AIAA-2004-2035

Handling Bifurcations in the Optimal Design of Transient Dynamic Problems Samy Missoum*, Samir Ben Chaabane♣, Bruno Sudret§ Ecole Supérieure d’Ingénieurs Léonard de Vinci, Paris La Défense, France Electricité de France, R&D Division, Site des Renardières, Moret-sur-Loing, France. Abstract This paper presents a methodology to tackle some of the difficulties encountered in the optimal design of transient dynamic problems. For this class of problems, the structural responses can be discontinuous due to numerous bifurcations. This characteristic makes gradient-based or response surface optimization techniques difficult to implement. This work (in progress) proposes an approach to define regions of the design space where the dynamic behavior is homogeneous and hence does not produce discontinuities. This is done by isolating points that correspond to unwanted bifurcations within boundaries that are defined explicitly in terms of the design variables. Using this method, a designer can obtain an optimal design with a prescribed dynamic behavior. The approach is applied to the design of a tube impacting a rigid wall. In addition, as the transient dynamic behavior is very sensitive to small variations of the design, reliability-based optimization is considered. I Introduction Currently, large-scale structural optimization problems can be solved efficiently with commercially available software. However, these problems are often limited to linear and simple nonlinear behaviors. For highly nonlinear problems such as those encountered in transient dynamics (e.g. crash), there are no efficient and systematic ways to perform an optimization. In the literature, two issues that hinder the optimization of highly nonlinear problems are often cited: • •

The cost of an analysis is usually prohibitive and is therefore difficult to include in an optimization loop with repeated analysis, Due to the numerical noise and convergence uncertainties often encountered in explicit dynamics codes, the sensitivities are very difficult to compute.

In order to overcome these difficulties, several authors have used metamodels and design of experiments (DOE) to optimize (or simply improve) their design-- especially for crashworthiness [1-4]. * Assistant Professor, Département Mécanique des Systèmes [email protected] ♣ Associate Professor, Département Mécanique des Systèmes [email protected] § Département Matériaux et Mécanique des Composants. [email protected]

The metamodel (or response surface) is used to replace the responses extracted from expensive simulations by a simple analytical model. The model, which is a function of the design variables, is built based on sampling points selected from the design space with a chosen design of experiments. In addition, response surfaces allow the removal of the numerical noise due to the simulations. Nonlinear dynamics problems also exhibit another major difficulty: the structural behavior is extremely sensitive to small variations of the design or to imperfections. The implications of these characteristics are twofold: •

•

Some responses of the system might be discontinuous with respect to small variations of the design due to the presence of numerous bifurcations and limit points, Deterministic optimal designs might be very unreliable as the “optimal design” is often located at the boundary of the infeasible domain in order to exploit “resources” to their best.

The discontinuities of responses prevent the blind use of gradient-based methods or response surfaces. These discontinuities are associated with clusters in the response space that translate into specific regions of the design space.

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The poor reliability of deterministic optimal designs makes it necessary to take sources of uncertainty into account during the optimization. Uncertainties appear on quantities such as material properties, dimensions or load conditions and are represented by probabilistic distributions such as normal or Weibull distributions. Due to its importance, reliability-based design optimization (RBDO) has triggered a strong interest in the optimization community [5-8].

Hypercube Sampling (LHS) is used as a design of experiments. The analysis is performed with the explicit code ANSYS / LS-DYNA. Based on the experiments, a region of the design space with no global buckling is identified with straight lines or ellipses. To perform the reliability-based optimization, the responses are approximated based on compactly supported radial basis functions (RBF). II Discontinuities and clusters: a simple example

In this research, we investigate an approach to identify clusters resulting from the response discontinuities and to define regions in the design space where the functions are continuous. Then, response surface techniques can be used in these regions in order to perform an optimization. In addition, as the regions are associated with various dynamic behaviors, the identification of specific regions allows the designer to specify a given behavior (e.g., no global buckling). The regions of the design space are separated through explicit functions of the design variables. These functions can be used as constraints for an optimization problem or a limit state function if reliability is considered. As a first approach in the present research, lines and ellipses are considered for the explicit region separating functions. For a reliability-based optimization, these functions allow defining explicitly the boundaries of an “unwanted” (failure) domain. It is then rather simple to calculate the probability of failure used in an RBDO, based on the distributions of the uncertain variables. In the first part of this paper, discontinuities, clusters and regions are presented based on a simple shallow truss problem with geometric nonlinearties. The clusters identification procedure is explained as well as the construction of the separating functions (lines and ellipses). It is followed by the description of a reliability-based optimization problem and the corresponding calculation of the probability of “failure” based on the defined explicit functions. In the second part of this paper, the design of a column subjected to a crash is considered. An optimization problem is defined with a constraint enforcing crushing along the main axis of the tube, thus avoiding global buckling. Indeed, maintaining a deformation along the axis results in a much better energy absorption than when global buckling occurs. In order to identify the region corresponding to crushing and to define the separating functions, Latin

As a demonstrative example exhibiting discontinuities, the classical symmetric shallow twobar truss (figure 1) is used [9]. The member crosssectional area is A and the applied force is F. The displacement under the point of application of the force is u.

F u

Figure 1. Two-bar truss The truss exhibits the typical snap-through behavior as depicted on the force / deflection (F(u)) diagram on figure 2. Two curves are plotted corresponding to trusses with cross sectional areas A and A + ∆ A. It can be seen that for the same load F, the system might exhibit snap-through (displacement u2) or not This can happen for (displacement u1). infinitesimally small variation ∆A, hence the displacement u is a discontinuous function of A. This might be a serious limitation if optimal design is considered.

F

A A+∆A

u1

u2

Figure 2. Force – deflection diagram. Two-bar truss. Another difficulty lies in the poor reliability of the optimal design of nonlinear problems. For the twobar truss, if we consider the case where the optimal design is such that the applied load is a limit load, then a small variation of force can make the system buckle. Hence, the “optimal” design becomes extremely unreliable.

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In order to get a good sense of the behavior of the truss, the displacement value is calculated for 400 values of the force and area chosen randomly. The couples (F, A) used are represented on figure 3 with the axes scaled based on the maximum values Fmax and Amax.

R

Figure 5. Division of the design space into two regions corresponding to clusters in the response space.

Figure 3. Couples (F, A) chosen for the study of the displacement The corresponding values of the displacements are plotted on figure 4. As the displacement is discontinuous with respect to the force and the area, two clusters are generated. The cluster that contains the circled dots is associated with a snap-through behavior.

If we want to optimize the two-bar truss enforcing no snap-through, then the design space is limited to the R region. The function separating the two regions (gR) can be used as a constraint for optimization and/or a limit state function (LSF) if uncertainties are considered. III Identification of clusters and definition of separating functions III.1 Cluster identification In order to identify the regions of interest in the design space, the clusters created by the discontinuities of a system responses have to be identified. There are statistical methods that are dedicated to finding clusters within a cloud of points. One of the most widely used methods is the K-means algorithm used in our work [10]. The basic idea of the method is to minimize the sum of the Euclidean distances of the points of a cluster to its centroïd. The number of clusters to be identified is an input parameter. In two-bar truss example, the cluster separation as depicted on figure 4 is obvious. There are cases where the separation is not so clear as will be seen in examples treated in this paper.

Figure 4. Displacement u with respect to force and area values. In the variable space (A, F) (figure 5), the set of points corresponding to the two clusters can be easily represented. The clusters result in two regions that can be separated, in this case, by a straight line. This separation, which is rather intuitive, can be demonstrated formally for this small problem, as the analytical solution is available.

III.2 Separating functions Once the clusters are identified, the corresponding regions are isolated in the variables space. This is done using simple geometric entities such as lines and ellipses that confine the points that belong to a cluster in a simply “shaped” domain. The construction of the separating functions is explained in the sequel with the following notation: N: Total number of sampling points S : Set of points with “unwanted” bifurcation points

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NS : Number of sampling points in S S : Set of points without “unwanted” bifurcation points

di → j = Pi − Pj 2 : Euclidean distance between points Pi and Pj D: Maximum Euclidean distance in S

{

D = max Pi − Pj 2 ; (Pi , Pj ) ∈ S 2

(L) is the line passing through A and B. Find the center C of [AB] and angle the θ (figure 7). Calculate b, the maximum distance from the points of S to (L). An example of ellipse is provided on figure 7. Step 2: Create the rotated ellipse of equation: T

g R (X) = ( X − Xc ) A R (X − Xc ) − 1 = 0

}

A, B: two most distant points in S

( {

( A, B) = Arg max Pi − Pj 2 ; (Pi , Pj ) ∈ S 2

(2)

T

with A R = R AR

})

 cos θ R=  sin θ

Algorithm for the definition of a linear separation function In a two dimensional space, the algorithm used for the linear separating functions is: Step 1: Find the maximum distance D and the corresponding points A and B. Step 2: Define the equation g(x1,x2)=0 of the line (L) going through A and B. Step 3: Create two lines (L1) of equation gR1(x1,x2)=0 and (L2) (gR2(x1,x2)=0) with the same slope as (L) so that:

∀ Pi ∈ S, g R1 (Pi ) ≤ 0 and g R 2 (Pi ) ≥ 0

(1)

An example of definition of linear separation function is provided on figure 6.

A

1/ α 2

-sin θ 

and A =  cos θ   0

0 



1/ β 2 

and α = D/2 and β=b Which is an ellipse of center C( XC ) and semi-axis α and β. Step 3: For all the points of S that are not included in the ellipse, modify β so that

∀ Pi ∈ S, g (Pi ) ≤ 0

(3)

For a point Pi that does not belong to the ellipse. Let e and f be the distances of Pi to its projection K onto (L) and the distance from K to C respectively (figure 8). In order for Pi to belong to the ellipse, the update is:   f 2  β = e2 / 1 −     α  

gR1

(4)

The final ellipse is such that all the points that belong to S are inside it.

gR2

B Figure 6. Example of linear separation functions

A α

It is noteworthy that when the two linear functions are defined as constraints, they cannot be used at the same time since the feasible space they define is disjoint. Two optimization problems have to be created with gR1 and gR2 as constraint respectively. The solutions of these two problems must be compared to find the optimum.

C

Algorithm for the definition of an elliptic separation function

Figure 7. Trial ellipse based on the most distant points.

β

θ B

In a two dimensional space, the algorithm is: Step 1: Find D and points A and B 4 American Institute of Aeronautics and Astronautics

Pi

K K

• Use of the reliability index β. In the standard normal space, β is the minimum distance from the origin to the limit state function. The closest point to the origin is referred to as the design point u* (figure 9). If the LSF is linear, the probability of failure is

K

e f

C

(L)

Figure 8. Modification of the ellipse to incorporate an outsider point. IV Reliability-based design optimization (RBDO)

The separating function defined in the previous section can be used as a limit state function for an RBDO problem formulated as follows: Min f (X)  X  s.t. g D (X) ≤ 0  P(g R (X) ≤ 0) < ε 

(5)

With X={ XD , XR } Where XD and XR are vectors of deterministic and random design variables respectively. gD is a vector of deterministic constraints and gR is a probabilistic constraint or limit state function (LSF). P(gR (X)≤0) is the probability of failure. ε is the maximum allowed probability of failure. It is interesting to further explain this so-called “optimization on random variables”. A deterministic optimal design would lead to an optimal design vector X*. When the structure is build according to this optimal design, it is likely that manufacturing imperfections appear, which would make the design unacceptable (i.e., the real structure falls in the unwanted region of the design space). In practice, those design parameters which are subject to non negligible uncertainties during manufacturing are modeled as random variables, whose mean value is updated in each iteration of the optimal design algorithm, whereas their standard deviation (or coefficient of variation) is prescribed in advance as a manufacturing tolerance. In this work, gR is the separating function given explicitly in terms of the variables. The probability of failure

P(g R (X) ≤ 0) = ∫

g R (X )≤ 0

f (X )dX XR

R

R

(6)

where f XR is the joint probability density function, can be calculated in several ways:

P(g R (X) ≤ 0) =1-Φ(β ) = Φ(-β )

(7)

where Φ is the cumulated normal distribution function. If the LSF is nonlinear, the probability of failure is approximated by:

P(g R (X) ≤ 0) ≈1-Φ(β ) = Φ(-β )

(8)

That is, the probability is approximated by replacing the original failure domain Ωf (where g R (X )≤0 ) by the half space tangent to the transformed LSF in the standard normal space (gu) at the design point. This approximation is the foundation of the first order reliability method (FORM)[11, 12]. The further the design point is from the origin, the better the approximation is. Another approximation is given by the second order reliability method (SORM) [11] that uses a second order approximation of the LSF around the design point.

gu (u) < 0

u2

Ωf Design point u* β

O

u1

Figure 9. Representation of the limit state function and reliability index in the standard normal space. These approaches are interesting when the LSF (or boundaries of the failure space) are not known explicitly with respect to the design variables. • Direct calculation of the probabilistic content When the limit state functions are known explicitly, it is more accurate to evaluate the probabilistic content of the failure space directly as the latter is also explicitly known. In the case of various linear LSFs, the probability of failure can be evaluated exactly based on the

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reliability indices associated to each individual LSF. Hence, this is applicable to our linear separation functions as in the example depicted in figure 6. Note that equation 7 is only valid if the random variables are transformed into the standard normal space. Another approach consists of solving the integral directly based on the known “shape” of the failure space. In our problem, this space is delimited by two lines or by an ellipse (figure 10).

objective of this work is to optimally design the tube so that no global buckling appears. That is, a constraint on the dynamic behavior has to be defined to enforce a crushing of the tube following its axis. The thickness t and the length L of the tube are chosen as basic design variables. Note that the section of the model has also been parameterized. However, a fixed rectangular section of height 50 mm and width 40 mm has been used.

The solution to the integral in equation 6 can be obtained numerically with the Monte Carlo approach. Based on the given distributions of the random variables, a very good approximate of the integral can be obtained efficiently with the Monte-Carlo approach. The advantage of this approach relies in the fact that there is no restriction on the type of probabilistic distributions and variable correlations used. As an add-on to this work in progress, we are investigating the formal (exact) calculation of the integral over an elliptic domain.

P=∫

ellipse

f X R ( X R )dX R

(9)

x2

x2

Ωf

Ωf

x1

x1

Figure. 11. Tube impacting a rigid wall. The analysis is performed with the explicit software ANSYS/LS-DYNA and the simulation time is 40 ms. Four masses of 15 Kg are located at the four corners at the rear of the tube. The tube is meshed with 3600 reduced integration Belytschko-Tsai shell elements. The tube can deform in various ways after impact onto the rigid wall. Here, the dynamic behavior is divided into two main categories: crushing (deformation along its axis) and global buckling. Two examples of these behaviors are given in figure 12 and 13.

Figure. 10. Failure domain delimited by lines and an ellipse The definition of explicit separation functions between the clusters allows the identification of a failure space with explicit boundaries for which the probabilistic content is rather easy to evaluate. V Transient dynamic problems

The methodology described above for the definition of the boundaries of a failure domain is applied to a transient dynamic problem. V.1 Problem description The problem considered is a tube impacting a rigid wall with a velocity of 15 m/s (figure 11). The

Figure. 12. Crushing of the tube

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V.4 Response study

To detect the design for which global buckling occurs, the absolute values of the maximum transverse displacements |Uxmax| and |Uymax| were stored. The sum |Uxmax| + |Uymax| is used as a response that encompasses the buckling in the x and y directions. This quantity should clearly exhibit a discontinuity compared to the case when there is crushing of the tube following its axis.

Figure. 13. Global buckling of the tube V.2 Design of experiments

The response is plotted in a 3D diagram with respect to the values of length and thickness (figure 15). For a clearer visualization of the point distribution, the response is projected on the (response, length) subspace (figure 16).

In order to sample the design space, the Latin Hypercube Sampling (LHS) technique [13,14] has been used .The ranges of the two variables are: L ∈ [0.3m , 1.0 m] t ∈ [1.0 mm, 5.0 mm]

The design of experiments, constituted of 100 points, is depicted in figure 14.

Figure. 15. Response plot with respect to the length and area.

Figure. 14. Design of experiments for the transient dynamic problem V.3 Deletion of invalid analysis

For each experiment performed, the ratio of hourglass energy over the total energy is stored. The hourglass energy is the work done by the artificial forces used to overcome the spurious modes that appear in elements with reduced integration. If the ratio is higher than 10%, the analysis is considered failed. Out of the 100 experiments, 7 failed this criterion and were not considered.

Figure. 16. Projected response plot. The points with the highest response value (i.e., sum of displacements) correspond to designs with global buckling. At this stage, one could impose an arbitrary limit on the response to select the points that “seem” without buckling. However, the use of the cluster identification technique (K-means) as described in III is a less arbitrary way of selecting the points.

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When using K-means with two clusters, the clusters identified are represented in figure 16. The circled dots correspond to points with potential global buckling.

 L  − 1.016  A  + 1.474 = 0 (10)     Lmax   Amax   L  − 1.016  A  + 0.949 = 0 (11) gR 2 ( A, L ) = −      Lmax   Amax 

g R 1( A , L ) = − 

If three clusters are to be identified, the result is given in figure 17. The green-circled dots are the points that correspond to a crushing following the axis. This set of points, automatically identified, is closer to the intuitive selection that one would have made based on a graphical inspection.

Figure. 18. Separating function based on an ellipse. Selection of points with three clusters.

gR1 Figure. 17. Response plot with three clusters In the case latter case, the distribution of “failure” points in the design space is given in figure 18.

gR2

Figure 19. Separating function based on straight lines In the case of two clusters selected, the ellipse gives a separating function that includes a moderate number of acceptable points.

Figure 18. Distribution of “failure” points in the design points. As a separating function, an ellipse is constructed automatically as described in III. However, in this case the use of an ellipse is far too conservative as a high number of acceptable designs (represented by the non-circled blue dots) are included in the ellipse (figure 18). If lines are used, the resulting separating functions are depicted in figure 19. The two line equations are:

Figure 20. Elliptic separation function. Selection of points based on two clusters. V.5 RBDO problem

The RBDO problem considered consist of finding the length L and the thickness t of the tube so that: 8 American Institute of Aeronautics and Astronautics

 V Min L,t   s.t. P((L, t ) ∈Ωf )  E/ET = 0.99  

≤ ε

(13)

Where V is the volume, E is the internal (absorbed) energy and ET is the total energy. Ωf is the domain limited by the two lines as defined by equations (10) and (11). The variable L is deterministic while the thickness follows a normal distribution with a mean defined as the current iterate of the optimization process and a standard deviation of σ=0.06 mm. The ratio absorbed energy over total energy is approximated by radial basis functions (RBF) with compact supports. This metamodeling technique is chosen because it allows grasping the behavior of complex responses. The approximation E of E/ET is given by: NS

E ( X) = ∑ aiϕ (r )

(14)

i =1

With ϕ (r ) = (1− r )4 (4 +16r +12r 2 + 3r 3 )   r ≥1 ϕ (r ) = 0

X − Pi r= R

r