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  1. Integrator
  2. Implicit

GSSSS

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Last updated 2 years ago

The Generalized Single Step Single Solve Unified Framework

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

References

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 ho1,∞ho_{1,\infty}ho1,∞​, ho2,∞ho_{2,\infty}ho2,∞​ and ho3,∞ho_ {3,\infty}ho3,∞​. They satisfy the following condition,

0⩽ρ3,∞⩽ρ1,∞⩽ρ2,∞⩽1.0\leqslant\rho_{3,\infty}\leqslant\rho_{1,\infty}\leqslant\rho_{2,\infty}\leqslant1.0⩽ρ3,∞​⩽ρ1,∞​⩽ρ2,∞​⩽1.

The syntax takes three spectral radii in arbitrary order, they are clamped between zero and unity, sorted and assigned to ho3,∞ho_{3,\infty}ho3,∞​, ho1,∞ho_{1,\infty}ho1,∞​ and ho2,∞ho_{2,\infty}ho2,∞​ 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

Newmark

U0

Classic Midpoint

U0/V0

Generalised Alpha

U0

WBZ

U0

HHT

U0

U0-V0 Optimal

U0/V0

New Midpoint

V0

Value

Value

Value

ho1,∞ho_{1,\infty}ho1,∞​
ho2,∞ho_{2,\infty}ho2,∞​
ho3,∞ho_{3,\infty}ho3,∞​
111
111
000
111
111
111
hohoho
hohoho
hohoho
hohoho
hohoho
000
hohoho
hohoho
1−ρ2ρ\dfrac{1-\rho}{2\rho}2ρ1−ρ​
hohoho
111
hohoho
111
111
000
Advances in Computational Dynamics of Particles, Materials and Structures
10.1002/nme.89
10.1002/nme.873