Mixed-boundary linear elasticity

Gallery

Model a two-dimensional linear-elastic square with a fixed left edge and traction-free remaining edges, then compare its original and deformed mesh.

Before you begin

Follow Get started and keep its eqiora-source checkout and .venv. From the folder containing both, add plotting support and save the computed displacement:

uv add 'eqiora[matplotlib]'
uv run python eqiora-source/examples/python/mixed_boundary_elasticity.py --displacement-png displacement.png

Create a $1\,\mathrm{m}\times1\,\mathrm{m}$ rectangle. The left edge has zero displacement; the right, lower, and upper edges have zero traction. A $16\times16$ Cartesian mesh covers the domain.

graph = eqiora.geometry.GeometryGraph()
rectangle = graph.rectangle(x_bounds=(0.0, 1.0), y_bounds=(0.0, 1.0))
geometry = graph.build(
    rectangle,
    named_topology={
        "body": rectangle.region,
        "x_lower": rectangle.boundaries[0],
        "x_upper": rectangle.boundaries[1],
        "y_lower": rectangle.boundaries[2],
        "y_upper": rectangle.boundaries[3],
    },
)

Geometry and mesh in the example script

The .eqi component defines displacement, material parameters, load potential, equilibrium, the body support and its four boundary supports. With displacement $\boldsymbol{u}$ , shear modulus $\mu$ , Lamé coefficient $\lambda$ , and length $\ell$ , its stress and load are

\boldsymbol{\sigma}(\boldsymbol{u}) =2\mu\,\operatorname{sym}(\nabla\boldsymbol{u}) +\lambda\,\operatorname{div}(\boldsymbol{u})\boldsymbol{I}, \qquad q=\frac{2\mu x}{\ell}.

The balance relation is

-\nabla\!\cdot\boldsymbol{\sigma}(\boldsymbol{u})-\nabla q=0.

Eqiora component source

Compile the model, then select Q1 elements and linear-solve settings.

model = eqiora.compile(
    path=files(eqiora).joinpath("examples", "mixed-boundary-elasticity.eqi"),
    geometry=geometry,
    entry="MixedBoundaryElasticity2d",
    bindings={
        "body": geometry.selection("body"),
        **{
            side: (geometry.selection(side), geometry.selection("body"))
            for side in ("x_lower", "x_upper", "y_lower", "y_upper")
        },
        "mu": 3.0, "lambda": 0.0, "length_scale": 1.0,
    },
)
plan = eqiora.resolve(
    model,
    mesh=mesh,
    spatial=eqiora.fem.Q1(),
    solve=eqiora.solve.Linear(...),
)

Compile and resolve in the example script

The example function runs the plan and returns it with the result:

return plan, eqiora.run(plan)

Select the displacement field:

displacement = result.output(plan.fields[0])

The plotting code calls the same function, so it does not repeat the geometry, physics, mesh, or solver settings.

Reference and deformed meshes for the 2D mixed-boundary elasticity example. — Units: mDisplacement at scale 1, with the undeformed reference mesh shown for comparison.View the plotting path

The dashed mesh is the undeformed geometry; the orange mesh shows it displaced at scale 1. Changing the plot scale does not change the computed displacement.

Compare the fixed left edge with the displacement toward the loaded interior. The equations use small-strain linear elasticity; a visually large deformation does not make this a large-deformation material model.

Next steps

Derive strain, stress, and equilibrium in the structural mechanics lessons, then follow the same model in the Python guide.

Mixed-boundary linear elasticity

Before you begin

Stage 1 Problem and boundaries

Stage 2 Eqiora model definition

Stage 3 Choose numerical settings

Stage 4 Run and inspect displacement

Stage 5 Displacement presentation

Stage 6 Reading the deformation

Next steps