Coupled multiphysics

Overview

Every physics so far solved for a single field — a displacement, a temperature, a potential. Real devices often couple two fields, so that one drives the other: a voltage that strains a crystal, a temperature gradient that builds a voltage, a fluid pressure that deforms a porous solid. These are coupled problems, and the same solve(Physics(), mesh, material; …) marker pattern handles them; the only difference is that a coupled solve returns a NamedTuple of fields.

Piezoelectricity

A piezoelectric material — quartz, PZT, PVDF — couples mechanics to electrostatics. Apply a voltage and it strains (the actuator effect); strain it and it charges (the sensor effect). The constitutive law adds a coupling term in both directions,

\[\boldsymbol{\sigma} = C\varepsilon - e^{T}E , \qquad D = e\,\varepsilon + \epsilon E ,\]

so stress $\boldsymbol{\sigma}$ depends on the electric field $E$ as well as the strain $\varepsilon$, and the electric displacement $D$ depends on the strain. Rasmah solves the coupled displacement–potential system as one linear block.

Piezoelectricity

julia> Piezoelectricity() isa AbstractPhysics
true

A worked example: a voltage-driven actuator

Apply a $100\ \text{V}$ potential across a PZT block and watch it strain:

m = tetrahedralize_box(1.0, 1.0, 1.0, 4, 4, 4)
left  = nodes_in_box(m, [0.0, 0.0, 0.0], [0.0, 1.0, 1.0])
right = nodes_in_box(m, [1.0, 0.0, 0.0], [1.0, 1.0, 1.0])

r = solve(Piezoelectricity(), m, pzt;
          fixed_u = left,
          fixed_φ = (left => 0.0, right => 100.0))
maximum(abs.(r.u))

1.56896928398627e-8 m

The voltage strains the crystal through the inverse piezoelectric effect. The potential itself ramps from $0$ to $100\ \text{V}$ across the block:

maximum(abs.(r.φ))

100.0 m^2·kg/(s^3·A)

vtk(m; field=r.φ, fieldname="φ", edges=false)

Thermoelectricity

A temperature gradient across a conductor also drives an electric potential — the Seebeck effect, which is how a thermocouple works. Rasmah couples the heat equation to the charge equation through the Seebeck coefficient $S$: a temperature difference $\Delta T$ builds a potential $S\,\Delta T$.

Thermoelectricity

julia> Thermoelectricity() isa AbstractPhysics
true

A worked example: the Seebeck voltage

Hold a conductor at $0\ \text{K}$ on one face and $100\ \text{K}$ on the other, with a Seebeck coefficient of $10^{-4}\ \text{V/K}$:

te = Material(youngs_modulus=200e9, poisson_ratio=0.3, thermal_conductivity=50.0,
              electrical_conductivity=1e7, seebeck_coefficient=1e-4)

r = solve(Thermoelectricity(), m, te;
          fixed_T = (left => 0.0, right => 100.0),
          fixed_φ = (left => 0.0))
maximum(abs.(r.φ))

0.010000000000000018 m^2·kg/(s^3·A)

The Seebeck voltage comes out $S\,\Delta T = 10^{-4} \cdot 100 = 0.01\ \text{V}$ — exactly the coefficient times the temperature drop:

vtk(m; field=r.T, fieldname="T", edges=false)

Poroelasticity

A fluid-saturated porous solid — soil, rock, biological tissue — deforms when its pore fluid is pressured, and the deformation squeezes the pores back. Biot's theory couples the solid displacement $u$ to the pore pressure $p$: pressure contributes an isotropic stress $-\alpha\,p\,I$ ($\alpha$ the Biot coefficient), and the fluid flows by Darcy's law with permeability $k$.

PoroElasticity # COV_EXCL_LINE

julia> PoroElasticity() isa AbstractPhysics
true

A worked example: pressuring a porous block

Pressurize one face of a porous solid and watch it swell:

rock = Material(youngs_modulus=200e9, poisson_ratio=0.3)

r = solve(PoroElasticity(), m, rock, 1e-12, 0.6;
          fixed_u = left,
          fixed_p = (left => 0.0, right => 1.0e5))
maximum(abs.(r.u))

6.294877595159258e-8 m

The $10^5\ \text{Pa}$ pore-pressure difference strains the solid through the Biot coupling, giving a displacement of $\approx 6 \times 10^{-8}\ \text{m}$.

Next steps

Coupled solid physics shares the block-solver machinery with the fluid solvers — the Stokes and Navier–Stokes markers — which are the subject of the fluid-flow chapter. The linear algebra underneath every coupled block solve is covered in Sparse linear algebra.