmetry plane
"...Pending"
#Variational problem
dU = TrialFunction(VU)
U_ = TestFunction(VU)
Wint = inner(sigma(dU, Delta_T), eps(U_)) * dx
aM = lhs(Wint)
LM = rhs(Wint)
A = assemble_matrix(form(aM), bcs = bcU)
A.assemble()
b = assemble_vector(form(LM))
apply_lifting(b, [form(aM)], bcs=[bcU])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcU)
#Create Krylov solver
solver = PETSc.KSP().create(A.getComm())
solver.setOperators(A)
#Set the null space
null_space = nullspace1.null_space(VU)
A.setNearNullSpace(null_space)
#solve
U = Function(VU)
solver.setMonitor(lambda _, its, rnorm: print(f"Iteration: {its}, rel. residual: {rnorm}"))
solver.solve(b, U.vector)
```
and below is the mesh for the half-model：
```auto
N = 31;
Circle(1) = {0, 0, 0, 7, 0, 2*Pi};
Transfinite Curve {1} = N Using Progression 1;
Curve Loop(1) = {1};
Plane Surface(1) = {1};
Extrude {0, 0, 5} {Surface{1};Layers {5};}
Physical Surface("bottom", 4) = {1};
Physical Surface("sym", 5) = {3};
Physical Surface("cylinder", 6) = {2};
Physical Volume("v", 7) = {1};
```
---
### Author:  [@Stein](https://fenicsproject.discourse.group/u/Stein)
#### Post date: [April 8, 2025, 11:26am UTC](https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406/2 "2025-04-08T11:26:57Z")
Symmetry conditions are typically just homogeneous natural conditions (zero heat flux, zero mechanical traction). For vector valued fields (e.g., displacement) there is the additional Dirichlet (essential) condition that the normal component is also zero.
You have implemented the latter through a z-dispacement of 0 at the symmetry plane. The first two should automatically be satisfied by virtue of the weak formulation.
So (without having looked at your code, but purely from your description) it sounds to me that you’re already doing things correctly.
---
### Author:  [@French\_fries](https://fenicsproject.discourse.group/u/French_fries)
#### Post date: [April 8, 2025, 12:27pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406/3 "2025-04-08T12:27:27Z")
Thank you for your prompt reply. It indeed seems to be the case, but I would like to know if there is a specific symmetry condition module in DolfinX to implement this functionality (instead of defining the displacement in the z-direction of the symmetry plane as 0 through code like I did).
---
### Author:  [@Stein](https://fenicsproject.discourse.group/u/Stein)
#### Post date: [April 8, 2025, 12:44pm UTC](https://fenicsproject.discourse.group/t/how-to-apply-symmetry-boundary-conditions-on-symmetry-planes-in-thermoelastic-calculations/17406/4 "2025-04-08T12:44:15Z")
I’m not sure if I correctly understand your question. FEniCS is problem agnostic, and “symmetry conditions” are problem dependent. So FEniCS does not have a particular ‘module’ for enforcing them. There is no universal strategy for enforcement other than what I describe here:
> [@Stein](#):
>
> Symmetry conditions are typically just homogeneous natural conditions (zero heat flux, zero mechanical traction). For vector valued fields (e.g., displacement) there is the additional Dirichlet (essential) condition that the normal component is also zero.