contique

Numeric continuation of equilibrium equations

Example

A given set of equilibrium equations in terms of x and lpf (a.k.a. load-proportionality-factor) should be solved by numeric continuation of a given initial solution.

Function definition

def fun(x, lpf, a, b):
    return np.array([-a * np.sin(x[0]) + x[1]**2 + lpf, 
                     -b * np.cos(x[1]) * x[1]    + lpf])

with it's initial solution

x0 = np.zeros(2)
lpf0 = 0.0

and function parameters

a = 1
b = 1

Run contique.solve and plot equilibrium states

Res = contique.solve(
    fun=fun,
    x0=x0,
    args=(a, b),
    lpf0=lpf0,
    dxmax=0.05,
    dlpfmax=0.05,
    maxsteps=80,
    maxcycles=4,
    maxiter=20,
    tol=1e-6,
)

For each step a summary is printed. This contains needed Newton-Rhapson iterations per cycle and an information about the control component at the beginning and the end of a cycle. Finally the lpf, control and equilibrium norm values are listed. As an example the ouput of Step 77 is shown below.

Begin of Step 77 
====================================================================

| Cycle | converged in                        | control component  |
|-------|-------------------------------------|--------------------|
| #   1 | Solution converged in  1 Iteration  | from   -2 to   -1  |
| #   2 | Solution converged in  2 Iterations | from   -1 to   -1  |

*final lpf value     = -5.376e-01
*final control value =  3.179e+00
*final equilibrium   =  1.949e-09 (norm)

Next, we have to assemble the results

X = np.array([res.x for res in Res])

and plot the solution curve.

import matplotlib.pyplot as plt

plt.plot(X[:, 0], X[:, 1], ".-")
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')