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The theory, applied on rod and line, is described in some detail on this page. For further details refer to the following reference:

H. Ekander, N.C. Perkins, B. Richards, Development of a simulation model for fly casting and application to overhead casting, Sports Engineering (2025) 28:2
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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: