ODE Package for E-MicroM contains classes to solve the Landau Lifshitz Systems by ODE integrations as Runge Kutta... More...

ODE Package for E-MicroM contains classes to solve the Landau Lifshitz Systems by ODE integrations as Runge Kutta...

All systems inherit classes from EMM_LandauLifschitzODE which implement the resolution of EMM_LandauLifschitzSystem . All the ODE implemented methods need to compute a new time step by calculating on error on solving $\frac{dM}{dt}=F(M)$ . The class EMM_ODETimeStep manages the time step computing. The different implemented integration methods are :

EMM_LandauLifschitzRK45 implements the solving of the Landau-Lifschitz system by the Runge-Kutta-Fehlman methods

The main method to integrate the Landau-Lifschitz system is EMM_LandauLifschitzODE::solveODE() :

From an initial magnetization field M, an initial time step $ dt=T/I $ (where T is the target time and I is the number of steppers) and initial energy $E_{old}=\infty$ it computes:

the magnetic excitation field H(M) and energy E(M,H)
the magnetization field time derivative $\frac{dM}{dt}=F(M,H)$
the energy time derivative $\frac{dE}{dt}= \left ( F(M,H),H \right )_{L^2}$
the energy time variation $\frac{\Delta E}{\Delta t}= \frac{E(M,H)-E_{old}}{dt \cdot N}$ where N is the number of magnetized cells
the equilibrium criterium $Q=min(\frac{dE}{N \cdot dt},\frac{\Delta E}{\Delta t})$ . If Q is less than EMM_LandauLifschitzSystem::getMinimumEnergyVariation() the relaxation process is stopped.
compute the new time step dt_{new}, magnetization field M(dt) and operator fields U(dt) (as displacement, velocity, constraint tensor...) EMM_LandauLifschitzODE::computeFieldsAtTime()

The computing of the new fields at time EMM_LandauLifschitzODE::computeFieldsAtTime() depends on the evolution scheme:

EMM_LandauLifschitzSystem::Os : computes only the operator fields when magnetization field is permanent : calls only EMM_LandauLifschitzSystem::computeOperatorsFieldsAtTime(), the time step is regular.
EMM_LandauLifschitzSystem::M : computes the new time stemp dt_{new}, the magnetization field M(t+dt) when operator fields are permanent EMM_Stepper::computeNewTimeStep(dt), the time step is adaptative.
EMM_LandauLifschitzSystem::MOs : computes the new time step dt_{new}, the magnetization field M(t+dt) and the operator fields at t+dt form M(t+dt)
EMM_LandauLifschitzSystem::OsM : computes the new time step dt_{new}, the operator fields at t+dt from M(t) and the magnetization field M(t+dt) Note that dt is adapted to decrease the error on M(t+dt). It is not neccesary the same as the time step given at the begin of the relaxation step except for regular time step

The computing of the new time step is done by EMM_ODETimeStep::computeNewTimeStep() :

for a given dt, integrates the magnetization field from t to t+dt EMM_LandauLifschitzODE::integrateMagnetizationFieldAtTime() by estimating the error and compute a best time step
if the error is acceptable EMM_ODETimeStep::setMaximumError() the time step adaptative process is stopped.
if the number of loops is greater then the EMM_ODETimeStep::setMaximumIterationsNumber(), the time step adaptative process is stopped and a warning is reported
on all other cases, recompute the adapative time step
EMM_LandauLifschitzODE::integrateMagnetizationFieldAtTime() integrates M between t and t+dt and
compute the error on M(t+dt)
computes a new time step

The magnetization field at t+dt is stored within the EMM_ODETimeStep class : EMM_ODETimeStep::getMagnetizationFieldAtTime().

The EMM_ODEClassFactory class manages the creation of the class of the package

The method EMM_ODETest class manages the tests of the package.