# Crashworthiness

Crashworthiness design using Topology Optimization.

Comments and questions can be sent to e-mail: cbwp2@eng.cam.ac.uk

Design problem and Objective function.

Modelling (Beam elements and plasticity).

Analysis (Implicit backward Euler algorithm) and Analytical Sensitivities (Direct differentiation).

Animations of structures optimized with clamped supports.

Animations of structures optimized with sliding supports.

Some of the results in the present page can also be found in Pedersen (2004,2002a,2002b). Related work is presented in Pedersen (2003a,2003b,2003c,2001a,2001b).

Design problem and Objective function.

In this example the car front in figure 1 will be designed with respect to crashworthiness using topology optimization. The acceleration and the displacement of the point just under the front passenger in the cabin is the subject of the optimization.

If one has the desired acceleration history given as a function of time, see figure 2, the following minmax-formulation can be used in the optimization,

where is an error estimate of each design points,

which describes the relative acceleration error between the actual response and the desired response in the chosen design points. *M* is the number of design points. The heights *h* of the beam elements are the design variables.

Because of the symmetry of the car front, only half the car front is used in the optimization. Figure 3 shows the ground structure containing two masses. In the optimization the rigid wall is either simulated as clamped or sliding supports. Furthermore the masses are assumed to be rigid. For simulating these rigid parts of the structure the heights of the beams where the masses are placed have a minimum height of 0.10 m.

**Figure 3:** Symmetric condition then the half of the car front of figure 1 is analysed. The passenger cabin and the engine are simulated as rigid masses *M _{1}* = 750 kg and

*M*= 75 kg. In the ground structure 205 nodes are connected by 292 elements. Left: clamped supports. Right: sliding supports.

_{2}

The design domain has a ground structure (see figure 3) consisting of beam elements with rectangular cross sections. The in-plane height for each beam is the design variable. Each beam has an imperfection and each beam is also subdivided into two elements to obtain more realistic buckling behaviour.

The plasticity of the beam elements is modelled using a plastic zone model.

In the analysis numerical damping is added to stabilize the numerical integration. The numerical damping dampens only the contributions in the response from the high eigenfrequencies.

But more important numerical damping also smoothes the sensitivities. When the response of the ground structure is highly oscillating, the sensitivities will have an even more pronounced oscillating behaviour. The oscillating sensitivities give often designs which end up in local minima. For obtaining more smooth sensitivities, numerical damping is added.

Animation a to p show eight optimized structures optimized using the two ground structures in figure 3 with different supports.

Animation a and i show the responses of two structures optimized using maximum numerical damping and 10 design points. It can be seen that when damping is added in the optimization, the optimized structures have a highly oscillating response with high acceleration peaks when they are analysed without damping (see animation b and j). This is because the effect on the response from the axial stiffness of the beam elements have been neglected when numerical damping is added.

The following describes a method to solve the problem of designs optimized with numerical damping having oscillating responses with high peaks. The main idea is to add 10 more design points to the optimization problem. These 10 design points have positions close to the existing 10 design points, to ensure that the slope of the actual response is close to the slope of the desired curve.

The responses of structure e and m optimized using 20 design points and numerical damping are close to the desired response. The optimized designs have also responses close to the desirable response when they are analysed without damping (see animation f and n) even though they are optimized with numerical damping.

The structures in animation a, e, i and m are then used as an initial starting guess for an optimization where no numerical is added. The responses of these optimized structures are given in animation d, h, l and p. In these designs the amplitudes of the highly oscillating response for no numerical damping are decreased. Furthermore, there are only small differences in the topologies for the designs obtained using maximum numerical damping and using no numerical damping. The designs optimized with no numerical damping (see animation d, h, l and p) have fewer beam elements in the designs compared with the designs optimized with maximum numerical damping (see animation b, f, j and n). Then it can be concluded that the small changes in the designs can remove the highly oscillating peaks in the response. In particular, the beam elements with low heights are removed so from a manufacturing point of view, these designs are also more desirable.

Modelling (Beam elements and plasticity).

See reference Pedersen (2004,2002a,2002b).

Analysis (Implicit backward Euler algorithm) and Analytical Sensitivities (Direct differentiation).

See reference Pedersen (2004,2002a,2002b).

For measuring the CPU-time for computing the analytical sensitivities relative to the CPU-time for the analysis of the plastic zone model, the initial ground structure in the right figure 3 is used as a test example. The CPU-time for the analytical sensitivities of one design variable is determined to be 2.2 % relative to the CPU-time for the analysis. So the total CPU-time for computing the sensitivities is 292*2.2% = 642% compared with the time for determining the equilibrium.

Animations of structures optimized with clamped supports.

Optimized structures for the design domain shown in the left figure 3.

In all the examples the material is assumed to be steel. The material parameters are: Young's modulus *E*=2.1*10^{11} N/m^{2}, yield stress is 5.1*10^{8} N/m^{2} N/m^{2} and hardening *h*'=0.05*E*. The thickness *b* for all the elements is equal to 0.05 m The initial design domain is described by beam elements of identical height.

Optimized using 10 design points and maximum numerical damping.

Animation a: Analysed with maximum numerical damping.

Animation b: Analysed with no numerical damping.

Optimized using 10 design points and no numerical damping.

(Previous structure applied as initial design).

Animation c: Analysed with maximum numerical damping.

Animation d: Analysed with no numerical damping.

Optimized using 20 design points and maximum numerical damping.

Animation e: Analysed with maximum numerical damping.

Animation f: Analysed with no numerical damping.

Optimized using 20 design points and no numerical damping.

(Previous structure applied as initial design).

Animation g: Analysed with maximum numerical damping.

Animation h: Analysed with no numerical damping.

Animations of structures optimized with sliding supports.

Optimized structures for the design domain shown in the right figure 3.

Optimized using 10 design points and maximum numerical damping.

Animation i: Analysed with maximum numerical damping.

Animation j: Analysed with no numerical damping.

Optimized using 10 design points and no numerical damping.

(Previous structure applied as initial design).

Animation k: Analysed with maximum numerical damping.

Animation l: Analysed with no numerical damping.

Optimized using 20 design points and maximum numerical damping.

Animation m: Analysed with maximum numerical damping.

Animation n: Analysed with no numerical damping.

Optimized using 20 design points and no numerical damping.

(Previous structure applied as initial design).

Animation o: Analysed with maximum numerical damping.

Animation p: Analysed with no numerical damping.

A Seat example.

Desinging a seat for a truck when is exposed to an explosion. Here, two load cases are considered: a quasi-static load case due to daily driving and a dynamic crashworthiness load case due to an explosion. See example.

Discussion.

This page shows results where the plastic zone model is used for crashworthiness optimization. The beam elements with rectangular cross sections have high axial stiffness so numerical damping is added to stabilize the Newmark scheme. The numerical damping have also an enormous smoothing effect upon the sensitivities.

If numerical damping is added in the optimization procedure the designs may not be valid when they are analysed with no numerical damping. This problem can partly be solved by adding extra design points.

Pedersen, C. B. W.: 2004, Crashworthiness Design of Transient Frame Structures using Topology Optimization, *Computer Methods in Applied Mechanics and Engineering*, 193(6-8): 653-678, 2004.

Pedersen, C. B. W.: 2003a, Topology Optimization of 2D-Frame Structures with Path Dependent Response, *International Journal for Numerical Methods in Engineering*, 57:1471-1501, 2003.

Pedersen, C. B. W.: 2003b, Topology Optimization Design of Crushed 2D-Frames for Desired Energy Absorption History, *Structural and Multidisciplinary Optimization*, 5-6:368-382, 2003.

Pedersen, C. B. W.: 2003c, Topology Optimization for Crashworthiness of Frame Structures, *International Journal of Crashworthiness*, 8:29-39, 2003.

Pedersen, C. B. W.: 2002a, Topology Optimization of Energy Absorbing Frames, *WCCM V - Fifth World Congress on Computational Mechanics*, Vienna, Austria.

Pedersen, C. B. W.: 2002b, *On Topology Design of Frame Structures for Crashworthiness*. Ph.D. thesis, Technical University of Denmark, 2002.

Pedersen, C. B. W.: 2001a, Topology Optimization with Respect to Dynamic Crushing, *2nd Max Planck Workshop on Engineering Design Optimization*, Nyborg, Denmark.

Pedersen, C. B. W.: 2001b, Topology Optimization of 2D-Frames for Crashworthiness, *4rd WCSMO*, Dalian, China.