contique 0.1.4

Numerical continuation of nonlinear equilibrium equations

contique

Numeric continuation of nonlinear equilibrium equations

Fig. 1 Archimedean spiral equation solved with contique

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 its 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=0.1,
    dlpfmax=0.1,
    maxsteps=75,
    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 72 
====================================================================

| Cycle | converged in                        | control component  |
|-------|-------------------------------------|--------------------|
| #   1 | Solution converged in  3 Iterations | from   -1 to   +2  |
| #   2 | Solution converged in  2 Iterations | from   +2 to   +2  |

*final lpf value     = -4.820e-01
*final control value = -5.739e-01
*final equilibrium   =  6.136e-11 (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], "C0.-")
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.plot([0],[0],'C0o',lw=3)
plt.arrow(X[-2,0],X[-2,1],X[-1,0]-X[-2,0],X[-1,1]-X[-2,1],
          head_width=0.075, head_length=0.15, fc='C0', ec='C0')
plt.gca().set_aspect('equal')

Fig. 2 Solution states of equilibrium equations solved with contique