Linköping University ORSAY Solid Mechanics Division LITH-IKP-PR 04/11--SE

Transcription

1 FIUPSO Linköping University Bat 620 Plateau du Moulon Mechanical Engineering Department ORSAY Solid Mechanics Division FRANCE SWEDEN LITH-IKP-PR 04/11--SE Person in charge in the FIUPSO: Mr. Denis SOLAS Person in charge in Linköping University: Mr. Larsgunnar NILSSON

2 AKNOWLEDGMENT I would like to thank Professor Larsgunnar Nilsson for having accepted me in his research project. Thanks to my entire colleague during this training courses and especially Jimmy Forsberg who helped me every day to do my work.

3 PREFACE This training period has been done at the division of Solid Mechanics, one of the departments of Mechanical Engineering in the University of Linköping in Sweden from the 4 th of May 2004 until the 28 th August Solid Mechanics is a basic subject in the field of engineering. Here, they study the thermo-mechanical behaviour of materials and the mechanics of solids and structures composed of these materials. The success of all industries making mechanical products depends on how well the engineers understand the properties of their products. To stay competitive, the industry must have a better understanding than the competitors (product s formability, product s functionality in its intended usage.). Their research in Solid Mechanics contributes to this knowledge and thus to the success of industry. In the division of Solid Mechanics research is performed in the areas of Contact and Impact Mechanics, Constitutive Modelling and Structural Dynamics. The funding of this department is provided by VR, VINNOVA, NFFP, SFS and different industrial parties. The software that I have used during my work are TRUEGRID as pre-processor, LS- DYNA as solver and LS-PRE/POST as post processor for the construction, solving and evaluation of an impact problem. Some of the works have been done with TRINITAS. TRUEGRID is a general purpose tool for creating a multiple-block-structured mesh. LS-DYNA is an explicit finite element program for the analysis of the non-linear dynamic response of three dimensional structures. On contrary, TRINITAS is software which makes static linear elastic finite element evaluations. For more details on these softwares, information are joined in annexe.

4 CONTENTS 1. Introduction [1], [2], [3] General topology Existing methods Linear elastic, static problem Topology optimization for crashworthiness, non linear problem Our concept The application problem Static interpretation of the load cases Identification of suitable material Initial conditions and material parameters The mass and the velocity of the rigid beam Influence of the Young s modulus Influence of the yield stress Influence of the density Influence of the hardening behaviour (etan) Influence of the hardening behavior and the yield stress Definition of simulation parameters Tsim influence The hourglass deformation mode The mesh influence Final model and parameters Topology optimization - one load case at a time Theory description Symmetric frontal loading case Offset frontal loading case Side loading case Topology optimization simultaneous evaluation of several load cases Theory description Results of the optimization for several load cases Topology optimization modification of the boundaries Theory description Results of the optimization Topology optimization - evaluation using a scaled approach Theory description Results for one loading case Side loading case Symmetric loading case Results for several loading cases simultaneous...45

5 8.4 Results for several loading cases with a different internal energy level Theory description Results Topology optimization Changing element thickness Theory description Thickness optimization for the symmetric load case Final thickness and topology optimization Results for the thickness optimization for the symmetric load case Results for the final topology optimization for the side load case Results for the topology optimization for several loading cases simultaneously Conclusions Future work Appendix Software TRUEGRID [21] LS-DYNA [22] TRINITAS [24] Bibliography Introduction [1], [2], [3] 1.1 General topology

6 Many countries have intensified their regulations on crashworthiness of vehicles, and in order to cope with this situation the automobile body has increased in weight. Therefore, weight reduction has become a major task in the development of new vehicles. It is desirable to develop optimization techniques which are capable of both improving the body strength and reducing the body weight. Basically, crashworthiness corresponds to everything that has to do with the car safety, and crashworthiness design includes the structure, the materials, the safety accessories, etc. This new wave of topology optimisation techniques can change the design process in the automotive industry by providing better structures, not only in the early stages of the process, but also as a technique to improve component designs in subsequent phases. It was after a paper of Bendsoe and Kikuchi in 1988 [4] that new techniques of topology optimisation started to be considered in other fields of automotive engineering. Their main contribution was the implementation of a methodology for topology optimization of continuum structures with arbitrary geometry (domain) for plate, shell and solid structures. The determinations of these new topologies for structural domains were more realistic than with previous method. It has been used to design structures subject to multiple kinds of physical phenomena such as static loads, free vibrations, forced vibrations, stress concentrations, and many others. Structural topology optimization for crashworthiness requires more research due to many complex phenomena and the simulation of these phenomena is difficult and takes time. These phenomena are explained in the next part. The work done until now ([5], [6], [7], [8]) is limited because of the lack of sensitivity analysis (as in [6]) or lack of comprehensive modelling of all phenomena in a crash (as in [7]). More opportunities for applications are still pending and require the interaction between academia, software companies and the industry research and product development organizations. Among the first publications on topology optimization applications in the automotive industry are Huang et al. [9] in 1993 and Yang and Chuang [10] in They implemented topology optimization software that used a commercial finite element method code to perform the structural analysis, and solved automotive design problems with a large number of degrees of freedom. Further developments are still needed. Structural topology optimization for crashworthiness design is in its infancy. The combination of nonlinearities that is present in a vehicle structural analysis simulating a collision is a difficult task, not only for the design, but also for the analysis. More studies are needed to circumvent these difficulties.

7 In the topology optimization, results are considered satisfactory when the shape obtained is better than the original design in one or more metrics, such as acceleration, deformation, energy absorbed, weight, etc. [18] 1.2 Existing methods There are many types of structural problems that can be encountered in an industrial application, from simple linear static problems like a bracket design, to nonlinear transient problems like a car design for crashworthiness. The simplest problem is the design for maximizing global stiffness, and the most complex, still unsolved, is the optimum structural topology design to maximize the absorption of kinetic energy during vehicle collisions. Several methodologies have been proposed to solve structural topology optimization as a material distribution problem. This is a brief description of the most relevant methods: - Methods that use composite materials to relax the space of solutions, along with homogenization techniques to compute average properties needed in the structural analysis (Bendsoe and Kikuchi, 1988) - Methods that use artificial materials instead of real composites as above, and need no homogenization techniques (Mlejnek and Schirrmacker, 1993) - Methods that use evolutionary approaches where the topology is obtained by deleting finite elements of the structural mesh as the iterative algorithm proceeds (Xie and Steven, 1997). - Methods that use directly the entries in the elasticity tensor of the material as design variables allowing the largest relaxation of the space of solutions. Homogenization techniques are not needed in this case (Bendsoe and al, 1994). All these methods have advantages and disadvantages, but all look for creating a new structural topology design in systematic form, with the help of computer tools. In classical structural optimization methods, gradients are needed to construct an approximation in order to approach the optimum. However, in the problem at hand, the gradient based methods are usually not an option since the construction of numerical approximations to the gradients is too expensive. In my work the third method was chosen because it is the simplest for a complex problem and in the problem at hand, gradient information cannot be determined for all the functions used in the optimization problem. The following sections present applications in the industry starting from the simpler problem to the more advanced.

8 1.2.1 Linear elastic, static problem The simplest of topology optimization problems is to minimize compliance in linear elastostatics. It is equivalent to maximize the global stiffness of the structure under a given load. Compliance minimization in linear elastostatics is the easiest problem to solve; it converges quickly, smoothly, and in most case results are intuitive. The optimization problem can be written after an FE discretization, as Minimize f i u i, i=1, number of degrees of freedom (ndof) C Such that c i v i V, i=1, number of finite element (nelem) K ij (c). u j (c) = f i, i, j = 1, ndof (1a) (1b) (1c) Where f i, is the external load; u i is the displacement due to the load and the equation (1a) represents the fact that the stiffest structure possible is the objective; c i is the design variable (volume fraction); v i is the volume of the finite element i; V is the total amount material that can be used to create the topology and this equation represents the fraction of volume available; K is the stiffness matrix of the structural system. To achieve this, the displacement can be changed but not the load which is defined in the beginning of the problem. Hence, a variation of the thickness of the element is allowed (larger or smaller thickness) to get the better structural stiffness. The (1c) equation represents the equilibrium equation for a static case. The convention of summation on repeated indices is applied. It is assumed in (1b) that there is one design variable (density) per finite element. More than 80% of structural topology design optimization problems in industry can be addressed by solving (1b). Several publications have addressed the stresses in topology optimization, among them, Cheng and Jiang [13] (in trusses), Yang and Chen [14] and Duysinx and Bendsoe [11]. It is interesting to bring a point presented in Bendsoe, et al. [15] indicating that the minimization of compliance, for single load case, produces designs whose mean stresses are minimized as well. Stresses have been used in the objective function as in Yang and Chen [14], or in the constraint functions as in Duysinx and Bendsoe [11]. In the former case, it was found that a linear combination of compliance and stresses in the objective resulted in better results, numerical stability, and faster convergence (Yang and Chen [14]) Duysinx and Bendsoe [11] showed that in order to consider local stresses as constraints a relaxation of the allowable stress (inspired on Cheng and Jiang [13]) needs to be introduced to guarantee a solution and eliminate singular results Topology optimization for crashworthiness, non linear problem

9 In crashworthiness analysis of vehicles there is a long list of complex phenomena: nonlinear materials (plasticity, hardening, etc); nonlinear geometry (large deformations and displacements, buckling); dynamics (inertial forces); surface contacts (including selfcontact of members) and strain rate effect due to the speed of the crash, among others. Firstly, about nonlinearities introduced in the constitutive law of material, one of the first publications dealing with this subject in the context of design optimization is Bendsoe and al., It used softening materials modeled by means of a new complementary energy principle introduced by Plaxton and Taylor in They demonstrated that the design of local properties of materials can be extended to a general class of analysis situations made of elastic/softening materials. The objective function in that work was the usual maximization of global stiffness. The work of Mayer et al. (1996) also included the use of elasto-plastic materials. However, since its main application was in structural crashworthiness, its presentation will be given later. In 1998, Maute et al. presented a treatment for maximizing the ductility (plastic deformation before failure) of the structures for a given range of prescribed displacements using elastoplastic materials in a two-dimensional elastostatic setting. The material was described by a yield function that included the deviatoric stresses and the yield stress with isotropic hardening/softening. The optimization problem was posed as (Maute et al., 1998): Maximize έ σ i di n, i=1, ndof C Subject to c i v i V, i=1, nelem (2) K ij (c). u j (c) = f i, i, j = 1, ndof Where έ is a prescribed strain, σ is the stress and ε the strain. It was shown in Maute et al. (1998) that the optimum topology considering nonlinear materials can be substantially different to the results when the material is linear elastic. Compared to the objective in equation (1), the objective of this equation formulation is characterized by the maximization of the internal energy. Also in 1998, Yuge et al. presented a topology optimization algorithm for twodimensional elasticity and shells using homogenization methods and applied it to the design of steel frame joints under static loads. In addition, large deformations were considered in the design of shells. They applied homogenization techniques to obtain the average properties of a microscopic porous material as was originally used by Bendsoe and Kikuchi in Again, they concluded that topologies for nonlinear materials differ from those obtained with linear materials. Next, contact is a very important phenomenon in the automotive industry. In a vehicle crash, when parts are collapsing due to the impact, structural surfaces enter in contact and produce a pattern of deformation governed by the contact phenomenon in many cases.

10 The literature in this subject, however, is limited due to the technical difficulty of the problem. The work of Klarbring et al. (1995), Kocvara et al. (1996) and the Applied Mechanics Review article by Hilding et al. (1999) are some of them. Until now, almost all topology optimization contact problem formulations do not account for friction which is a very important phenomenon. The mathematical representation of a contact surface is given by an unequally which is simply included in the compliance minimization problem as: Minimize f i u i, i=, ndof C Such that c i v i V, i=1, nelem (3) g m 0, m=1, ncc (contact condition) K ip (c). u j (c) = f i, i, j = 1, ndof Where g m is the distance of this surface to the associated node. In our case, the contact is treated by using penalty methods. It means that a penetration in the contact area is allowed and a fictitious spring is used to separate the two parts after the impact. Furthermore, for the dynamic nonlinear problem, such as in a collision problem, some techniques of optimizing structures using a method which minimizes objective functions directly have recently been report. [16], [17]. However, even through the computers are fast, problems still remain for convergence to an optimal solution. In order to improve this situation for dynamic nonlinear analysis, a structural method using FEM was developed (by the Society of Automotive Engineers of Japan). This method defines the optimality criteria as in the linear analysis of a fully stressed design, and indirectly finds an optimal solution. It is based on the concept that making each plastic strain value of all shell elements almost equal is effective in the weight reduction of such a structure. One of the most recent complexities added to structural topology optimization problems is the consideration of large deformations. Bruns and Tortorelli (1998) presented a paper on topology optimization considering large displacements but small deformations with some examples in compliant mechanisms. Buhl et al. (1999) presented some results for compliance minimization considering both, large displacements and deformations. They concluded that the effect of the nonlinearities can be substantial in some cases, and that multiple load cases are beneficial to obtain sound topology designs. Finally, the entire phenomena which appear in crashworthiness have been studied separately, but not together. In the area of crashworthiness design, some initial work has been done (Mayer et al. (1996) and Diaz and Soto (1999)), but these are preliminary

11 investigations. Other publications worth mentioning here are Arora et al. (1999), Knap and Holnicki-Szulc (1999), Yamakawa et al. (1999) and Marzec and Holnicki-Szulc (1999). The optimization problem for kinetic energy absorption can be set in many different ways. One of them, which reflect the two main constraints in the problem, i.e. accelerations and deformations, is: Minimize T ü A 2 dt C δ B-C δ (4) c i v i V, i=1, nelem M ij ü j + A ij ú j + K ij u j = f i, i, j=1, ndof Where ü A is the acceleration at the structural point A, δ B-C is the relative displacement between points B and C in the structure, δ is a prescribed upper bound, T is the total time of the event. The acceleration is proportional to the loads exerted to the structure and its minimization is therefore required to protect the goods. On the other hand, to decrease acceleration a softer structure is needed, yielding large deformations. These deformations can cause intrusion into the container and eventually exert damage to the goods. In other words, deformations and accelerations are conflicting constraints. An upper bound on deformations was then included to capture this conflict. The second constraint is the isoperimetric constraint on material used. What makes equation 4 a difficult problem is the physics (the analysis), not the design part. The author has not found any reference where this optimization problem has been addressed considering all phenomena. In addition, there are other issues to be resolved before attempting to solve equation 4. If the goal is to pose the topology optimization problem as a material distribution problem, it is necessary to find the relation between the design variable, density of material, and the characteristics of the material such as stiffness, yielding stress, strain energy, unloading stiffness, strain rate behavior, etc. For linear systems, theses relations have been obtained mathematically or prescribed heuristically, and have worked successfully. For nonlinear transient dynamic problems, as vehicle collisions, theses relations are still unknown. 1.3 Our concept The objective is to find an optimal structure, where as much as possible of the material used for the construction also is used in the absorption of energy. This can be motivated by both weight and cost aspects of the detail. If the material is used during a crash event, it will undergo plastic deformation. Hence, the internal energy will increase in areas which absorb energy. Areas which do not absorb any energy will have a low internal energy value and it is our assumption that if the elements in this area are removed, a better construction is obtained. However, there are many other questions as

12 well. Does the fictitious material represent the behavior of a real material? How much plastic strain can be allowed in an element before rupture in our fictitious material, etc. 2. The application problem My work is about the structural topology optimization of an underrun protection device, which will be added under/behind the bumper of a truck. This piece will play the role of

13 energy absorber during a crash event. See below a scheme of the ground structure and the loading situation. Fixed beam under the truck Third load case (static) Bumper of the truck, top view Energy absorber Symmetric load case, velocity v 0 Tyre Offset load case, velocity v 0 Lack of internal energy, delete element Symmetry axis Max energy Topology optimization is about how to distribute the material in a structure. In our strategy, the internal energy is used as a measure where material is needed. It consists of obtaining the internal energy as homogeneous and constant as possible in a material in order to have an efficient use of this. The second purpose is to make the structure as light as possible with an optimal shape. 3. Static interpretation of the load cases

14 Firstly, an interpretation of similar static load cases has been done. TRINITAS software [24] was used in order to simulate these cases. These optimizations are independent of time. The structures obtained after topology optimization of the front load case, the offset load case and the side load case with TRINITAS, after approximately 20 iterations. Figure 3.1: Front load case Figure 3.2: Offset load case Figure 3.3: Side load case 4. Identification of suitable material

15 Firstly, TRUEGRID pre-processor [21] was used to create a Finite Element (FE) model. In this step, the parameters of the material were defined (Young s modulus, yield strength, hardening behaviour, Poisson ratio, density) as well as some boundary conditions and initial conditions. In order to obtain a reasonable plastic deformation and internal energy, these parameters are modified and then the FE model simulation is carried out as described in Section 4.1. The results of these first simulations see figures to and 4.3.1, where the level of the internal energy is represented by the dark colour. Concerning the FE simulation, some parameters have to be defined and modified. In these simulations a high IE is desired representing large plastic deformation and a nonlinear behaviour. The evolution of the internal energy for a variation of time and control of the hourglass, see on figures from to 4.2.3, is described in section Initial conditions and material parameters The mass and the velocity of the rigid beam In a first step, a mass and a velocity were assigned to the structure which should sustain this impact event. These two parameters generate kinetic energy which mostly will be changed into internal energy in the energy absorber during the impact event. A study of the influence of the velocity of this beam on the deformation was done and the values attributed to them were 25 m/s for the velocity and 1300 kg for the mass Influence of the Young s modulus In a first consideration, a material is characterized by different laws. It can be elastic, elastoplastic, etc (Figure ). An elastic material is defined by the Young s modulus, denoted E, and the Poisson s ration. The stress of the deformed material increases proportionally to the increase of the strain.

16 Figure : Three different material behaviors The hardening plasticity is a more complex material model than the other. [20] Figure shows the influence of the Young s modulus on the internal energy distribution and on the plastic deformation of the energy absorber. Figure : IE, E = 0,5e 9 N/m 2 Figure : IE, E = 50 e 9 N/m 2 When the Young modulus increases, the internal energy increases in the energy absorber, and hence larger plastic deformations occur.

17 4.1.3 Influence of the yield stress Figure : IE, σ y = 50 e 6 N/m 2 Figure : IE, σ y = 100 e 6 N/m 2 Figure : IE, σ y = 100 e 6 N/m 2 Figure : IE, σ y = 5 e 6 N/m 2 A comparison between the figures and together and and together shows that when the yield stress decreases, the deformation of the energy absorber increases and the distribution of the internal energy of the absorber is modified. This is due to the relation between the three previous parameters: the internal energy IE: Σ i=1 N element IE = σ i ε i. IE is constant in the energy absorber (there is no variation of the mass or the velocity), but not in an element, so when the yield stress decreases, the deformation of the energy absorber should increase.

18 4.1.4 Influence of the density Figure : IE, ρ = 5400 Figure : IE, ρ = 4800 When the density value decreases, the internal energy seems to focus in a smaller area but it is not visible in these two pictures due to the black and white colour. The variation of the internal energy distribution is between the two legs Influence of the hardening behaviour (etan) Figure : IE, etan =50 N/m 2 Figure : IE, etan =6000 N/m 2 The hardening behaviour seems to be a parameter with little influence on the deformation or on the internal energy. However, the influence of plastic hardening depends strongly on the initial yield stress.

19 4.1.6 Influence of the hardening behavior and the yield stress Figure : IE, etan = 5 e 6 N/m 2 Figure : IE, etan = 20 e 6 N/m 2 σ y = 5 e 6 N/m 2 σ y = 20 e 6 N/m 2 An increase in deformation is observed when the two parameters are decreased simultaneously. But when these values are too low, the deformation is too high and some distortions in the mesh around the boundaries near the impact appear. Consequently, two solution strategies are required to avoid such a situation: A modification of the mesh is necessary and an increase of the hardening (etan), as illustrated in the Figure Definition of simulation parameters Tsim influence Figure : IE, tsim = 0,025s Figure : IE, tsim = 0, 03s Tsim is a parameter which determines the response time of the FE simulation. It depends only on how long our event is to be studied.

20 4.2.2 The hourglass deformation mode The hourglass deformation mode is a problem that must be considered during a simulation, since it generates a distortion in the mesh due to the under-integrated 2D elements used. Hourglassing derives its name from the fact that the deformed element literally resembles an hourglass. An hourglass mode is a special case of kinematics modes or spurious zeroenergy modes see [19], to suppress this spurious behaviour various stabilization (hourglass control) techniques are used. Due to reduced integration one-point volume respective in-plane integration, new artificial deformations may develop, linked to the zero-energy modes. The volume, respective shell, finite elements deform according to hourglass shapes (Figure ). Figure : One hourglass deformation mode. In order to control these purely numerical deformations, hourglass resisting forces are added for cases when they are excited. Then, in the mechanical energy balance, it appears an hourglass energy that is linked to the hourglass resisting forces against formation of hourglass modes. [20] In order to study the influence of the hourglass on the internal energy level, the deformation and the mesh, several simulations were done. Basically, for control and consequently reducing the hourglass, different method could be applied (ihq) and for each method, a coefficient is allotted (qh). When this coefficient decreases, a mesh less distorted, a lower hourglass energy level and a decrease of the deformation are observed. In this Figure , a high hourglass is observed and there is too much deformation in the mesh.

21 Figure : Hourglass deformations Parameters etan 10000; sigy 20e+06; Poisson ratio 0.3; Young 50e+9; rho 4800; tsim 0.035; massc 1300; THIC 0.02; VEI -25; Left 20; boundary1 12; Middle 40; boundary2 12; Right 20; Up 20; Down 20; Hourglass control LSDYOPTS ihq 4 Qh 0.001; The mesh influence Figure : First mesh Figure : Decrease of the mesh When the mesh is increased, the deformation of the energy absorber evolutes in the same way but the hourglass increases as well. Consequently, an hourglass control should be done. A correct hourglass control does not influence the internal energy distribution. Furthermore, the more the mesh is refined, the longer the computing time of the simulation will be, but at the same time the results are better due to a better resolution of the problem.

22 4.3 Final model and parameters Figure 4.3.1: Final internal energy repartition Figure 4.3.2: Plot of the different characteristic of the structure

23 Parameters mesh etan 20e+06; left 25; sigy 20e+06; boundary1 10; Poisson ratio 0.3; middle 35; Young 50e+9; boundary2 10; rho 4800; right 25; tsim 0.035; up 15; massc 1300; down 5; THIC 0.02; down2 15; VEI -25; Hourglass control LSDYOPTS ihq 4 qh 0.005; Figure 4.3.3: Final parameters of the models In conclusion, Young modulus, yield strength, hardening and density are closely related. For the density, the value was selected heuristically using the notion that a more deformable material is often characterized by a lower density. Finally, the yield strength and hardening influenced the internal energy distribution.

24 5. Topology optimization - one load case at a time 5.1 Theory description An LS-DYNA input file, input.k, is established using the pre-processor TRUEGRID [21]. In this, another file include.k is integrated. A program in Perl was written with the aim of updating the FE model. During the optimization process this program is established in order to analyze the maximum, minimum and the average of the internal energy of each finite element of the energy absorber during the deformation. After determining these values, some of the finites elements with the lowest value of IE are deleted from the structure with the aim to optimize it. The working schedule of the simulation of one load case is described below. PREPROCESS TRUEGRID - shell mesh (2D) - definition of the materials parameters - hourglass control - time parameters Input.k Include.k LS-DYNA Solving the impact problem Extract IE LS-PRE/POST Analyse: - max IE - min IE - ave IE Perlscript Update model, topology optimization

25 Now, the three load cases were studied: - symmetric front load - offset front load - side load In this optimization procedure, the final process is considered either when a homogenous internal energy level is obtained, when too many holes appear or when elements near boundaries are deleted. Furthermore, when some elements (which have a low internal energy) are deleted, the mass of the energy absorber decreases but not the total internal energy. The internal energy density focuses in a smaller part, since the frontal underrun protection device still have to absorb all the kinetic energy of the impact problem. 5.2 Symmetric frontal loading case Firstly, a percentage of the internal energy was entered only to delete a few elements. The value of the internal energy considered after the execution of the Perl program is [percentage*(max IE - min IE) + min IE]. This percentage is not the same in all of the load cases. The percentage was modified with the aim to see how it influences the optimisation process as well as the final result. The different results obtained for each simulation with different percentages of the IE fraction are given in figures to Figure 5.2.1: Initial topology Figure 5.2.2: First iteration, 3.5% (Max IE= Min IE= Use IE= )

26 Figure 5.2.3: Second iteration, 3.5% Figure 5.2.4: Fourth iteration, 3.5% (Max IE= (Max IE= Min IE= Min IE= Use IE = ) Use IE= ) Figure 5.2.4: Initial topology Figure 5.2.5: First iteration, 1% Figure 5.2.6: Ninth iteration, 1% Figure 5.2.7: 19 th iterations, 1%

27 Figure 5.2.8: 32 nd iteration, 1% Figure 5.2.9: Initial topology Figure : First iteration, 2.5% Figure : Fourth iteration, 2.5% Figure : Ninth iteration, 2.5%

28 Figure : 14 th iteration, 2.5% The percentage has an influence in the number of iteration needed to get the optimal structures but not on the final shape of the structure. For all percentage values, the same shape seems to be obtained at the end of the optimization. Furthermore, it could be observed that there is no link between the values of the max, min of the internal energy between two iterations. They changed but never in the same way, i.e. they can increase or decrease between two iterations. 5.3 Offset frontal loading case The results for the second load case, the offset frontal loading case, are shown in figures to Figure 5.3.1: Initial topology Figure 5.3.2: First iteration, 3.5%

29 Figure 5.3.3: Third iteration, 3.5% Figure 5.3.4: Initial topology Figure 5.3.5: First iteration, 2.5% Figure 5.3.6: Third iteration, 2.5% Figure 5.3.7: Fifth iteration, 2.5% _

30 Figure 5.3.7: Initial topology Figure 5.3.8: First iteration, 1% Figure 5.3.9: Sixth iteration, 2.5% Figure : Seventh iteration, 2.5% Figure : Eighth iteration, 2.5% The observations are the same as for the other cases. The percentage used does not change the final shape of the optimization. The optimization has deleted elements which connect to the other boundary condition, see figures (5.3.3) and (5.3.11). If this connection is wanted the optimization should be stopped before. But this new topology could be a better structure for this load case. 5.4 Side loading case Finally, the optimization and the study of the influence of the percentage were done for the third load case, i.e. the side loading case.

31 This load case is actually a static load case modelled as a dynamic load case with the same mass and initial velocity of the rigid beam that impacts the energy absorber as in the previous two load cases. The results of the optimization are shown in figures to Figure 5.4.1: Initial topology Figure 5.4.2: First iteration, 2.5% If the internal energy is concentrated in one place, the rest of the structure will not have much energy so many elements will be removed in one iteration and an incoherent structure will be obtained, see Figure The boundaries disappear in the structure, since too many elements are deleted in one iteration because of the high value of the percentage compare to the IE distribution in the model. Basically, using a too high value on the IE percentage results in a loss of the connection in the absorber between the supports and loading area on the absorber. Hence, the structure would only accelerate due to the impact. Figure 5.4.3: Initial topology Figure 5.4.4: First iteration, 0.001%

32 Figure 5.4.5: Fourth iteration, 0.001% Figure 5.4.6: Fifth iteration, 0.005% Figure 5.4.7: 12 th iteration, 0.001% Figure 5.4.8: 14 th iteration, 0.005% Some problems could be observed with the mesh in Figure This illustrates the fact that after a number of iteration, the shape obtained is not coherent, so it is important to choose a good IE percentage in order to get a coherent structure at the end. In conclusion, the IE percentage used for element deletion is dependant on the IE distribution. Furthermore, for the same distribution, a similar structure will be obtained, whatever the number of iterations is done to obtain this result using a sufficiently small value of the percentage.

33 6. Topology optimization simultaneous evaluation of several load cases 6.1 Theory description Next, the objective was to optimize the topology of the energy absorber for several load cases simultaneously. The IE percentage was chosen with the aim to obtain a coherent structure in less time. In this optimization, it is important that the three load cases have the same importance. They should have the same kind of internal energy level, otherwise elements which are important for one load case might be deleted since they are not deformed in another load case. The work schedule for the optimization procedure with several load cases simultaneously is given below. PREPROCESS TRUEGRID - shell mesh (2D) - definition of the materials parameters - hourglass control - time parameters Input.k Include.k LS-DYNA Simulation of the internal energy for all load cases (3) Extract IE LS-PRE/POST Analyze: - max IE - min IE - ave IE Sum IE if there are several load cases Perlscript Update model, topology optimization

34 6.2 Results of the optimization for several load cases On the figures below, the internal energy distribution is illustrated in black for the high values and in white for the low values. Initial state: Figure and 6.2.2: First shape and repartition of the IE for the frontal loading case Figure and 6.2.4: First shape and repartition of the IE for the frontal offset loading case Figure and 6.2.6: First shape and repartition of the IE for the side loading case

35 First iteration: 1% Figure 6.2.7: Topology optimization Figure 6.2.8: IE, case 1 Figure 6.2.9: IE, case 2 Figure : IE, case 3 Sixth iteration: 1% Figure : Topology optimisation Figure : IE, case 1

36 Figure : IE, case 2 Figure : IE, case 3 Ninth iteration: 1.5% Figure : Topology optimisation Figure : IE, case1 Figure : IE, case 2 Figure : IE, case 3

37 12 th iteration: 2% Figure : Topology optimisation Figure : IE, case 1 Figure : IE, case 2 Figure : IE, case 3 14 th iteration: 2.5% Figure : Topology optimisation Figure : IE, case 1

38 Figure : IE, case 2 Figure : IE, case 3 So, for several load cases, a new shape was obtained based on the influence from each load. The distribution of the internal energy of the element is not the same since each case has to be considered. But in this optimization, the IE percentage has a lower impact compared to the single loading case. During the simulation, some holes appear in the structure generating some troubles in the mesh but also an incoherent structure. These holes lead to a new IE distribution around them.

39 7. Topology optimization modification of the boundaries 7.1 Theory description In order to decrease the singularities of the mesh near the boundaries, the rigid beam and the boundaries were modified. Their corners were rounded off. In fact, in numerical simulation, sharp corners must be avoided because they introduced high stress and plasticity. It would be better with a small radius using small elements, but it is too expansive. 7.2 Results of the optimization The internal energy distribution is illustrated in the figures below for the symmetric loading case and the IE percentage is chosen in order to obtain a good topology in a minimum of computing time. Figure 7.2.1: Initial topology Figure 7.2.2: First iteration, 2% Figure 7.2.3: Fourth iteration, 1% Figure 7.2.4: Seventh iteration, 1.5%

40 Figure 7.2.5: Eighth iteration, 1.5% Figure 7.2.6: 12 th iteration, 1% Firstly, a new distribution of the internal energy is observed at the beginning and not at the end. This is due to the round corners. In fact, the figures 7.2.3, and show the importance of the choice of the percentage. If a higher percentage is chosen at the beginning, some elements would be deleted whereas they are useful in the topology. Consequently, a new structure will be obtained. Basically, a better optimization is obtained by this modification but the same structure is obtained at the end.

41 8. Topology optimization - evaluation using a scaled approach 8.1 Theory description In the beginning of the last optimization (considering all load cases at once), the fact that the three load cases have the same level of IE was assumed. This assumption is based on that the IE is introduced in the material by the impact of the rigid beam which in all of the cases has the same mass and velocity. So, these three load cases could be analysed simultaneously and they have the same importance in the optimization. Now, the aim is to know if several load cases could be optimize simultaneously even if their internal energy level is not the same. So, a scaled approach is studied. The program is based on the fact that firstly the IE will be scaled with the maximum value of the internal energy for each load case. Secondly, the resulting IE is summed and elements deleted depending on this level. In a first step, the scaled optimization was done for the symmetric load case and for the side load case alone and after they are optimized simultaneously. Afterwards, a new case is created which corresponds to a pressure applies on the side, and the optimization was done with the symmetric load case simultaneously in order to check the argument of the normalisation. The work schedule for the scaled optimization is shown below.

42 PREPROCESS TRUEGRID - shell mesh (2D) - definition of the materials parameters - hourglass control - time parameters Input.k Include.k LS-DYNA Solving the impact problem Extract IE using LS-PRE/POST Analyze: - max IE, scaled - min IE - ave IE Sum scaled IE if there are several load cases Perlscript Update model, topology optimization 8.2 Results for one loading case Side loading case In this first part, the general observation from a scaled approach on the internal energy distribution for the third load case is studied, again with the new round boundaries.

43 Figure : Initial topology Figure : First iteration, 0.001% Figure : Second iteration, 0.007% Figure : Third iteration, 0.001% Figure : Sixth iteration, 0.005% Figure : Seventh iteration, 0.005% In conclusion, these figures show a slight modification of the final shape but the general topology is still conserved.

44 8.2.2 Symmetric loading case The optimization is done with the round boundaries and for a medium percentage. The internal energy distribution is represented in the figures below. Figure : Initial topology Figure : First iteration, 1% Figure : Fifth iteration, 2% Figure : Seventh iteration, 2.5% Figure : 11 th iteration, 3% Figure : 15 th iteration, 3.5%

45 Figure : 16 th iteration, 6% Once again, the same topology, that is the part 7, Figure without scaled approach, is still observed at the end of the optimization. The normalisation, as could be expected, does not modify something in the final result. 8.3 Results for several loading cases simultaneous The normalisation for the symmetric (case 1) and the side load case (case 3) simultaneous, the internal energy being represented and the new boundaries models are shown below. Figure 8.3.1: Initial topology Figure 8.3.2: Initial topology Case 1 Case 3

46 Figure 8.3.3: First iteration, 1% Figure 8.3.4: First iteration, 1%, Case 1 Case 3 Figure 8.3.5: Third iteration, 1% Figure 8.3.6: Third iteration, 1% Case 1 Case 3 Figure 8.3.7: Fifth iteration, 2% Figure 8.3.8: Fifth iteration, 2% Case 1 Case 3

47 Figure 8.3.9: Seventh iteration, 3% Figure : Seventh iteration, 3% Case 1 Case 3 Figure : Eighth iteration, 3% Figure : Eighth iteration, 3% Case 1 Case 3 Again, the same structure is obtained at the end of the topology optimization as in part 5.2, Figure for the symmetric case and in part 5.4, Figure for the side case but it is obtained faster than without the normalisation. 8.4 Results for several loading cases with a different internal energy level Theory description A new simulation in which the loading case was modified was done. In this simulation, a pressure and not an impact was applied on the side of the energy absorber. The intensity of the pressure was determined such that the structure remained in the elastic domain in the initial iteration.

48 8.4.2 Results The optimization with the pressure case (case 2) and the symmetric loading case (case 1), simultaneously, is shown in figures to The internal energy distribution is still represented on the figures. In the figures below, two load cases with two different internal energy levels are observed. The internal energy level for the pressure case is lower than the internal energy level for the symmetric load case. Consequently, the dark fringe colour does not correspond at the same internal energy value for each load cases. Figure 8.4.1: Initial topology Figure 8.4.2: Initial topology Case 1 Case 2 Figure 8.4.3: First iteration, 1% Figure 8.4.4: First iteration, 1% Case 1 Case 2 Figure 8.4.5: Third iteration, 2% Figure 8.4.6: Third iteration, 2% Case 1 Case 2

49 Figure 8.4.7: Fifth iteration, 3% Figure 8.4.8: Fifth iteration, 3% Case 1 Case 2 Figure 8.4.9: Sixth iteration, 6% Figure : Sixth iteration, 6% Case 1 Case 2 Figure : Seventh iteration, 8% Figure : Seventh iteration, 8% Case 1 Case 2 Firstly, it is observed that the topology got a similar structure after the optimization even if the side loading case is replaced by a pressure with a lower internal energy level. In the pressure case, the internal energy level is lower (/1000) than in the others cases. But since the internal energies from the different loading cases are scaled, the optimization could be done at the same time. As it is illustrated in the figures below ( and ) even for a high percentage the elements with a low internal energy

50 (corresponding to the elements on which the pressure is applied) are not deleted during the optimization. Figure : Initial topology Figure : Initial topology Case 1 Case 2 Figure : First iteration, 6% Figure : First iteration, 6% Case 1 Case 2 In conclusion, the normalisation allows a combination between all kind of loading cases, the kind and the level of the internal energy do not intervene in the simulation. It leads to a real topology optimization.

51 9. Topology optimization Changing element thickness 9.1 Theory description In a new approach, the influence of the thickness of the element is evaluated because it influences the weight of the structure and also the internal energy distribution. The modification of the thickness generates a variation of the plastic deformation and consequently a variation of IE provided that the load path remains the same. In order to study the influence of the thickness of the mesh on the optimisation, the optimization procedure was modified. In this new procedure, the thickness will be increase and consequently the deformation will be reduced if the IE of the element is higher than an average value of IE. If the element IE is lower than the average, the thickness will be decreased and consequently the deformation increased. Explanation of the program: Max IE t increase Min IE Min IE t decrease t decrease Min IE Figure 9.1.1: Repartition of IE t decrease Figure 9.1.2: Variation of the thickness

52 decrease IE increase IE increase IE increase IE Figure 9.1.3: Final result of the repartition of IE after the variation of the thickness The working procedure for the thickness optimization is shown below.

53 PREPROCESS TRUEGRID - shell mesh (2D) - definition of the materials parameters - hourglass control - time parameters Input.k Include.k LS-DYNA Simulation of the internal energy Extract IE LS-PRE/POST Analyze: - max IE - min IE - ave IE Sum IE if there are several load cases Perlscript Update model, topology optimization, thickness variation 9.2 Thickness optimization for the symmetric load case Firstly, the average value of IE used for determining the thickness of the element was the IE value of the previous iteration. Consequently, this value was not constant and some oscillations in the thickness appear and it could lead to an absence of thickness (Figure 9.2.4). Results from different percentage of IE for the symmetric loading case are shown below.

54 Figure 9.2.1: Initial thickness Figure 9.2.2: First iteration, 25% Figure 9.2.3: Third iteration, 25% Note that the black fringe colour corresponding to the lower thickness and the white to the higher thickness for these simulations. Figure 9.2.5: Initial thickness Figure 9.2.6: First iteration, 50%

55 Figure 9.2.7: Fourth iteration, 50% This thickness optimization shown in Figure did not converge so a new average was determined. 9.3 Final thickness and topology optimization Instead of using the average value of IE in each iteration, the IE criterion is the value of the first iteration for one case. For several cases, it is the average of the sum of the IE value at the first iteration. The results from the simulation are shown below Results for the thickness optimization for the symmetric load case

56 Figure : Initial thickness Figure : First iteration, 25% Shell element thickness Figure : Fourth iteration, 25% Shell element thickness Note that the black fringe colour corresponding to the lower thickness and the white to the higher thickness for the simulations above. The optimization converges towards a value of the element thickness after a few iterations. The number of iterations depends on the percentage of IE applied for the optimization. Compared to the previous updating scheme, the same influence of the percentage was observed. In addition, an analysis of the variation of the thickness during the iterative optimization shows that the thickness value does not really change after the first iteration for each element. In the optimization process, a maximum and a minimum value for the thickness are defined. Many of the elements located at the front of the device, where the rigid beam impacts the energy absorber, as well as those located near the rear fixed boundary conditions take on the high limit value for the thickness in the first iteration. The low limit value of the thickness is given to the elements located outside of the two legged structure (see Figure for an example of the two legged structure). In the middle of the energy absorber, a small variation of the thickness could be observed. So, it is not necessary to do many iterations for the thickness optimization Results for the final topology optimization for the side load case After the thickness optimisation, elements were deleted. The case 3 with round boundaries, thickness optimization, and topology optimization is shown below.