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GSSSS

The Generalized Single Step Single Solve Unified Framework

The GSSSS approach unifies various time integration methods in a single framework.

References

  1. Advances in Computational Dynamics of Particles, Materials and Structures
  2. 10.1002/nme.89
  3. 10.1002/nme.873

There are quite a few papers on this topic by the same group of authors. Similar contents can be found in a number of papers. The implementation is based on a unified predictor multi-corrector representation. It is sufficiently general so that both elastic and elastoplastic systems can be analyzed. The implementation is documented in details in Section 14.3.4 (Eqs. 14.280 --- 14.296) of the first reference.

It is strongly recommended to give the references a careful read as GSSSS is very elegant if you wish to learn more about the advances in computational dynamics.

Syntax

Both U0 and V0 families are available.

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integrator GSSSSU0 (1) (2) (3) (4)
integrator GSSSSV0 (1) (2) (3) (4)
# (1) int, unique integrator tag
# (2) double, spectral radius (order does not matter)
# (3) double, spectral radius (order does not matter)
# (4) double, spectral radius (order does not matter)

The optimal scheme (see table below) only requires one spectral radius, one can use the following command to use the optimal scheme.

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integrator GSSSSOptimal (1) [2]
# (1) int, unique integrator tag
# [2] double, spectral radius, default: 0.5

Remarks

The framework has three parameters to be defined, namely \(\rho_{1,\infty}\), \(\rho_{2,\infty}\) and \(\rho_ {3,\infty}\). They satisfy the following condition,

\[ 0\leqslant\rho_{3,\infty}\leqslant\rho_{1,\infty}\leqslant\rho_{2,\infty}\leqslant1. \]

The syntax takes three spectral radii in arbitrary order, they are clamped between zero and unity, sorted and assigned to \(\rho_{3,\infty}\), \(\rho_{1,\infty}\) and \(\rho_{2,\infty}\) to compute internal parameters. Users can thus assign three valid radii without worrying about the order.

A number of commonly known methods can be accommodated in the framework. For example:

Method Family Value \(\rho_{1,\infty}\) Value \(\rho_{2,\infty}\) Value \(\rho_{3,\infty}\)
Newmark U0 \(1\) \(1\) \(0\)
Classic Midpoint U0/V0 \(1\) \(1\) \(1\)
Generalised Alpha U0 \(\rho\) \(\rho\) \(\rho\)
WBZ U0 \(\rho\) \(\rho\) \(0\)
HHT U0 \(\rho\) \(\rho\) \(\dfrac{1-\rho}{2\rho}\)
U0-V0 Optimal U0/V0 \(\rho\) \(1\) \(\rho\)
New Midpoint V0 \(1\) \(1\) \(0\)