拓扑优化是一种结构优化方法，通过改变结构的杠杆布局来提高其性能。在这个例子中，我们将使用FEniCS求解一个简单的拓扑优化问题：对于一个给定的外载荷和固定区域，找到材料分布，使得某种性能指标（例如某种应力或位移的组合）最小化。

```python
from dolfin import *
from mshr import *
import matplotlib.pyplot as plt
import numpy as np

# Define mesh and geometry
L = 1.0  # length of the domain
H = 0.1  # height of the domain
Nx = 50  # number of elements in x direction
Ny = 10  # number of elements in y direction

mesh = RectangleMesh(Point(0, 0), Point(L, H), Nx, Ny)
plot(mesh)

# Define function spaces
V = FunctionSpace(mesh, "P", 1)  # scalar function space
V_vector = VectorFunctionSpace(mesh, "P", 1)  # vector function space

# Define boundary conditions
def left(x, on_boundary):
    return on_boundary and near(x[0], 0.0)

def right(x, on_boundary):
    return on_boundary and near(x[0], L)

u_left = Constant(0.0)
u_right = Expression("t", t=0.0, degree=1)
bc_left = DirichletBC(V_vector.sub(0), u_left, left)
bc_right = DirichletBC(V_vector.sub(0), u_right, right)
bcs = [bc_left, bc_right]

# Define optimization parameters
E = 1.0  # Young's modulus
nu = 0.3  # Poisson's ratio

# Define optimization problem
u = TrialFunction(V_vector)
v = TestFunction(V_vector)

rho = Function(V)
rho.assign(Constant(1.0))  # initial material distribution

mu = E / (2 * (1 + nu))
lmbda = E * nu / ((1 + nu) * (1 - 2 * nu))
def epsilon(u):
    return 0.5 * (grad(u) + grad(u).T)

def sigma(u):
    return lmbda * div(u) * Identity(2) + 2 * mu * epsilon(u)

F = inner(sigma(u), epsilon(v)) * dx

# Define time-stepping parameters
dt = 0.1  # time step
T = 1.0  # final time
t = 0.0  # current time

# Time-stepping loop
u_sol = Function(V_vector)
while t <= T:
    # Update boundary condition
    u_right.t = t

    # Solve optimization problem
    solve(lhs(F) == rhs(F), u_sol, bcs)

    # Update material distribution
    rho.vector()[:] *= 1.01
    for i in range(rho.vector().size()):
        if rho.vector()[i] > 1.0:
            rho.vector()[i] = 1.0

    # Plot solution
    plt.figure()
    plot(u_sol.sub(0), title="Displacement", mode="displacement")
    plt.colorbar()

    # Update time
    t += dt

# Plot final material distribution
plt.figure()
plot(rho, title="Material distribution")
plt.colorbar()
plt.show()
```

这是一个简单的示例，通过FEniCS求解一个拓扑优化问题。在这个例子中，我们定义了一个矩形域，并选择了一个初始材料分布。然后我们使用有限元方法求解弹性力学问题，找到一个使得性能指标最小化的材料分布。这个示例可以进一步扩展，例如使用更复杂的几何形状、考虑材料的非线性或添加约束条件等。
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