Composite Overwrapped Pressure Vessel (COPV)

A filament-wound pressure vessel modeled as a layered Shell: a metallic liner overwrapped with helical carbon-fiber plies and loaded by internal pressure. One octant of the vessel is meshed and closed with symmetry planes.

Material and laminate

An isotropic aluminium liner combined with an orthotropic carbon/epoxy overwrap. The liner is much thicker than the individual composite plies, which are wound in a ±15° helical pattern. Laminate.plot() shows the resulting stacking sequence.

import torch

from torchfem import Laminate
from torchfem.data import get_data
from torchfem.io import import_mesh
from torchfem.materials import (
    IsotropicElasticityPlaneStress,
    OrthotropicElasticityPlaneStress,
)

torch.set_default_dtype(torch.float64)

cfrp = OrthotropicElasticityPlaneStress(
    E_1=176800.0,
    E_2=10300.0,
    nu_12=0.23,
    G_12=4800.0,
    G_13=4800.0,
    G_23=3000.0,
    rho=1.6e-9,
)

alu = IsotropicElasticityPlaneStress(
    E=72000.0,
    nu=0.33,
    rho=2.7e-9,
)


p = torch.deg2rad(torch.tensor(15.0))
m = torch.deg2rad(torch.tensor(-15.0))
layup = Laminate(
    materials=[alu, cfrp, cfrp, cfrp, cfrp],
    thicknesses=[4.0, 0.5, 0.5, 0.5, 0.5],
    angles=[torch.tensor(0.0), p, m, p, m],
)

layup.plot()

Geometry, boundary conditions and loading

One octant of the vessel — a dome-capped half cylinder — is imported from a VTU file and assigned the laminate. The default orientation (global x, the vessel axis) projects onto each element as the meridional direction, so ply angles are measured from the meridian.

Three symmetry planes close the octant: the cut at x = 0 and the y = 0 and z = 0 planes. Each plane fixes the out-of-plane translation and the two in-plane rotations. Internal pressure is applied as outward nodal forces, with every element contributing p · area / 3 along its outward normal to each of its three nodes.

# Import mesh
path = get_data("copv.vtu")
copv = import_mesh(path, layup)
nodes = copv.nodes
elements = copv.elements


# Symmetry at x
x_symm = nodes[:, 0] > -0.1
copv.constraints[x_symm, 0] = True
copv.constraints[x_symm, 4] = True
copv.constraints[x_symm, 5] = True
# Symmetry at y
y_symm = nodes[:, 1] < 0.1
copv.constraints[y_symm, 1] = True
copv.constraints[y_symm, 3] = True
copv.constraints[y_symm, 5] = True
# Symmetry at z
z_symm = nodes[:, 2] < 0.1
copv.constraints[z_symm, 2] = True
copv.constraints[z_symm, 3] = True
copv.constraints[z_symm, 4] = True

# Internal pressure acting along the outward element normals
pressure = 10.0  # internal gauge pressure [MPa]
surface = torch.ones(copv.n_nod, dtype=torch.bool)
copv.forces[:, 0:3] = copv.integrate_surface_load(surface, pressure)

# Show
copv.plot(bcs=True)

Solve

A single linear solve, since all plies are linear elastic.

u, f, sigma, _, _ = copv.solve(aggregate_integration_points=False)

Results

The nodal radial expansion is recovered by projecting the displacement onto the radial direction.

radius = torch.sqrt(nodes[:, 1] ** 2 + nodes[:, 2] ** 2)
u_radial = (u[:, 1] * nodes[:, 1] + u[:, 2] * nodes[:, 2]) / radius.clamp(min=1e-6)

copv.plot(u=10.0 * u[:, :3], node_property={"Radial expansion [mm]": u_radial})

Per-ply stresses

In the element frame the in-plane axes are meridional (x', the winding reference) and hoop (y'). The aluminium liner carries most of the hoop load, while the helical CFRP plies carry the meridional/axial load.

n_layers = copv.section.n_layers
n_simpson = copv.section.n_simpson

meridional = sigma[:, :, 0, 0].reshape(n_layers, n_simpson, -1)
hoop = sigma[:, :, 1, 1].reshape(n_layers, n_simpson, -1)
labels = ["Al liner", "CFRP +15°", "CFRP -15°", "CFRP +15°", "CFRP -15°"]
for k, label in enumerate(labels):
    print(
        f"{label:10s}: Max. meridional : {meridional[k].max():6.2f} MPa,"
        f" Max. hoop : {hoop[k].max():6.2f} MPa"
    )

Al liner  : Max. meridional : 122.23 MPa, Max. hoop : 241.08 MPa
CFRP +15° : Max. meridional : 192.07 MPa, Max. hoop :  36.06 MPa
CFRP -15° : Max. meridional : 191.28 MPa, Max. hoop :  36.53 MPa
CFRP +15° : Max. meridional : 193.49 MPa, Max. hoop :  37.31 MPa
CFRP -15° : Max. meridional : 191.93 MPa, Max. hoop :  37.92 MPa