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Alpha Shape Based Design Space Decomposition for Island Failure Regions in Reliability Based Design

Harish Ganapathy a,*, Palaniappan Ramub, Ramanathan Muthuganapathyc

aGraduate Student Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai-36, India E-mail address: harishganapathy@yahoo.com * Corresponding author 207, Material Processing Section – Workshop, Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai 600036, India Phone: +91-9994493540

bAssistant Professor Department of Engineering Design, Indian Institute of Technology Madras, Chennai-36, India E-mail address: palramu@iitm.ac.in cAssociate Professor Department of Engineering Design, Indian Institute of Technology Madras, Chennai-36, India E-mail address: mraman@iitm.ac.in Abstract

Treatment of uncertainties in structural design involves identifying the boundaries of the

failure domain to estimate reliability. When the structural responses are discontinuous or

highly nonlinear, the failure regions tend to be an island in the design space. The boundaries

of these islands are to be approximated to estimate reliability and perform optimization. This

work proposes Alpha (α) shapes, a computational geometry technique to approximate such

boundaries. The α shapes are simple to construct and only require Delaunay Tessellation.

Once the boundaries are approximated based on responses sampled in a design space, a

computationally efficient ray shooting algorithm is used to estimate the reliability without

any additional simulations. The proposed approach is successfully used to decompose the

design space and perform Reliability based Design Optimization of a tube impacting a rigid

wall and a tuned mass damper.

Key words: Alpha shape, reliability, island failure region, design space decomposition,

optimization

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1 Introduction

Often times, structural 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

expense (Kutaran et al. 2002) 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,

researchers (Sobieszczanski et al. 2001; Gu 2001) resort to metamodels (/surrogates) based

on design of experiments (DoE) to optimize their design.

Metamodels are emulators that replace expensive simulation by simple algebraic

functions. Also, they help in removing the numerical noise associated with computer

simulations. However, metamodels are not suitable when the response is highly nonlinear and

discontinuous as in transient dynamic problems. In addition, metamodels might lead to

erroneous failure probability estimates (Ramu et al. 2008).

In reliability studies, the boundaries of the failure domain are expressed using explicit

separation functions in terms of design variables, in the design space. These are also called as

limit states (Melchers 1999). Analytical approaches 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. Missoum et al. (2007) used a convex hull approach to

approximate the boundaries of such an island failure domain. There could also be multiple

such islands of failure in the design space.

In Missoum et al. (2004, 2007), 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 a particular 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 another cluster as well. 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

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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.

Basudhar, Missoum and their co-workers, through a series of papers addressed the problem

of decomposing the design space using a method called Support Vector Machines (SVM)

from statistical learning theory. Basudhar et al. (2008b) used SVM to construct explicit limit

state functions of disjoint failure regions and used it in reliability based design. Basudhar et

al. (2008a) proposed an adaptive sampling scheme to construct an accurate failure domain

boundary with less function evaluations. Dribusch et al. (2010) adopt the same idea with a

multifidelity approach and used it in solving an aero elastic problem that has disjoint failure

regions. Basudhar et al. (2010) present an improved adaptive algorithm where they also

addressed the phenomenon of locking in SVM. They also note that kernel selection plays an

important role in the application of SVM. Lacaze and Missoum (2013) combine kriging and

SVM for solving RBDO problems in a sequential two level scheme. First, they approximate

the objective function and failure domain by kriging and SVM respectively. Then, using

subset simulation technique (Au and Beck 2001), they evaluate the probability of failure and

sensitivity information. Finally, they propose a novel max-min algorithm and refine the

failure domain locally for better estimates. Jiang et al. (2011) decompose the design space

using SVM while the variables are correlated and solve a reliability problem. Basudhar et al.

(2012) use Probabilistic SVM to solve efficient global optimization (EGO) problems.

Basudhar and Missoum (2009) develop an approach to adaptively refine the SVM locally in a

design space and use this approach to solve reliability based design optimization problems.

Lin et al. (2013) extend the idea of SVM to parallelize SVM boundary estimation. Hao et al.

(2012) construct the limit state using SVM and sample adaptively around the boundary to

refine it. Song et al. (2013) combine SVM with kriging and generate a virtual sample near the

classification boundary to increase the classification accuracy. They successfully demonstrate

it on a series of large variable problem. Yang and Hsiesh (2013) developed a framework PS2:

Particle Swarm Optimization (PSO), Subset Simulation and SVM, to address RBDO with

discrete design parameters. The SVM classifier provides information to PSO about the

feasibility of solutions while the PSO optimal solution serves as training points for the SVM.

Thus they work cooperatively and achieve better accuracy. Haldar and Farag (2010) propose

an evaluation method which requires few samples as against many samples for estimating

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reliability of time dependent loading. Here, they combine efficient factorial design schemes

with analytical approach such as FORM to accomplish this.

Based on the above discussions, locating the discriminant boundary is one of the major

tasks in the raw data. Prominent approaches for this problem come from either statistical

learning theory or from computational geometry. In the former, SVM has played a primary

role for identifying the boundary where as in the latter, prominent constructions such as

convex hull and Gabriel graph (which is a subset of proximity graphs) have played key role.

As we introduce Alpha (α) shape as another computational geometry construction for the

boundary detection, it is worthwhile to compare with other approaches. Typical factors that

are employed while comparing these approaches are as follows (Zhang and King 2002):

• Worst-case Complexity

• Set membership classification

Formulation of the SVM for detecting the boundary is done using optimization techniques.

Arriving at the optimum solution for SVM takes O(n5 log n) in the worst case, whereas

Gabriel graph takes about O(n3) for boundary detection (Zhang and King 2002). Also,

parameter setting is another key factor for the success of SVM. In order to increase the

classification accuracy, Kim et al. (2003) proposed an ensemble of SVM along with boosting

(or bagging) and de Freitas et al. (1999) propose a sequential SVM using Kalman filtering

and observe that it is a better alternative to solving the quadratic optimization problem. On

the other hand, computation of α shape takes only O(n log n) in the worst case. In comparison

with SVM that depends on many parameters and require optimization to solve it, α shape

depends on only a single parameter α. The value of α depends on the nature of the problem.

Often, once the classification of data is done, any new data point can be classified using

the existing polygonal boundary (convex hull or alpha shape), rather than evaluating the new

point. This problem, termed as point classification problem in the field of computational

geometry, is a powerful one. This can be solved using ray-shooting algorithm that runs

typically in logarithmic time (Kalos and Martin 1998) essentially justifying the use of

computational geometry techniques. Here, we propose to use the ray shooting algorithm to

estimate reliability. It is to be noted that is a very powerful approach for reliability estimation

as no additional function evaluations are required.

This work proposes to use the α shapes to decompose the design space. α shapes have

their roots in computational geometry and are a generalization of convex hulls. In that

perspective this will be a different paradigm on solving the island failure region types of

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problems and its performance is compared with one such computational geometry technique,

the convex hull. Similar to Missoum et al. (2004) clustering techniques are used to identify

clusters in the design space. Once clusters are identified, α shapes are used to form the

boundary of the clusters and hence the boundary of the failure domain. Similar to convex

hull, the walls of the α 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. α shapes are a broad generalization of convex hull and can be derived

from Delaunay triangulation. The developments are well understood and powerful algorithms

are available to address higher dimensional problems as well.

Rest of the paper is organized in the following manner: Section 2 describes α shapes and

how it can be used to decompose design space with multiple islands. Two numerical

examples are used to demonstrate the island boundary estimation using α shapes in section 3.

Section 4 discuss the reliability estimation and Reliability based Design for the two examples.

Discussions on the limitations of the proposed approach and scope for improvements are

presented in section 5 ahead of conclusions.

2 Identification of clusters and design space decomposition using α shapes

2.1 Cluster identification

To carry out any design study, the design space needs to be decomposed into safe and

failure regions. To accomplish this, the design space is explored using a Design of

Experiment (DoE) and the evaluated responses are grouped using clustering techniques.

Statistical techniques are available to find clusters, given a cloud of points. This work uses

the K-means algorithm to identify the clusters. The number of clusters needs to be input

apriori and this sometimes is a limitation. However, there are adaptive K-means algorithms

that find the optimal number of clusters in an iterative fashion. The basic idea of K-means

algorithm is to minimize the sum of Euclidean distances of the points of the cluster to its

centroid. The boundaries of the identified clusters decompose the design space into regions of

interest. Simple entities like lines, ellipses and convex hulls were used to construct the

boundaries in earlier works (Missoum et al. 2004; Missoum et al. 2007). However,

sometimes points belonging to the same cluster might be available in multiple patches in a

design space. When such clusters contain points that correspond to failure, we refer it as

island failure region. In such situations, the decision functions used in earlier work (Missoum

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et al. 2004; Missoum et al. 2007) do not work well. This work uses the α shapes to obtain the

boundaries of such island regions.

2.2 α shapes and α hulls

Edelsbrunner et al. (1983) introduced the concept of α hulls as a natural generalization of

convex hulls. The positive α hull of a set of points (say, {p1, p2, . . . ,pn}) is the intersection

of all closed discs (termed as α discs) with radius Rα (where Rα=1/α) that contains all the

points. Negative α hull, on the other hand, is the intersection of all closed compliments of

discs that contains all the points. Essentially, negative α hull is generated by point pairs that

can be touched by an empty disc of radius Rα. In this work, the term α hull represents

negative α hull. More details on α hull is available in Edelsbrunner et al. (1983) ,α shapes are

widely used in many applications which are reviewed in Edelsbrunner (2010).

α hulls have curved edges (passing through point pairs) resembling the curved disc

periphery. α shape is obtained when the point pairs having curved edges is replaced by

straight lines. The difference between α shapes and hull is presented in Figure 1. In Figure 1,

the dashed circle represents the disc of radius 1/α, the red arcs form the hull boundary and the

blue line represents the boundary of α shape between two α nodes (or simply called a point

in design space). The structure of α shape solely depends on the α value. For a same set of

points the shape differs with α. With reliability estimation in perspective, we use α shapes in

this work. That is, the α shapes can be represented as linear boundaries which directly adapt

itself to analytical approaches like First Order Reliability Method (FORM).

α shape is elegant and efficient to compute. α shape has been shown to be very closely

related to popular computational geometry structures – Voronoi diagram and Delaunay

triangulation. A region is a Voronoi region of a point p1, if the points in the region are as

close to p1 as to any other point in the set. Voronoi diagram is the union of all the Voronoi

regions of all the points in the set. Figure 2 shows the Voronoi diagram of a set of points.

Let G = {V,E} represent a graph G with nodes V and the set of edges connecting the

vertices in V be denoted as E. Let each Voronoi region be represented as a node (using its

corresponding point). Connecting the nodes by a straight line only if they share a Voronoi

edge results in a graph. The graph is a triangulation, termed as Delaunay triangulation (DT).

Since the nodes and edges are arrived out of Voronoi diagram (VD), this graph is called dual

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and hence DT is a dual graph of VD. DT is shown superimposed with VD in Figure 3 (do

note that Voronoi diagrams in Figures 2 and 3 are for different sets of points).

As the centers of α disc has been shown to lie on the Voronoi diagram, and VD for a set of

points can be computed efficiently, the α shape of a set of points can also be computed

efficiently (Edelsbrunner et al. 1983). α shape has found applications in wide variety of fields

including shape reconstruction (Edelsbrunner and Ernst, 1994) molecular modeling (Wilson

et al. 2009) etc.

Figure 1. Alpha shape and hull

However, selection of an optimal α, the radius of the disc is a challenge. Mandal and

Murthy (1997) suggest a way to find α through minimal spanning tree approach as in Eq 1.

ln

α = (1)

 

Figure 2. Voronoi diagram of a set of points

Figure 3. Voronoi diagram and Delaunay triangulation

 

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Where l = length of minimal spanning tree of nodes; n = No of nodes. Packer et al. (2011)

developed a classification engine by overlaying different alpha shapes and use the correlation

in feature to decide on an alpha. The α obtained is for a given set of points. Under multiple

islands case, the α might take different values for different islands. In addition, α shapes

suffer from formation of multi degree edges (encircled) as shown in Figure (4a), multi degree

nodes (encircled) as in Figure (4b) and multiple patches as in Figure (4c). A complex α shape

with all the above mentioned drawbacks is shown in Figure (4d).

This work proposes the following algorithm to identify islands and to select suitable α to

avoid issues stated in Figure 4.

1. Given a DoE, responses are evaluated at all points and classified into two groups (say

safe and failure cloud).

2. Delaunay Tessellation is performed on complete set of failure point cloud. The

maximum and minimum length of Delaunay edges is recorded. The minimum length

 

Figure 4. Unfavorable featuresinα shapes. (a) Multi degree edge (b) Multi degree node (c) Inner loops and multiple patches (d) Combination of (a), (b) and (c)

 

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is assumed to be the initial radius of empty disc Rα and the failure point cloud is

subjected to alpha shape encapsulation.

3. The obtained alpha shape is checked for presence of multi degree edges and zero area

patches. Multi degree edges are formed when boundaries of two circles touch each

other and pass through the same point. If Rα is increased, it is possible to avoid the

multi degree edges and zero area patches. Therefore, Rα is incremented (here we use

1%) until there is area convergence and the shape is void of zero area patches. At this

stage the design space might comprise of different α shapes (for example see Figure

5b) corresponding to the islands. These shapes might have multi-degree nodes and

internal loops that need to be processed to obtain clear boundary.

4. For each α shape, the points are extracted and re-processed. After Delaunay

Tesselation the maximum edge length is assumed as the radius and used to reconstruct

the α shape. This avoids the formation of internal loops and multi-degree nodes.

Finally the resultant design space will be comprised of different α shapes

corresponding to different islands.

It is to be noted that each α shape in a design space corresponds to independently defined

disc radius Rα. Each α shape can either be convex or concave, depending on the distribution

of points of a particular island and radius of empty disc (Rα).

In order to show the difference between convex hull and α shape in approximating the

boundaries of an island region in a design space, an artificial design space with three islands

is considered. Figures 5a and 5b show how convex hull approach and α shape approach

would approximate the boundaries of points belonging to different behavior. It is clear that α

shape bounds the islands more appropriately than the convex hull.

(a)

(b)

Figure 5.Approximations of boundaries of multiple islands in design space. (a) Convex Hull (b) Alpha Shape

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3 Numerical examples for island boundary estimation

In this section, α shapes are used to decompose the design space. The examples

considered are the nonlinear transient dynamic example treated in Missoum et al. (2007) and

a tuned mass damper example presented in Chen et al. (1999).

3.1 Nonlinear Transient Dynamic example

The problem considered is a tube impacting a rigid wall with a velocity of 15 m/s (Figure 6).

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 al. (2007) for additional details. LHS design of

experiment is used to sample the design space. The ranges of the two variables are given in

Table 2.

Figure 6. Tube impacting a rigid wall. Two modes of energy absorption

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x  

y  

z  

L  

t  

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Table 1. Details of the Transient Dynamic Example

Table 2. Ranges of t and L

The DoE with 100 points is depicted in Figure 7. 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 8. 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 Figure 9.

   

Design Variables Thickness t and length L

Height (mm) 50

Width (mm) 40

Simulation Time (ms) 40

Elements 3600 Belytschko– Tsai shell

Variable Min Max

t (mm) 1 5

L (mm) 30 100

Figure 7. LHS DoE  

Figure 8. 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 α shapes is used as the decision function. The convex hull boundary from

Missoum et al. (2007) is also provided in figure 10 for comparison. Figure 11 and Figure 12

show the α shape after the preliminary iteration and further processing using the proposed

algorithm. It is to be noted that in Figure 11, the α shape obtained is with internal loops. It is

then processed to obtain the final α shape as in Figure 12. The SVM boundary is also

presented in Figure 12 for comparison purpose. SVM was used with Gaussian Radial Basis

Function kernel and a scaling factor of 0.35. The reported accuracy is 93.5%. This need not

be the best SVM classifier one can get. Here it is presented to show that alpha shapes can

perform equally well in classification. Comparison of Figures 10 and 12 clearly show that the

processed α shape provides a precise and less conservative approximation of the failure

domain than the convex hull.

   

Figure 9. Distribution of failure and acceptable points in the design space

(length, thickness)  

Figure 10. Convex hull approach to provide distinct boundary

 

Figure 11. Alpha shape with internal loops  

Figure 12. Boundaries from final Alpha shapes and SVM

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3.2 Tuned Mass-Damper

The Tuned Mass-Damper (Chen et al. 1999) presented in Figure 13 is treated here. The

amplitude of vibration depends on

mRM

= , the mass ratio of the absorber to the original system

ζ , the damping ratio of the original system

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nr ωω

= , ratio of the natural frequency of the original system to the excitation frequency

21

nr ωω

= , ratio of the natural frequency of the absorber to the excitation frequency

11n

KM

ω = , ratio of the stiffness of original system to its mass

22n

Km

ω = , ratio of the stiffness of absorber system to its mass

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

2

2 22 2 22

2 2 21 1 2 1 2 1 1 2

11

1 1 1 1 1 11 4

ry

Rr r r r r r r r

ζ

⎛ ⎞− ⎜ ⎟⎝ ⎠

=⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − − + + −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦

(2)

This example treats 1r and 2r 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 14. There are two peaks where the normalized amplitude reached

undesirable vibration levels. The corresponding contour plot is presented in Figure 15 with

two islands of failure. α shapes are used here to decompose the failure region for the above

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design space. Figure16.a and b shows the final α shapes with and without the design points.

The α shapes were obtained through a LHS DoE of 600 samples. It can be observed that the

α shape in Figure 16 cannot be obtained using a convex hull.

Figure 13. Tuned vibration absorber  

Figure 14. Normalized amplitude vs r1and r2

Figure 15. Contour of the normalized amplitude  

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(a) (b)

While constructing the α shapes, the following are observed:

1) Computing individual α shape with the maximum radius in second step (proposed

approach in previous section) gives better result than computing it using the entire

set of points.

2) α shapes are not a complete solution to the challenges introduced by the convex hull

approach. That is, even α shapes confine some acceptable behavior points into the

unwanted behavior patch.

3) The α 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 α shape can be carried out. Such a study will let us optimize the

number of samples that are required.

4 Reliability based Design Optimization (RBDO)

Reliability based Design Optimization is one of the advantageous methods in structural

design because of its ability to account for the unavoidable effects of uncertainty. Since it’s

numerically involving, over the past few decades researchers have developed different

approximate reliability methods, advanced simulation techniques and surrogate based

concepts to address the challenges. Valdebenito MA and Schuëller GI (2010) provide a

review on the different approaches in RBDO. They note that for system reliability analysis

with large number of variables and non-linear functions, simulation approaches are the

Figure 16. Alpha shapes for the failure zones in the tuned mass damper design space. (a) with DoE points (b) comparison of exact boundaries with alpha shape approximation

 

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natural choice while approximate reliability estimation approaches are suitable for component

reliability analysis with near linear limit state functions. Youn and Choi (2004) also discuss

the errors introduced while dealing with non-linear limit state functions. In order to reduce

the computational effort in simulation approaches, score function based re-weighting

schemes are usually used (Kleijnenand and Rubinstein, 1996, Fonseca et al. 2007, Lee et al.

2011). Lin et al. (2014) propose an alternate approach to use the kernel density estimation to

estimate the sensitivity of probabilistic responses. In this section, we discuss estimating the

reliability after the alpha shapes are constructed and thereafter using the estimated reliability

for RBDO.

4.1 Reliability estimation

Once the boundaries are approximated using the α shapes, it is straightforward to account

for the uncertainties. When the design space is sampled with random realizations, the failure

probability is computed as the ratio of the number of samples that fall inside the boundary to

the total number of samples.

The problem of whether a point in design space is inside an island or not is equal to the

following classical problem in the field of computational geometry – whether a point is

inside/outside a given polygon. As there can be more than one island, this query has to be

addressed with respect to all the identified islands (polygons). To solve this problem, we use

the classical ray shooting technique (de Berg 1993) to find whether a point in design space is

inside an island or not. The algorithm is explained in Figure 17.

Figure 17. Ray shooting algorithm. (a) Point p is inside the polygon (b) Point q is outside

the polygon

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Ray shooting algorithm finds whether a point is inside an inland or outside by computing

the number of intersections a ray emerging out of the design point makes with the boundaries

of the polygon. If a point is inside an island as point p in Figure 17a, the ray projected in any

direction like R1 and R2 makes odd number of intersections before it reaches the design space

boundary. If a point is outside the island as point q in Figure 17b, the rays in any direction

like S1 and S2 make even number of intersections before it reaches the design space boundary.

In spite of the formulation of the problem being simple, it has posed lot of issues to

implement it efficiently. Though a naïve approach is possible using an exhaustive search, this

will result in exorbitant amount of time and hence heuristics are employed to improve the

efficiency. In this paper, the algorithm that uses simple co-ordinate checks combined with

winding number has been employed (Hormann and Agustos 2001).

It is to be noted that this is a powerful approach to estimate failure probability because no

additional response evaluation is done after the boundary is obtained from the initial DoE and

reliability estimation is reduced to a computational geometry problem. It is shown in

Missoum et al. (2007) that it is advantageous to work in the reliability index space than

failure probability space. Reliability index and failure probability are related as: ( )fP β=Φ − ,

where Φ is the standard normal cumulative distribution function and β is the reliability index.

Through the ray shooting algorithm discussed above, the reliability can be estimated at design

points in the DoE.

4.2 RBDO of the nonlinear transient dynamic problem

The RBDO formulation for the problem presented in 3.1 consisted of finding the length L

and thickness t for which the volume is minimized:

( ),

1target

. : - Prob(( , ) ) 3

0.99

L t

f

T

Min V

s t L t

EE

β−Φ ∈Ω > =

=

(3)

Where V is the volume, fΩ is the failure domain and targetβ is the target reliability index.

Here it is taken as 3, which corresponds to a failure probability of 0.001. The design variables

L and t follow a normal distribution with the mean defined as the current iterate of the

optimization process and a standard deviation of 0.02×Lmax for L and 0.06 ×tmax for t.

Missoum et al. (2007) considers L to be deterministic. The second constraint is that the

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energy ratio, T

EE

need to be equal to 0.99 where E is the internal energy (absorbed) and ET is

the total energy.

As discussed in the previous sections, once the design space is decomposed using the α

shapes, reliability index of each point in the DoE can be estimated using the ray shooting

algorithm. It is to be noted in Figure 12 that the space between the two islands and many

other points away from the α shape boundary have zero failure probability theoretically

because no realization of t falls within the failure zone delimited by the α 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.

The energy constraint is evaluated using a Polynomial Response Surface. A quadratic

polynomial is fit to the energy ratio. The details of the response surface are provided in

Table 3. The details of error metrics are provided in appendix 1. It can be observed that the

surrogate is reasonably good. Since the spread of reliability index across the design space is

not very smooth, that constraint is evaluated using a kriging surrogate. A kriging model with

first order polynomial and spline correlation functions were constructed using the surrogate

toolbox (Viana 2011). As suggested by Acar (2013), different correlation function and

polynomial order were used and the ones with best cross validation metric were picked. The

results are presented in Table 4. Based on these response surfaces, the optimization in Eq 3 is

carried out using Sequential Quadratic Programming (SQP) and the results are presented in

Table 5. It can be clearly observed that α shapes are advantageous in approximating the

island failure regions and hence finding a better design in terms of volume. The optimal

values were used to run a validation FE simulation and it was observed that the energy ratio

was 0.99.

Table 3. Error metrics for the polynomial response surface – energy ratio

Metrics Values

R2 0.8913

R2adjusted 0.8851

RMSE 0.0747

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Table 4. Error metrics for kriging – reliability index

Table 5. RBDO results for the transient dynamic example

Method t (mm) L(mm) Volume (mm3) Reliability Index Energy Ratio

Convex hull 4.80 645.26 498037 3.09 0.99

Alpha shape 2.84 656.80 314566 4.75 0.99

4.3 RBDO of the Tuned Mass Damper

The RBDO formulation for the problem presented in 3B consists of finding

,

target

. : 2.25m MMin R

s t β β> = (4)

In order to achieve the numbers presented in 3.2, (r1, r2=N(1,0.025), R = 0.01, ζ =0.01), the

stiffness are considered random and follow a normal distribution such as:

K1= N (10, 2.5)

K2= N(0.1,0.025).

This randomness contributes to 1r and 2r being random. We consider ω = 0.316. A

decomposed 1r and 2r space is presented in Figure 18. This space will vary depending on m

and M. A simple uniform grid of m and M are considered with the following

limits:M=[90:10:110] , m=[0.9:0.1:1.1]. The decomposed space of 1r and 2r for each point in

the M-m grid is presented in Figure 17. The allowable vibration limit is 35. For each of the m-

M combination, using 600 samples in a LHS DoE, boundaries are approximated in 1r and 2r

space. Upon obtaining the boundaries, the reliability estimates can be found using the ray

shooting algorithm. It is to be noted that while using the ray shooting algorithm, the random

realizations that fall outside the design space are neglected. Once the reliability indices are

found for the M-m grid, aquadratic polynomial response surface of the reliability indices is

constructed whose metrics are provided in Table 6. The fitted response surface is used for

constraint evaluation in the process of optimization given by Eq 4. The results of

Metrics Values

PRESS_RMS 1.39

R2pred 0.99

20    

optimization are presented in Table 7. From the metrics presented in Table 6, the index

estimated as 2.25 using the response surface can vary anywhere between 2.09 and 2.4. The

reliability was estimated with 1e5 samples (as against 600 in the proposed approach) and the

optimum values presented in Table 7. Reliability index was found to be 2.11. This results in a

6.6% error in the final estimate which translates to 30% error in failure probability. The

comparison of the relative errors is presented in Table 8. Since one knows the variability in

the estimate, if required, one can adjust the target index such that even with the lower bound

of the deviation, one can achieve the required reliability index. Another logical option would

be to use better surrogates which might mean a choice of surrogate or more samples in the

initial DoE.

 

21    

Má

m

ê 0.9 1.0 1.1

90

100

110

In the subfigures, x axis and y axis denote ‘r1’ and ‘r2’ respectively

Figure 18. Decomposed r1 and r2 space for different combinations of m-M

Table 6. Error metrics for the Reliability Index response surface

R2 0.9258

R2adjusted 0.8022

RMSE 0.21

22    

Table 7. RBDO results for the tuned mass damper example

M 90.53

m 1.06

R 0.012

Rel Index 2.25

Table 8. Relative errors between predicted and actual probabilities of failure

Method Probability of Failure Reliability Index

Response surface 0.0122 2.25

Actual (MCS-1e5 Sample) 0.0174 2.11

Relative error (%) 29.89 6.6

5 Discussion:

The performance of the proposed approach is dependent on the DoE, the dimension of the

problem and other factors. In this section, we discuss its limitations in the current form,

extension to higher dimension and the scope for further improvement.

5.1 Dependency on the DoE

It is to be noted that the island approximation using alpha shapes is only as good as the initial

DoE selected. For cases like the Tuned Mass Damper, large number of samples are required

to capture the narrow islands. Therefore, selection of DoE plays a vital role in avoiding the

error owing to sampling. Even popular designs like LHS leave large chunks of design space

unsampled for moderate dimensions. Goel et al. (2008) propose to combine different criteria

to choose the right DoE and demonstrate improvements in the trade-offs between noise and

bias error by combining a model-based criterion, like the D-optimality criterion, and a

geometry-based criterion, like LHS. Also, there are adaptive sampling techniques that are

proposed (Ramu and Krishna 2012, Song et al. 2013) which will certainly reduce the number

of samples required. Another possible improvement can come from using low discrepancy

sampling strategy (Ganapathy and Ramu 2013).

In the Tuned Mass Damper example, the DoE consists of 600 samples which is large

for a 2D problem. It is evident from Figure 18 that some islands could be very narrow and

when the design space is populated with relatively less samples, it is unlikely to approximate

23    

the islands well. In order to demonstrate the advantage of Alpha shapes, for a given DoE with

600 samples, we approximate the island with alpha shapes and compare it with the island

approximated through a surrogate based approach. Kriging was used as the surrogate.

A kriging surrogate was fitted to the case of M=90, m=0.9. A second order regression model

and spherical correlation model was used. Different correlation function and polynomial

orders were used and the ones with best cross validation metric were picked. The details of

the fit are provided in Table 9.

Table 9. Metrics for the kriging model fitted to Eq. 2

The range of the vibration amplitude is 57. It can be observed that the kriging model is good

with a large predicted R2 and PRESS_RMS is less than 5% of the range. The exact function,

kriging approximation along with theirs and Alpha shape’s overlapped contours are provided

in Figure 19. It can be clearly seen that kriging approximation misses the left island and when

this model is used for estimating the failure probability, there will be large errors. This

phenomenon of small errors in surrogate fitting amplifying into large errors in failure

probability estimation was discussed in (Ramu et al. 2008).

However, for a larger island, one might not need 600 samples. Adaptive sampling

techniques as noted earlier can necessarily do better in this situation. A surrogate based

approach approximates the response function and uses it for optimization and reliability

estimates. The shapes approach focus on extraction of contours of interest (limit state). If the

designer suspects an island and in reality there is none, there is nothing to lose other than few

sample evaluations. Hence this approach can be considered as insurance against bad

predictions.

Metrics Values

PRESS_RMS 2.64

R2pred 0.99

24    

(a)

(b)

(c)

(d)

(e)

Figure 19. Function view for the (a) Exact case (b) Kriging approximation. Top view of the (c) Exact case (d) Kriging approximation

(e) Overlapped contours of the encircled region in (c)

25    

5.2 Extension to higher dimensions

Though we have demonstrated implementation of α-shapes in R2, the concept of α-shapes is

applicable in any dimension `d’, d > 2. In R2, it is shown that the entire Delaunay

triangulation will have fewer than 6n simplices (here simplices imply points, lines, or

triangles constituting a Delaunay triangulation). This implies that the number of computations

have to be done for at most 6n times, and hence linear in complexity. It can be further noted

that, as α shapes are a subset of simplices for most α, this number is typically much lesser.

For dimension d = 3, it has been shown that the number of simplices (here this term also

includes tetrahedrons apart from points, lines and triangles) depends on the distribution of

points, the worst case arising in cyclic polytopes in Rd+1 (Ziegler, 1995). The number of such

cyclic polytopes has shown to be at most constant times n 𝑑/2 simplices. The expected

number of simplices has been argued to be some constant times n, where the constant

depends on the dimension (Dwyer R. A, 1988). For the particular case of d = 3, for a well

distributed points on a smooth surface, the number of simplices have been shown to be bound

by n log2 n (Attali D et al. 2003). It is to be noted that this number is for Delaunay

triangulation and hence it is much lesser for an α-shape in three dimensions. A linear number

of simplices have also been shown to be possible for an α-shape in R3. This implies that the

number of computations is still linear in R3. The above arguments hold good for d > 3 as

well. Hence, the overall approach discussed in the paper is applicable for any dimension `d’.

Efficient sampling techniques (Ebeida et al. 2014) can be used to address the curse of

dimensionality for the Delaunay triangulation leading to less computationally expensive

alpha shapes.

The shapes concept is applied to a 3D problem adopted from Basudhar and Missoum (2010)

and presented in Equation 5.

( ) ( ) ( )2 2 21 2 3 1 2 32 2 3 1 0x x x x x x− + + + − + =           (5)  

Figure 20 shows the function and the corresponding alpha shape. In such problems, the

adaptive sampling approaches will be very advantageous because the sampling need to be

concentrated only around the corners.

26    

Figure 20. Exact and alpha shape approximation of Eq. 5

6 Conclusion

This work proposed an α shape based approach to decompose island failure regions. An

iterative algorithm was developed to select an appropriate α. Once the boundaries of the

failure regions are identified, they are used to estimate reliability index or failure probability

using the ray shooting algorithm. Two reliability based optimization problems were solved to

demonstrate the advantage of the proposed approach. It was shown that the proposed

approach works well in decomposing islands failure regions and can be used directly for

propagating uncertainties in Reliability based Design Optimization. The current work used

the entire DoE points to construct the boundaries. However, an adaptive approach can also be

used which is likely to get the same boundary at a fraction of the computational cost.

Discussions regarding efficiency of the alpha shapes and its scalability to higher dimensions

are discussed.

Alpha shape

Exact function

27    

Acknowledgment Thanks are due to members of the CAD lab and Mr.Sivashankar, Department of Engineering Design, IIT Madras. The authors thank Professor Samy Missoum, University of Arizona and Dr.Anirban Basudhar, Livermore Software Technology Corporation for their comments and suggestions. The authors thank Mr. Ryan Asher John for help with the simulations. Support from IIT Madras for the summer fellowship program is appreciated here. Mr. Harish Ganapathy worked on this paper in one such fellowship during his undergraduate study at SCSVMV University, Kanchipuram, India. References

Acar E (2013) Effects of the correlation model, the trend model, and the number of training points on the accuracy of kriging metamodels. Expert Syst 30: 418-428 Attali D, Boissonnat JD, Lieutier A. (2003) Complexity of the Delaunay triangulationof points on surfaces: the smooth case. Proc. 19th Ann. Sympos. Comput. Geom, 201–210. Au KS, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16:263–277 Basudhar A, Dribusch C, Lacaze S, Missoum S (2012) Constrained efficient global optimization with probabilistic support vector machines. Struct and Multidiscip Optim46:201-221

Basudhar A, Missoum S (2008a) Adaptive explicit decision functions for probabilistic design and optimization using support vector machines. Comput and Struct 86:1904–1917 Basudhar A, Missoum S (2009) Local update of support vector machine decision boundaries. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, And Materials Conference Basudhar A, Missoum S (2010) An improved adaptive sampling scheme for the construction of explicit boundaries.Struct and MultidiscipOptim42:517-529 Basudhar A, Missoum S, Sanchez AH (2008b) Limit state function identification using support vector machines for discontinuous responses and disjoint failure domains. ProbabEngMech 23:1–11 Chen S, Nikolaidis E, Cudney HH (1999) Comparison of probabilistic and fuzzy set methods for designing under uncertainty.Proceedings, AIAA/ASME/ASCE/AHS/ASC Structures, structural dynamics, and materials conference and exhibit. 2860-2874 de Berg M (1993) Ray shooting, depth orders and hidden surface removal. Lecture Notes in Computer Science:703, Springer, New York

28    

de Freitas N, Milo M, Clarkson P, Niranjan M, Gee A (1999) Sequential support vector machines. In neural networks for signal processing IX—Proceedings of the 1999 IEEE Signal processing society workshop Dribusch C, Missoum S, Beran P(2010) Amultifidelity approach for the construction of explicit decision boundaries: application to aeroelasticity.Struct and MultidiscipOptim 42:693–705 Dwyer RA, (1988) Average-case analysis of algorithms for convex hulls and Voronoi diagrams. Ph. D. thesis, Report CMU-CS-88-132, Carnegie-Mellon Univ., Pittsburgh, Pennsylvania. Ebeida, MS, Mitchell SA, Awad MA, Park C, Swiler LP, Manocha D and Wei LY (2014). Spoke Darts for Efficient High Dimensional Blue Noise Sampling. arXiv:1408.1118. Edelsbrunner H (2010) Alpha shapes-a survey. Tessellations in the Sciences27 Edelsbrunner H, Ernst PM (1994) Three-dimensional alpha shapes. ACM Trans. Graph 13:43-72 Edelsbrunner H, Kirkpatric DG, Seidal R (1983) On the shape of a set of points in the plane. IEEE Trans on Inf Theory 29:551-559 Fonseca J R, Friswell M I, Lees AW (2007) Efficient robust design via Monte Carlo sample reweighting. Int. J. Numer. Meth.Engng., 69: 2279–2301. Ganapathy H, P.Ramu (2013) A low discrepancy sampling strategy and alpha shapes for design space decomposition in reliability studies, First Indian Conference in Applied Mechanics 2013, Chennai, India Goel T, Haftka RT, Shyy W, Watson LT (2008), Pitfalls of using a single criterion for selecting experimental designs. Int. J. Numer Meth.Eng., 75: 127–155. Gu L (2001) A comparison of polynomial based regression models in vehicle safety analysis. ASME Design engineering technical conferences – design automation conference, Pittsburgh Haldar A, Farag R (2010) A novel reliability evaluation method for large dynamic engineering systems.2nd International conference on reliability, safety and hazard. Hao HY, Qiu HB, Chen ZZ, Xiong HD (2012) Reliability analysis method based on support vector machinesclassification and adaptive sampling strategy, Adv Mater Res 544: 212-217 Hormann K, Agathos A (2001) The point in polygon problem for arbitrary polygons. ComputGeom 20: 131-144

29    

Jiang P, Basudhar A, Missoum S, (2011) Reliability assessment with correlated variables using support vector machines, 52nd AIAA/ASME/ASCE/AHS/ASC Structures, structural dynamics and materials conference proceedings Kalos SL, Marton G (1998) Worst-case versus average case complexity of ray-shooting. Computing 61:103-131 Kim HC, Pang S, Je HM, Kim D, Yang BS (2003) Constructing support vector machine ensemble. Pattern recogn 36: 2757-2767 Kleijnen JPC, Rubinstein RY (1996) Optimization and sensitivity analysis of computer simulation models by the score function method.Eur J Oper Res 88:1–15 Kutaran H, Eskandarian A, Marzougui D, Bedewi NE (2002) Crashworthiness design optimization using successive response surface approximations. Comput Mech 29:409-421 Lacaze S, Missoum S (2013) Reliability-based design optimization using kriging and support vector machines. Proceedings of the 11th International conference on structural safety & reliability, New York Lee I, Choi KK, Noh Y, Zhao L, Gorsich D (2011) Sampling-based stochastic sensitivity analysis using score functions for RBDO problems with correlated random variables. J Mech Des 133(2):21003 Lin K, Basudhar A, Missoum S (2013) Parallel construction of explicit boundaries using Support vector machines. Eng Computations 30:132-148 Lin SP, Shi L, Yang RJ. (2014). An alternative stochastic sensitivity analysis method for RBDO. Struct Multidisc Optim, 49(4), 569-576. Mandal DP, Murthy CA (1997) Selection of alpha for alpha-hull in R2. Pattern Recogn, 30:1759-1767 Melchers RE (1999) Structural Reliability Analysis and Prediction, Wiley Missoum S, Benchaabane S, Sudret B (2004) Handling bifurcations in the optimal design of transient dynamic problems. 45th AIAA/ASME/ASC/AHS/ASC Structures, structural dynamics and materials conference, Palm Springs, CA Missoum S, Ramu P, Haftka RT (2007) A convex hull approach for the reliability-based design optimization of nonlinear transient dynamic problems. Comput Methods in ApplMech and Eng 196: 2895-2906 Packer E, Tzadok A, Kluzner V (2011) Alpha-shape based classification with applications to optical character recognition. International conference on document analysis and recognition.

30    

Ramu P, Kim NH, Haftka RT (2008) Error amplification in failure probability estimates of small errors in response surface approximations. Trans of Soc of AutomotEng 116:182-193 Ramu P, Krishna. M, (2012) A Variable-fidelity and convex hull approach for limit state identification and reliability estimates. 12th AIAA Aviation technology, integration, and operations (ATIO) and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, Indiana Sobieszczanski SJ, Kodiyalam S, Yang RJ (2001) Optimization of car body under constraints of noise, vibration, and harshness (NVH), and crash.Struct and MultidiscipOptim 22:295-306 Song H, Choi KK, Lee I, Zhao L, Lamb D. (2013). Adaptive virtual support vector machine for reliability analysis of high-dimensional problems. Struct and Multidiscip Optim, 47(4): 479-491. Valdebenito MA, Schuëller GI (2010) A survey on approaches for reliability-based optimization.Struct and MultidiscipOptim 42.5: 645-663. Viana FAC (2011) Surrogates Toolbox User's Guide, Gainesville, FL, USA, Version 3.0 ed, available at https://sites.google.com/site/srgtstoolbox Wilson JA, Bender A, Kaya T, Clemons PA (2009) Alpha shapes applied to molecular shape characterization exhibit novel properties compared to established shape descriptors. CheInf and Model 49: 2231-2241 Yang IT, Hsieh YH (2013) Reliability-based design optimization with cooperation between support vector machine and particle swarm optimization. Eng with Comput, 29(2): 151-163. Youn BD, Choi KK (2004) An investigation of nonlinearity of reliability-based design optimization approaches. Mech des 126(3) 403-411. Zhang W, King I, A (2002) Study of the relationship between support vector machine and gabriel graph, International Joint Conference on Neural Networks. Ziegler GM. (1995) Lectures on Polytopes. Springer-Verlag, New York.

31    

Appendix A: Error Metrics

1. R2:

The coefficient of multiple determinations is defined as:

( )2

2 1

2

1

ˆ1

( )

n

i iin

ii

y yR

y y

=

=

−= −

−

∑

∑ (A.1)

where iy is the actual value at the ith design point, ˆiy is the predicted value at the ith design

point, and y the mean of the actual response. 2R is a measure of the amount of reduction in

the variability of y obtained by using the response surface. 20 1R≤ ≤ . A larger value of 2R

is desirable for a good response surface. But, a larger 2R does not necessarily guarantee a

good response surface. Thus, this estimate should be used in conjunction with other error

estimates to gauge the quality of the response surface. 2R continuously increases with

addition of terms irrespective of whether the additional term is statistically significant.

2. Adjusted R2:

The adjusted coefficient of multiple determinations is defined as

2 211 (1 )adjnR Rn p−

= − −−

(A.2)

wherenis the number of design points, and p is the number of regression coefficients.

Unlike 2R , 2adjR decreases when unnecessary terms are added. Hence, 2

adjR along with 2R can

be used to comment on the quality of response surface and the presence of unnecessary terms

in the response surface.

3. Root-Mean-Square Error (RMSE):

The root-mean-square error, RMSE, and the predicted RMS errors are defined, respectively,

as

32    

2

1

ˆ( )RMS

n

i iiy y

n=

−=∑

(A.3)

4. Prediction Error Sum of Squares (PRESS):

The prediction error sum of squares provides error scaling. To estimate the PRESS, an

observation is removed at a time and a new response surface is fitted to the remaining

observations. The new response surface is used to predict the withheld observation. The

difference between the withheld observation and the computed response value gives the

PRESS residual for that observation. This process is repeated for all the observations and the

PRESS statistic is defined as the sum of the squares of the n PRESS residuals. When

polynomial response surfaces are used, the repetitive estimate of PRESS residuals can be

obviated by using the following expression:

2

1PRESS

1

ni

i ii

eE=

⎛ ⎞= ⎜ ⎟

−⎝ ⎠∑ (A.4)

where ( ) 1T T−=E X X X X and X is the Grammian matrix ( )ˆ =y Xb , and b is the coefficient

vector. Data points at which iiE are large will have large PRESS residuals. These

observations are considered high influence points. That is, a large difference between the

ordinary residual and the PRESS residual will indicate a point where the model fits the data

well, but the model built without that point has a poor prediction. A RMS version of PRESS

allows us to compare the PRESS_RMS with the RMS errors. This permits us to explore the

influence that few points might have on the entire fit. The PRESS_RMS is expressed as:

PRESSPRESS_RMS=n

(A.5)

PRESS can be used to estimate an approximate 2R for prediction as:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=n

1i

2_

i

2pred

yy

PRESS1R (A.6)

The denominator in Eq. (A.6) is referred to as total sum of the squares.