1

Structural optimization of the PSEM GP-16 monocoque prototype using laminated composites João Alves joao.oliveira.alves@tecnico.ulisboa.pt Instituto Superior Técnico, Universidade de Lisboa, Portugal May 2017 Abstract The subject of this thesis is structural optimization using composite materials. These materials are used in a wide range of applications, most noticeably in the automotive and aerospace industries. The goal of this thesis is the development of an optimization algorithm that maximizes torsional stiffness of a monocoque computational model (structural shell) by choosing the best layup for each area while meeting weight restrictions, keeping the structure symmetric and making the output result easy to fabricate by creating large layup zones. A modified DMO (Discrete Material Optimization) method is used and a specialized genetic algorithm is built, employing a newly developed crossover type that resorts to NASTRAN Finite Element Method commercial program to perform structure evaluations. MATLAB is used as a platform to the genetic algorithm but also as an interface between the optimizer and NASTRAN. The computational model obtained meets the required specifications, proving the effectiveness of not only the genetic optimizer but also the developed crossovers ability to create large enough layup zones to provide ease of fabrication. Keywords: Composite Materials, DMO Method, Genetic Algorithm, Zone-Based Crossover, Shell, Finite Element Method, Structural Optimization, MATLAB, NASTRAN

Introduction

The Projecto de Sustentabilidade Eléctrica Móvel (Mobile Electric Sustainability Project in English) or PSEM aims at the development, fabrication and competition with high-efficiency manned electric vehicles. Originally created to participate in the Shell Eco-Marathon race the project now competes at the Greenpower series of events. Although the race event has changed, the main goal stays the same. The rules of the competition are so that every team is forced to use the same powertrain, motor and batteries, which are quite limited in terms of energy density, power and efficiency so a low-drag aerodynamic body and a lightweight chassis are the key for a good performing car. The chosen material for the construction of the monocoque is the carbon-fiber reinforced with an epoxy resin matrix with foam core, due to its high specific-strength [1] and availability from sponsors. A

resin infusion manufacture processes was chosen due to the unavailability and high cost of pre impregnated fibers, this method was deemed to have the best results when compared to other wet fabrication methods, particularly the wet layup method. Goals

Due to the nature of the race event in which the GP16 prototype will run, weight becomes a high priority feature of the build second only to aerodynamics. As said before the chassis of the racing prototype will be made out of carbon fiber-reinforced polymer. Given the nature of the material and its orthotropic properties, the orientation and number of layers of the layup will have a direct impact on the stiffness of the structure, allowing an increase of stiffness for the same quantity of material for a certain orientation and number of layers of carbon fiber-reinforced polymer.

2

Aside from static loads such as the pilot, motor, batteries and car weight, it’s important to design the chassis for race conditions and the various dynamic loads present during the race. An effective way of guaranteeing a good performance against dynamic racetrack loads is to increase the chassis torsional stiffness [2] in order to deal with the maximum loads from side force, braking force, bump, aerodynamic loads, engine torque reaction, etc. With the described above the objective of this thesis is to develop a computational model that allows the user to determine to optimal layup distribution and orientation in order to maximize overall stiffness while complying with the following requirements: • Static loads • Torsional loads • Maximum weight • Symmetry of the structure • Ease of fabrication In order to meet the overall goal of this thesis, a NASTRAN (finite element solver) computational model will be built using Siemens NX (Graphic interface program for various finite element solvers). This commercial program is used due to the present sponsorship by Siemens PLM to the PSEM project. A computational optimization algorithm will also be implemented using MATLAB (numerical computational software), this algorithm will work together with the NASTRAN solver to calculate the optimal solution to the above problem. The structure will be optimized using the DMO method in order to choose the layup for each element to maximize the stiffness. A zone creating algorithm will be implemented to generate large areas that share the same layup to keep the structure easy to fabricate.

Fig. 1- monocoque structural body

Optimization Problem

The problem goal is to maximize the global stiffness of the structure, [D]. This stiffness can be related with the structure compliance, N, which is the work done by the applied loads on a structure with a stiffness defined by [D]. Since the work of a load can be described by the same load multiplied by the displacement, the compliance can be written as: E F GHI (1) where [G] is the displacement vector and [I] is the applied load vector. The dynamics of the structure can be defined by the following relation: I F GHD (2) thus the compliance of the same structure can be written as: E F GHDG (3) From eq. (3) it’s implied that for the same load vector I, the structure stiffness [D] is inversely proportional to the complianceE, then a problem of minimizing a structures compliance becomes a problem of maximizing its stiffness and vice-versa. Considering that a system’s total strain energy can be written as [3]: K F 12 GHDG (4) where K is the total strain energy of a system, [G] is the displacement vector and [D] is the global stiffness matrix. From equations (3) and (4) is clear that: E F 2K (5) the problem of minimizing the structures compliance is equivalent of minimizing the total strain energy and maximizing its stiffness: min: E ⇒ OPQ: K ⇒ ORS: [D] (6)

3

Structure parameterization

Using the DMO method [4-6] where a structural element can be defined as a weighted sum of different materials from a certain set. Being the matrix [D]T its stiffness matrix and U the number of materials, [D]T can be written as: [D]T F V WXT[D]XT

YXZ[ ∶ 0 ] WXT ] 1 (7)

with [D]XT being the stiffness matrix of a structural element ^ with a certain material P. The WX represent the weighs of each material, these will determine how much a stiffness matrix defined by a certain material ([D]XT), will influence the stiffness matrix [D]T of the element. Using the Finite Element Method (FEM) to divide the structure in a _ number of finite subdomains, each one being described by a stiffness matrix [D]T , the structure is parametrized in the following form, with the number of project variables being the number of finite elements (_) times the number of possible layups (U).

Fig. 2- Example of parametrization of a set of finite elements

Constraints

Various constraints are implemented to this problem in order to comply with the project requirements, these are: • Maximum weight The maximum weight constraint is of the inequality kind and can be written in the following form: weight ] maxweight (8)

and the constraint violation can be written as: max (W_Pbcd e ORSW_Pbcd, 0) (9)

• Structural response symmetry Chassis torsion exists in both ways as track loads vary symmetrically when cornering left turns or right turns. Maximizing torsional stiffness in one way does not necessarily guarantee the same torsional stiffness on the opposite way and consequent structural response symmetry due to other loads being applied simultaneously, so a symmetry constraint must be added.

Fig. 3- Possible cases of torsional chassis load

4

The need to evaluate both torsional load cases turn this into a multi-objective optimization problem with two objective functions and an equality constraint between then that takes the following form: K[ F Kg (10) With K[ and Kg being the strain energy values for both torsional load cases. If there is no need for the objective function to be differentiable: |K[ e Kg| (11)

• Material weight functions sum In the DMO method, the sum of all material weight functions for each element can never be higher than 1 since the representation of each material cannot in each subdomain be higher than itself, so, a constraint to the project variables must be added:

V WXT F 1YXZ[ ; jkl RQm _U_O_Qd ^ (12)

If there is no need for the objective function to be differentiable: V nVoWXT e 1oY

XZ[ pqTZ[ (13)

Objective function

For simplification, the multi-objective function problem will be converted into a single objective function. This is done by averaging both strain energy results from the two different torsional cases: K F K[2 + Kg2 (14) With K being the average strain energy of the torsional cases K[ andKg. By minimizing S and by implementing the constraint in eq. (11) it is ensured that the strain energy results from both torsional cases are minimized equally, thus maximizing stiffness and maintaining the structure symmetry. By adding the constraints violations present in eq. (9), eq. (11) and in eq. (13) to the objective function in eq. (14), the final objective function of the problem is obtained. This can be written as the minimization of a function of the project variables. s[, sg and st being the respective penalties of each constraint. These penalties influence how much the violation of each constrain will alter the progression of the optimization algorithm used [7], as it tries to minimize the constraint portion of the equation when one of them is violated.

Optimization algorithm

In order to minimize the objective function (eq. (15)), described in the previous chapter, an optimization algorithm that finds its minimum must be devised and implemented.

This algorithm is nothing more than an iterative process where the structure is repeatedly analyzed until an optimal solution is found. This process can be described in the following diagram:

min juWXTv F OPQ wxK[uWXTv2 + Kg(WXT)2 y + s[ × uoK[(WXT) e Kg(WXT)ov + sg× {ORSu0, W_Pbcd(WXT) e ORSW_Pbcdv|} + st × V nVoWXT e 1oY

XZ[ pqTZ[

(15)

5

Fig.4 - Iterative optimization diagram A commercial FEM solver will be used for structure an objective function evaluation taking the computational model and input project variables and outputting mass and strain energy of both torsional cases. The solver used will be NX NASTRAN, which is used as a part of its graphic user interface (GUI), Siemens NX. Construction of computational

model

For the construction of the FEM mesh, the CQUAD4 element will be used, which as opposed to its quadrangular CQUAD8 counterpart, it has 4 nodes and 6 degrees of freedom (DOF) for each one [8], resulting in a lower complexity and lower analysis time. Due to its quadrangular shape, there will be specific points in the structures

geometry, pointed edges and other small features, where the mesh buildup would result in the heavy distortion of this element. In these cases, NX will automatically add a new element to this zones, a triangular element called CTRIA3, which has 3 nodes, and like the CQUAD4, 6 DOF for each node. With the geometric properties defined, the 2D shell physical properties can be chosen. The physical property type is set to “Laminate” which enables the NX Ply Sketcher [9]. To calculate each lamina properties, the rule of is used. The type of fiber that will be used is T-800 high modulus carbon fiber which has the following mechanical properties: Tab. 1- Relevant properties of T-800 carbon fiber

ρ (��/��) E

(GPa) ν Areal weight (g/ ��) 1754 290 0.35 200

This carbon fiber cloth will be vacuum-infused with epoxy resin, this process typically results in a laminate with around 60% of fiber/resin ratio [10]. The Epoxy resin used has the following properties: Tab. 2- Relevant properties of epoxy resin ρ (��/��) E (GPa) ν 1200 4.5 0.4

A foam core will also be used in some of the layups. This core is made of Rohacell structural foam which bears the following mechanical properties: Tab. 3- Relevant properties of Rohacell structural foam ρ (��/��) E (MPa) G(MPa) ��(��) 75 105 45 7

6

The properties of each lamina can then be calculated[13]: Tab. 4– Properties of each ply of T-800 Carbon fiber with 40% epoxy resin ��(���) 175.8 ��(���) 10.99 ��� 0.37 ��� 0.0231 ���(���) 3.913 ��(��) 0.19 With the composite properties are defined the models mesh collectors can be constructed using the different layups. Four different layups where chosen to be used in the structure and these are the following: • A 2 layer layup in a [0/90] • A 4 layer layup in the [0/90]� • A 4 layer sandwich panel with Rohacell structural foam core in [0/90]� • A 6 layer sandwich laminate in a [90/�45]� configuration with Rohacell

With the stacking sequences and lamina properties defined, the mesh collectors can be created. Additional elements that contribute to the overall structure stiffness are also added to the computational model, these are: • Front chassis braces • Rear roll-bar • Rear axle Since these elements are made from solid parts of 7075-T6 aluminum and they have a far superior rigidity than the monocoque, they can be simplified as Rigid-Body Elements (RBE) of two types:

The front braces and the roll bar due to their fixture type or their length will be modeled as not having moment constraints (3 DOF) and only having translation constraints, for this the RROD element type is used [11]. The rear-axle, because it’s firmly bolted on the chassis will be modeled as having all 6 DOF constrained. For this component, an RBAR, 500mm in length element type will be used [11].

Structural constraints are added:

Fig. 6 – Front suspension fixture points represented in FEM constraints

Loads are also added to simulate, pilot and batteries weight, motor pull, and most important torsional loads on the rear axle for both torsional cases:

Fig. 7 -“Positive” and ”Negative” torsional Subcases

Fig. 5–2D mesh and RBE elements

7

Genetic Algorithm

In the simulation .dat file, it is possible to see where the each elements type material and nodes are defined, this follows the following format: element type, element number, material, node 1, node 2, node 3, and node 4.

CQUAD4 3894 1 1898 1902 1903 1899

For the problem formulation through the DMO method, there are WXT design variables, with P F 1, … , _ possible layup configurations and ^ F 1, … , U number of elements. However, since it is only needed to identify one possible layup for each element that corresponds to when for each element ^ all WXT F 0 except one, which is equal to 1. This allows an easy and simple implementation if we set: WXT F 1 ⇒ WT� F P (16) where P is the layup chosen for that element. By doing this, then we have only on design variable for each element which takes only integer values�1,2, … , U�, identifying the chosen layup. Then this is easily introduced in the NASTRAN input data file just by changing the material selector for each element. To find the optimal solution a genetic-type algorithm is then used, with each of its individual solutions being of the following form:

Fig. 8 – gene for of any given solution A random initial population of size � is generated with the format described in Fig. 8 and with each individual chromosome being a random number between[1: U]. Each individual of the population is evaluated using NASTRAN, and sorted from the highest fitness (lowest objective function value) to the lowest.

The individuals with the highest fitness values (%qYX�q� × �) are paired up and selected for crossover. These pairs of individuals will generate the next improved population and are set in the following form: 12↓� ��[[�[g �g[�gg⋮�[� ⋮�g�

  : �[~G�[1, (QqYX�q�× �; )]� , �[ ∈ £ ; �g~G�[1, (QqYX�q�× �; )]� �g ¤ �[ , �g∈ £

(17) After the pairs are set, two types of crossovers can be implemented by the user. One is a traditional random crossover that overwrites 50% of the chromosomes of �g over �[ and operates in the following form:

Fig. 9 – Example of function of the random crossover However, due to the randomness nature of this crossover, it leads to a solution that is better but not practical, with the different layups being placed in an unorganized fashion throughout the shell:

Fig. 10 – solution obtained using a random crossover A new type of crossover is developed in order to create large zones of layups on the shell.

8

This crossover needs to be able to identify the neighboring elements of a certain chromosome of �[ and then imprint the chromosomes of �g on �g corresponding to these neighbor elements when they have the same layup as the element corresponding to the chromosome of �[ being analyzed. To do this, a connectivity matrix first needs to be created, this matrix has in each row, that corresponds to each element, all the other elements that are in contact with it.

CQUAD4 3894 1 1898 1902 1903 1899

For each one of the elements, the nodes are checked on the simulation .dat file and compared with the nodes of all the other elements. If an element (_g) has a node in common with the element being analyzed (_) then the number of (_g) is added to the row corresponding to the element (_[). This results in a matrix with a _ number of rows with all the connected elements. This matrix has as its number of columns, the number of elements connected to the element with most connections (O), since not all elements have as many connections, the entries of the matrix will be 0 for the elements that have number of connections ¥ O

¦T,§ F ¨¦[[ … ¦[©⋮ ⋱ ⋮¦q[ … ¦q©« (18) This new type of crossover can be implemented, called Zone-Based Crossover (ZBC) in which the chromosomes in individual �[are sequentially imprinted with the genes of �g if the following condition is met: For a chromosome ¬ W′T �® F W′T �¯ ↔ W′± �® F W′²³´ �® (19) This is done sequentially for each chromosome ¬ F 1 → ^ Eq. (19) means that, for a chromosome ¬ in �[, corresponding to a given element, if another chromosome ¬′ in �g corresponding to an element that is in contact from matrix [¦], has the same material to the element that corresponds to ¬, then ¬′ is imprinted in �[ on the same position. This operation is done, starting on the first chromosome, corresponding to element 1 and ending on chromosome _, corresponding to the last element.

Fig. 11 – Example of ZBC function at the start in gene 1. This is repeated for all genes in ascending order

9

To minimize the probability of the genetic algorithm converging to a local minimum [7], genetic mutations are introduced. This is done by randomly changing a certain percentage of chromosomes of a certain number of individuals from the elite group. This introduces new genetic material to the elite group, independently of the algorithm progression governed by the fitness function. To stop the optimization two stopping criteria can be used, maximum number of generations or convergent stop which stops the optimization when the solutions are no longer significantly improving: ¬cRO�PkQ jPdQ_¶¶§ e ¬cRO�PkQ jPdQ_¶¶§·[¬cRO�PkQ jPdQ_¶¶§] ¶d_� ∶ Q F 1,2,3, …

(20) While the ZBC creates zones that get more and more defined in each generation, when the stopping criteria is reached there might still be isolated elements that have no elements around them with the same material, in this cases an algorithm is needed to merge these isolated spots with the zones around them.

Fig. 12 – Generic operation of the merging algorithm All chromosomes of the champion solution are analyzed and for each element the amount of elements around it with the same material are counted, if this number is zero then the merging algorithm makes the element corresponding to this chromosome have the same material that is most common around it. Running the optimizer

All of the optimization where made using the following system specifications:

Tab. 6– PC system specifications

¸�¹ Intel i7 – 4790k – 4.7GHz – 8 threads º»�¼½¾ Kingston HyperX – 8Gb – 1866MHz – Single channel ¿�½À Á½�» Crucial SSD 240GB R/W 540/490 Mb/s ÃÄ Windows 7 Ultimate – 64bit �½�Å��ÆÇ ¸�½À Radeon R9 290x 4Gb The following penalty values where used: Tab. 7–Penalty values used �È 100 �� 6 × 10É The following genetic algorithm parameters where used: Tab. 8– Genetic algorithm parameters ÊË�Ì»½ ¼Í �ÊÀ�Â�ÀË�ÎÇ �Ê � żÅËÎ�Ï�¼Ê 450 % ¼Í »Î�Ï» �ÊÀ�Â�ÀË�ÎÇ 20% ��Ð ÊË�Ì»½ ¼Í �»Ê»½�Ï�¼ÊÇ 12 % ¼Í �ËÏ�Ï»À »Î�Ï» �ÊÀ�Â�ÀË�ÎÇ 20% % ¼Í �ËÏ�ÏÀ �»Ê»Ç (ͼ½ »�Æ� �ÊÀ�Â�ÀË�Î) 20% Ѽʻ Ò�Ç»À ¸½¼ÇǼ»½ yes ÄϼÅÅ�Ê� ƽ�Ï»½��

Max generations ÓË�Ì»½ ¼Í ÍËÊÆÏ�¼Ê »Â�ÎË�Ï�¼ÊÇ 9000 After the optimization had been completed, the following results were obtained: Tab. 9– Optimization Results �¼Ï�Î ¼ÅÏ���Ô�Ï�¼Ê Ï��» 22162s - ~6h ϼÏ�Î ÇϽËÆ˽» ��ÇÇ 4.25kg Õ»½��» ÇϽ��Ê »Ê»½�¾ ¼Í ̼Ï� Æ�Ç»Ç 7.27Ö ÄϽ��Ê »Ê»½�¾ ¸�Ç» È 7.2668Ö ÄϽ��Ê »Ê»½�¾ ¸�Ç» � 7.2764Ö ×�¼Î�Ï�¼Ê ¼Í ƼÊÇϽ��ÊÏ È (��ÇÇ)

None (below target) ×�¼Î�Ï�¼Ê ¼Í ƼÊÇϽ��Ê � (Ǿ��»Ï½¾) Minimal (0.0096Ö)

10

Analyzing Results The results obtained are imported to NX for graphic representation

Fig. 13 – Visual representation of the layup distribution obtained after optimization After a solution is obtained from the optimizer it serves as a guide for creating a new model with a refined mesh for FEM analysis to see if modifications are needed and to test any eventual improvements. Conclusions In conclusion the optimizer developed can return a solution that is between required parameters and can completed in an acceptable amount of time. This optimizer also has the advantage being able to be operated by a user that has very little knowledge about laminated composites and optimization and only possesses basic understanding of FEM model building in NX. Future Development

• Transition from scripted language to a compiled language. • Automatic layup creation • Multi-objective optimization • Experimental laminate properties

References

[1]. Gay, D. Matériaux Composites. s.l. : Hermes, 1991. [2]. Miliken W. F, Miliken, D. L. Race Car Vehicle Dynamics. s.l. : SAE International, 1994. [3]. Gavin, H. P. Review of strain energy methods and introduction to stiffness matrix methods of structural analysis. North Carolina : Duke University, Department of Civil and Environmental Engineering, 2012. [4]. Stegmann, J. Analysis and optimization of laminated composite shell structures. Aalborg University : PhD Thesis, Institute of Mechanical Engineering, 2005. [5]. Lund, E., Stegmann, J. Discrete material optimization of general composite shell structures. Topological Design Optimization of Structures, Machines and Material: Status and Perspective. s.l. : Springer, 2005, pp. 89-98. [6]. On structural optimization of composite shell structures using a discrete constitutive parametrization. Lund, E., Stegmann, J. 9, 2005, Wind Energy, pp. 109-124. [7]. Arora, J. S. Introduction to Optimum Design. Iowa City : Elsevier, 2012. [8]. PLM, Siemens. NX Nastran 10 Quick Reference Guide. Plano, TX : Siemens, 2014. [9]. —. NX Laminate Composites, Student Guide. s.l. : Siemens Product Lifecycle Management Software Inc., 2012. [10]. Wanberg, J. Composite Materials, Fabrication Handbook #2. Stillwater, MN : Wolfgang, 2010. [11]. PLM, Siemens. Nastran Users Guide. Plano, TX : Siemens PLM, 2014. [12]. Bendsoe, M. P., Sigmund, O. Topology Optimization: Theory, Methods, and Applications. s.l. : Springer, 2003. [13]. Reddy, J. N. Mechanics of laminated composite plates and shells, theory and application. s.l. : CRC Press, 2004.