4. El complejo en el contexto regional
4.2. Vertiente occidental
4.2.3. Población y actividades económicas
8.1
Introduction
Energy absorption structures are employed where collision may cause serious consequences such as injury or fatality to humans and damage to vehicles (Johnson and Reid 1978; 1986; Jones 1989; Lu and Yu 2003). When such a structure is subjected to collision, the exter- nal kinetic energy is dissipated, to a great extent, by its large, plastic deformation. Design optimization of energy absorption structures is of special interest in the automotive indus- try. There has been some research carried out on optimizing parameters (e.g. dimensions) of tubular structures using structural optimization techniques (Lust 1992; Avalle et al. 2002; Jansson et al. 2003). It is noted, however, that there has been very limited work on topology optimization of energy absorption structures despite its great potential. The work by Pedersen (2003; 2004) on crashworthiness design deals with discrete frame structures. In this chapter, we present the more challenging work of topology optimization of continuum structures for energy absorption using a hard-kill BESO method (Huang et al. 2007).
8.2
Problem Statement for Optimization of Energy
Absorption Structures
Topology optimization problems of energy absorption structures usually have certain con- straints, such as limits on the force and the deformation (see Figure 8.1). Typically, a maxi- mum allowable crushing distance is prescribed, so as to retain sufficient space for survival of the occupants or other important devices. At the same time, a high level of force is required in order to dissipate a large amount of energy. However, the maximum force should not be too large, as it might be beyond the tolerance level of the occupants. In other words, the energy absorption structure should be neither too stiff, which may exceed the force limit; nor too compliant, which may exceed the allowable crushing distance. Therefore, an ideal
Evolutionary Topology Optimization of Continuum Structures: Methods and Applications Xiaodong Huang and Mike Xie C
2010 John Wiley & Sons, Ltd
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152 Evolutionary Topology Optimization of Continuum Structures
Stiff design
Cr
ushing force
Max allowable force Max crushing distance
Feasible optimal design
Flexible design
Crushing distance
Figure 8.1 Typical load-displacement curve and design constraints for energy absorption structure.
energy absorption structure should possess a rectangular force-displacement relationship (see Figure 8.1), although this is practically unachievable.
To obtain the most efficient energy absorption design, one may maximize the total absorbed energy per unit volume (E/V ) within the prescribed limits for the force and the displacement. Thus, the optimization problem can be formulated using the elements as the design variables as Maximize f (x)= E V (8.1a) Subject to Fmax= F∗ (8.1b) Umax= U∗ (8.1c) xj ∈ {0, 1} j = 1, · · · , M (8.1d)
where F is the external force and U is the displacement. F∗and U∗are the allowable maximum force and displacement. The binary design variable xjdeclares the absence (0) or presence (1)
of an element. M is the total number of elements in the design domain.
To simulate the crushing behaviour of a structure, nonlinear finite element analysis is conducted by gradually increasing the displacement of impact points from 0 to the maximum allowable crushing distance, U∗. Therefore, the maximum displacement constraint of Equation (8.1c) is easily satisfied. To satisfy the maximum force constraint, the sensitivity information of the maximum force needs to be calculated in conventional optimization methods. However, this is a difficult task for energy absorption structures with geometrical and material nonlinearities. In this chapter it is assumed that the force-displacement curve does not change significantly before and after an element is eliminated from the design domain as shown in Figure 8.2 and that the maximum force will decrease or increase as the total volume of the structure decreases
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Optimal Design of Energy Absorption Structures 153
Before element removal
After element removal
Displacement F orce ∆ Fi ∆ Fi–1 Fi–1 Ui–1Ui Un Fi
Figure 8.2 Force-displacement curves before and after removing an element for sensitivity analysis.
or increases. Therefore, the force constraint of Equation (8.1b) will be heuristically satisfied by varying the total volume of the structure. As a result, only the sensitivity of the objective function in Equation (8.1a) needs to be considered. This will be discussed below.
8.3
Sensitivity Number
8.3.1
Criterion 1: Sensitivity Number for the End Displacement
The variation of the objective function with respect to the change in design variable x is f (x) = 1 V E − E VV (8.2) According to the principle of energy conservation, the total strain energy is equal to the external work. When the structure is crushed to the end displacement U∗(which is equal to Un), the
total strain energy E and the total external work W can be obtained from Figure 8.2 as
E= W = lim n→∞ 1 2 n i=1 UTi − U T i−1 (Fi+ Fi−1) (8.3) It is noted that the above expression is identical to the defined mean compliance of nonlinear structures in Chapter 7 (see Equation 7.2).
When the jth element is completely removed from the system, according to the sensitivity analysis in Chapter 7, the variation of the external work can be approximately expressed by the final strain energy of the jth element as
E = −Ej
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154 Evolutionary Topology Optimization of Continuum Structures
where Enjis the total strain energy of the jth element when U= U∗. Meanwhile, the variation
of the total volume can be easily calculated by
V = −Vj (8.5)
where Vj denotes the volume of the jth element. Substituting Equations (8.4) and (8.5) into
Equation (8.2), the variation of the objective function can be rewritten as f (x) = 1 V Vj V E− E j n (8.6) From the above equation, a nondimensional elemental sensitivity number is defined by dividing the variation of the objective function by E/V as
αj n = Vj V − Enj E (8.7)
The elemental sensitivity number provides an estimate of the relative ranking of each element in terms of its effect on the objective function if it is removed. Note that the elemental sensitivity number can be positive or negative, which implies that the objective function may decrease or increase when an element is removed. To maximize the objective function, those solid elements with the highest positive sensitivity numbers should be switched to void; at the same time, those void elements with the lowest negative values should be changed to solid.
8.3.2
Criterion 2: Sensitivity Number for the Whole Displacement History
In general, the actual crushing distance of a structure varies with the amount of external kinetic energy and may not always reach the maximum allowable displacement U∗in a collision. In order to produce an efficient design that is suitable for any magnitude of impact within the allowable limit, the above definition of the sensitivity number can be extended to multiple crushing distance cases. For different crushing distances, the removal of an element has different effects on the objective function. An overall sensitivity number should be defined, which can be used to estimate the effect of the removal of an element on the external works for all crushing distance cases. Thusαj = n i=1 αj i (8.8)
where n is the number of crushing distance cases. The above sensitivity number can be found by gradually increasing the crushing distance from 0 to the maximum. In this chapter, we divide the maximum crushing distance into 10 even design points. The sensitivity number for each case is calculated once the crushing distance reaches the corresponding design point. Then, the topology evolves according to Equation (8.8). As a result, the final topology has high efficiency for absorbing energy across the whole range of the design points.
It is seen that criterion 1 focuses on the maximum energy absorbed per unit volume at the end displacement; whereas criterion 2 considers the energy absorption in the whole defor- mation history. For criterion 1, the aim is to absorb the largest amount of energy (at the end
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Optimal Design of Energy Absorption Structures 155
displacement) with the least amount of material. Therefore, the design performance can be measured by
e1=
W
V (8.9)
To meet criterion 2 most effectively, the structure should consistently sustain a high level of load below the allowable maximum crushing force throughout the whole displacement history. In this case, the design performance can be measured by
e2=
W Wmax
(8.10) where Wmax= F∗U∗is the maximum absorbed energy of an ideal energy absorption structure.