Sizing procedure and optimization with OpenMDAO (Student version)

Sizing procedure and optimization with OpenMDAO (Student version)#

Written by Marc Budinger (INSA Toulouse) and Scott Delbecq (ISAE-SUPAERO), Toulouse, France

The objective of this notebook is to learn how to implement a sizing code and use a simple numerical optimization to find the optimal design of the system. The system studied is the TVC EMA of the VEGA launcher.

import numpy as np
import scipy.optimize
from scipy import log10
from math import pi

Objectives and specifications#

The objective is to select the reduction ratio of a gear reducer in order to minimize the mass of the motor.

The application have to ensure at nozzle level :

a max torque \(T_{load}\) of \(48 kNm\) and a max acceleration of \(\dot{\omega}_{max}=811 °/s²\)
a max speed \(\omega_{max}\) of 9.24 °/s
a max magnitude \(\alpha_{max}\) of 5.7 °

We will give here an example based on a linear actuator with a preselected roller screw (pitch of 10 mm/rev). We assume here, for simplification, the efficiency equal to 70%.

EMA components:

EMA

We first define the specifications and assumptions for the sizing:

# Specifications
angular_magnitude_max = 5.7 * pi / 180  # [rad]
max_dyn_torque = 48e3  # [N.m]
max_speed_rot = 9.24 * pi / 180  # [rad/s]
max_acc_rot = 811 * pi / 180  # [rad/s²]

# Assumptions
pitch = 10e-3 / 2 / pi  # [m/rad]
nu_screw = 0.7  # [-]

# Security coefficient for mechanical components
k_sec = 2

We then define the main characteristics for the components for the scaling laws:

# Motor
T_mot_guess_max_ref = 13.4  # [N.m]
W_mot_max_ref = 754  # [rad/s]
J_mot_ref = 2.9e-4 / 2  # [kg.m²]
M_mot_ref = 3.8  # [kg]

# Rod end
F_rod_max_ref = 183e3  # [N]
M_rod_ref = 1.55  # [kg]
L_rod_ref = 0.061  # [m]

# Screw
M_nut_ref = 2.1  # [kg]
Ml_screw_ref = 9.4  # [kg/m]
D_nut_ref = 0.08  # [m]
L_nut_ref = 0.12 * 0.08 / 0.09  # [m]
F_screw_max_ref = 135e3  # [N]

# Bearing
M_bearing_ref = 5.05  # [kg]
L_bearing_ref = 0.072  # [m]
F_bearing_max_ref = 475e3  # [N]

Sizing code#

The sizing code is defined here in a function which can give an evaluation of the objective and of the constraints function of design variables.

The design variables of this sizing code are :

the reduction ratio of the reducer
an oversizing coefficient for the selection of the motor used to tacke an algebraic loop
the positions (\(d_1\) and \(d_2\)) of the actuator anchorages

New design variables Mechanical Loads

The objective is the global mass of the actuator.

The constraints which should be positives are here:

the speed margin, ie. the motor doesn’t exceed its maximum speed
the torque margin, ie. the motor doesn’t exceed its maximum torque
the length margin, ie. the global length of the actuator doesn’t exceed the distance between anchorage points

import openmdao.api as om

class LeverArm(om.Group):
    """
    Actuator model.
    """

    def setup(self):

        self.add_subsystem(
            "lever_arm",
            om.ExecComp(
                "lever_arm = ((-(-0.9744 * d1 - 1.372) * (0.2248 * d1 - 0.3757) * ((0.2248 * d1 - 0.3757) ** 2 + (-0.9744 * d1 + d2 - 1.172) ** 2) ** (-0.5) + (0.2248 * d1 + 0.9823) * ((0.2248 * d1 - 0.3757) ** 2 + (-0.9744 * d1 + d2 - 1.172) ** 2) ** (-0.5)* (-0.9744 * d1 + d2 - 1.172))** 2) ** 0.5",
                d1=0.0,
                d2=0.0,
            ),
            promotes=["*"],
        )

class Actuator(om.Group):
    """
    Actuator model.
    """

    def setup(self):

        self.add_subsystem(
            "actuator_length",
            om.ExecComp(
                "actuator_length = ((0.2248 * d1 - 0.3757) ** 2 + (-0.9744 * d1 + d2 - 1.172) ** 2) ** 0.5",
                d1=0.0,
                d2=0.0,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "stroke",
            om.ExecComp(
                "stroke = angular_magnitude_max * 2 * lever_arm",
                angular_magnitude_max=angular_magnitude_max,
            ),
            promotes=["*"],
        )

class LoadSpeed(om.Group):
    """
    Load and speed model.
    """

    def setup(self):

        self.add_subsystem(
            "max_speed",
            om.ExecComp("max_speed = max_speed_rot * lever_arm", max_speed_rot=max_speed_rot),
            promotes=["*"],
        )

        self.add_subsystem(
            "max_load",
            om.ExecComp("max_load = max_dyn_torque / lever_arm", max_dyn_torque=max_dyn_torque),
            promotes=["*"],
        )

class Forces(om.Group):
    """
    Stall and mechanical forces model.
    """

    def setup(self):

        self.add_subsystem(
            "max_stall_force",
            om.ExecComp("max_stall_force = T_mot_guess / pitch * reduction_ratio", pitch=pitch),
            promotes=["*"],
        )

        self.add_subsystem(
            "max_mech_force",
            om.ExecComp("max_mech_force = k_sec * max_stall_force", k_sec=k_sec),
            promotes=["*"],
        )

class Motor(om.Group):
    """
    Motor model.
    """

    def setup(self):

        self.add_subsystem(
            "M_mot",
            om.ExecComp(
                "M_mot = M_mot_ref * (T_mot_guess / T_mot_guess_max_ref) ** (3 / 3.5)",
                M_mot_ref=M_mot_ref,
                T_mot_guess_max_ref=T_mot_guess_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "J_mot",
            om.ExecComp(
                "J_mot = J_mot_ref * (T_mot_guess / T_mot_guess_max_ref) ** (5 / 3.5)",
                J_mot_ref=J_mot_ref,
                T_mot_guess_max_ref=T_mot_guess_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "W_mot",
            om.ExecComp(
                "W_mot = W_mot_max_ref * (T_mot_guess / T_mot_guess_max_ref) ** (-1 / 3.5)",
                W_mot_max_ref=W_mot_max_ref,
                T_mot_guess_max_ref=T_mot_guess_max_ref,
            ),
            promotes=["*"],
        )

class RodEnd(om.Group):
    """
    Rod end model.
    """

    def setup(self):

        self.add_subsystem(
            "M_rod",
            om.ExecComp(
                "M_rod = M_rod_ref * (max_mech_force / F_rod_max_ref) ** (3 / 2)",
                M_rod_ref=M_rod_ref,
                F_rod_max_ref=F_rod_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "L_rod",
            om.ExecComp(
                "L_rod = L_rod_ref * (max_mech_force / F_rod_max_ref) ** (1 / 2)",
                L_rod_ref=L_rod_ref,
                F_rod_max_ref=F_rod_max_ref,
            ),
            promotes=["*"],
        )

class Nut(om.Group):
    """
    Nut model.
    """

    def setup(self):

        self.add_subsystem(
            "M_bearing",
            om.ExecComp(
                "M_bearing = M_bearing_ref * (max_mech_force / F_bearing_max_ref) ** (3 / 2)",
                M_bearing_ref=M_bearing_ref,
                F_bearing_max_ref=F_bearing_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "M_screw",
            om.ExecComp(
                "M_screw = Ml_screw_ref * (max_mech_force / F_screw_max_ref) ** (2 / 2) * actuator_length / 2",
                Ml_screw_ref=Ml_screw_ref,
                F_screw_max_ref=F_screw_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "D_nut",
            om.ExecComp(
                "D_nut = D_nut_ref * (max_mech_force / F_screw_max_ref) ** (1 / 2)",
                D_nut_ref=D_nut_ref,
                F_screw_max_ref=F_screw_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "L_nut",
            om.ExecComp(
                "L_nut = L_nut_ref * (max_mech_force / F_screw_max_ref) ** (1 / 2)",
                L_nut_ref=L_nut_ref,
                F_screw_max_ref=F_screw_max_ref,
            ),
            promotes=["*"],
        )

class Bearing(om.Group):
    """
    Bearing model.
    """

    def setup(self):

        self.add_subsystem(
            "M_nut",
            om.ExecComp(
                "M_nut = M_nut_ref * (max_mech_force / F_screw_max_ref) ** (3 / 2)",
                M_nut_ref=M_nut_ref,
                F_screw_max_ref=F_screw_max_ref,
            ),
            promotes=["*"],
        )

        self.add_subsystem(
            "L_bearing",
            om.ExecComp(
                "L_bearing = L_bearing_ref * (max_mech_force / F_bearing_max_ref) ** (1 / 2)",
                L_bearing_ref=L_bearing_ref,
                F_bearing_max_ref=F_bearing_max_ref,
            ),
            promotes=["*"],
        )

class MotorTorqueReal(om.Group):
    """
    Real motor torque model.
    """

    def setup(self):

        self.add_subsystem(
            "T_mot_real",
            om.ExecComp(
                "T_mot_real = max_load * pitch / reduction_ratio / nu_screw + J_mot * max_acc_rot * lever_arm * reduction_ratio / pitch",
                pitch=pitch,
                nu_screw=nu_screw,
            ),
            promotes=["*"],
        )

Optimization with SLSQP algorithm#

We will now use the opmization algorithms of the Scipy package to solve and optimize the configuration. We will first use the SLSQP algorithm without explicit expression of the gradient (Jacobian).

The first step is to give an initial value of optimisation variables:

# Optimization variables
# Reduction ratio
reduction_ratio_init = 1  # [-]
reduction_ratio_min = 0.1  # [-]
reduction_ratio_max = 10  # [-]

# Oversizing coefficient for multidisciplinary coupling
k_oversizing_init = 1  # [-]
k_oversizing_min = 0.2  # [-]
k_oversizing_max = 5  # [-]

# Anchorage positions
d1_init = 0  # [m]
d1_min = -80 / 100  # [m]
d1_max = 80 / 100  # [m]

d2_init = 0  # [m]
d2_min = -20 / 100  # [m]
d2_max = 20 / 100  # [m]

# Initial values vector for design variables
parameters = np.array((reduction_ratio_init, k_oversizing_init, d1_init, d2_init))

MDF formulation#

MDF

class MotorTorque(om.Group):
    """
    Real motor torque model.
    """

    def setup(self):

        self.add_subsystem(
            "T_mot_guess",
            om.ExecComp(
                "T_mot_guess = max_load * pitch / reduction_ratio / nu_screw + J_mot * max_acc_rot * lever_arm * reduction_ratio / pitch",
                pitch=pitch,
                nu_screw=nu_screw,
            ),
            promotes=["*"],
        )

class ObjectiveConstraints(om.Group):
    """
    Objective and constraints model.
    """

    def setup(self):

        self.add_subsystem(
            "objective",
            om.ExecComp("objective = M_mot + M_bearing + 2 * M_rod + M_screw + M_nut"),
            promotes=["*"],
        )

        self.add_subsystem(
            "C1",
            om.ExecComp("C1 = W_mot - reduction_ratio * max_speed / pitch", pitch=pitch),
            promotes=["*"],
        )

        self.add_subsystem(
            "C3",
            om.ExecComp("C3 = actuator_length - stroke - L_nut - L_bearing - 2 * L_rod"),
            promotes=["*"],
        )

class SystemMDF(om.Group):
    """
    Overall system model with MDF formulation
    """

    def setup(self):

        self.add_subsystem("lever_arm", LeverArm(), promotes=["*"])
        self.add_subsystem("actuator", Actuator(), promotes=["*"])
        self.add_subsystem("load_speed", LoadSpeed(), promotes=["*"])

        # We have to do something here regarding the cycle
        cycle = self.add_subsystem("cycle", om.Group(), promotes=["*"])
        cycle.add_subsystem("motor_torque", MotorTorque(), promotes=["*"])
        cycle.add_subsystem("forces", Forces(), promotes=["*"])
        cycle.add_subsystem("motor", Motor(), promotes=["*"])
        # We had a solver
        cycle.nonlinear_solver = om.NonlinearBlockGS(maxiter=100)

        self.add_subsystem("rod_end", RodEnd(), promotes=["*"])
        self.add_subsystem("nut", Nut(), promotes=["*"])
        self.add_subsystem("bearing", Bearing(), promotes=["*"])
        self.add_subsystem("motor_torque_real", MotorTorqueReal(), promotes=["*"])
        self.add_subsystem("objective_constraints", ObjectiveConstraints(), promotes=["*"])

import openmdao.api as om
import time

prob = om.Problem()
prob.model = SystemMDF()

prob.driver = om.ScipyOptimizeDriver()
prob.driver.options["optimizer"] = "SLSQP"
# prob.driver.options['maxiter'] = 100
prob.driver.options["tol"] = 1e-8

prob.model.add_design_var("reduction_ratio", lower=reduction_ratio_min, upper=reduction_ratio_max)
prob.model.add_design_var("d1", lower=d1_min, upper=d1_max)
prob.model.add_design_var("d2", lower=d2_min, upper=d2_max)

prob.model.add_objective("objective")
prob.model.add_constraint("C1", lower=0)
prob.model.add_constraint("C3", lower=0)

# Ask OpenMDAO to finite-difference across the model to compute the gradients for the optimizer
prob.model.approx_totals()

prob.setup()
prob.set_solver_print(level=1)

# Initialization of design variables
prob.set_val("reduction_ratio", reduction_ratio_init)
prob.set_val("d1", d1_init)
prob.set_val("d2", d2_init)

start = time.time()
prob.run_driver()
end = time.time()

=====
cycle
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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 4 iterations

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NL: NLBGS Converged in 7 iterations

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NL: NLBGS Converged in 7 iterations

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NL: NLBGS Converged in 7 iterations

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NL: NLBGS Converged in 12 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 6 iterations

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NL: NLBGS Converged in 12 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 11 iterations

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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 11 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 8 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 10 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 10 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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cycle
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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 10 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 8 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations

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NL: NLBGS Converged in 9 iterations
Optimization terminated successfully    (Exit mode 0)
            Current function value: 10.915557692240649
            Iterations: 13
            Function evaluations: 15
            Gradient evaluations: 13
Optimization Complete
-----------------------------------

/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/scipy/optimize/_optimize.py:284: RuntimeWarning: Values in x were outside bounds during a minimize step, clipping to bounds
  warnings.warn("Values in x were outside bounds during a "

print("Objective:")
print("     Total mass = %.2f kg" % (prob.get_val("objective")))
print("Design variables:")
print("     reduction_ratio =  %.2f" % prob.get_val("reduction_ratio"))
print("     d_1 =  %.2f m" % prob.get_val("d1"))
print("     d_2 =  %.2f m" % prob.get_val("d2"))
print("Performances:")
print("     Stroke = %.2f m" % prob.get_val("stroke"))
print("     Max load = %.0f N" % prob.get_val("max_load"))
print("     Stall load = %.0f N" % prob.get_val("max_stall_force"))
print("Components characteristics:")
print("     Lever arm = %.2f m" % prob.get_val("lever_arm"))
print("     Actuator length = %.2f m" % prob.get_val("actuator_length"))
print("     Motor mass = %.2f kg" % prob.get_val("M_mot"))
print("     Max Tem = %.2f N.m" % prob.get_val("T_mot_real"))
print("     Rod-end mass = %.2f kg" % (2 * prob.get_val("M_rod")))
print("     Rod-end length = %.2f m" % prob.get_val("L_rod"))
print("     Screw mass = %.2f kg" % prob.get_val("M_screw"))
print("     Nut mass = %.2f kg" % (2 * prob.get_val("M_nut")))
print("     Nut length = %.2f m" % prob.get_val("L_nut"))
print("     Bearing length = %.2f m" % prob.get_val("L_bearing"))
print("Constraints (should be >= 0):")
print("     Speed margin: W_mot-reduction_ratio*max_speed/pitch= %.3f" % prob.get_val("C1"))
print(
    "     Length margin: actuator_length-stroke-L_nut-L_bearing-2*L_rod =  %.3f"
    % prob.get_val("C3")
)
print("Calculation time:\n", end - start, "s")

Objective:
     Total mass = 10.92 kg
Design variables:
     reduction_ratio =  5.23
     d_1 =  -0.33 m
     d_2 =  0.20 m
Performances:
     Stroke = 0.27 m
     Max load = 35719 N
     Stall load = 53830 N
Components characteristics:
     Lever arm = 1.34 m
     Actuator length = 0.79 m
     Motor mass = 4.52 kg
     Max Tem = 16.39 N.m
     Rod-end mass = 1.40 kg
     Rod-end length = 0.05 m
     Screw mass = 2.96 kg
     Nut mass = 2.99 kg
     Nut length = 0.10 m
     Bearing length = 0.03 m
Constraints (should be >= 0):
     Speed margin: W_mot-reduction_ratio*max_speed/pitch= -0.000
     Length margin: actuator_length-stroke-L_nut-L_bearing-2*L_rod =  0.299
Calculation time:
 0.4518289566040039 s

om.n2(prob)

NVH formulation#

You now have to implement the NVH formulation based on the previous lab.

MDF

class MotorTorqueGuess(om.Group):
    """
    Guess of motor torque model.
    """

    def setup(self):
        

  Cell In[21], line 7
    
    ^
IndentationError: expected an indented block

class ObjectiveConstraints(om.Group):
    """
    Objective and constraints model.
    """

    def setup(self):

        self.add_subsystem(
            "objective",
            om.ExecComp("objective = M_mot + M_bearing + 2 * M_rod + M_screw + M_nut"),
            promotes=["*"],
        )

        self.add_subsystem(
            "C1",
            om.ExecComp("C1 = W_mot - reduction_ratio * max_speed / pitch", pitch=pitch),
            promotes=["*"],
        )

        self.add_subsystem(
            "C3",
            om.ExecComp("C3 = actuator_length - stroke - L_nut - L_bearing - 2 * L_rod"),
            promotes=["*"],
        )

class SystemNVH(om.Group):
    """
    Overall system model with NVH formulation
    """

    def setup(self):

  Cell In[23], line 6
    def setup(self):
                    ^
IndentationError: expected an indented block

import openmdao.api as om
import time

prob = om.Problem()
prob.model = SystemNVH()

prob.driver = om.ScipyOptimizeDriver()
prob.driver.options["optimizer"] = "SLSQP"
# prob.driver.options['maxiter'] = 100
prob.driver.options["tol"] = 1e-8

prob.model.add_design_var("reduction_ratio", lower=reduction_ratio_min, upper=reduction_ratio_max)
prob.model.add_design_var()
prob.model.add_design_var("d1", lower=d1_min, upper=d1_max)
prob.model.add_design_var("d2", lower=d2_min, upper=d2_max)

prob.model.add_objective("objective")
prob.model.add_constraint("C1", lower=0)
prob.model.add_constraint()
prob.model.add_constraint("C3", lower=0)

# Ask OpenMDAO to finite-difference across the model to compute the gradients for the optimizer
prob.model.approx_totals()

prob.setup()
prob.set_solver_print(level=0)

# Initialization of design variables
prob.set_val("reduction_ratio", reduction_ratio_init)
prob.set_val("k_oversizing", k_oversizing_init)
prob.set_val("d1", d1_init)
prob.set_val("d2", d2_init)

start = time.time()
prob.run_driver()
end = time.time()

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[24], line 5
      2 import time
      4 prob = om.Problem()
----> 5 prob.model = SystemNVH()
      7 prob.driver = om.ScipyOptimizeDriver()
      8 prob.driver.options["optimizer"] = "SLSQP"

NameError: name 'SystemNVH' is not defined

We can print of the characterisitcs of the problem before optimization with the intitial vector of optimization variables:

print("Objective:")
print("     Total mass = %.2f kg" % (prob.get_val("objective")))
print("Design variables:")
print("     reduction_ratio =  %.2f" % prob.get_val("reduction_ratio"))
print("     k_oversizing =  %.2f" % prob.get_val("k_oversizing"))
print("     d_1 =  %.2f m" % prob.get_val("d1"))
print("     d_2 =  %.2f m" % prob.get_val("d2"))
print("Performances:")
print("     Stroke = %.2f m" % prob.get_val("stroke"))
print("     Max load = %.0f N" % prob.get_val("max_load"))
print("     Stall load = %.0f N" % prob.get_val("max_stall_force"))
print("Components characteristics:")
print("     Lever arm = %.2f m" % prob.get_val("lever_arm"))
print("     Actuator length = %.2f m" % prob.get_val("actuator_length"))
print("     Motor mass = %.2f kg" % prob.get_val("M_mot"))
print("     Max Tem = %.2f N.m" % prob.get_val("T_mot_real"))
print("     Rod-end mass = %.2f kg" % (2 * prob.get_val("M_rod")))
print("     Rod-end length = %.2f m" % prob.get_val("L_rod"))
print("     Screw mass = %.2f kg" % prob.get_val("M_screw"))
print("     Nut mass = %.2f kg" % (2 * prob.get_val("M_nut")))
print("     Nut length = %.2f m" % prob.get_val("L_nut"))
print("     Bearing length = %.2f m" % prob.get_val("L_bearing"))
print("Constraints (should be >= 0):")
print("     Speed margin: W_mot-reduction_ratio*max_speed/pitch= %.3f" % prob.get_val("C1"))
print("     Torque margin: T_mot_guess-T_mot_real= %.3f " % prob.get_val("C2"))
print(
    "     Length margin: actuator_length-stroke-L_nut-L_bearing-2*L_rod =  %.3f"
    % prob.get_val("C3")
)
print("Calculation time:\n", end - start, "s")

Objective:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[25], line 2
      1 print("Objective:")
----> 2 print("     Total mass = %.2f kg" % (prob.get_val("objective")))
      3 print("Design variables:")
      4 print("     reduction_ratio =  %.2f" % prob.get_val("reduction_ratio"))

File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/openmdao/core/problem.py:517, in Problem.get_val(self, name, units, indices, get_remote)
    490 def get_val(self, name, units=None, indices=None, get_remote=False):
    491     """
    492     Get an output/input variable.
    493 
   (...)
    515         The value of the requested output/input variable.
    516     """
--> 517     if self._metadata['setup_status'] == _SetupStatus.POST_SETUP:
    518         abs_names = name2abs_names(self.model, name)
    519         if abs_names:

TypeError: 'NoneType' object is not subscriptable

om.n2(prob)

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
Cell In[26], line 1
----> 1 om.n2(prob)

File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/openmdao/visualization/n2_viewer/n2_viewer.py:637, in n2(data_source, outfile, path, values, case_id, show_browser, embeddable, title, display_in_notebook)
    635 # grab the model viewer data
    636 try:
--> 637     model_data = _get_viewer_data(data_source, values=values, case_id=case_id)
    638     err_msg = ''
    639 except TypeError as err:

File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/openmdao/visualization/n2_viewer/n2_viewer.py:510, in _get_viewer_data(data_source, values, case_id)
    506     raise TypeError(f"Viewer data is not available for '{data_source}'."
    507                     "The source must be a Problem, model or the filename of a recording.")
    509 data_dict = {}
--> 510 data_dict['tree'] = _get_tree_dict(root_group, values=values)
    511 data_dict['md5_hash'] = root_group._generate_md5_hash()
    513 connections_list = []

File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/openmdao/visualization/n2_viewer/n2_viewer.py:254, in _get_tree_dict(system, values, is_parallel)
    249 tree_dict['is_parallel'] = is_parallel
    251 children = [_get_tree_dict(s, values, is_parallel)
    252             for s in system._subsystems_myproc]
--> 254 if system.comm.size > 1:
    255     if system._subsystems_myproc:
    256         sub_comm = system._subsystems_myproc[0].comm

AttributeError: 'NoneType' object has no attribute 'size'