FIUPSO Linköping University Bat 620 – Plateau du Moulon … · 2005. 2. 7. · continuum structures with arbitrary geometry (domain) for plate, shell and solid structures. The

FIUPSO Linköping University Bat 620 – Plateau du Moulon Mechanical Engineering Department 91405 ORSAY Solid Mechanics Division FRANCE SWEDEN

LITH-IKP-PR—04/11--SE Person in charge in the FIUPSO: Person in charge in Linköping University: Mr. Denis SOLAS Mr. Larsgunnar NILSSON

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.

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 4th of May 2004 until the 28th August 2004. 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.

CONTENTS 1. Introduction [1], [2], [3] ..............................................................................................5

1.1 General topology .................................................................................................5 1.2 Existing methods .................................................................................................7

1.2.1 Linear elastic, static problem.......................................................................8 1.2.2 Topology optimization for crashworthiness, non linear problem................8

1.3 Our concept........................................................................................................11 2. The application problem............................................................................................12 3. Static interpretation of the load cases ........................................................................13 4. Identification of suitable material..............................................................................14

4.1 Initial conditions and material parameters.........................................................15 4.1.1 The mass and the velocity of the rigid beam .............................................15 4.1.2 Influence of the Young’s modulus ............................................................15 4.1.3 Influence of the yield stress ......................................................................17 4.1.4 Influence of the density .............................................................................18 4.1.5 Influence of the hardening behaviour (etan)..............................................18 4.1.6 Influence of the hardening behavior and the yield stress ..........................19

4.2 Definition of simulation parameters .................................................................19 4.2.1 Tsim influence ...........................................................................................19 4.2.2 The hourglass deformation mode ..............................................................20 4.2.3 The mesh influence....................................................................................21

4.3 Final model and parameters...............................................................................22 5. Topology optimization - one load case at a time.......................................................24

5.1 Theory description.............................................................................................24 5.2 Symmetric frontal loading case .........................................................................25 5.3 Offset frontal loading case.................................................................................28 5.4 Side loading case ...............................................................................................30

6. Topology optimization – simultaneous evaluation of several load cases..................33 6.1 Theory description.............................................................................................33 6.2 Results of the optimization for several load cases.............................................34

7. Topology optimization – modification of the boundaries .........................................39 7.1 Theory description.............................................................................................39 7.2 Results of the optimization ................................................................................39

8. Topology optimization - evaluation using a scaled approach ...................................41 8.1 Theory description.............................................................................................41 8.2 Results for one loading case ..............................................................................42

8.2.1 Side loading case .......................................................................................42 8.2.2 Symmetric loading case.............................................................................44

8.3 Results for several loading cases simultaneous .................................................45

8.4 Results for several loading cases with a different internal energy level ...........47 8.4.1 Theory description.....................................................................................47 8.4.2 Results .......................................................................................................48

9. Topology optimization – Changing element thickness .............................................51 9.1 Theory description.............................................................................................51 9.2 Thickness optimization for the symmetric load case.........................................53 9.3 Final thickness and topology optimization........................................................55

9.3.1 Results for the thickness optimization for the symmetric load case..........55 9.3.2 Results for the final topology optimization for the side load case ............56

9.4 Results for the topology optimization for several loading cases simultaneously 57

10. Conclusions ...........................................................................................................60 11. Future work............................................................................................................61 12. Appendix ...............................................................................................................62

13.1 Software.............................................................................................................62 13.1.1 TRUEGRID [21] .......................................................................................62 13.1.2 LS-DYNA [22] .......................................................................................62 13.1.3 TRINITAS [24] .........................................................................................63

13. Bibliography ..........................................................................................................65

1. Introduction [1], [2], [3] 1.1 General topology

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 1994. 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.

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.

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 fiui, i=1, number of degrees of freedom (ndof) (1a) C Such that civi ≤V, i=1, number of finite element (nelem) (1b) Kij (c). uj (c) = fi, i, j = 1, ndof (1c) Where fi, is the external load; ui is the displacement due to the load and the equation (1a) represents the fact that the stiffest structure possible is the objective; ci is the design variable (volume fraction); vi 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.

1.2.2 Topology optimization for crashworthiness, non linear problem

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 self-contact 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., 1995. It used softening materials modeled by means of a new complementary energy principle introduced by Plaxton and Taylor in 1993. 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 σi din, i=1, ndof C Subject to

civi ≤V, i=1, nelem (2) Kij (c). uj (c) = fi, 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 two-dimensional 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 1998. 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.

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 fiui, i=, ndof C Such that civi ≤V, i=1, nelem (3) gm≤ 0, m=1, ncc (contact condition) Kip (c). uj (c) = fi, i, j = 1, ndof Where gm 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

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 üA

2 dt C δB-C≤ δ’ (4) civi ≤V, i=1, nelem Mij üj + Aij új + Kij uj = fi, i, j=1, ndof

Where üA is the acceleration at the structural point A, δB-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

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

energy absorber during a crash event. See below a scheme of the ground structure and the loading situation.

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

Bumper of the truck, top view

Energy absorber

Symmetric load case, velocity v0

Offset load case, velocity v0

Third load case (static)

Tyre

Max energy

Lack of internal energy, delete element

Symmetry axis

Fixed beam under the truck

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

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 4.1.2 to 4.1.6 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 non-linear behaviour. The evolution of the internal energy for a variation of time and control of the hourglass, see on figures from 4.2.1 to 4.2.3, is described in section 4.2.

4.1 Initial conditions and material parameters 4.1.1 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.

4.1.2 Influence of the Young’s modulus In a first consideration, a material is characterized by different laws. It can be elastic, elastoplastic, etc (Figure 4.1.2.1). 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.

Figure 4.1.2.1: Three different material behaviors

The hardening plasticity is a more complex material model than the other. [20] Figure 4.1.2.2 shows the influence of the Young’s modulus on the internal energy distribution and on the plastic deformation of the energy absorber.

Figure 4.1.2.2: IE, E = 0,5e9 N/m2 Figure 4.1.2.3: IE, E = 50 e9 N/m2 When the Young modulus increases, the internal energy increases in the energy absorber, and hence larger plastic deformations occur.

4.1.3 Influence of the yield stress

Figure 4.1.3.1: IE, σy = 50 e6 N/m2 Figure 4.1.3.2: IE, σy = 100 e6 N/m2

Figure 4.1.3.3: IE, σy = 100 e6 N/m2 Figure 4.1.3.4: IE, σy = 5 e6 N/m2 A comparison between the figures 4.1.3.1 and 4.1.3.2 together and 4.1.3.3 and 4.1.3.4 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 = σi εi. 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.

4.1.4 Influence of the density

Figure 4.1.4.1: IE, ρ = 5400 Figure 4.1.4.2 : 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”.

4.1.5 Influence of the hardening behaviour (etan)

Figure 4.1.5.1 : IE, etan =50 N/m2 Figure 4.1.5.2: IE, etan =6000 N/m2 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.

4.1.6 Influence of the hardening behavior and the yield stress

Figure 4.1.6.1: IE, etan = 5 e6 N/m2 Figure 4.1.6.2: IE, etan = 20 e6 N/m2 σ y = 5 e6 N/m2 σy = 20 e6 N/m2 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 4.1.6.2.

4.2 Definition of simulation parameters 4.2.1 Tsim influence

Figure 4.2.1.1: IE, tsim = 0,025s Figure 4.2.1.2: 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.

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 zero-energy 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 4.2.2.1).

Figure 4.2.2.1: 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 4.2.2.2, a high hourglass is observed and there is too much deformation in the mesh.

Figure 4.2.2.2: Hourglass deformations

4.2.3 The mesh influence

Figure 4.2.3.3: First mesh Figure 4.2.3.4: 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.

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;

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

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.

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;

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.

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

PREPROCESS TRUEGRID - shell mesh (2D) - definition of the materials parameters - hourglass control - time parameters

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 5.2.2 to 5.2.13.

Figure 5.2.1: Initial topology Figure 5.2.2: First iteration, 3.5% (Max IE= 571326270 Min IE= 3590450.8 Use IE= 23461204.472)

Figure 5.2.3: Second iteration, 3.5% Figure 5.2.4: Fourth iteration, 3.5% (Max IE= 822256770 (Max IE= 524162180 Min IE= 248247.64 Min IE= 448891.66 Use IE = 29018545.9226) Use IE= 18778856.7519) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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%

Figure 5.2.8: 32’nd iteration, 1% _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Figure 5.2.9: Initial topology Figure 5.2.10: First iteration, 2.5%

Figure 5.2.11: Fourth iteration, 2.5% Figure 5.2.12: Ninth iteration, 2.5%

Figure 5.2.13: 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 5.3.1 to 5.3.11.


Figure 5.3.3: Third iteration, 3.5% _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ ___ _ _ _ _ _ _ _ _ _


Figure 5.3.6: Third iteration, 2.5% Figure 5.3.7: Fifth iteration, 2.5% _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _


Figure 5.3.9: Sixth iteration, 2.5% Figure 5.3.10: Seventh iteration, 2.5%

Figure 5.3.11: 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.

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 5.4.1 to 5.4.8.

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 5.4.2. 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 5.4.5: 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 5.4.6. 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.

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.

Input.k Include.k

LS-DYNA Simulation of the internal energy for all load cases (3)


Analyze: - max IE - min IE - ave IE


Sum IE if there are several load cases


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 6.2.1 and 6.2.2: First shape and repartition of the IE for the frontal loading case

Figure 6.2.3 and 6.2.4: First shape and repartition of the IE for the frontal offset loading case

Figure 6.2.5 and 6.2.6: First shape and repartition of the IE for the side loading case

First iteration: 1%

Figure 6.2.7: Topology optimization Figure 6.2.8: IE, case 1

Figure 6.2.9: IE, case 2 Figure 6.2.10: IE, case 3 Sixth iteration: 1%

Figure 6.2.11: Topology optimisation Figure 6.2.12: IE, case 1

Figure 6.2.13: IE, case 2 Figure 6.2.14: IE, case 3 Ninth iteration: 1.5%

Figure 6.2.15: Topology optimisation Figure 6.2.16: IE, case1

Figure 6.2.17: IE, case 2 Figure 6.2.18: IE, case 3

12th iteration: 2%


Figure 6.2.21: IE, case 2 Figure 6.2.22: IE, case 3 14th iteration: 2.5%


Figure 6.2.25: IE, case 2 Figure 6.2.26: 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.

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 7.2.3: Fourth iteration, 1% Figure 7.2.4: Seventh iteration, 1.5%

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, 7.2.5 and 7.2.6 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.

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.

8.2 Results for one loading case 8.2.1 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.

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


Figure 8.2.1.1: Initial topology Figure 8.2.1.2: First iteration, 0.001%

Figure 8.2.1.3: Second iteration, 0.007% Figure 8.2.1.4: Third iteration, 0.001%

Figure 8.2.1.5: Sixth iteration, 0.005% Figure 8.2.1.6: Seventh iteration, 0.005% In conclusion, these figures show a slight modification of the final shape but the general topology is still conserved.

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 8.2.2.1: Initial topology Figure 8.2.2.2: First iteration, 1%

Figure 8.2.2.3: Fifth iteration, 2% Figure 8.2.2.4: Seventh iteration, 2.5%

Figure 8.2.2.5: 11’th iteration, 3% Figure 8.2.2.6: 15’th iteration, 3.5%

Figure 8.2.2.7: 16’th iteration, 6% Once again, the same topology, that is the part 7, Figure 7.2.6 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

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

Figure 8.3.9: Seventh iteration, 3% Figure 8.3.10: Seventh iteration, 3% Case 1 Case 3

Figure 8.3.10: Eighth iteration, 3% Figure 8.3.11: 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 5.2.13 for the symmetric case and in part 5.4, Figure 5.4.8 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 8.4.1 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.

8.4.2 Results

The optimization with the pressure case (case 2) and the symmetric loading case (case 1), simultaneously, is shown in figures 8.4.1 to 8.4.12. 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 8.4.3: 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

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 8.4.10: Sixth iteration, 6% Case 1 Case 2

Figure 8.4.11: Seventh iteration, 8% Figure 8.4.12: 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 (8.4.15 and 8.4.16) even for a high percentage the elements with a low internal energy

(corresponding to the elements on which the pressure is applied) are not deleted during the optimization.


Figure 8.4.15: First iteration, 6% Figure 8.4.16: 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.

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:

Figure 9.1.1: Repartition of IE Figure 9.1.2: Variation of the thickness

t increase

t decrease

t decrease

t decrease

Max IE

Min IE

Min IE

Min 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.

decrease IE

increase IE

increase IE

increase IE

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.

Input.k Include.k

LS-DYNA Simulation of the internal energy


Analyze: - max IE - min IE - ave IE

Perlscript Update model, topology optimization,

thickness variation

Sum IE if there are several load cases


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%

Figure 9.2.7: Fourth iteration, 50% This thickness optimization shown in Figure 9.2.7 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.

9.3.1 Results for the thickness optimization for the symmetric load

case

Figure 9.3.1.1: Initial thickness Figure 9.3.1.2: First iteration, 25% Shell element thickness

Figure 9.3.1.3: 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 5.2.4 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.

9.3.2 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.

Figure 9.3.2.1: Initial shape Figure 9.3.2.2: First iteration, 25% Thickness

Figure 9.3.2.3: Fifth iteration, 0.1% Figure 9.3.2.4: Seventh iteration, 0.1%

Deletion

Figure 9.3.2.5: Ninth iteration, 0.1%

A new repartition of IE due to the change of the thickness was observed and a new shape was obtained as it is explained in Section 1.3.

9.4 Results for the topology optimization for several loading cases

simultaneously

Optimization combining two loading cases using the thickness of the elements as design variables is done. The two load cases used are the 1 and the 3. The results from the internal energy distribution after an iteration changing the thickness of the element are

shown below.

Figure 9.4.3.1: first iteration, Thickness optimization, 25% After iterations, deletion of elements was done as in the optimizations performed in Chapter 5.

Figure 9.4.3.2: Fourth iteration, 1% Figure 9.4.3.3: Fourth iteration, 1% Case 1 Case 3

Figure 9.4.3.4: Sixth iteration, 2% Figure 9.4.3.5: Sixth iteration, 2% Case 1 Case 3

Figure 9.4.3.6: Seventh iteration, 3.5% Figure 9.4.3.7: Seventh iteration, 3.5% Case 1 Case 3

Figure 9.4.3.8: Ninth iteration, 5% Figure 9.4.3.9: Ninth iteration, 5% Case 1 Case 3

Figure 9.4.3.10: Tenth iteration, 5% Figure 9.4.3.11: Tenth iteration, 5% Case 1 Case 3 The final result of the optimization is a new topology, when the value of the thickness was modified in the first iteration (for the comparison, see the part 8.3) This new shape is more compact, fewer elements are deleted and the new structure is lighter than the structure from the thickness optimization only.

10. Conclusions Firstly, a comparison between the FE software used can be made: TRINITAS software is faster and easier in use during a simulation than LS-DYNA and it gives the same results at the end for the optimization scheme used. So for these schemes the static and the dynamic approaches give the same topology after optimization. In, TRINITAS a linear elastic, static problem is solved and in LS-DYNA a non-linear impact problem is solved. To conclude on the topology optimization, different Perl programs as the element deletion, the thickness optimization and the energy normalization have been developed and executed in order to see and improve the structural topology. At the end, a topology, where all the elements seem to absorb the same level of energy is obtained for all the optimization procedures. The study on the scaled approach has shown that several load cases with a different internal energy level could be evaluated in the same optimization procedure. More precisely, these different programs and optimizations have shown the importance of the choice of a rule for deletion of elements, e.g. a percentage of IE, and this parameters influence on the final shape. This influence is linked with the distribution of the internal energy in the energy absorber. For the same distribution a similar shape is obtained whatever the percentage of IE used but for a different distribution, a different topology is observed. Finally, the study of the influence of the thickness of the element gives a new kind of topology optimization. This thickness parameter converges towards a value fixed by the user from the first iteration. Thereafter a new distribution of the internal energy is observed and consequently a new shape is obtained for the final structure of our energy absorber.

11. Future work Different studies could be performed after this preliminary work. Some suggestions are:

- A stop criterion for the optimization process has to be defined in order to make it automatic. This criterion could be represented by different criteria like the kind of topology expected, the internal energy distribution, etc.

- A correlation to a real material should be done. Our fictitious material, is it

representative of a real material? Furthermore, topology optimization with the parameters of a real material has to be done. Which parameters values are the best? What kind of internal energy level or deformation is really expected? These are some of the questions which should be answered in order to know which material to choose and how to distribute it.

- It would be interesting, for the symmetric loading case, to perform more

iterations of the topology optimization. Perhaps after several iterations and mesh refinements some buckling appears in the small “legs” when the impact event occurs. Hence, this formulation might go into a buckling problem as well.

- The influence of the mesh on the final results could also be studied.

12. Appendix

13.1 Software

13.1.1 TRUEGRID [21] TrueGrid® is a Registered Trademark of XYZ Scientific Applications, Inc. Firstly, TRUEGRID, TG, is completely interactive. TRUEGRID is based on a powerful technique known as the projection method. The projection method allows faces, edges and nodes of the mesh to be directly placed on surfaces and edges and nodes of the mesh to be placed along curves.

TRUEGRID behaves in different ways depending on what commands have been issued so far. The processes of creating a mesh and creating geometry are separated within TG. One may observe different windows.

- the control phase: this is the initial phase of the code where output options are chosen, material models defined, and where parameters governing the final mesh output file are set. No graphics capabilities are available in the control phase.

- The part phase: this is entered as soon as a block mesh is created. Three new windows appear. It is in this phase where the mesh is constructed by positioning, projecting, deleting, zoning, refining and smoothing parts of the mesh. Boundary conditions for a part can also be specified in the part phase.

- The merge phase: this is where parts are assembled into one model by merging, gluing, nodes together that are within a specified tolerance of each other.

13.1.2 LS-DYNA [22] The program is owned and developed by Livermore Software Technology Corporation (LSTC) based in Livermore, California. LS-DYNA is widely used by the automotive industry to an analyze vehicle designs. LS-DYNA accurately predicts a car's behaviour in a collision and the effects of the collision upon the car's occupants. With LS-DYNA, automotive companies and their suppliers can test car designs without having to build and experimentally test a prototype, thus saving time and expense. LS-DYNA is a general purpose transient dynamic finite element program capable of simulating complex real world problems. LS-DYNA accurately predicts the stresses and deformations experienced by the materials, and determine if the material will fail. LS-DYNA supports adaptive remeshing and will refine the mesh during the analysis, as necessary, to increase accuracy and save time.

LS-PRE/POST is an advanced interactive program for preparing input data for LS-DYNA and processing the results from LS-DYNA analyses. The user interface is intuitive and easy to use. All data and menus are designed in a logical and efficient way to minimize number of mouse clicks and operations. LS-PRE/POST runs on all Unix/Linux workstations and MS/Windows computers utilizing the OpenGL graphics standard to achieve fast rendering and XY plotting. More information is given in the LS-DYNA theoretical manual. [23]

13.1.3 TRINITAS [24] TRINITAS is an integrated graphical environment for linear finite element analysis. This program system is under continuous development at the division of Solid Mechanics, Linköping University, and, at present, it is used in educational environments as well as for practical computations in structural and solid mechanics. TRINITAS capabilities can be summarized as follow: Mathematical foundations Boundary value problems Parabolic and hyperbolic initial value problems Eigenvalue problems Implemented algorithms Full, selective (and B-bar) integration technique Direct solution technique operating on sky-line stored matrices Generalized Newmark method for hyperbolic transient elasticity problems Generalized trapezoidal rule for parabolic transient heat transfer problems Subspace iteration technique for eigenvalue problems User features Based on a "What You See Is What You Get" technique Fully graphically driven behind Hierarchical User Interface No unnecessary figures needed for transforming a model idealization from your mind to the computer and the finite element algorithm, just use the mouse a number of tools designed for supporting repeated analysis sequences Support for geometry modeling, boundary condition definition, meshing, analysis and result evaluation

Research activities Shape optimization based on analytical Brookman derivatives Adaptive h-refinement techniques and shape optimization Topology optimization Application of general Object-Oriented data base technique in the TRINITAS finite element environment Implementation Techniques Object-Based program decomposition The entire code is new and written from scratch Dynamic memory allocation through a new special purpose data base implementation A general output interface embedding both X and PostScript

13. Bibliography [1] Ciro A. Soto, “Structural Topology Optimization: from minimizing compliance to maximizing energy absorption”, Int. J. Vehicle Design, Vol.25, Nos ½ (special issue), 2001 [2] Ciro A. Soto, “Applications of Structural Topology Optimization in the Automotive Industry: Past, Present and Future”, WCCM, July 2002, Austria [3] Takashi Ebisugi, Hideharu Fujita, Gozo Watanabe, “Study of Optimal Structure Design method for Dynamic Nonlinear Problem” JSAE Review, 1998 [4] Bendsoe, M.P., Kikuchi, N., “Generating Optimal Topologies in Structural Design using a Homogenization Method”, Comp. Meth. In Applied Mech and Engin, 71, (1988) [5] Yuge, K., Iwai, N., Kikuchi, N., „Topology Optimization Algorithm for Plates and Shells Subjected to Plastic Deformations”, Proc. 1998 ASME Design Engineering Technical Conference, paper DETC98/DAC-5603, 1998 [6] Soto, C.A., “Structural Topology Optimization for Crashworthiness Design by Matching Plastic Strain and Stress Levels”, ASME Design Engineering Tech. Conferences, sept 2001, Pittsburgh, PA [7] Pedersen, C.B.W., “Topology Optimization of 2D-Frame Structures with Path Dependent Response”, International Journal for Numerical Methods in Engineering, 2002 [8] Mayer, R.R., Kikuchi, N., Scott, R.A., “Applications of Topology Optimization Techniques to structural Crashworthiness”, Int. J. Num. Meth. Engrg, vol 39, 1996 [9] Huang, J., Walsh, T., Mancini, L., Wlotkowski, M., Yang, R.J., Chuang, C.H., “A New Approach for Weight Reduction in Truck Frame Design,” SAE 1993 Transaction, J. of Commercial Vehicles, 1993, [10] Yang, R.J., Chuang, C.H., “Optimal Topology Design Using Linear Programming”, Computers and Structures, 1994 [11] Duysinx, P., Bendsoe, M.P., “Topology Optimization of Continuum Structures with Local Stress Constraint”, Int. J. Num. Meth. Engrg, 1998 [12] Lipton, R., “Homogenization of Stress Fluctuations, Failure Criteria and design of Functionally Graded Materials for Strength and Stiffness”, submitted to J. of the Mechanics and Physics of Solids, 2002 [13] Cheng, G.D. and Jiang, Z., “Study on Topology Optimization with Stress Constraints”, Engineering Optimization, 1992 [14] Yang, R.J., Chen, C.J., “Stress-Based Topology Optimization”, Structural Optimization, vol 12, 1996 [15] Bendsoe, M.P., Diaz, A., Kikuchi, N., “topology and Generalized Layout Optimization of Elastic Structures”, Proceedings of the NATO Advanced Research Workshop on Topology Design of Structures, Eds. M. Bendsoe and C.A. Mota Soares, Sesimbra, Portugal, June 1992 [16] Nagai, K. and Igarashi, M., “Application of Structural Optimization Method to Crashworthiness”, Suzuki Tech. Rev., vol 22, 1996

[17] Kuboshima, T. and Ishizuka, “An Approach to the Optimizing Structures with PAN-OPT, PUCA ’96, 1996 [18] Ciro A. Soto, “Optimal Structural Topology Design for Energy Absorption: A Heuristic Approach, ASME 2001 Design Engineering Technical Conference and Computers and Information in Engineering Conference, Pittsburgh, PA, sept 2001 [19] D. Kosloff and G. Frazier, T Belytschko and co-workers, W.K. Liu, 0. P Jacquotte, J. T. Oden, N. Kikuchi and others. [20] http://iamlasun8.mathematik.unikarlsruhe.de/parallel/skript/node147.html [21] TRUEGRID training manual [22] http://www.lstc.com/master.htm [23] http://www.dynamore.de/download/manual/ls-dyna_theory_manual.pdf [24] http://ohio.ikp.liu.se/service_trin.htm

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FIUPSO Linköping University Bat 620 – Plateau du Moulon … · 2005. 2. 7. · continuum structures with arbitrary geometry (domain) for plate, shell and solid structures. The