alpha shape based design space decomposition for …raman/ahull_reliability.pdf · 1" "...
Post on 09-Apr-2018
218 Views
Preview:
TRANSCRIPT
1
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
2
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
3
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
4
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
5
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
6
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
7
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
8
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)
9
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
10
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
[Type a quote from the document or the summary of an interesting point. You can position the text box anywhere in the document. Use the Text Box Tools tab to change the formatting of the pull quote text box.]
[Type a quote from the document or the summary of an interesting point. You can position the text box anywhere in the document. Use the Text Box Tools tab to change the formatting of the pull quote text box.]
x
y
z
L
t
11
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|
12
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
13
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
11
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
14
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
15
(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
16
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
17
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
18
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
19
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.
top related