ESI LIVE 2023
INNOVATION TALKAdvanced reduced order modeling for more effective structural crashoptimization
Gavin England, Senior Product Marketing Manager, ESI Group
Fatima Daim, Team Leader Computational Science, ESI GroupIn this article, we're diving into ESI’s game-changing technology called ADMORE. This isn't just anothertech tool—ADMORE represents a leap forward in how engineers approach structural crash optimization.
Imagine having a tool that not only speeds up the design process but also pinpoints the most crucialfactors for achieving optimal safety and performance in vehicle structures. That's exactly what ADMORE
does. It transforms complex simulations into more manageable, insightful, and actionable data. But before
we get into the nitty-gritty and see a real-life use case from our partnership with Renault, let's set thescene with some background on why this technology is such a breakthrough.

What is the current dilemma automakers face when developing new vehicle concepts?

To optimize vehicle safety against performance, many unknowns are considered and assessed. To
ensure occupant and pedestrian safety, and adhere to ever-stricter regulations, those unknown variables
include many different crash scenarios and impact positions as well as an increasing diversity of humanmodels. At the same time, to achieve the best vehicle performance at minimal cost and in the shortest
possible lead times, variables such as different design configurations, component size options, materialtypes, thicknesses, and the impact of the range of component tolerances need to be considered. These
1|9variables usually affect each other as they change, so they shouldn't be assessed in isolation. Somecombinations of variable values will have a big influence on the design, either positively or negatively,
whilst others won’t. When you consider the different permutations of these combined variables, there arepotentially many thousands of possible outcomes. To fully explore those unknowns, to assess every
combination of variables using simulation, to deliver the best product, is simply not possible. The numberof simulation runs and design iterations is limited by time and cost.
So how do you know which of these unknowns have the biggest influence on the design, and, therefore,
which variable values should you focus those precious simulation runs on to achieve the best crashsimulation results whilst optimizing vehicle performance? This is the challenge facing many automotive
manufacturers.

To solve this, ESI developed a unique solution, called ‘ADMORE’. What is it about andhow does it help solve this problem?
To help solve this and other similar problems, ESI has
developed ADMORE, an intelligent model order
reduction technology. ADMORE quickly guides
engineers to the variables that have the most influenceon the product, and so are the most important toassess, so that they know exactly where to focus theirsimulation runs. It does this using advanced modelorder reduction, driven by intelligent machine learningtechnology. It is a powerful decision-support tool forengineers.
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ADMORE enables engineers to build an intelligent parametric crash simulation model and carryout real-time design space exploration to quickly examine the impact of changing multiple
parameters, to assess which have the greatest influence on crash scenarios. For example,
which combination of parameter values is likely to optimize the strength-to-weight ratio of thepart of the vehicle, or, conversely, could potentially cause a safety or regulatory issue in a crashsituation?
Whether those parameters are material thicknesses of structural components, limits ofmanufacturing tolerances, or different impact positions, ADMORE enables fast parametric
design space exploration to quickly assess those unknowns. It provides quick insight forengineers, who will have a much clearer understanding of the direction they should take thoselimited number of simulation runs to deliver the most meaningful, beneficial crash simulation
results.
ADMORE enables both live manual and automated design space exploration to quickly andeasily determine the priority load cases and design parameters to analyze in more detail. The
nature of ADMORE, with its intelligent automation and real-time assessment capabilities, meansthat it can be used by both experienced and non-experienced engineers, helping to democratize
simulation tasks.
ADMORE optimizes simulation cost v accuracy. It enables a better, more optimized productdesign for the same investment and effort.

With regards to crash simulation, what are the details on this specific use case?
ADMORE can be applied to many different use cases, particularly where there are multiple combinationsof variables to assess. For crash simulation, it brings value in three main areas:
2|9Firstly, it enables the development of a highperforming vehicle whilst ensuring safety regulations
are easily met. Being able to focus simulation runs on
the most critical scenarios results in improved safety
outcomes. Having a better understanding of theevolving vehicle design enables more informed
development decisions to be made. And thedemocratization of simulation activities frees specialist
engineers to work on other high-value activities, for amore optimized vehicle design.
Secondly, it helps reduce vehicle development time. Lead times are cut due to the decrease in highfidelity crash simulations required, new vehicle configurations can be assessed for safety much faster,
democratization of the simulation processes helps release additional specialist engineering capacity, andautomation further helps cut the need for time-consuming, laborious tasks.
And, finally, it helps to decrease development costs. The need for expensive physical testing and highfidelity simulations is further reduced, and the improved understanding of the evolving design reduces therisk of potentially costly unforeseen downstream errors.

Looking at the mechanics behind ADMORE: How is a high fidelity model typically used insimulation and how is optimization study commonly carried out?
Let us start by introducing the classical numerical simulation scenario, which we refer to as the highfidelity model.
This High Fidelity model (HFM) describes the evolutionof a physical model, given by discretized partial
differential equations (PDE) denoted here as sigma, ona mesh M during time interval T. This evolution is
described by the variable u at each space point x andeach instant t, starting from the initial condition
denoted Uin. In crash simulation, U is thedisplacement, velocity, or acceleration. In general, forsuch simulations, what interests us is the performanceof this physical system denoted here by O, which
depends on the variable U. In the crash simulation,these performances are a set of intrusions measuring the occupant safety during the crash.
The performances are, in general, obtained by post-processing the variable U. They are usually very
quick to calculate once U is computed. What is expensive, however, is the computation of the U variable
itself, which in the case of the finite element method, is given as a combination of shape functions
denoted Ni. The coefficients of this combination are computed at each time step by solving a linearsystem. in the case of implicit time discretization. Or under a set of linear systems imposed by the CFL
condition in the case of an explicit scheme, which is the case in crash simulation. This combination isstrongly linked to the fineness of the mesh. The finer the mesh, the higher the computational cost.

3|9This complexity increases further when we consider
design parameters, denoted here as P. These couldbe, for example, the varying thicknesses of structuralparts. Therefore, the variable U depends also on
those parameters. In structural car optimization, weneed to evaluate the performance for eachconfiguration of parameters.

When carrying out an optimization study, the classical workflow typically looks something like this:
1. We first develop a design of experiments (DOE)
where some key parameter combinations aredefined, denoted here as P1, P2, etc.
2. We then launch several high-fidelity modelcomputations to obtain the related variable U andthen use post-processing to obtain the related
performances
3. Next, we construct the response surfaces basedon these performances. This surface is built based on
an algebraic tool to interpolate between theperformances related to DOE.0
4. Finally, we use interpolation on the response
surfaces to compute the performances required by the optimization algorithm

How using ADMORE, can a reduced order model be built?
By introducing reduced order models, we aim to
drastically reduce the description of the variable U.
Instead of looking for U as the combination of a large
number of shape functions, a reduced order model(ROM) builds U as a combination of a small numberof functions, denoted here by Si, known as reduced
basis or modes. The response surfaces reduce thecomplexity of the performances, which are functions
dependent on the variable U. Essentially, the reducedorder model tackles the cause of the computational
complexity in the calculation of U.
To define such reduced bases, we can use a variety of methods. In this presentation, we will focus on justtwo, the Proper Generalized Decomposition (PGD) and Column-Row Decomposition (CUR) methods.

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Let’s, first, summarise how the reduced order model
will be used in an optimization workflow:
1. We start by developing a design of experiments,
where some key parameter combinations are defined,denoted here as P1, P2, and so on
2. We then launch the high-fidelity model computations
to obtain the related variables U(P1), P2, etc.
3. The reduced order model is then built, which allows
the computation of the variable U for any combinationof parameters
4. Then we use the reduced order model to approximate the variable U for any combination ofparameters and post-process it to obtain the related performances required by the optimization algorithm.

With regards to the sPGD method, how can this approach be applied to an optimization
study?
sPGD stands for sparse Proper Generalized Decomposition and was developed by ESI’s ScientificDirector Prof. Francisco Chinesta. This method considers the parameters as extra coordinates of space
and time. It also approximates the variable U as a combination of a separated representation of thespace, time, and parameter coordinates. This combination contains a reduced number of functions, ormodes, depending separately on space, time, and parameters.
How do we do that? Well, once the design ofexperiments is defined and the related high-fidelity
model computation is launched, we collect, for eachcomputation of parameter combination, the variable U
as a matrix. To illustrate this decomposition, wedevelop a 3D representation of this - one axis forspace, one for time, and one for the parameters. ThesPGD method collects these modes iteratively. For thefirst mode, we look for 3 vectors whose product best
approximates the cube. For the second mode, thethree vectors whose product best approximates theresidue are found, and so on. We continue until we reach a good approximation by collecting N reducedorder model modes representing the matrices U of the high-fidelity model.

What is the second reduced order model method “CUR” about and how can we apply
this approach to an optimization study?
The second method is based on the CUR
decomposition usually used to compress matrices.
For example, for a matrix denoted X, we approximate
it by the products of three matrices C, U, and R. C isa set of selected columns of X, R is a set of selectedrows, and U is a small matrix combining thesecolumns and rows. This also means that each cell in
X can be explained or approximated by a combinationof a product of one column and one row.

5|9For a parametric study, we extend the CUR decomposition to approximate Xp by regression for any
parameter combination outside the DoE. Re-CUR comes from the phrase ‘Regression CUR’.
From high-fidelity simulations, we collect the Xp
matrices containing displacements of the variable U.
We apply the Empirical Interpolation Method to all theXp matrices to collect columns and rows denoted,
respectively, C and R. Each Xp is approximated by aCUR decomposition where the link with parameters
is supported by the matrix Up. To interpolate for any
other combination of parameters, we have tointerpolate the Up matrices. To achieve this, we canuse a linear regression, such as a least squares orlasso method, or a nonlinear regression via machinelearning. In this study, we used the random forest method.

Is there a real-life case study of applying ADMORE to a specific crash simulation case?
There’s a use case where we deployed this approach for crash simulation, specifically to assess vehicleside member reinforcement methods. This is based on work we carried out in partnership with Renault.
This is an example of a Mobile Progressive
Deformable Barrier or MPBD test. Here we used ahigh-fidelity model with 6 million nodes and 8 million
elements. We are interested in 31 different intrusions
to measure and assess occupant safety, and, bearing
in mind that one high-fidelity simulation took 12 hours
on 144 processors, this number of quantities ofinterest will have a significant effect on the simulation
performance.

The optimization considered in this particular study isa reinforcement method. This method is applied
during the project vehicle development when theperformance of the nominal design, determined by
the crash development team, deviates from thetargets and no longer satisfies the safety criteria.
This deviation is typically due to constraints imposedby different service teams involved in the project
development. This method looks to reinforce orreduce local zones in the structure using triggers
such as safety targets.
In the case of the model in our study typically theside member is divided into 23 local zones to be
weakened or reinforced by varying each thickness
zone by +- 0.2 mm. These local zones are thosemarked in blue and red.

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In practice, and to identify the local zone to reinforce or reduce, the engineer starts by varying each zone
one at a time using high-fidelity models to find a solution. In this case that equates to 46 high-fidelity
simulation runs, which takes a significant amount of time and computing cost. If any solution is found fromthose 46 runs, it is sent to the optimization team for further work.

What was the goal of this study with Renault?
The objective of this study is to use ADMORE to accelerate this process and find a solution throughsignificantly fewer high-fidelity computations. For this optimization problem, with those 23 parameters,
high-fidelity simulation runs need to be first carried out to build the reduced-order model. Using sPGD, 24runs are required, or, using ReCUR, just 12 runs are needed. We then carry out an iterative, orcombinatorial, exploration of the zones to reinforce or to weaken to meet the safety targets:••

In iteration one, we explore configurations that reduce or weaken one zone at a time. So, forthose, 23 zones, 46 configurations need to be assessed, using the two reduced-order modelingmethods, sPGD and ReCUR. Among these predictions, we select the best candidate giving thebest performance and we launch a related high-fidelity simulation to validate this candidate.
In iteration two we explore configurations that reduce or weaken two zones at a time, which
requires 1012 predictions to be carried out with the two ROMs. We, again, choose the candidate
giving the best performance and launch the relevant high-fidelity simulation.

How does the solution look like in the software?
In this video you can see the reduced order modelusing sPGD, where our 23 parameters can be varied
using dynamic sliders. The sliders enable theparameters to be varied, changing the thicknesses ofthe 23 zones from -0.2 mm, weakening the zone, to0.2mm, reinforcing the zone. The sliders are all on
zero at the start, which represents the nominal case,so we can see the shape of the nominal side
members. Moving the sliders for one value toanother shows how the side members behave anddeform with reinforcing or weakening of the zones.

What are the results that Renault got from this work?
This image shows the quality of the reduced ordermodel built via the sPGD method. We have asection view of the car during the crash simulation,comparing the high-fidelity in blue and, in red, thepredicted results using sPGD. The view on the left
shows a simulation run where the parameters are
set to the same as one of the 24 configurations
already used in the design of experiments training.
The view on them left uses parameter configurations
outside of those used in the design of experiments
stage.
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If we focus on the frame where the intrusions are at the maximum, we can see that the predictions arequite good, specifically on the footwell area where the intrusions are measured for the occupant safety.
The image below shows a comparison between, in blue, the high fidelity computations of the nominal
which does not satisfy the safety targets, and, in red, the high fidelity computations of the best candidates
identified by the sPGD model.
• On the left, showing iteration 1, where we reduceor weaken one zone at the time, zone 1 wasselected to be weakened.
• On the right, for iteration 2, where we reduce orweaken two zones at the time, the adjacentzones 1 and 2 were selected to be weakened.
When focusing on the frame where the intrusions
are at the maximum, we can clearly see thedifference in deformation.

Similar to the previous example, where we used the sPGD method, this shows the quality of the reducedorder model built, this time, via the ReCUR method. As before, we have a section view of the car during
the crash simulation, comparing the high-fidelity in blue and, in red, the predicted results using ReCUR.
The view on the left shows a simulation run where
the parameters are set to the same as one of the 12
configurations already used in the design ofexperiments training. The view on them left uses
parameter configurations outside of those used in
the design of experiments stage.
If we focus on the frame where the intrusions are at
the maximum, where they are measured, we cansee that the predictions are, again, good, specificallyon the footwell area where the intrusions aremeasured for the occupant safety.
Again, as with the previous sPGD example, here we see a comparison between, in blue, the high fidelity
computations of the nominal which does not satisfy the safety targets, and, in red, the high fidelity
computations of the best candidates identified by the ReCUR model.
• On the left, showing iteration 1, where we reduceor weaken one zone at the time, zone 5 wasselected to be weakened.
• On the right, for iteration 2, where we reduce orweaken two zones at the time, the adjacentzones 5 and 6 were selected to be weakened.
When focusing on the frame where the intrusions
are at the maximum, we can clearly see thedifference in deformation where new collapse
modes were detected by weakening zones 5 and 6
which appear to be critical.

8|9What are the conclusions that were reached during this study?
During their presentation at ESI Live 2023, Fatima summarized the main project success as follows: “Firstand foremost, we are happy that it was confirmed that Renault plans to use these methods in car
development projects.”
There are three main outcomes that were reached so far:
• Optimisation studies for crash simulations using reduced order models significantly reduces thecomputational time and costs compared to a classical response surface optimisation approach• The two reduced order model methods considered for this particular real-life industrial case wereboth effective in optimising side members, though actual engineering teams will use different
solution, for example triggers.
• Both methods have successfully identified the front of the side member as candidate zones to beweakened.

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