Numeric Optimization of Metallic Component With Deformation

in the Plastic Regime

Hugo Jorge Goncalves Marqueshugo.marques@ist.utl.pt

Instituto Superior Tecnico, Lisboa, Portugal

May 2014

Abstract

This work reports the topology optimization made to a metallic component that deforms plasticallyduring loading. The optimization had the objective of minimizing this component’s weight.

The part in study is a roll hoop of a formula one car that must be subjected to static tests in orderto evaluate its strength. To do so, a force of 119KN is applied by a circular steel pad, and to accountwith this loading method, geometric non-linear analysis with contact interaction are performed withthe help of commercial codes.

The manufacturing and testing procedures were also closely followed in this work.Keywords: Topology optimization, Contact analysis, Non-linear material model, OptiStruct, Geo-metric non-linear analysis

1. Introduction

The roll hoop is a safety structure placed behindand above of a formula 1 driver’s head. It’s mainfunction is to protect the driver in case the car acci-dentally rolls to an upside down position, but it alsoserves as an air intake for cooling the car’s engineplaced beyond and bellow its position. Therefore,this component not only has to have a certain aero-dynamic shape, as it also must be strong enoughto withstand several times the weight of the car, inorder to account with the potential scenario wherethe car is accelerated towards the ground in the in-verted position.

Nevertheless, because formula 1 is such a com-petitive sport, is crucial for the car to be made ofefficient and lightweight parts, or otherwise it willbe impossible to achieve interesting results. Bear-ing in mind the aerodynamic requirements that rulemodern cars, a formula 1 car is designed under sim-ilar premises to the ones airplanes are.

Due to its aerodynamic requirements, the shapeof the roll hoop is reasonably complex. Due to thisconstraint it has been decided that the lightest andeasiest way of manufacturing this part was to 3Dprint it with a high strength metallic alloy. Fortu-nately 3D printing allows for almost complete free-dom in the design, potentiating the creation of veryefficient structures.

1.1. ObjectiveConsidering the facts here named, the objective ofthis work is to topologically optimize a given de-sign domain (or starting point geometry) in orderto obtain an optimum geometry that minimizes theweight of this component, without compromisingthis strength.

The optimization will be performed by the useof finite elements analysis (FEA) commercial codessuch as OptiStruct/Radioss [1] and Abaqus [2].

The optimization must obey the regulationsestablished by the Federation Internationale del’Automobile (FIA), the governing body of the 2012Formula One Championship.

2. Technical RegulationsPreviously to the beginning of the season, the FIApublishes the technical regulations that will governthe competition during the upcoming year. All thecars competing in the event must obey these regu-lations and pass all the required testing establishedin the same.

Specific geometric and strength rules are estab-lished for roll structures in which the roll hoop isincluded.

2.1. Geometric ConstraintsIn essence, the geometric constraints must respectthe dimensions described in figure 1. If this di-mension are respected and the roll hoop is strongenough, the drivers head and hands will not touchthe ground in case the car is in the inverted po-

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sition. The roll hoop (which in figure 1 is namedprincipal roll structure) must also have a minimumenclosed structural cross section of 10000mm2, invertical projection, across a horizontal plane 50mmbelow its highest point. The projected area mustnot exceed 200mm in length or width and may notbe less than 10000mm2 below this point.

Figure 1: Geometric constraints.

2.2. Strength RequirementsRegarding its strength, the roll hoop must undergoa static test where a combined load of 119,16KN isapplied on its top by a rigid and circular at pad.The pad has has a diameter of 200mm and it’s sur-face is perpendicular to the loading direction. Thisload is the combination of 50KN in the lateral direc-tion (perpendicular to the car’s traveling direction),60KN longitudinally (opposite to the car’s travelingdirection) and 90KN vertically in downwards direc-tion.

The load must be applied in less than 3 minutesand maintained for 10 seconds, while deformationunder loading must be less than 25mm measuredalong the loading direction. Structural failure islimited to 100mm bellow the top of the structuremeasured vertically. The testing must be made withthe roll hoop mounted on top of the survival cell(monocoque) while this is supported in its undersideand sides.

Is also mandatory to present detailed calculationdemonstrating that the roll hoop also withstandsthe same load when the longitudinal component isapplied forwards (60KN applied in the car’s travel-ing direction).

The roll hoop also serves as hoisting point for thesituations where the car must be lifted by a crane.The FIA does not provide any loads cases to studythis situation, leaving the teams the responsibilityof doing it. In our case we consider the situationwhere a fully loaded car, (driver and full fuel tank)with an approximate weight of 820Kg, suffers a 3Gacceleration (29.4m=s2) in the upwards direction.Thus the load case where a 24.2KN force is appliedin the underneath of the roll hoop is also considered.

With the regulations is mind, an optimizationproblem as been set up as the following chaptersdescribe.

3. Methodology

As said before the numeric tools used in this workwere Radioss/OptiStruct and Abaqus. So here isbrief description of the reasons why.

3.1. Radioss and OptiStruct

Radioss is a Nastran based finite element (FE)solver. From the user point of view, and consideringthe results outputted by both softwares, Radioss isvery similar to the more popular Nastran. Radiossis used in the majority of the linear static analysisperformed in this work and comes integrated in thesoftware package HyperWorks 10 (version used inthis work), that also includes the topological opti-mizer Optistruct.

OptiStruct uses the analytical capabilities of Ra-dioss to compute responses for optimization. Theresults of Radioss are analyzed by OptiStruct whereobjective and constraints are evaluated, and newdesign variables are defined. With this new vari-ables a new solution to the problem is computed byRadioss and the process is repeated until an opti-mum solution is found.

For Topology optimization the method used byOptiStruct is the very popular Solid Isotropic Ma-terial with Penalization (SIMP) [3]. In this method,to each element in the design domain is attributed adensity value ρ ranging from 0 to 1, that will take atool in elements stiffness matrix according to equa-tion 1,

K(ρ) = ρpK, p > 1. (1)

In here, K and K represent the element originalstiffness matrix and penalized stiffness matrix, re-spectively, and p is a penalization factor that willmitigate intermediate densities [4], which by experi-ence has been set equal to 2. Low density elementshave very low stiffness and behave like voids, andthis way the material removal is simulated withoutremoving elements from the mesh.

3.2. Abaqus

Considering the extreme load to be applied, it isknown that the roll hoop deforms considerably andthat the stresses get dangerously close to the mate-rial’s yield and ultimate strength. Besides, it is alsoimportant not to forget the effect of the contact be-tween the loading pad and the roll hoop. For thosereasons a geometric non linear analysis with mate-rial non-linear behaviour and contact interference isperformed.

Although capable of performing geometric nonlinear analysis, Radioss and Nastran do not havesuch a good performance in this kind of studies [5].For this reason the FE solver used for geometric nonlinear analysis is Abaqus. This solver is famous forits capabilities on the non-linear domain, not onlybecause of its accuracy but also due to the reliable

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algorithms that automatically help stabilizing thecalculus process.

Contact Formulations

Abaqus provides the user with several contact for-mulations based on a choice of a contact discretiza-tion and a tracking approach. The correct choicedepends on the type of contact we are dealingwith and on the computational cost. Indepen-dently of the user’s choice, a set of master and slavenodes/surfaces must be defined in order to identifythe elements that will be in contact.

Abaqus offers two possible contact discretizationmethods: node-to-surface and surface-to-surface.

In node-to-surface discretization the nodes fromthe slave surface interact with their correspondingprojection on the master surface. The slave nodesare then constrained not to penetrate the mastersurface, although the opposite may occur if themesh of the slave surface is not at least as refinedas the master [2].

In surface-to-surface discretization the shape ofboth master and slave surfaces are considered tomodel the contact interference, since contact direc-tion is based on an average normal of the slave sur-face region surrounding the slave node. Thus, withthis method, contact forces are distributed in anaverage region of slave surface, and not individualnodes as in node-to-surface discretization [2].

The choice between these methods depends onhow precise/detailed we desire our analysis to be.Surface-to-surface discretization usually providesa smoother and more accurate stress distributionthan node-to-surface, since contact forces are notconcentrated at individual nodes as in node-to-surface discretization.

Because contact forces are distributed in an aver-age region of slave surface, surface-to- surface inter-action is also less sensitive to mesh refinements andattenuates the differences between slave and mas-ter surfaces. However, surface-to-surface discretiza-tion requires more constraints per node, which canincrease the solution cost specially in models withlarge contact areas, or several contact surfaces thatinteract between each other (a surface that simulta-neously acts as salve and master for different pairsof surfaces).

Considering that in node-to-surface there is thenatural tendency to concentrate the load on indi-vidual nodes leading to higher peak stresses, thisdiscretization method becomes more conservativewhich often is a desired characteristic in engineer-ing applications. Since it is also computationallycheaper to run, this was the discretization methodchosen in this work.

Contact Tracking Approaches

The contact modulation must be supported by atracking approach that simulates the sliding be-tween the master and slave surfaces. For this,Abaqus also provides two possibilities. Finite-sliding and small-sliding

Finite sliding allows for separation, sliding androtation of the interacting surfaces as the relativemotion goes on. For the particular case where finite-sliding is used together with node-to-surface dis-cretization, it is necessary for the master surfaceto have continuous surface normals at all points, orotherwise, slave nodes may get stuck at the pointswith this normal discontinuities, leading to conver-gence problems. To prevent this from happening,Abaqus automatically smooths the surface normalsat the required locations, but the user can still man-ually control the level of smoothing.

Small-sliding is a approach better suited for prob-lems where the motion between the slave and mas-ter surface is small or even null. However, this doesnot invalidate large deformation and rotation (geo-metric non-linear behaviour) as long as sliding be-tween slave and master surface is kept small.

In small-sliding a given slave node will always in-teract with the same subset of nearest master nodesthroughout the entire analysis. While still in theundeformed shape, for a given slave node, a group ofnear master nodes is chosen to interact with. Thesenodes will define a local tangent plane, and all thecontact interaction between the salve node and themaster surface, such as load transfer and sliding,happens on this plane. The local tangent plane re-mains fixed relatively to the master surface, but fornon-linear geometric analysis its position is updatedin order to follow the rotation and deformation ofthe master surface.

Small-sliding is better suited for problems whererelative sliding is within half of the element lengthfor relatively plane surfaces. Since the loading padand roll hoop are both flat and parallel in the con-tact area and sliding is expected to be practicallynull, small-sliding was the tracking approach cho-sen for this work. Specially when considering thatsmall-sliding is computationally cheaper to run,once is not required to track for possible contact inother regions of the master surface when the localtangent planes have all been defined.

Non-linear Material ModelAbaqus provides several different formulations tomodel material plastic behaviour such as rate-dependent strains and temperature dependence.But for our situation we are dealing with anisotropic material which will be slowly loaded at alow temperature. Thus, we can reduce our focus totemperature and strain-rate independent material

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models.For our case, the material elastic modulus was

defined linear until yield, and null from then on.As if stiffness modulus E2 and E3 from figure 2where zero. The yield criteria used is the Misesyield surface [1], [2] and [6], which in brief words,means that the yield is considered when von Misesstresses are equal or greater than the yield stress.

σ

E3

E1

ε

E2

Figure 2: Material plasticity approximated by por-tions of linear behaviour.

4. Calculus ResultsHere we present the most relevant calculus made tothe the roll hoop, that eventually lead to the finaland optimum geometry obtained for the roll hoop.

4.1. Topological Optimization Run

The initial geometry can be seen in figures 3 and4. This is the starting geometric point for the op-timization run, and is the combined result of theconstraints defined in the regulations, manufactur-ing constraints, aerodynamic functions and interac-tion with the monocoque.

343.6

176.0

93.3(*)7

39.8

(*)Distances between the beginning ofthe fillet radius

207.5

26.3

Figure 3: Roll Hoop dimensions (dimensions inmm) - Side view.

The design domain used in the optimization runis the area colored green in figure 5. The elements in

9.7

288.6

148.8

172.7

Figure 4: Roll Hoop dimensions (dimensions inmm) - Front view.

grey and yellow colour are near the load applicationarea and naturally tend to concentrate the loads.For this reason they are excluded from design spacein order to guaranty the this region remains solid.

Figure 5: Roll hoop design domain colored green.

Then the optimization run was setup in the fol-lowing manner: the objective function defined wasto minimize the roll hoops mass, and displacementconstraints were set as a maximum displacement of20mm in the loaded nodes. Regulation impose amaximum deformation under loading of 25mm but,as precaution measure and in order not to get toclose to this limit, 20mm is the maximum allowedin the optimization.

The finite element model (FEM) was mainlymade of 2mm wide CTETRA elements (triangu-lar pyramids of four vertices’s), which are not asdesirable as the hexagonal CHEXA ones [7] [8] (ele-ments in shape of a cube). However since the geom-etry of the roll hoop is considerably complex, it wasmandatory to use the Hypermesh [1] automesh fea-ture that can only generate CTETRA elements for3D meshes. Regardless, mesh refinements was goodenough to overcome this drawback and allowed forgood optimization results as demonstrated in figure6 where an element density plot is shown. In this

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figure, blue colored elements represent zero or nearzero density elements, while the red colour is usedfor elements of density 1.

(a) Side view

(b) Frontal view

(c) Bottom view

Figure 6: Optimization results. Elements density.

Since the optimization was done without bucklingconsiderations, the front legs would easily buckle ifit were made so thin, and it must also be taken inconsideration that some checkerboard results hap-pened in the inner walls of the roll hoop’s entrance.For this reason some reinforcements should be madein these areas in order to account with the optimiza-tion imperfections.

Nevertheless, based on this output the design of

an optimum geometry took place and lead to the fi-nal geometry presented in the following subsection.

4.2. Geometric Non-linear Analysis With MaterialPlastic Behaviour

After the topology optimization, several linearstatic analysis were performed to various geometriesin order to incrementally improve the roll hoop de-sign to its function. After all this iterations, thefinal geometry is the one represented in figures 7.In essence, the strategy adopted to create this ge-ometry was to hollow the original geometry fromfigure 4 and add reinforcing ribs between the twoskins. Afterwards holes were created in order tofurther remove material.

(a) Frontal view

(b) Side view

(c) Bottom view

Figure 7: Geometry from final iteration.

If we hide the outer skin, the layout of the rein-forcing ribs is the one represented in figure 8.

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Figure 8: Reinforcing ribs between the outer andinner skin.

The primary union of the roll hoop to the mono-coque is by an adhesive, however to make the unionstronger, the two parts are also bolted together inthe locations highlighted in figure 9

diameter =4.8mm

diameter =5.4mm

Figure 9: Holes for bolted unions.

The FEM used in this stage had an elementlength of 1.5mm instead of 2mm as in the opti-mization run. The reason for this is to assure agood enough element density through the thicknessof this geometry thiner walls. Apart from this therest of the model is very similar to the one from theoptimization run.

A Von Mises stress plot obtained from the geo-metric non-linear analysis with contact interactiona plastic material behaviour can be seen in figures10. These stress plots are a combination of the high-est stress levels obtain for the load cases where the119,16KN are applied in the forward and rearwarddirection.

In this figures the higher stresses are representedby the hot colours, and the red areas represent theareas where the yield stress has been overshoot.

The areas marked with (a) are the locationswhere the load is applied, and since the laod is so

(e)

(b1)

(a)

(d)

(a) View 1

(b2)

(c)(d)

(b) View 2

(c) View 3

Figure 10: Von Mises stresses for geometric non-linear analysis.

high, is almost impossible to prevent the stressesfrom overshooting the yield threshold. However thematerial plasticity in these areas is not prohibitivelyhigh and not a matter of concern.

In area (d) there are several small areas undermaterial plasticity, but their small size does not rep-resent a threat to the structural integrity of the rollhoop. In the inner skin however, the areas with thestress level higher than the materials yield value arebigger and more threatening. But a more carefulinspection reveals than the ribs behind this region

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remain in healthy levels of stress, which gives us theconfidence to accept the results.

Regions (b1) and (b2) also suffers from materialplasticity but is in similar conditions to region (d),as the affected regions is not to large and under-neath is supported by a rib with stress values wellbellow the yield threshold.

Areas (c) and (e) are a natural points of loadconcentration due to their sharp edged geometry. Alocal change in the geometry would be necessary toproperly solve the problem. However at this stageof the optimization time constraints did not allowedfor such, and since the areas above the yield aresmall and do not overshoot the material’s ultimatestrength allowable, this situation was accepted.

To better perception on how far the yield stresshas been passed, figures 11 plot the maximum prin-cipal plastic strains. Areas in blue represent 0%plastic strains and the element with the higheststrain is plotted in red with a value of 14.5%. Forthe material in question the maximum allowablestrain is 15.6% but the 14.5% registered are onlyfor a couple of elements that are too deformed andthus do not output faithful results.

Ignoring the excessively deformed elements themore realistic maximum strain detected in this plotsis about 12-13% in small areas near the load appli-cation points. A value still reasonably bellow thelimit.

4.3. Hoisting Load Case and Buckling Analysis

The hoisting load cases referenced in section 2 and abuckling analysis were also performed to make sureall the scenarios were studied.

For the hoisting load case the stress levels werevery low and represented no danger at all for the rollhoop’s integrity. Concerning buckling, the eigen-values were considerably high and represented noworries as reported in table 1.

Eigenvalue Load Case2.55 Forward load-3.54 Rearward load3.74 Forward load-4.94 Rearward load5.44 Rearward load

Table 1: Eigenvalues from buckling analysis.

5. Testing Results

In this section we present the results from the statictests made to the roll hoop in order to certify it forcompetition. Unfortunately the roll hoop did notpassed the test at the first attempt, due to reasonsexplained in the following.

Figure 11: Maximum principal plastic strains ob-tained from geometric non-linear analysis.

5.1. First Attempt

Considering the calculations presented in chapter4, it was decided to advance with the fabricationof the roll from figures 7. Unfortunately an errorduring the manufacturing process of the first rollhoop resulted in significant dent on the roll hoop’sleading edge as demonstrated by figure 12. In thisfigure, is clear that approximately 1mm of materialis missing near one the most critical areas of the rollhoop.

As a result from this imperfection the roll hoopfailed the required test for a load of 116KN out of119KN (97.5% of the total load), and two cracksopened in the area where the stresses were knownto be higher (see figure 13).

5.2. Second Attempt

Faced with this problem, a repair was urgent in or-der to approve the roll hoop for competition. At thetime, covering the top of the roll hoop’s leg by weld-ing a thin patch of material seemed to be the bestsolution. So with this idea in mind the FE modelfrom the last iterations was modified in order tosimulate this repair and evaluate its viability.

In this model a layer of shell elements with athickness of 2mm was added in the area were the

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Figure 12: Dented roll hoop.

Figure 13: Cracks formed during testing.

patch was going to be welded (figure 14), and alinear static analysis was ran since it is enough toevaluate the effect of the repair.

Shell elementscovering hole

Figure 14: Repair FEM.

The results from the analysis can be seen in fig-ure 15 where a Von Mises stress comparison be-tween the results from the previous model and thisrepair are presented. From this figure is clear thatthe stress reduction in this area is such, that theplasticity issues no longer exist. Which is a veryimportant factor to consider, since the welding pro-cesses weakens the strength of the alloy. knowingthat this improvement comes at a weight increaseof 16g, this solution seemed appropriate and it wasdecided to advance with the repair.

(a) Model from last iteration

(b) Model from repair

Figure 15: Von Mises stress comparison betweenlast iteration and repair.

Since this formula one team was competing withtwo cars the second roll hoop, with no imperfec-tions, was rushed out of production and a patchwelded in the vicinity of the original failure loca-tion. This proved to be a bad decision, as the weldwas not as robust as it should have been and led toa failure at even lower loads (90 kN), right in themiddle of the weld as shown in figure 16.

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Figure 16: Crack created during the second at-tempt.

5.3. Third Attempt

At this point was mandatory for the roll hoop topass the static test since another failure could jeop-ardize the participation of the team in the 2012Formula One championship. Thus it was decidedto make a more drastic repair.

This time, the leading edge of the roll hoop wassaw and a solid ring of material was welded in itsplace, which resulted in the geometry from figures17 and 18. To have a better perception of how bigthis reinforcing ring is, figure 19 presents the sec-tion increase in the lower area of the leading edge.To handle better with bending, in the middle ofthe ring (half way between the bottom and top ofthe ring) the section is even bigger as it measures26.9x15.0mm instead of 17.0x15.0mm.

Figure 17: Second repair geometry I.

Figure 18: Second repair geometry II.

11.7

11.5

(a) Geometry from last iteration.

17.0

15.0

(b) Geometry from second repair.

Figure 19: Leading edge section increase with sec-ond repair (dimensions in millimeters).

To validate this repairing strategy, a new FEMwas created from scratch with the same basic char-acteristics as the ones from previous iterations. Re-sults from this model are displayed on figure 20 forthe forward load and in figure 21 for the rear loadcase.

From figure 20 is clear how drastically the stresseshave reduced for the forward load load case. Thefrontal ring is now absorbing the majority of the

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Figure 20: Von Mises stress for forward load loadcase.

Figure 21: Von Mises stress for rearward load loadcase.

load, and since it has such massive section, materialplasticity practically doesn’t happen.

For the rearward load situation material plastic-ity still occurs but with the same conditions as thelast iteration reported in section 4. So as previouslyreported the plasticity levels are within the materialstrength, and this result is considered acceptable.

Considering the FEM results here presented itwas decided to advance with the repair, and thistime the roll hoop was capable of withstanding the119.2KN load and passed the test. This solutioncame at a considerable cost in the weight of the rollhoop, but taking in account the circumstances itwas mandatory to make sure that this componentpassed the required tests. And although the rollhoop increased its mass in 15.9%, it only representsan increase of 0.4Kg in the car weight, which is ne-glectful when compared with the car’s total weightof approximately 800Kg.

6. ConclusionsThe focus of this work was to, from a starting pointgeometry, create the lightest structure possible thatcould withstand an extremely high load withoutbreaking. This goal has been reached, even thoughan unfortunate defect alien to this study during themanufacturing process did not allow the prof of thispoint.

With hindsight the welded patch of the first re-pair should have never been added since the sec-ond roll hoop did not presented manufacturing de-

fects and thus it would be able to withstand the119KN load. But faced with the original failure itwas thought to be better to add a local reinforce-ment. This was an incorrect call, which has furthercomplicated the supply problems being faced.

Regardless of the drawbacks faced during thetesting of the roll hoop, the final result obtained wasstill competitive specially if considered the shorttime it took to develop.

Table 2 resumes the results obtained along thisiterative process for a better perspective of the evo-lution along the iterations.

Weight [Kg]Iteration 1 (optimization starting point) 6.21Iteration 2 2.16Iteration 3 2.47Iteration 4 2.62Iteration 5 2.71Iteration 6 2.51First repair 2.52Second repair 2.91

Table 2: Weight evolution along the iterative pro-cess.

AcknowledgementsThis work would not be possible without the agree-ment and consent of both the company OptimalStructural Solutions lda and the Scientific Center ofAeronautic and Aerospace Technologies of the Su-perior Technical Institute. For this reason, I wouldlike to express my deepest appreciation to both.

References[1] Hyperworks Manual Version 10. Altair Engi-

neering.

[2] Abaqus 6.8 Manual. Dassault Systemes.

[3] M. P. Bendsøe. Optimal shape design as a mate-rial distribution problem. Structural Optimiza-tion, 1989.

[4] M. P. Bendsøe and O. Sigmund. Topology Op-timization. Theory, Methods and Applications.Springer, 2003.

[5] Accuracy and performance comparison neinas-tran vs. msc.nastran vs. abaqus vs. marc. NoranEngineering, 2004.

[6] ANSYS 12.0 Manual. ANSYS Inc.

[7] D. L. Dewirst and P. M. Grinsel. The ctetra(10)element. Ford Motor Company.

[8] E. Kevin Arden. Evalutaion od msc’s parabolictetrahedron finite element. SEAWOLF Hull En-gineering.

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