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SecondTM Class

Overview​

SecondTM is the standard second-order transformation in CPBD. It consumes a SecondOrderParams object and applies both the linear and quadratic parts of the map.

This is not "a TransferMap plus a correction". SecondTM is its own transformation family with its own runtime algorithm.

Atom Contract​

For a no-edge element, the atom must provide:

  • create_second_order_main_params(energy, delta_length)

For an edge-aware element with has_edge=True, the atom must additionally provide:

  • create_second_order_entrance_params(energy, delta_length)
  • create_second_order_exit_params(energy, delta_length)

SecondTM.from_element(...) binds to those hooks.

Parameter Object​

SecondTM expects a SecondOrderParams object with:

  • R: first-order part of the map
  • B: additive offset vector
  • T: second-order tensor
  • tilt: roll angle
  • dx, dy: offsets used by the underlying nonlinear model

If tilt != 0, SecondTM uses get_rotated_R() and get_rotated_T().

Mathematical Action​

In schematic form, SecondTM applies:

Xf=RXi+T(Xi,Xi)+BX_f = R X_i + T(X_i, X_i) + B

where the quadratic term is evaluated by the internal second-order multiplication routine.

Runtime Flow​

SecondTM.from_element(element)
    -> element.create_second_order_*_params(...)
    -> SecondOrderParams
    -> use R, T, B
    -> apply second-order map

Important Methods​

from_element(cls, element, tm_type=TMTypes.MAIN, delta_l=None, **params)​

Builds a SecondTM by connecting the second-order hook family of the atom.

t_apply(energy, X, U5666=0.0)​

Gets the current SecondOrderParams, rotates the map if needed, applies the second-order multiplication, and adds B.

map_function(X, energy)​

Calls t_apply(...) and returns the updated coordinates.

Use in OCELOT​

Many element families expose SecondTM as an optional active tracking method. It is especially useful when the first-order map is not sufficient but a dedicated kick or field-integration algorithm is not required.