CoSApp tutorials on multimode systems

Multimode ODE

In this example, we show a system defining a simple Ordinary Differential Equation, with an event-driven discontinuity.

from cosapp.base import System


class ScalarOde(System):
    """System representing ODE df/dt = F(t)"""
    def setup(self):
        self.add_inward('df')
        self.add_transient('f', der='df')


class MultimodeScalarOde(System):
    def setup(self):
        self.add_child(ScalarOde('ode'), pulling=['f', 'df'])
        self.add_outward_modevar('snapped', False)
        self.add_event('snap')  # event defining mode var `snapped`

    def transition(self):
        if self.snap.present:
            self.snapped = True

from cosapp.drivers import RungeKutta
from cosapp.recorders import DataFrameRecorder

s = MultimodeScalarOde('s')

s.snap.trigger = "f > 0.547"

driver = s.add_driver(
    RungeKutta(order=2, time_interval=(0, 1), dt=0.1)
)
driver.add_recorder(DataFrameRecorder(includes=['f', 'df', 'snapped']), period=0.1)

# Set scenario such that df/dt changes with mode
driver.set_scenario(
    init = {'f': 0},
    values = {'df': '2 * t if not snapped else 0'},
)
s.run_drivers()

# Retrieve recorded data
data = driver.recorder.export_data()
data = data.drop(['Section', 'Status', 'Error code'], axis=1)

# Plot results
import plotly.graph_objs as go

def get_trace(col):
    return go.Scatter(
        x = data['time'],
        y = data[col],
        name = col,
        mode = "lines+markers",
    )

go.Figure(
    data = [
        get_trace('f'),
        get_trace('df'),
    ],
    layout = go.Layout(
        xaxis = dict(title="Time"),
        height = 450,
        hovermode = "x",
    ),
)

Show recorded data

Note the double entry at snap time. When snapped is False, the solution is \(f(t) = t^2\), since the derivative is \(2t\), with initial condition \(f(0) = 0\). Event time is therefore \(t_{\rm snap} = \sqrt{0.547}\).

from math import sqrt
print(f"{sqrt(0.547) = }")

sqrt(0.547) = 0.7395944834840239

data

	Reference	df	f	snapped	time
0	t=0.0	0.000000	0.000	False	0.000000
1	t=0.1	0.200000	0.010	False	0.100000
2	t=0.2	0.400000	0.040	False	0.200000
3	t=0.3	0.600000	0.090	False	0.300000
4	t=0.4	0.800000	0.160	False	0.400000
5	t=0.5	1.000000	0.250	False	0.500000
6	t=0.6	1.200000	0.360	False	0.600000
7	t=0.7	1.400000	0.490	False	0.700000
8	t=0.73959448348402	1.479189	0.547	False	0.739594
9	s.snap	0.000000	0.547	True	0.739594
10	t=0.8	0.000000	0.547	True	0.800000
11	t=0.9	0.000000	0.547	True	0.900000
12	t=1.0	0.000000	0.547	True	1.000000