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The theory in 2D, applied on rod and line, is
described in some detail here.

Coordinates for the co-moving system *(s,n)* and the inertial system *(x,y)* are presented below left, and
force definitions for an element below right:

**Equations:**

Newtons equation:

Kinematic constraints: ** **

Bending moment:

**Nomenclature:**

mass per unit
length

force (), bending moment

force per unit length, (fluid drag, gravity…)

bending stiffness, material relaxation time

** **curvature, , rate of change,

velocity (), angular velocity,

**Solution
procedure:**

7 differential equations and 7 unknowns:

BCs external and internal between materials:

Discretization:

· Generalized alpha in time.

· Space centered.

Iterative solution procedure on each time
step:

• Banded matrix for the solution vector:

• The error vector gives the Jacobian:

• The Jacobian matrix is inverted:

• The solution vector is updated:

• Iterate until:

**Coordinates**

The physical coordinates are calculated as
follows (at each time step).

**Energies and work**

Energies and work are additional output
computed at each time step on each domain *i* e.g. rod, fly line, leader.

Potential, kinetic and strain energies are
computed as follows:

Work by drag forces, material damping, and
caster input are computed as follows: