human
3. Affine decomposition
For this problem the affine decomposition is straightforward. Indeed, owing to the definitions of $\Lambda_1(\boldsymbol{\mu})$ and $\Lambda_2(\boldsymbol{\mu})$, we have:
$$
a(\boldsymbol{u}, \boldsymbol{v}.
\boldsymbol{\mu}) = \sum_{p=0}^7 \underbrace{\mu_{\color{red} p}}{\Theta^{a}{\color{red} p}(\boldsymbol{\mu})} \underbrace{\int_{\Omega_{\color{red}{p + 1}}}
\left{\lambda_1(1)\;\text{tr}\left[\nabla_{S}\boldsymbol{u}\right]\;\text{tr}\left[\nabla_{S}\boldsymbol{v}\right] + 2\;\lambda_2(1)\;\nabla_{S}\boldsymbol{u} : \nabla_{S}\boldsymbol{v} \right}  d\boldsymbol{x}}{a{\color{red} p}(\boldsymbol{u}, \boldsymbol{v})} +\
\underbrace{1}{\Theta^{a}{\color{red} 8}(\boldsymbol{\mu})} \underbrace{\int_{\Omega_{\color{red} 9}}
\left{\lambda_1(1)\;\text{tr}\left[\nabla_{S}\boldsymbol{u}\right]\;\text{tr}\left[\nabla_{S}\boldsymbol{v}\right] + 2\;\lambda_2(1)\;\nabla_{S}\boldsymbol{u} : \nabla_{S}\boldsymbol{v} \right}  d\boldsymbol{x}}{a{\color{red} 8}(\boldsymbol{u}, \boldsymbol{v})}\
$$
$$
f(\boldsymbol{v}.
\boldsymbol{\mu}) = 
\sum_{p=8}^{10} \underbrace{\mu_{\color{red} p}}{\Theta^{f}{\color{red}{p-8}}(\boldsymbol{\mu})} \underbrace{\int_{\Gamma_{\color{red}{p-6}}} \boldsymbol{v} \cdot \mathbf{n}}{f{\color{red}{p-8}}(\boldsymbol{v})}.
$$
We will implement the numerical discretization of the problem in the class
class ElasticBlock(EllipticCoerciveProblem):
by specifying the coefficients $\Theta^{a}(\boldsymbol{\mu})$ and $\Theta^{f}_(\boldsymbol{\mu})$ in the method
def compute_theta(self, term):
and the bilinear forms $a(\boldsymbol{u}, \boldsymbol{v})$ and linear forms $f_(\boldsymbol{v})$ in
def assemble_operator(self, term):"
Please write code following the instructions in jupyter notebook style.
assistant
#Here is code to 3.

class ElasticBlock(EllipticCoerciveProblem):

    # Default initialization of members
    def __init__(self, V, **kwargs):
        # Call the standard initialization
        EllipticCoerciveProblem.__init__(self, V, **kwargs)
        # ... and also store FEniCS data structures for assembly
        assert "subdomains" in kwargs
        assert "boundaries" in kwargs
        self.subdomains, self.boundaries = kwargs["subdomains"], kwargs["boundaries"]
        self.u = TrialFunction(V)
        self.v = TestFunction(V)
        self.dx = Measure("dx")(subdomain_data=self.subdomains)
        self.ds = Measure("ds")(subdomain_data=self.boundaries)
        # ...
        self.f = Constant((1.0, 0.0))
        self.E = 1.0
        self.nu = 0.3
        self.lambda_1 = self.E * self.nu / ((1.0 + self.nu) * (1.0 - 2.0 * self.nu))
        self.lambda_2 = self.E / (2.0 * (1.0 + self.nu))

    # Return custom problem name
    def name(self):
        return "ElasticBlock"

    # Return theta multiplicative terms of the affine expansion of the problem.
    def compute_theta(self, term):
        mu = self.mu
        if term == "a":
            theta_a0 = mu[0]
            theta_a1 = mu[1]
            theta_a2 = mu[2]
            theta_a3 = mu[3]
            theta_a4 = mu[4]
            theta_a5 = mu[5]
            theta_a6 = mu[6]
            theta_a7 = mu[7]
            theta_a8 = 1.
            return (theta_a0, theta_a1, theta_a2, theta_a3, theta_a4, theta_a5, theta_a6, theta_a7, theta_a8)
        elif term == "f":
            theta_f0 = mu[8]
            theta_f1 = mu[9]
            theta_f2 = mu[10]
            return (theta_f0, theta_f1, theta_f2)
        else:
            raise ValueError("Invalid term for compute_theta().")

    # Return forms resulting from the discretization of the affine expansion of the problem operators.
    def assemble_operator(self, term):
        v = self.v
        dx = self.dx
        if term == "a":
            u = self.u
            a0 = self.elasticity(u, v) * dx(1)
            a1 =...