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.

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.

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 ${\rho }_{1,\mathrm{\infty }}$, ${\rho }_{2,\mathrm{\infty }}$ and ${\rho }_{3,\mathrm{\infty }}$. They satisfy the following condition,

$$0⩽{\rho }_{3,\mathrm{\infty }}⩽{\rho }_{1,\mathrm{\infty }}⩽{\rho }_{2,\mathrm{\infty }}⩽1.$$

The syntax takes three spectral radii in arbitrary order, they are clamped between zero and unity, sorted and assigned to ${\rho }_{3,\mathrm{\infty }}$, ${\rho }_{1,\mathrm{\infty }}$ and ${\rho }_{2,\mathrm{\infty }}$ 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 ${\rho }_{1,\mathrm{\infty }}$	Value ${\rho }_{2,\mathrm{\infty }}$	Value ${\rho }_{3,\mathrm{\infty }}$
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	$\rho$	$\rho$	${\displaystyle \frac{1-\rho }{2\rho }}$
U0-V0 Optimal	U0/V0	$\rho$	$1$	$\rho$
New Midpoint	V0	$1$	$1$	$0$