Automatic Time Step Size Calculation in Transient Analysis

In transient analysis, QuickField is now capable to automatically calculate and adjust the time step size for the integration process.

To calculate the initial time step size, the following conservative estimate is used:

Δt₀ = min (ξ² / 4α) ,

where ξ is the "mesh size" (diameter of a mesh element) and
α = λ / ρ·C - for problems of heat transfer,
α = 1 / μ·g - for electromagnetic problems.

The ratio ξ² / 4α is evaluated in all the mesh elements in the model, and the smallest value is used as an initial time step size.

As the solution progresses, the time steps are adjusted automatically by an adaptive time stepping scheme.

The next time step is adjusted by

Δt_k+1 =k·Δt_k

where k is a scaling factor varying from 0.25 to 4.0 (with discrete values of 0.25; 0.5; 1.0; 2.0; 4.0) and dependent on behavior of potential and its time derivative, as well as all the time- and coordinate-dependent sources and boundary conditions in the model.

The two factors are taken into account when choosing the value of k:

• The norm of time derivative variation on previous time step in all mesh nodes:

  Δūn = 2

  ||ūn - ūn-1|| ||ūn|| + ||ūn-1||

• The inverse of characteristic time:

  ωn =	{Δūn}T·{Fn - Fn-1} {Δūn}T·[KTn]·{Δūn}

In thermal analysis, {F_n} is the heat flow vector associated with conduction, convection, and radiation. In transient magnetics, {F_n} is a vector of induction. u is a potential value, and K_T is a stiffness matrix in the finite-element analysis.

The actual value of scaling factor k is chosen based upon two dimensionless characteristics: Δū_n and 2π/Δt_nω_n, by means of predetermined proprietary threshold tables, and the smaller is considered to be used for the next time step size, thus guaranteeing to produce smooth and accurate time dependency in every spatial point of the model.