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# AMS and Multi-Domain method in RADIOSS

Jean-Pierre Bobineau, Lionel Morançay

► To cite this version:

Jean-Pierre Bobineau, Lionel Morançay. AMS and Multi-Domain method in RADIOSS. 11e colloque national en calcul des structures, CSMA, May 2013, Giens, France. hal-01717077

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Submitted on 25 Feb 2018

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# AMS and Multi-Domain method in RADIOSS

Jean-Pierre Bobineau<fnref n="1" />, Bertrand Maurel<fnref n="2" />, Lionel Morançay <fnref n="3" /> *

**Abstract**— RADIOSS Explicit is on top of the industrial FE codes which are used for crash simulation as well as for fluid-structure interaction or stamping process simulation.

One upon the main concerns of RADIOSS development is to improve its computational performances while providing a high result quality.

The “Advanced Mass Scaling” method (AMS) allows to increase the critical time step without modifying the rigid body translational momentum of the system without significantly affecting its low frequencies. It is well adapted to problems for which the high frequency energy range is low, especially the quasi-static simulations. Nevertheless it is sometimes applied within the field of crash simulation efficiency and accuracy of AMS is demonstrated with a roof crush case.

On the other hand, the Multi-Domain method is well adapted to problems where the mesh size and the critical time step of different areas of the model (also called “domains”) are significantly different. Each domain is then solved, based on its own time step, and a coupling algorithm supporting multi-time scales insures the kinematic continuity at the interface between domains, as well as the stability of the scheme. The efficiency and accuracy of the Multi-Domain method is illustrated through a robustness study of a frontal crash against a pole.

**Keywords** — explicit method, « advanced mass scaling », AMS, Multi-Domain, multi-time scale, parallelism, crash

## 1. Introduction

Let the discretized equation of motion

$$M\ddot{u} + C\dot{u} + Ku = F_{ext} \quad (1)$$

The centered difference explicit scheme - which corresponds to Newmark scheme □=1/2, □=0 - is conditionally stable:

$$\Delta t_{critical} = \frac{2}{\omega_{max}}, \text{ where } \squaremax \text{ is the highest Eigen frequency of the system[1]} \quad (2)$$

Assuming there is no damping, the current acceleration at time t is calculated by solving:

$$M\ddot{u}(t) = F^{ext} - F^{int}, \quad \text{avec} \quad F^{int} = Ku(t) \quad (3)$$

The use of a lumped mass matrix instead of the consistent mass matrix avoids solving simultaneous equations for solving (3).

<footnote n="1">1 ALTAIR Engineering Inc., jpb@altair.com</footnote>

<footnote n="2">2 ALTAIR Development France, bertrand@altair.com</footnote>

<footnote n="3">3 ALTAIR Development France, lionel@altair.com</footnote>

<footnote>* Corresponding Author</footnote>
Commercial explicit codes generally provide a way of raising the time step by adding (diagonal) mass to the nodes. This method has several drawbacks, namely the non-conservation of the rigid body translational momentum, and the modification of the low frequencies of the system.

On the other hand, mesh sizes in crash simulation tend to become smaller. A 5mm mesh size is commonly used, possibly less,in order to model local phenomena like ruptures. Even considering that commercial explicit codes are highly parallel - benchmarks realized with RADIOSS in 2012 demonstrated scalability up to 4,000 cores with an industrial model of 15 Million elements whose size was 2mm - there is a main interest to develop methods allowing to increase the time step while preserving the result accuracy.

## 2. Selective Mass Scaling

### 2.1. Basics

Olovsson and al. [2,3] proposed to add additional terms to the lumped mass, aiming at decreasing the highest frequencies of the system, while affecting the lowest frequencies of the system as little as possible.

Setting:

$$ \bar{M} = M + \Lambda, \text{ avec } \Lambda = \alpha K \quad (4) $$

Eigen frequencies are modified, however, low frequencies are less affected:

$$ \bar{\omega}^2 = \frac{\omega^2}{1 + \alpha\omega^2} \quad (5) $$

Instead of assembling the stiffness matrix at each time step (due to the system non-linearity), they proposed to assemble □□□ from constant elementary matrices so that the rigid body translational acceleration of an element is not modified. For instance, in a 4-node shell element:

$$ \lambda_e = \frac{\mu_e}{12} \begin{bmatrix} 3 & -1 & -1 & -1 \\ -1 & 3 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ -1 & -1 & -1 & 3 \end{bmatrix} \quad (6) $$

where $\mu_e$ is chosen in order to raise the elementary time step.

Equation (3) can be solved with a conjugate gradient method up to some tolerance on the residue. Practically, the scheme remains stable with a tolerance of about 10-3 to 10-5, so that a limited number of iterations insufficient.

An original implementation in RADIOSS so-called “Advanced Mass Scaling” was designed to obtain important performance gains for varied industrial cases.

### 2.2. Roof Crush

The stamping process simulation and more generally, the quasi-static analysis, are particularly well-suited to the use of Selective Mass Scaling, since the energy content in the high frequency range is very small.
We present the results of a roof crush simulation, based on the TAURUS public domain; the speed of the impactor is 1m/s for a total displacement of 200mm. We compare the results which were obtained with:

An imposed time step of 0.5s and diagonal added mass (/DT/NODA/CST) – we checked a posteriori that the mass increase was small enough;

"Advanced Mass Scaling" and a time step of 10s (/DT/AMS).

Figure 1 compares the obtained deformations with the two methods, and figure 2 compares the reaction force on the impactor. Elapsed times on 16 cores where:

33,4 hours for 402,044 cycles with diagonal added mass

4,2hoursfor 20,146 cycles with “Advanced Mass Scaling”, for a greater than 7speedup.

<figure><img src="image_2.png" /><figcaption>Fig. 1: deformations</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: reaction force on the impactor</figcaption></figure>

# 3. Multidomains

## 3.1. Aim of the method

The Multi-Domain method is another solution to significantly improve the performances of explicit computations while preserving the physics of the problem.

This method is well suited to problems where some parts of the model have a very low time step compared to the rest of the model. These parts can be computed based on their low time step in a separate domain - so-called "subdomain" - while the rest of the model is computed using a much higher time step.
Considering for example a model with Nt elements, made of two domains whose respective time step and number of elements are respectively noted dt1, dt2 (with dt2< dt1), and N1, N2, and assuming the average cost per cycle and per element is the same for both domains, then, the following formula gives a the speed-up estimation provided by the Multi-Domain method:

$$ \alpha = \frac{Elapsed\_time_{monodomain}}{Elapsed\_time_{multi-domain}} = \frac{N_t}{N_1 \frac{dt_2}{dt_1} + N_2} = \frac{1}{\frac{dt_2}{dt_1} + \frac{N_2}{N_t} \left(1 - \frac{dt_2}{dt_1}\right)} \quad (7) $$

Thus, the speedup can be very high, especially if the relative size of the subdomain is small and the time steps ratio is large.

## 3.2. Multi time scale method

The coupling method in RADIOSS belongs to the family of “non-overlapping” domain decomposition FETI methods. Proposed by Combescure and Herry [4], the method used in RADIOSS is suitable for both non-matching time and space interfaces.

Whereas RADIOSS Multi-Domain computations actually use matching space interfaces, non-matching time interface is a key issue as completely independent time scales must be used. Indeed, for non-linear explicit computations, the time steps of both domains are evolving continuously making the use of a constant time step ratio and the synchronization of the domains very difficult.

## 3.3. Usability

Setting of Multi-Domain computation has been done very convenient for the user who only needs to specify a list of parts within a classical RADIOSS input deck. Then a subdomain is automatically created by the initialization sequence of RADIOSS(figure3) which extracts the subdomain, determines the common nodes and all the connections between domains and sets all the needed coupling parameters.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Flow chart</figcaption></figure>

<caption>Fig. 4: Coupling scheme</caption>
The Multi-Domain computation starts by launching two RADIOSS computations (one for each domain) plus the coupling program RAD2RAD. This program will compute the coupling forces and ensure the synchronization of the RADIOSS processes. RAD2RAD will determine along the time which RADIOSS process has to be activated and will put the other one in a sleeping mode. Thus, all the CPU resource is alternatively used by RAD2RAD and the 2 RADIOSS processes (figure 4).

Moreover, this solution has been adapted to highly parallel computations, where a big number of processors are used, which requires the parallelization of the RAD2RAD executable and of the communications between processes.

# 4. Robustness analysis

## 4.1. Problem description

We study the behavior of the front beam of the TAURUS public model when impacting a pole at 5m/s. The steel beam is represented using a Johnson-Cook elastic-plastic material [5] and a yield stress equal to 800 MPa. We also set the Johnson-Cook failure criterion, with

$$D_1 = 0,174 \quad D_2 = 0,0559 \quad D_3 = 3,9025 \quad D_4 = D_5 = 0$$

The beam has been refined at its center in order to accurately model its rupture and buckling against the pole. The mesh size in the central part is about 1mm.

<figure><img src="image_5.png" /><figcaption>Fig. 5: Global view</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 6: Beam mesh</figcaption></figure>

We compare Multi-Domain computation with respect to the standard computation. Using the standard computation, a minimum time step of 0,1µs is used. Using Multi-Domain, the time step of the central part of the bumper beam is set to 0,1µs while the time step of the rest of the model is set to 0,5µs.

We compare the following results:

* Section forces through the left and right rails (which are highlighted in figure 5)

* Accelerometers are filtered using an SAE 180 Hz filter

* Deformation and rupture modes of the central part of the bumper beam

* The performances using SPMD version of RADIOSS on 48 cores

A sensitivity analysis is performed in both cases (10 runs of each case are launched using a 1µm random noise magnitude on the all node coordinates).
## 4.1. Results

In the figures below, the results of standard computation are presented in red, and Multi-Domain in blue.

<figure><img src="image_7.png" /><figcaption>Fig. 7: Beam deformation</figcaption></figure>

Results obtained with multi-domain are very close compared to the standard simulation. In both cases, the buckling mode (figure 7) is well captured, and the same (horizontal) crack appears on the beam reinforcement. These results do not show dispersion along the sensitivity analysis.

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8: Sections & Accelerometers</figcaption></figure>

The section forces and accelerometers signals show some dispersion along the sensitivity analysis, but the Multi-Domain results are quite close to the standard computation ones.

The “speed-up” compared to the standard simulation is 2.6 with Multi-Domain.

Conclusion

AMS allows increasing the time step without changing the rigid body translational momentum, neither affecting the lowest frequencies of the system. It is very well adapted to quasi-static simulations where it demonstrates to provide a very high quality of the results and very much efficiency.

Multi-domains method is suitable for cases where the time step of some parts is very much low with respect to the rest of the model. It can be applied particularly to the cases where a fine mesh is needed locally to accurately model some phenomena such as rupture, etc.

References

[1] T.J.R. Hugues, The finite element method: linear static and dynamic finite element analysis, Prentice Hall
Engelwood Cliffs New Jersey 0763, 1987.

[2] L.Olovsson, K. Simonsson, M. Unosson, Selective mass scaling for explicit finite element analyses, International Journal for Numerical Methods in Engineering, 63, 1436-1445, 2005.
[3] L.Olovsson, K. Simonsson, Iterative solution technique in selective mass scaling, Communications in Numerical Methods in Engineering, 22, 77-82, 2006.

[4] A.Combescure,A.Gravouil,B.Herry, An algorithm to solve transient structural non-linear problems for non-matching time-space domains, Computers & Structures, 81, 1211-1222, 2003

[5] G.Johnson, W.Cook "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures", 7th Symposium on Ballistics, The Hague, 1983.