Aggregation of the equations on the boundary

[luAndre00]

I am trying to solve an optimal control problem constrained with the stokes equation. After writing the Lagrange system of equation I get many boundary conditions:

` def dirichlet1ad(input, output): return output.extract(['zx'])

def dirichlet2_ad(input_, output_):
    return output_.extract(['zy'])

def neumann1_ad(input_, output_):
    return -output_.extract(['r']) + 0.1 * grad(output_, input_, components=['zx'], d=['x'])

def neumann2_ad(input_, output_):
    return output_.extract(['zy'])

def dirichlet1(input_, output_):
    return output_.extract(['vx']) - input_.extract(['y'])

def dirichlet2(input_, output_):
    return output_.extract(['vy'])

def neumann1(input_, output_):
    return -output_.extract(['p']) + 0.1 * grad(output_, input_, components=['vx'], d=['x'])

def neumann2(input_, output_):
    return output_.extract(['vy'])`

With

conditions = {
    'gamma_above': Condition(location=CartesianDomain({'x': xrange, 'y': 2, 'mu': murange}),
                             equation=SystemEquation([dirichlet1, dirichlet2, dirichlet1_ad, dirichlet2_ad], reduction = "none")),
    # Dirichlet
    'gamma_left': Condition(location=CartesianDomain({'x': 0, 'y': yrange, 'mu': murange}),
                            equation=SystemEquation([dirichlet1, dirichlet2, dirichlet1_ad, dirichlet2_ad], reduction = "none")),
    # Dirichlet
    'gamma_below': Condition(location=CartesianDomain({'x': xrange, 'y': 0, 'mu': murange}),
                             equation=SystemEquation([dirichlet1, dirichlet2, dirichlet1_ad, dirichlet2_ad], reduction = "none")),
    # Dirichlet
    'gamma_right': Condition(location=CartesianDomain({'x': 1, 'y': yrange, 'mu': murange}),
                             equation=SystemEquation([neumann1, neumann2, neumann1_ad, neumann2_ad], reduction = "none")),  # Neumann
    'D': Condition(location=CartesianDomain({'x': xrange, 'y': yrange, 'mu': murange}),
                   equation=SystemEquation([momentum_x, momentum_y, continuity,
                                            momentum_ad_x, momentum_ad_y, continuity_ad], reduction = "none"))
}

I was wondering if it is correct to define the boundaries as a System of equation with reduction = None, because I have problems on the boundary conditions and I do not understand why.

Another doubt I had concerns the definition of the momentum equation: I wrote them considering one components at a time, so I have momentum_x, momentum_y and not only one equation of the momentum with length 2. The same is true for the momentum of the adjoint variables. My question is: is it different if I define only one momentum with length 2? Should I get the same results?

Thank you for you help!

[dario-coscia]

Hi! Can you provide in latex form the problem you are trying to solve? What time of problem you are getting with the boundaries (code error or not converging)?

[luAndre00]

$$ \begin{cases} \displaystyle-0.1\Delta z(x,\mu) + \nabla r(x,\mu) = [x_2 - v_1(x,\mu),0]^T \ \displaystyle\nabla\cdot z(x,\mu) = 0 \ \displaystyle\alpha u(x,\mu) = z(x,\mu) \ \displaystyle-0.1\Delta v(x,\mu) + \nabla p(x,\mu) = f(x,\mu) + u(x,\mu) \ \displaystyle\nabla\cdot v(x,\mu) = 0\ \displaystyle z_1(x,\mu) = z_2(x,\mu) = 0 \ \displaystyle v_1(x,\mu) = x_2 \text{and} v_2(x,\mu) = 0 \ \displaystyle-r(x,\mu) + 0.1\frac{\partial z(x,\mu)}{\partial n} = 0 \text{and} z_2(x,\mu) = 0 \ \displaystyle-p(x,\mu) + 0.1\frac{\partial v(x,\mu)}{\partial n} = 0 \text{and} v_2(x,\mu) = 0
\end{cases} $$

sorry for some problem of visualization, I don't know how to fix them. Actually at the moment I got a good solution of this problem but I had to use eq 3 to change all z into u and solve the problem not considering z. The reconstruction of z was made after redefining the forward of the net imposing the third equation strongly. The problem I had before was that the boundary conditions were not converging. I don't know if there was something about PINA that I was not considering/out of my knowledge, or if these results (solving with u plus z = bad, solving only using u = good) are totally reasonable.

Thank you for your help!