Integrators

class classes.Integrator.EM_pH

Energy_Momentum-Integration scheme for PH mechanical system

• not derived from variational principle

• uses discrete gradient for ext. potential and internal potential

• uses mixed variables (e.g. for strains) and has port-Hamiltonian structure

• more info: https://doi.org/10.1002/pamm.202300144

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.Integrator

Abstract Integrator class

Property Summary

DT

time step size

INDI_VELO

whether integrator deals with independent velocities AND momenta

LM0

initial value for Lagrange multipliers

NAME

integrator name

NT

number of time steps

PARA

integrator parameters

T_0

start time

T_END

end time

compute_potential_from_mixed_quantity

whether potential is computed from mixed quantity or not

hasPARA

whether integrator features parameters

has_enhanced_constraint_force

whether constraint force is enhances

nVARS

number of unknown variables

t

time

class classes.Integrator.EMG_noCons

Energy_Momentum-Integration scheme for ODE

• not derived from variational principle

• taken from Gonzales 1996

• uses standard midpoint gradient for ext. potential and discrete gradient for internal potential

• takes account of non-constant mass-matrices

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_std

GGL integration scheme for constrained DAE

• based constraints on position and momentum level

• independent momenta variables (Livens approach)

• standard stabilisation scheme by Gear, Gupta & Leimkuhler applied to the constrained symplectic Euler B method

• not derived from variational principle: constraints evaluated at t_{n+1} and ODE-RHS for q at t_{n+1} and for p at t_n

• rather bad performance

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.MP_std

Midpoint-Integration scheme for standard constrained DAE

• based only on constraint on position level

• independent momenta variables (Hamilton Potryagin approach)

• not derived from variational principle but simply evaluates RHS at t_{n+1/2}

Method Summary

compute_resi_tang(zn1, zn, this_system, time_n)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.Lagrange_top_ODE

Integration scheme for Lagrange top

• based on angular momentum and director d3 (ODE approach)

• not derived from variational principle

• used for NODY publication(https://doi.org/10.1007/s11071-023-08522-7), taken from Bobenko & Suris, 1999

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_VI_mod

Variational integration scheme for GGL-like constrained DAE

• based on constraint on position and momentum level

• independent momenta variables (Livens approach)

• derived from variational principle

• symplectic

• constraints are enforced at t_{n+1}

• more info: https://doi.org/10.1007/s11044-023-09889-6

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.EMS_ggl

Energy_Momentum-Integration scheme for constrained DAE with GGL principle

• based only on constraints on position and velocity level

• independent momenta variables (Hamilton Potryagin approach)

• not derived from variational principle

• basic approach from Gonzales 1999 here also applied to constraint on velocity level

• uses standard gradient for ext. potential and discrete gradient for internal potential and constraints

• more info: https://doi.org/10.1007/s11071-023-08522-7

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.EML_noCons_cyclic

Energy_Momentum-Integration scheme for ODE with cyclic coordinate

• not derived from variational principle

• uses Livens equations of motion

• uses discrete gradient for ext. potential and internal potential

• takes account of non-constant mass-matrices

• splits up for cyclic coordinates! import for conservation of corresponding conjugate momenta

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.CSE_B

Variational integration scheme for constrained DAE

• based on constraint on position level

• generalization of the symplectic Euler B method

• independent momentum variables

• constraints are enforced at t_{n+1}

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.MP_noCons

Midpoint-Integration scheme for unconstrained dynamics (ODEs)

• based on momenta and positions (Hamiltonian approach)

• not derived from variational principle but simply evaluates RHS at t_{n+1/2}

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.EMS_std

Energy_Momentum-Integration scheme for standard constrained DAE

• based only on constraint on position level

• not derived from variational principle

• taken from Gonzales 1999, Hamiltonian framework

• uses standard gradient for ext. potential and discrete gradient for internal potential and constraint

Method Summary

compute_resi_tang(zn1, zn, this_system, time_n)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_VI_mod_visc

Variational integration scheme for GGL-like constrained DAE

• based on constraint on position and momentum level

• independent momenta variables (Livens approach)

• derived from variational principle

• symplectic

• constraints are enforced at t_{n+1}

• takes into account non-conservative viscous forces

• more info: https://doi.org/10.1007/s11044-023-09889-6

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

rearrange_unknowns(~, this_simulation, this_system)

v_n is an unknown of this scheme, has to be shifted backwards by 1 after computation

class classes.Integrator.GGL_VI_theta_A

Runge-Kutta typed scheme for GGL-like constrained DAE

• based on constraint on position and velocity level (GGL-stabilisation)

• independent momentum variables (Hamilton Potryagin approach)

• derived from variational principle

• symplectic method

• more info: https://doi.org/10.1007/s11071-023-08522-7

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.EML_noCons

Energy_Momentum-Integration scheme for ODEs

• not derived from variational principle

• uses Livens equations of motion

• uses discrete gradient for ext. potential and internal potential

• takes account of non-constant mass-matrices

Method Summary

compute_resi_tang(zn1, zn, this_system, time_n)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.Rattle

Variational integration scheme for GGL-like constrained DAE

• based on constraint on position and velocity level (GGL-stabilisation)

• constraints are enforced at t_{n+1}

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_theta_mod

One-stage-theta-method (1) for GGL-like constrained DAE

• based on constraint on position and velocity level (GGL-stabilisation)

• independent momentum variables (Hamilton Potryagin approach)

• derived from one-stage thetat method (constraints have been modified)

• still from masters thesis, not used further

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_VI

Variational integration scheme for GGL-like constrained DAE

• based on constraint on position and momentum level

• independent momenta variables (Livens approach)

• derived from variational principle by Peter Betsch (1st attempt for new ‘GGL-functional’

• not symplectic

• constraints are enforced at t_{n+1}

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.EML

Energy_Momentum-Integration scheme for standard constrained DAE

• not derived from variational principle

• uses Livens equations of motion

• uses discrete gradient for ext. potential, internal potential and constraint gradients

• G-equivarient versions for constraint and int. pot. gradients

• takes account of non-constant mass-matrices

Method Summary

compute_resi_tang(zn1, zn, this_system, time_n)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.GGL_VI_theta_B

Runge-Kutta typed scheme for GGL-like constrained DAE

• based on constraint on position and velocity level (GGL-stabilisation)

• independent momentum and velocity variables (Hamilton Potryagin approach)

• derived from variational principle

• from symplectic-theta-framework

• more info: https://doi.org/10.1007/s11071-023-08522-7

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.MP_noCons_Livens

Midpoint-Integration scheme for standard constrained DAE

• based only on constraint on position level

• independent momenta variables (Livens approach)

• not derived from variational principle but simply evaluates RHS at t_{n+1/2}

Method Summary

compute_resi_tang(zn1, zn, this_system)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1

class classes.Integrator.MP_ggl

Midpoint-Integration scheme for standard constrained DAE

• based on constraint on position and velocity level (GGL-stabilisation)

• independent momenta variables (Hamilton Potryagin approach)

• not derived from variational principle but simply evaluates RHS of ODE at t_{n+1/2}

• constraints are enforced at t_{n+1}

Method Summary

compute_resi_tang(zn1, zn, this_problem)

Computes residual vector & tangent matrix

Parameters:

• zn1 – state vector for next time step

• zn – state vector at current time step

• this_system – System object

Returns:

[ResidualVector, TangentMatrix] for the Newton’s method to update zn1