# Equilibrium of System with Piecewise Linear Springs (SPRINGCHAIN)#

`springchain.py`

```"""
## GAMSSOURCE: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_springchain.html
## MODELTYPE: QCP

Equilibrium of System with Piecewise Linear Springs (SPRINGCHAIN)

This model finds the shape of a hanging chain consisting of
N springs and N-1 nodes. Each spring buckles under compression and each
node has a weight hanging from it. The springs are assumed
weightless. The goal is to minimize the potential energy of the
system.

We use rotated quadratic cone constraints to model the extension
of each spring.

M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret,
Applications of second-order cone programming, Linear Algebra and its
Applications, 284:193-228, November 1998, Special Issue on Linear Algebra
in Control, Signals and Image Processing.
"""

from __future__ import annotations

import math
import os

import pandas as pd
from gamspy import (
Card,
Container,
Equation,
Model,
Ord,
Parameter,
Problem,
Sense,
Set,
Sum,
Variable,
)

def main():
cont = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
)

N = 10
NM1 = 9

# Set
n = Set(
cont,
name="n",
records=[f"n{str(i)}" for i in range(N + 1)],
description="spring index",
)

# Data
a_x = 0
a_y = 0
b_x = 2
b_y = -1
L0 = 2 * math.sqrt((a_x - b_x) ** 2 + (a_y - b_y) ** 2) / N
g = 9.8
k = 100

m = Parameter(
cont,
name="m",
domain=n,
records=pd.DataFrame([[f"n{i}", 1] for i in range(1, NM1 + 1)]),
description="mass of each hanging node",
)

# Variable
x = Variable(
cont, name="x", domain=n, description="x-coordinates of nodes"
)
y = Variable(
cont, name="y", domain=n, description="y-coordinates of nodes"
)
delta_x = Variable(cont, name="delta_x", domain=n)
delta_y = Variable(cont, name="delta_y", domain=n)
unit = Variable(cont, name="unit")
t_L0 = Variable(cont, name="t_L0", domain=n, type="Positive")
t = Variable(
cont,
name="t",
domain=n,
type="Positive",
description="extension of each spring",
)
v = Variable(cont, name="v", type="Positive")

# Equation
delta_x_eq = Equation(cont, name="delta_x_eq", domain=n)
delta_y_eq = Equation(cont, name="delta_y_eq", domain=n)
cone_eq = Equation(cont, name="cone_eq")

pot_energy = (
Sum(n.where[Ord(n) > 1 & (Ord(n) < Card(n))], m[n] * g * y[n]) + k * v
)
delta_x_eq[n] = delta_x[n] == x[n] - x[n.lag(1)]
delta_y_eq[n] = delta_y[n] == y[n] - y[n.lag(1)]

link_L0[n] = t_L0[n] == L0 + t[n]
t_L0[n] ** 2 >= delta_x[n] ** 2 + delta_y[n] ** 2
)

cone_eq[...] = 2 * v * unit >= Sum(n.where[Ord(n) > 1], t[n] ** 2)

spring = Model(
cont,
name="spring",
equations=cont.getEquations(),
problem=Problem.QCP,
sense=Sense.MIN,
objective=pot_energy,
)

x.l[n] = ((Ord(n) - 1) / N) * b_x + (Ord(n) / N) * a_x
y.l[n] = ((Ord(n) - 1) / N) * b_y + (Ord(n) / N) * a_y

x.fx["n0"] = a_x
y.fx["n0"] = a_y
x.fx[f"n{N}"] = b_x
y.fx[f"n{N}"] = b_y
unit.fx[...] = 1

spring.solve()

assert math.isclose(spring.objective_value, -185.4461, rel_tol=0.001)

if __name__ == "__main__":
main()
```