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American Institute of Aeronautics and Astronautics
1
Alpha Shape Based Design Space Decomposition for Island
Failure Regions in Reliability Based Design
Harish Ganapathy
1
Department of Metallurgical and Materials Engineering, IIT Madras, Chennai-600036, India.
Palaniappan Ramu2 * and Ramanathan Muthuganapathy
Department of Engineering Design, IIT Madras, Chennai-600036, India
Nomenclature
= natural frequency of the excitation frequency
= natural frequency of the absorber
= natural frequency of the original system
f(x) = objective function
gd = vector of deterministic constraints
gr = limit state function
L = Length of tube
l = length of minimal spanning tree
m = mass of absorber
M = mass of original system
n = No of nodes
Pftarget = maximum allowed or target probability of failure
R = the mass ratio of the absorber to the original system
r1 = ratio of the natural frequency of the original system to the excitation frequency
r2 = ratio of the natural frequency of the absorber to the excitation frequency
t = Thickness of tube
Uxmax = displacement in x-axis
Uymax = displacement in y-axis
ζ = the damping ratio of the original system
I. Introduction
tructural optimization involves repeated calls to Finite Element (FE) simulation to compute the objective
function or constraint(s). The simulations are run at each design point the optimizer visits in the design space.
Though recent developments in commercial FE software allow solving large scale highly nonlinear structural
problems, in an optimization framework it becomes infeasible due to challenges such as computational expense6
associated with repeated simulations and sensitivity computation. These problems only aggravate when probabilistic
approaches such as reliability based design are considered to account for uncertainties. In such situations,
researchers7-9
resort to metamodels based on design of experiments to optimize their design.
Metamodels replace expensive simulation by simple algebraic functions. The metamodels are fitted to
responses evaluated at design points selected through a Design of Experiment (DOE). Also, they help in removing
the numerical noise associated with computer simulations. However, metamodels are not suitable when the response
is highly non linear and discontinuous as in transient dynamic problems9. The advantage of using metamodels is that
they reduce the computational effort and provide an algebraic function for the boundary for the failure domain
which allows for direct integration of uncertainties.
In reliability studies, the boundaries of the failure domain in the design space need to be expressed using
explicit separation functions in terms of design variables. These are also called as limit states. Analytical approaches
1 Graduate Student, Department of Metallurgical and Materials Engineering, [email protected].
2 Assistant Professor, Department of Engineering Design, [email protected]. *Corresponding Author
3 Assistant Professor, Department of Engineering Design, [email protected]
S
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2
such as First Order Reliability Method (FORM) approximate the failure region as half plane but there are chances
that the failure region is an island in the design space. There could also be multiple such islands of failure in the
design space. Missoum et al2 used a convex hull approach to approximate the boundaries of such an island failure
domain.
In Missoum et al1-2
, the discontinuous response is used to identify the regions of unwanted behavior by
identifying the clusters in the design space. They used the K-means algorithm to identify the clusters. Once clusters
are formed, a convex hull is wrapped around the cluster that corresponds to unwanted behavior. The walls of the
convex hull form the boundary of the domain and can be represented using multiple linear functions. These
boundaries serve as explicit limit functions in terms of design variables. A limitation of the convex hull approach is
that, in order to preserve the convexity property, the convex hull might enclose points belonging to acceptable
behavior. This can be rectified to a certain extent by performing additional response evaluation around the
boundaries, only at the expense of more computational power. Sometimes, the cluster of unwanted behavior appears
as disjoint patches. That is, the points of unwanted behavior form multiple islands amidst points of acceptable
behavior. In such cases, the convex hull in order to preserve convexity approximates the disjoint patches of failure
as a continuous patch leading to an incorrect boundary of the failure domain. It is desirable to develop an approach
that can handle multiple islands well, with limited simulations.
This work proposes to use the alpha shapes to decompose the design space. Similar to Missoum et al1
clustering
techniques are used to identify clusters in the design space. Once clusters are identified, alpha shapes are used to
form the boundary of the clusters and hence the boundary of the failure domain. One fundamental difference from
the convex hull approach is that, alpha shapes enclose only the points belonging to a particular patch. Similar to
convex hull, the walls of the alpha shape can also be approximated using linear functions which will serve as limit
states for reliability studies and allow straightforward inclusion of uncertainties in the design process. Rest of the
paper is organized in the following manner: Section 2 describes alpha shapes and how it can be used to decompose
design space with multiple islands. Two demonstrative examples using alpha shapes are discussed in the section 3
and finally summary of the work is provided in Section 4.This version of the paper discusses using alpha shapes for
decomposing the design space. The final version will include reliability estimation and optimization for design space
with island failure zones.
II. Alpha Hull and Alpha Shape Edelsbrunner et al
3 introduced the concept of alpha-hulls as a natural generalization of convex hulls. An alpha hull
of a set of points is the space generated, edges constructed by point pairs that can be touched by an empty disc of
radius alpha. Alpha hulls have curved edges resembling the curved disc periphery. When these curved edges are
replaced by straight lines they are called alpha shape. The difference between alpha shapes and hull is presented in
Figure 1. In Figure 1, the red disc represents the disc of radius alpha and forms the hull boundary and the blue line
represents the boundary of alpha shape between two alpha nodes (or simply called a point in design space). The
structure of alpha shape solely
depends on the alpha value. For a
same set of points the shape
differs with alpha. Since we are
interested in linear limit state
functions, we would use alpha
shapes in this work.
Alpha shape is elegant and
efficient to compute. However,
selection of an optimal alpha, the
radius of the disc is a challenge.
Mandal and Murthy4 suggest a
way to find alpha through
minimal spanning tree approach
as in Eq 1.
(1)
Where l=length of minimal spanning tree of nodes;
n= No of nodes;
Figure 1. Difference between Alpha Shapes and Hull.
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The alpha obtained is for a given set of points. Under multiple islands case, the alpha might take different values for
different islands. In addition, alpha shapes suffer from formation of Multi degree edges(encircled) as shown in
Figure (2a), multi degree nodes (encircled) as in Figure (2b) and multiple patches as in Figure (2c). A complex alpha
shape with all the above mentioned drawback is shown in Figure (2d).
In this work, the optimal alpha is selected using the following algorithm:
1. Delaunay Tessalation is performed on the set of design points. The maximum and minimum length of Delaunay
edges are recorded.( Delaunay tessalation: A tessellation obtained by connecting a pair of points p.q S with a line
segment if a circle C exists that passes through p and q and does not contain any other site of S in its interior or
boundary. The edges of DT(S) are called Delaunay edges. The resultant is Delaunay triangulation ). This minimum
edge length is assumed to be the initial alpha value.
2. The alpha is tuned from its minimum value and is subjected to a delta increment in each loop (here we use 1%)
and alpha shape is recomputed until we get a shape without lines (i.e. multi degree alpha edges or zero area patches).
At this stage the design space might comprise of different alpha shapes with multi-degree alpha nodes, alpha shapes
with multiple patches (internal loops) etc as shown in Fig 2d. The alpha shape(s) obtained in this stage is to be
processed to obtain clear boundary.
3. Vertices of each different alpha shape are extracted separately. For the points belonging to each alpha shape
alone, Delaunay tessellation is performed. The maximum edge length is assumed as the radius and used to
Figure 2. Challenges associated with Alpha shapes. (a) Multi degree edges (b) Multi degree nodes (c) Inner loops
and multiple patches (d) A combination of (a), (b) and (c)
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reconstruct that particular alpha shape. This helps in avoiding the internal loops and multi-degree nodes. Finally the
resultant design space will be comprised of different alpha shapes corresponding to different island failure regions.
Each alpha shape corresponds to its own alpha value.
An artificial design space with multiple islands is considered. Figures 3a and 3b show how both convex hull
approach and alpha shape approach would approximate the points belonging to unwanted behavior. It is clear that
alpha shape bounds the islands more appropriately than the convex hull.
III. Numerical examples for island boundary estimation In this section, alpha shapes are used to decompose the design space. The examples considered are the nonlinear
transient dynamic example treated in Missoum et al2 whch is a good example of island failure domain and a tuned
mass damper example.
A.Nonlinear Transient Dynamic
The problem considered is a tube impacting a rigid wall with a velocity of 15 m/s (Fig 4). The tube crash can
occur in two ways:
(i) Along the axis of the tube, called crushing
(ii) Global buckling
Crushing is preferable to global buckling as the former is a better energy absorption mode. The objective of this
work is to optimally design the tube so that no global buckling appears. The details of the example are presented in
Table 1.
The reader is referred to Missoum et al2 for further details. LHS design of experiment is used to sample the
design space. The ranges of the two variables are L - [300 mm; 1000 mm]; t - [1:0 mm; 5:0 mm];
Table 1. Details of the Transient Dynamic Example
Figure 3. Approximations of boundaries of multiple islands (unwanted behavior) in design space. (a) Convex
Hull (b) Alpha Shape
Design Variables Thickness t and length L
Height 50 mm
Width 40 mm
Software Ansys LS Dyna
Simulation Time 40 ms
Elements 3600 Belytschko– Tsai shell
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The design of experiments, constituted of 100 points, is depicted in Fig 5. Four vertices of the domain were added to
the design of experiments. Therefore, the total number of sampling points is 104. |Uxmax|+|Uymax| is recorded and
plotted in Figure 6. The points with the highest response value (i.e., sum of displacements) correspond to designs
with global buckling. The circled dots correspond to points with potential global buckling. The clusters in the
response space translate into corresponding sets of failure and acceptable points in the design space as represented in
Fig. 7.
Figure.4. Tube impacting a rigid wall. Two modes of energy absorption.
Figure 5. LHS Design of Experiments.
Figure 6. Response plot: |Uxmax|+|Uymax|
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Decision functions are constructed in the design space to define the boundaries of the failure domain. Here,
the alpha hull is used as the decision function. The convex hull boundary from Missoum et al2 is also provided for
comparative purpose. Figure.8 show the convex hull approach to the clustered points. Fig.9 and Fig.10. show the
alpha shape after the preliminary iteration and further processing respectively. It is to be noted that in Fig.9. the
alpha shape obtained is with internal loops and then processed to get efficient alpha shape as in Fig.10.
Comparison of Figs. 8 and 10 clearly show that the processed alpha shape provides a much more precise and
less conservative definition of the failure domain than the convex hull based approach.
B. Tuned Mass-Damper
Figure 11 illustrates the tuned damper system presented in Ref 5. It consists of the single degree of freedom
system and a dynamic vibration absorber to reduce the vibrations. The original system is externally excited by a
harmonic force. The absorber serves to reduce the vibration.
Figure 7. Distribution of failure and acceptable
points in the design space (length, thickness)
Figure 10. Final Alpha shapes
Figure 9. Alpha shape with internal loops
Figure 8. Convex hull approach to provide distinct
boundary
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The amplitude of vibration depends on
, the mass ratio of the absorber to the original
system
ζ, the damping ratio of the original system
, ratio of the natural frequency of the original
system to the excitation frequency
, ratio of the natural frequency of the absorber to
the excitation frequency
The amplitude of the original system normalized by the amplitude of its quasi static response and is a function of
four variables expressed as (Eq 2)
(2)
This example treats r1 and r2 as random variables. They follow a normal distribution N(1,0.025) and R = 0.01,
ζ=0.01. The normalized amplitude of the original system is plotted in Figure 12. There are two peaks where the
normalized amplitude reached undesirable vibration levels. The corresponding contour plot is presented in Figure
13. There are two islands of failure.
MAKE IT r1 and r2 in Fig 13
Considering 500 sampling points the alpha
shape approach is applied here to decompose the the
failure region for the above design space. Fig.14. a and
b shows the preliminary and final alpha shapes. It can
be observed that the alpha shape in Fig 14 cannot be
obtained using convex hull.
MAKE IT r1 and r2 in figure 14 – figure axes
Figure 11. Tuned vibration absorber
Figure 12. Normalized amplitude vs r1 and r2
Figure 13. Contour of the normalized amplitude
r1
r2
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Discussions:
1)It is observed from various cases that obtaining each of the individual alpha shape with the maximum radius in
second step gives better result than computing for whole set of points.
2) Alpha shapes are not a complete solution to the challenges introduced by the convex hull approach. That is,
even alpha shapes confine some acceptable behavior points into the unwanted behavior patch.
3) The alpha shape(s) is dependent on the number of sample points. The more the points, the better is the
approximation. However, convergence study of the area encompassed by an alpha shape can be carried out. Such a
study will let us optimize the number of samples that are required.
IV. Reliability estimates based on boundaries approximated by alpha shapes
Once the boundaries are approximated using the alpha shapes, it is straight forward to account for the uncertainties.
For each design point in the space considered, samples are generated with the point as mean and a defined co-
efficent of variation. Failure probabilty estimate for each point is the ratio of sum of the points that fall within the
area approximated by alpha shapes to the total number of sampled points. It is shown in Missoum et al [2] that is
advantageous to work in the reliability index space than failure probability space. Reliability index and failure
probability are related as: Pf =(-), where is the standard normal cumulative distribution function and is the
reliability index. It is to be noted that with sample evaluations limited to the initial DOE, the reliability index of each
of the point in the design space is obtained. This is possible because the projection of response contour in design
space is made available by the alpha shape approach. The reliability indices obtained in such a fashion are presented
in Figure 15 for the transient dynamic problem discussed in the previous section. Here, L and t are the design
variables. While L is deterministic, t follows a normal distribution with mean as the current iteratre and 0.06x tmax as
the standard deviation. It is to be noted that the space between the two islands and many other points away from the
alpha shape boundary have zero failure probability theoretically because no random sample of t falls within the
failure zone delimited by the alpha shapes. However, for the purpose of optimization, the zero failure probability
points are considered as high reliability points and are replaced with a reliability index of 4.75.
A. Reliability Estimation:
The widely used methods to compute the failure probability are Monte Carlo Simulations (MCS) or moment-based
methods such as the FORM. Here, we propose to use MCS. MCS requires the statistical distributions of the random
parameters. Then, the uncertainties are propagated and failure probability estimated as ratio of number of sample
that violates the limit state to the total number of samples used.
Figure 14. Alpha shapes for the failure zones in the tuned mass damper design space.
(a) Preliminary (b) Final
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The boundaries of the alpha shape can be replaced by explicit equations in terms of the design variables. This allows
representing the failure domain with explicit boundaries. Each equation is a limit state equation and a point is
considered to be failed if it violates all the limit state functions simultaneously. To find out whether an MCS sample
is inside an alpha shape, a standard ray-shooting algorithm is implemented. The ray shooting algorithm works on the
principle of shooting a ray from a point in any direction and by measuring the number of intersections the ray
makes, one can find whether the point is inside or outside the polygon. Here, a point within the alpha shape would
intersect the alpha shape only at one point before it hits the boundary of the design space. A point outside the alpha
shape will intersect twice before hitting the boundary of the design space.
Figure 15. Transient dynamic problem. Reliability indices for the design points.
V. Reliability-based design optimization (RBDO)
The RBDO problem consisted of finding the deterministic length L and t for which the volume is minimized:
,
. : Prob(( , ) )
0.99
L t
f
T
Min V
s t L t
E
E
(3)
Where V is the volume, E is the internal (absorbed) energy and ET is the total energy. 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 =0.06 x tmax mm. . The target failure probability is 1x10-3
.
To include the energy ratio as a constraint in the RBDO problem, it is fit with a response surface that not only
removes the numerical noise but also prevents the repetitive calls to costly transient dynamic simulations. Another
response surface could be used to approximate the probability of failure which is also known to be very noisy.
However, due to acute variations of the probability of failure, it is usually recommended to fit the reliability index
instead. Both response surfaces are fitted in the (L, t) space with second order polynomials.
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The response surface for the energy ratio is presented in Figure 16. The surface in the figure shows the quadratic
surface and the blue dots represent the values of exact energy ratio. The fitted response may not capture the actual
energy ratio exactly, but it follows the trend and represents the region of interest (E/ET 0.99) well. Similarly, for
the reliability index, a second order polynomial is fit and the surface of the approximated reliability index represents
the surface of the exact reliability index well. The approximated reliability index and the energy ratio are used in the
process of optimization. The results are presented in Table 2. The accuracy of the energy ratio and the reliability
index response surfaces are given in Table 3 based on the error measures.
Figure 16. Response surface of (a) approximated energy and (b) approximated reliability index
Table 2: Optimal design for the transient dynamic problem Alpha shape separation function
*100,000 Samples
Table 3: Error metrics for the response surfaces
It can be observed from the figures that the response surfaces were locally fine and we were able to get the
optimal solution. However, with more dimension, the visualization is not possible and better surrogate fitting and
error metrics like PRESS need to be used. However, the results in Table 2 clearly demonstrate the advantage of
using the alpha shape to bound the failure space.
Optimum Failure
probability*
Energy Ratio
t(mm) L(mm) V(mm3)
Convex Hull 4.74 563.8 430418.6 0.001 0.99
Alpha Shape 2.62
493 219115.6 1.13e-6 0.99
Metrics Reliability
index
Energy
ratio
R2 0.90 0.8913
R2 adj 0.8846 0.8851
RMSE 0.074 0.078
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VI. Summary In RBDO designs, the design space needs to be decomposed into failure and safe domains. Once decomposed the
boundary of the failure domain is used for reliability estimates. There are cases that are discussed in this paper
where the failure zone is an island in the design space. It is challenging to identify multiple such islands and create
their boundaries. An alpha shape based approach is proposed in this paper that will help one construct the boundary
of a failure zone. The examples discussed in the paper show that the alpha shape based approach can approximate
the boundary of the failure region better than other approaches in the literature, especially for multiple island case.
Once that is done, propogation uncertainties and obtaining reliability estimates are straight forward using MCS. A
RBDO problem is formulated and solved demonstrating the advantage of alpha shape approach.
Acknowledgment
This work is supported through the NFRG grant from Indian Institute of Technology Madras. Authors thank
Mr.BarathRam, Mr.Viswanath, and Mr. Jiju P Nair, members of the CAD and Geometric modeling lab for their
support in this work. Harish Ganapathy (first author) worked on this paper as a summer 2011 fellow at the Indian
Institute of Technology Madras.
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