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Figure 1 Configuration of a Typical Crash Box

Figure 2 shows the Deformation VS Reaction Force of the crash box under impact loading. Figure 3 shows the deformation of the box simulated by LS-DYNA. We will now use SmartDO to optimize the design.

Figure 2 Deformation VS Reaction Force of a Typical Crash Box Under Impact Loading

Figure 3 Crash Deformation of a Typical Crash Box Simulated by LS-DYNA

Software
The following software will be used for this problem.
(1) LS-PrePost 2.3 for pre- and post-processing (http://www2.lstc.com/lsprepost.htm).
(2) LS-DYNA for explicit transient finite element analysis (http://www.lstc.com)
(3) SmartDO for system integration and design optimization (http://www.fea-optimization.com/optm/index_e.htm )

Modeling Details and Problem Formulation
In order to solve this problem by Numerical Design Optimization, we will have to "formulate" the problem such that it can be fit into the solution mechanism of design optimization software, such as SmartDO. Here we will model and formulate this problem as follows

Units and Material Properties
The units used in the problem is mm-g-msec. The material used for the crash box is Aluminum EN AW 1200 O UNI EN 573-3, with the following parameters

Young's Modulus = 70000 MPa
Tensile Yield Stress = 82 MPa
Poisson's Ratio = 0.33
Density = 2.71E-3 g/mm^3
Tangent Modulus = 2000 MPa

The material formulation is assumed to be piecewise-linear plasticity. Material failure/rupture is not considered here. The thickness of the box is 2 mm.

Parametric Modeling
The crush box is decided by 6 (shaping) parameters, namely X(1) to X(6) as shown in Figure 4. This model was built in LS-PrePost 2.3, including its geometry, mesh and all the LS-DYNA input keywords. Currently it is not very straightforward to build the shaping parametric model with LS-PrePost. However with the Pre-Processor for Embedded Tcl/Tk (PET) in SmartDO, the users can actually apply Tcl/Tk on the input batch file of LS-PrePost 2.3 (even if if doesn't support Tcl/Tk at all !). Therefore with careful arrangement, a usual input batch modeling file in LS-PrePost can be easily transferred into a parametric model with SmartDO.

Figure 4 Parametric Model of the Crush Box

Design Optimization Formulation
As mentioned above, there are three major tasks in optimizing a crush box : maximum reaction force, energy absorption and maximum deformation. They are formulated as

Find : X(1)~X(6)
To Minimized : maximum reaction force of the tube
Subjected to :

Energy Absorption > 9.156 E6
Maximum Deformation < 241 mm

With

20 < X(1) < 80
20 < X(2) < 80
20 < X(3) < 80
10 < X(4) < 80
10 < X(5) < 80
10 < X(6) < 80

Initial Design

X = < 20, 80, 80, 75, 80, 80 >

Note that, here the constraints of energy absorption and maximum deformation are taken from the initial design. That is, we want SmartDO to improve the design as much as we can based on the initial configuration.

Design Optimization
SmartDO successfully optimizes the problem without much difficulty. The details are explained below.

Final Result
SmartDO comes up with the final configuration with the design variables of

X = < 20, 68, 68, 80, 10, 80 > (the initial design is X = < 20, 80, 80, 75, 80, 80 > )

Figure 5 shows the configuration of the Initial Design (Left) and the Optimal Design (Right).

Figure 5 Configuration of the Initial Design (Left) and the Optimal Design (Right) by SmartDO

The energy absorbed by the new design is around 9.16E6, the maximum reaction force is about 1.05E5 (compared with 1.22E5 for the initial design), and the maximum deformation is about 241 mm. The optimal design reduced about 20% of the maximum reaction force in the tube, and the energy absorption and deformation remains almost the same. Figure 6 shows the curves of Deformation VS Reaction Forces for both the Initial and the Optimal Design by SmartDO.

As we can see in Figure 6, SmartDO tries to lower the peak force yet still providing the same energy absorption, by re-arranging the reaction curve. Actually in some publication, the initial design was already considered optimal, but SmartDO can still reduce about 20% of its peak reaction force.

Figure 6 Deformation VS Reaction Force for Both Initial Design and the Optimal Design by SmartDO

Figure 7 shows the crushed deformation of the optimal design simulated by LS-DYNA

Figure 7 The Crushed Deformation of the Optimal Design Simulated By LS-DYNA

Computational Expense
One interesting thing to be observed would be the computational effort taken by SmartDO. As mentioned before, crashworthiness simulation is usually very time-consuming. Therefore if the optimization package needs too much finite element analysis, the whole process will be very unpractical.

For the example shown here, we have used the Robust Genetic Algorithms (RGA) in SmartDO (see these publications for reference). Since in the Genetic Algorithms we always have to discretize the design variables, we have chosen 8 points (2^3) for each design variables. If a DOE is to be used, it will requires the computations of 8^6=262144 sample points. For the current example, however, SmartDO has only taken 63 finite element analysis to achieve the optimal design shown here.

Conclusion and Remarks
In this issue of the SmartDO eNews, we have shown you how we can perform crashworthiness optimization with SmartDO. One thing important is that, the computational effort taken by SmartDO is very inexpensive. It is obvious that, SmartDO is a practical tools for industrial crashworthiness optimization.

For details about SmartDO, please visit our website at http://www.fea-optimization.com/optm/index_e.htm.