alpha shape based design space decomposition for island failure regions...

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

mailto:[email protected]

mailto:[email protected]

mailto:[email protected].


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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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