40
2 Preliminaries
The absolute values of the amplification functions are displayed as function of the
time step size in Fig. 2.9. From these the stability properties of the various time
integrators are easily assessed:
Euler Forward
:
conditionally stable for < 2,
Euler Backward
:
A-stable and L-stable,
Heun Trapezoidal/Midpoint :
conditionally stable for < 2,
Trapezoidal/Midpoint
:
A-stable
.
The sequence y
n only converges monotonically to zero if the amplification function
satisfies 1 > a((t) > 0, otherwise, i.e. if the amplification function satisfies 0 >
a((t) > −1, the sequence y
n converges in an oscillatory manner.
2.2.2 Finite Element Method
Although not explicitly used in the sequel, some basics of the one-dimensional finite
element method shall here be outlined briefly in order to give some context for the
computational material models considered in this treatise. Thereby, it is particularly
illuminating to showcase how the algorithmic update of the stress and its linearization,
as outlined in the sequel of the Catalogue in much detail for each computational
0
0.5
1
1.5
2
0
.5
1
Δt
|a(Δt)| Euler Forward
|1 − Δt|
Conditionally Stable
0
50
100
150
200
0
0.5
1
Δt
|a(Δt)| Euler Backward
|1 + Δt|
−1
A- and L-Stable
0
0.5
1
1.5
2
0
.5
1
Δt
Conditionally Stable
|a(Δt)| Heun
|1 − Δt +
1
2
Δt
2 |
0
50
100
150
200
0
0.5
1
Δt
A-Stable
|a(Δt)| Midpoint/Trapezoidal
|1 −
1
2
Δt||1 +
1
2
Δt|
−1
Fig. 2.9 Modulus of the amplification function |a((t)| for the algorithmic time integrators applied
to ˙
y(t) = −y. Time integrators are A-stable for |a((t)| < 1 and, in addition, L-stable for |a((t →
∞)| → 0