python - 如何在 OpenMDAO 中求解方程组？

Question

我尝试使用 OpenMDAO 求解简单的隐式方程。方程如下所示，
x*z + z - 4 = 0
y = x + 2*z

当 x = 0.5 时，解为 z = 2.666667，y = 5.833333。

对于这种情况，我使用了如下所示的代码，

from __future__ import print_function

from openmdao.api import Component, Group, Problem, Newton, ScipyGMRES

class SimpleEquationSystem(Component):
    """Solve the Equation 
            x*z + z - 4 = 0
            y = x + 2*z

       Solution: z = 2.666667, y = 5.833333 for x = 0.5
    """

    def __init__(self):
        super(SimpleEquationSystem, self).__init__()


        self.add_param('x', 0.5)
        self.add_state('y', 0.0)
        self.add_state('z', 0.0)
        self.iter=0

    def solve_nonlinear(self, params, unknowns, resids):
        """This component does no calculation on its own. It mainly holds the
        initial value of the state. An OpenMDAO solver outside of this
        component varies it to drive the residual to zero."""
        pass

    def apply_nonlinear(self, params, unknowns, resids):
        """ Report the residual """
        self.iter+=1
        x=params['x']
        y = unknowns['y']
        z = unknowns['z']


        resids['y'] = x*z + z - 4
        resids['z'] = x + 2*z - y

        print('y_%d' % self.iter,'=%f' %resids['y'], 'z_%d' % self.iter, '=%f' %resids['z'])
        print('x' ,'=%f' %x, 'y', '=%f' %y, 'z', '=%f' %z)

top = Problem()
root = top.root = Group()
root.add('comp', SimpleEquationSystem())


# Tell these components to finite difference
root.comp.deriv_options['type'] = 'fd'
root.comp.deriv_options['form'] = 'central'
root.comp.deriv_options['step_size'] = 1.0e-4
root.nl_solver = Newton()
root.ln_solver = ScipyGMRES()

top.setup()
top.print_all_convergence(level=1, depth=2)
top.run()
print('Solution x=%.2f, y=%.2f, z=%.2f' % (top['comp.x'], top['comp.y'], top['comp.z']))

我根据Solving an Implicit Relationship with a Newton Solver写了一段代码方法。为了运行这段代码，我得到了这样的解决方案，

##############################################
Setup: Checking root problem for potential issues...

No recorders have been specified, so no data will be saved.

The following parameters have no associated unknowns:
comp.x

The following components have no connections:
comp

Setup: Check of root problem complete.
##############################################

y_1 =-4.000000 z_1 =0.500000
x =0.500000 y =0.000000 z =0.000000
y_2 =-4.000000 z_2 =0.499900
x =0.500000 y =0.000100 z =0.000000
y_3 =-4.000000 z_3 =0.500100
x =0.500000 y =-0.000100 z =0.000000
y_4 =-3.999850 z_4 =0.500200
x =0.500000 y =0.000000 z =0.000100
y_5 =-4.000150 z_5 =0.499800
x =0.500000 y =0.000000 z =-0.000100
   [root] LN: GMRES   1 | 0 0
   [root] LN: GMRES   1 | Converged in 1 iterations
y_6 =-inf z_6 =-inf
x =0.500000 y =inf z =-inf
y_7 =-inf z_7 =-inf
x =0.500000 y =inf z =-inf
y_8 =-inf z_8 =-inf
x =0.500000 y =inf z =-inf
y_9 =-inf z_9 =-inf
x =0.500000 y =inf z =-inf
y_10 =-inf z_10 =-inf
x =0.500000 y =inf z =-inf
   [root] LN: GMRES   1000 | nan nan
   [root] LN: GMRES   1000 | Converged in 1000 iterations
y_11 =nan z_11 =nan
x =0.500000 y =nan z =nan
[root] NL: NEWTON   2 | nan nan (nan)
[root] NL: NEWTON   2 | FAILED to converge after 2 iterations
Solution x=0.50, y=nan, z=nan
C:\ProgramData\Anaconda3\Anaconda3\lib\site-packages\openmdao\core\system.py:750: RuntimeWarning: invalid value encountered in subtract
  resultvec.vec[:] -= cache2

您知道如何通过各自的迭代来解决这个问题吗？并告诉我，如何为这种情况构造solve_nonlinear 和apply_nonlinear。

Answer 1

当您为自变量添加 Indepvarcomp 时，问题似乎就解决了:

root.add('p1', IndepVarComp('x', 0.5))
root.add('comp', SimpleEquationSystem())
root.connect('p1.x', 'comp.x')

然后它很快收敛。

[root] NL: NEWTON   1 | 2.73692411e-11 6.7894731e-12 (6.41396046825)
[root] NL: NEWTON   1 | Converged in 1 iterations

我同意你不必添加它才能工作，这可能是一个错误。我已经证实这个问题已经在我们最新的 OpenMDAO 开发版本中得到修复。