Adding shear heating operator

This is to mainly add the shear heating term for TALA which is needed for the TerraNeo app hyteg#243 (closed), which is very similar to the Stokes form but still cannot be readily used in the energy equation as the function spaces are different.

I did a simple integration test (math) for the IdentityMap which looks fine. Have also added the literature references in the comments of the form for cross checking.

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assigned to @pponkumar

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• d891b84e - Tabulating the affine factor in blending part

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As both the contractions

$\tau:\dot{\epsilon}$

and

$\tau:\nabla u$

yield the same result (only applies for 3D), I guess we could stick with the most minimal version,

$$\tau:\nabla u$$

import sympy as sp

x, y, z = sp.symbols('x y z')

u = sp.Function('u')(x, y, z)
v = sp.Function('v')(x, y, z)
w = sp.Function('w')(x, y, z)

uvw = sp.Matrix(
    [
        [u], [v], [w]
    ]
)

grad_u = sp.diff(uvw, x).row_join(sp.diff(uvw, y)).row_join(sp.diff(uvw, z))
sym_grad_u = (grad_u + grad_u.T) / 2
div_u = grad_u.trace() * sp.eye(3)

eps_dot = sym_grad_u - sp.Rational(1, 3) * div_u
tau = 2*eps_dot

tau_eps_dot = sp.simplify(sum(sp.matrix_multiply_elementwise(tau, eps_dot)))
tau_grad_u = sp.simplify(sum(sp.matrix_multiply_elementwise(tau, grad_u)))

sp.simplify(tau_eps_dot - tau_grad_u)

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• 8e27a3f8 - Update shear heating form

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• a79c274a - Update shear heating form with simple contraction

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merged

mentioned in commit ea1f2072