Added a balanced field length example (#533)

* Added a multi-phase balanced field example case
* moved most parameters to the trajectory level, but restarts with trajectory parameters appears to be broken.
* trajectory simulation now records inputs to capture parameter values
* Added new test case to verify that default "val" for states, controls, polynomial controls, parameters, and times are sticking.
* added utility function to reshape default values to the proper shape

dymos/examples/balanced_field/balanced_field_ode.py

@@ -0,0 +1,122 @@
+import numpy as np
+import openmdao.api as om
+
+
+class BalancedFieldODEComp(om.ExplicitComponent):
+    """
+    The ODE System for an aircraft takeoff climb.
+
+    Computes the rates for states v (true airspeed) gam (flight path angle) r (range) and h (altitude).
+
+    References
+    ----------
+    .. [1] Raymer, Daniel. Aircraft design: a conceptual approach. American Institute of
+    Aeronautics and Astronautics, Inc., 2012.
+    """
+    def initialize(self):
+        self.options.declare('num_nodes', types=int)
+        self.options.declare('g', types=(float, int), default=9.80665, desc='gravitational acceleration (m/s**2)')
+        self.options.declare('mode', values=('runway', 'climb'), desc='mode of operation (ground roll or flight)')
+
+    def setup(self):
+        nn = self.options['num_nodes']
+
+        # Scalar (constant) inputs
+        self.add_input('rho', val=1.225, desc='atmospheric density at runway', units='kg/m**3')
+        self.add_input('S', val=124.7, desc='aerodynamic reference area', units='m**2')
+        self.add_input('CD0', val=0.03, desc='zero-lift drag coefficient', units=None)
+        self.add_input('CL0', val=0.5, desc='zero-alpha lift coefficient', units=None)
+        self.add_input('CL_max', val=2.0, desc='maximum lift coefficient for linear fit', units=None)
+        self.add_input('alpha_max', val=np.radians(10), desc='angle of attack at CL_max', units='rad')
+        self.add_input('h_w', val=1.0, desc='height of the wing above the CG', units='m')
+        self.add_input('AR', val=9.45, desc='wing aspect ratio', units=None)
+        self.add_input('e', val=0.801, desc='Oswald span efficiency factor', units=None)
+        self.add_input('span', val=35.7, desc='Wingspan', units='m')
+        self.add_input('T', val=1.0, desc='thrust', units='N')
+
+        # Dynamic inputs (can assume a different value at every node)
+        self.add_input('m', shape=(nn,), desc='aircraft mass', units='kg')
+        self.add_input('v', shape=(nn,), desc='aircraft true airspeed', units='m/s')
+        self.add_input('h', shape=(nn,), desc='altitude', units='m')
+        self.add_input('alpha', shape=(nn,), desc='angle of attack', units='rad')
+
+        # Outputs
+        self.add_output('CL', shape=(nn,), desc='lift coefficient', units=None)
+        self.add_output('q', shape=(nn,), desc='dynamic pressure', units='Pa')
+        self.add_output('L', shape=(nn,), desc='lift force', units='N')
+        self.add_output('D', shape=(nn,), desc='drag force', units='N')
+        self.add_output('K', val=np.ones(nn), desc='drag-due-to-lift factor', units=None)
+        self.add_output('F_r', shape=(nn,), desc='runway normal force', units='N')
+        self.add_output('v_dot', shape=(nn,), desc='rate of change of speed', units='m/s**2',
+                        tags=['state_rate_source:v'])
+        self.add_output('r_dot', shape=(nn,), desc='rate of change of range', units='m/s',
+                        tags=['state_rate_source:r'])
+        self.add_output('W', shape=(nn,), desc='aircraft weight', units='N')
+        self.add_output('v_stall', shape=(nn,), desc='stall speed', units='m/s')
+        self.add_output('v_over_v_stall', shape=(nn,), desc='stall speed ratio', units=None)
+
+        # Mode-dependent IO
+        if self.options['mode'] == 'runway':
+            self.add_input('mu_r', val=0.05, desc='runway friction coefficient', units=None)
+        else:
+            self.add_input('gam', shape=(nn,), desc='flight path angle', units='rad')
+            self.add_output('gam_dot', shape=(nn,), desc='rate of change of flight path angle',
+                            units='rad/s', tags=['state_rate_source:gam'])
+            self.add_output('h_dot', shape=(nn,), desc='rate of change of altitude', units='m/s',
+                            tags=['state_rate_source:h'])
+
+        self.declare_coloring(wrt='*', method='cs')
+
+    def compute(self, inputs, outputs, discrete_inputs=None, discrete_outputs=None):
+        g = self.options['g']
+
+        # Compute factor k to include ground effect on lift
+        rho = inputs['rho']
+        v = inputs['v']
+        S = inputs['S']
+        CD0 = inputs['CD0']
+        m = inputs['m']
+        T = inputs['T']
+        h = inputs['h']
+        h_w = inputs['h_w']
+        span = inputs['span']
+        AR = inputs['AR']
+        CL0 = inputs['CL0']
+        alpha = inputs['alpha']
+        alpha_max = inputs['alpha_max']
+        CL_max = inputs['CL_max']
+        e = inputs['e']
+
+        outputs['W'] = W = m * g
+        outputs['v_stall'] = v_stall = np.sqrt(2 * W / rho / S / CL_max)
+        outputs['v_over_v_stall'] = v / v_stall
+
+        outputs['CL'] = CL = CL0 + (alpha / alpha_max) * (CL_max - CL0)
+        K_nom = 1.0 / (np.pi * AR * e)
+        b = span / 2.0
+        fact = ((h + h_w) / b) ** 1.5
+        outputs['K'] = K = K_nom * 33 * fact / (1.0 + 33 * fact)
+
+        outputs['q'] = q = 0.5 * rho * v ** 2
+        outputs['L'] = L = q * S * CL
+        outputs['D'] = D = q * S * (CD0 + K * CL ** 2)
+
+        # Compute the downward force on the landing gear
+        calpha = np.cos(alpha)
+        salpha = np.sin(alpha)
+
+        # Runway normal force
+        outputs['F_r'] = F_r = m * g - L * calpha - T * salpha
+
+        # Compute the dynamics
+        if self.options['mode'] == 'climb':
+            gam = inputs['gam']
+            cgam = np.cos(gam)
+            sgam = np.sin(gam)
+            outputs['v_dot'] = (T * calpha - D) / m - g * sgam
+            outputs['gam_dot'] = (T * salpha + L) / (m * v) - (g / v) * cgam
+            outputs['h_dot'] = v * sgam
+            outputs['r_dot'] = v * cgam
+        else:
+            outputs['v_dot'] = (T * calpha - D - F_r * inputs['mu_r']) / m
+            outputs['r_dot'] = v