4. B and C models

4.1. Characteristics of modeling

B and C models are the same except that B modeling uses DIS_TR elements while C modeling uses 2D_ DIS_TR elements.

This modeling allows, in addition to a new use of modal recombination, the validation of direct integration at adaptive steps.

Discrete element of stiffness in translation and rotation

Characteristics of the elements:

DISCRET:	with nodal masses	M_ TR_D_N	M_ TR_N
	and stiffness matrices	K_ TR_D_L	K_ TR_L
	and damping matrices	A_ TR_D_L	A_ TR_L

Boundary conditions and blocked directions:

in all nodes	DDL_IMPO:	(TOUT: “OUI” DY: 0., DZ: 0.) (TOUT: “OUI” DRX: 0. DRY: 0 DRZ: 0)
at the end nodes		(GROUP_NO: AB DX: 0.)

Integration schemes tested in this release:

• Integration by modal recombination with the Euler schema.

• Integration by direct integration with algorithm ADAPT_ORDRE2, no maximum time \({10}^{-3}s\).

• Integration by modal recombination with schema RUNGE_KUTTA_32, with a relative error tolerance of \({10}^{-3}s\) and a maximum time step of \({10}^{-3}s\).

• Integration by modal recombination with schema RUNGE_KUTTA_54, with a relative error tolerance of \({10}^{-3}s\) and a maximum time step of \({10}^{-3}s\).

4.2. Characteristics of the mesh

Number of knots: 10

Number of meshes and types: 9 SEG2

4.3. Tested sizes and results

Transient by modal recombination with algorithm EULER

Time	Reference
0.09	4.02 E—5
0.18	4.22 E—6
0.27	3.89 E—5
0.37	5.98 E—6
0.46	3.73 E—5
0.54	7.14 E—6
0.63	3.64 E—5
0.72	8.07 E—6
0.81	3.58 E—5
0.9	8.76 E—6
0.99	3.52 E—5
1.08	—3.08 E—5
1.18	3.02 E—5
1.27	—2.88 E—5
1.36	2.80 E—5
1.45	—2.65 E—5

Transient by direct integration with algorithm ADAPT_ORDRE2

Time	Reference
0.09	4.02 E—5
0.18	4.22 E—6
0.27	3.89 E—5
0.37	5.98 E—6
0.46	3.73 E—5
0.54	7.14 E—6
0.63	3.64 E—5
0.72	8.07 E—6
0.81	3.58 E—5
0.9	8.76 E—6
0.99	3.52 E—5
1.08	—3.08 E—5
1.18	3.02 E—5
1.27	—2.88 E—5
1.36	2.80 E—5
1.45	—2.65 E—5

Transient by modal recombination with algorithm RUNGE_KUTTA_32

Time	Reference
0.09	4.02 E—5
0.18	4.22 E—6
0.27	3.89 E—5
0.37	5.98 E—6
0.46	3.73 E—5
0.54	7.14 E—6
0.63	3.64 E—5
0.72	8.07 E—6
0.81	3.58 E—5
0.9	8.76 E—6
0.99	3.52 E—5
1.08	—3.08 E—5
1.18	3.02 E—5
1.27	—2.88 E—5
1.36	2.80 E—5
1.45	—2.65 E—5

Transient by modal recombination with algorithm RUNGE_KUTTA_54

Time	Reference
0.09	4.02 E—5
0.18	4.22 E—6
0.27	3.89 E—5
0.37	5.98 E—6
0.46	3.73 E—5
0.54	7.14 E—6
0.63	3.64 E—5
0.72	8.07 E—6
0.81	3.58 E—5
0.9	8.76 E—6
0.99	3.52 E—5
1.08	—3.08 E—5
1.18	3.02 E—5
1.27	—2.88 E—5
1.36	2.80 E—5
1.45	—2.65 E—5

4.4. notes

The A and B models lead to the same results.

The relative minima (\(t=0.18,0.54,\dots\)) do not have very good precision during the excitation phase with a \(\Delta t=0.001\) step.