<html>
<title>Isight - Tutorial - Adapter Duct</title>
<head>
	<link rel="stylesheet" href="../EStyle.css" type="text/css">
	<link rel="stylesheet" href="../ETables.css" type="text/css">

</head>
<a href="http://www.simulia.com/products/sim_opt.html"><img src="../logo.gif"></a>


<body>

<h1>Tutorial - Sizing an adapter duct</h1>

<p>In this example, we will size the thickness of an engine
duct that is mounted on an aircraft (<a href="#figure1">Figure 1</a>). This thickness has to exceed a
specified minimum value. In doing so, we have to meet FAA requirements: the
duct structure should withstand FAA emergency loads (<a href="#table1">Table 1</a>), and, the structure
should be designed with a safety factor of 2. The material of the duct is
INCONEL625 and minor material variation is expected (<a href="#table2">Table 2</a>). </p>
<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure1"></a> <a href=""images/image002.jpg">
<img width=500 src="images/image002.jpg"> 
Figure 1: Simplified model of the adapter duct used in an aircraft
</a></div>

<p>&nbsp;</p>
<a name="table1"></a>
<table>
 <th colspan=2> Table 1: FAA emergency load conditions for Adapter Duct design</th>
 <tr>
  <td>
	<b>Emergency Landing Acceleration X (g) </b>
  </td>
  <td>
	  9.00 
  </td>
 </tr>

 <tr>
  <td>
  <b>Emergency Landing Acceleration Y (g)</b>
  </td>
  <td>
  -3.00
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Emergency Landing Acceleration Z (g)</b></p>
  </td>
  <td>
  6.00
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Engine Pressure (PSI) </b></p>
  </td>
  <td>
  32.00
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Preliminary Nozzle Weight (lbs) </b></p>
  </td>
  <td>
  250.00
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Preliminary Thrust Reverser Weight (lbs) </b></p>
  </td>
  <td>
  270.00
  </td>
 </tr>
</table>




<p align=center>&nbsp;</p>

<p align=center>&nbsp;</p>
<a name="table2"></a>
<table>
 <th colspan=2> Table 2: Material specifications </th>
 <tr>
  <td>
  <p><b>Material </b></p>
  </td>
  <td>
  <p align=center>Inconel 625</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Young's Modulus (psi) </b></p>
  </td>
  <td>
  <p align=center>3.15E+07</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Poisson's Ratio </b></p>
  </td>
  <td>
  <p align=center>0.2800</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Density (lbm/in^3) </b></p>
  </td>
  <td>
  <p align=center>0.3040</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Yield Strength (psi) </b></p>
  </td>
  <td>
  <p align=center>82,000.00</p>
  </td>
 </tr>
</table>

<p align=center>&nbsp;</p>

<p>&nbsp;</p>

<p>At the same time the duct is designed, other teams are
working on the thrust reverser and nozzle design. These parts are attached to
the duct and influence the loads. Unfortunately the design &amp; analysis of this
part is not yet complete but the duct has to be sized nevertheless. It was
found that the thrust reverser structure depends on high fidelity simulations
of the loads which will not be available until after the duct delivery
deadline. The nozzle weight will depend on the amount of acoustic treatment
that needs to be added. That detail will only be available after flight
testing. Right now only preliminary weights are available for these parts, but
we know the typical error of these methods based on previous programs.</p>

<h2>Step 1  Creating an Isight subflow for analysis</h2>

<p>We begin the sizing task by trying to understand the design
space of the adapter duct. In order to do so, we first create an Abaqus model
with pertinent analysis steps that take material properties, loads and
dimensions as inputs and calculate the maximum stress and mass of the duct as
shown in <a href="#figure2">Figure 2</a>. The nominal dimensions of the duct are given 
in <a href="#table">Table 3</a>. </p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure2"></a>
<a href="images/image003.jpg"> <img width=700 
src="images/image003.jpg">Figure 2: Abaqus model and results for the adapter duct.</a></div>
<p>&nbsp;</p>
<p>&nbsp;</p>

<a name="table3"></a>
<table>
 <th colspan=2>Table 3: Nominal dimensions</th>
 <tr>
  <td>
  <p><b>Thickness Duct (in)</b></p>
  </td>
  <td>
  <p>0.065</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Thickness Fwd. Flange (in)</b></p>
  </td>
  <td>
  <p>0.15</p>
  </td>
 </tr>
 <tr>
  <td>
  <p><b>Thickness Aft flange (in)</b></p>
  </td>
  <td>
  <p>0.15</p>
  </td>
 </tr>
</table>

<p>&nbsp;</p>

<p>An Isight model (<a href="#figure3">Figure 3</a>) is then created that uses an Abaqus
component to execute the adapter duct model. The input parameters for the Abaqus component are:
<ol> <li>Density_1_MassDensity</li>
<li>Dsload_1_PICKEDSURF19_Mag</li>
<li>Elastic_1_PoissonRatio</li>
<li>Mass_"_M7__PICKEDSET18_NOZZLE_WGHT__"_1_MassMag</li>
<li>Mass_"_M8__PICKEDSET15_THRUST_REVERSER_WGHT__"_1_MassMag</li>
<li>Shell_Section__PICKEDSET11_INCONEL625_1_Thickness</li>
<li>Shell_Section__PICKEDSET12_INCONEL625_1_Thickness, and</li>
<li>Shell_Section__PICKEDSET9_INCONEL625_1_Thickness.</li>
</ol> Likewise, the output parameters for the Abaqus component are: <ol><li>Step_1__Mass, and</li><li>Step_1__MISESMAX__max</li></ol>
The Isight model also includes a Calculator
component that calculates the Safety factor(<a href="#figure4">Figure 4</a>).</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure3"></a><a href="images/image004.jpg"><img width=700
src="images/image004.jpg">Figure 3: Isight model to execute Abaqus analysis and compute factor of safety</a></div>

<p>&nbsp;</p>

<div style="width:400; font-size:80%; text-align:center;"><a name="figure4"></a></a><a href="images/image005.jpg"><img width=400 
src="images/image005.jpg">Figure 4: The equation to compute the factor of safety using the Calculator component.</a></div>

<p>&nbsp;</p>

<h2>Step 2  Understand the design space</h2>

<p>In this example, executing the Abaqus model takes more than
a few seconds. Therefore, we will try to create an approximation of the
subflow. Approximations can be created in the Design Gateway or the Runtime
Gateway. In this example, we shall create approximations in the Runtime Gateway.
But before we can build approximations, we first need to decide the number of
samples and the type of sampling best suited for our problem. 
<table>
<th> Note 1: Selecting a DOE method</th>
<tr><td>
Isight has the following
DOE techniques that can be used for generating samples:</p>
<ul>
<li><b>Box-Behnken</b>  Suitable when corners of the design space are to be avoided</li>

<li><b>Central Composite</b>  A "star" like sampling technique centered around the nominal design
popularly used to build polynomial response surface models, but number of
points grows exponentially with number of factors.</li>

<li><b>Full Factorial</b>  generates samples with all possible combination of factor levels,
resulting in an exponential number of sample points and therefore only suitable when
very few factors or very few levels are involved.</li>

<li><b>Fractional Factorials</b>  as name implies, only a fraction of the full factorial set is
generated but still applicable only when very few factor levels are involved.</li>

<li><b>Latin Hypercube</b>  here the number of levels for each factor is equal to the number of
points and random combination of levels is selected to generate the sample set,
but each level has exactly one sample point. Latin hypercube allows more level
combinations to be studied but some regions of the design space may not have
any samples.</li>

<li><b>Optimal Latin Hypercube</b>  uses measures such as "entropy" to generate a Latin hypercube
sample that covers the entire design space uniformly but generating this sample
can take more than a few minutes when number of factors is large.</li>

<li>Various <b>Orthogonal Arrays</b>  are a subset of fractional factorial samples that maintain "orthogonality"
among the factors and interactions and are especially useful for constructing orthogonal
polynomial approximations.</li>
</ul>
</td></tr></table>
<p>&nbsp;</p>

<p>Since we are interested in identifying a suitable
approximation technique in order to explore the design space, we can choose techniques
such as Latin Hypercube or Optimal Latin Hypercube that are suitable for all the
approximations. Ideally, one may choose Optimal Latin Hypercube because of its
uniformity of samples resulting in better approximations <a href="#reference1">[1]</a>, but Latin
Hypercube will suffice for this example. </p>

<p>&nbsp;</p>

<p>The design space for exploration will be +/- 20% around the nominal design point. Since this 
approximation model will also be used for performing Six Sigma studies, potential random variables
along with design variables will be used as factors for the DOE. It is important to identify correctly
all inputs of an approximation; new set of samples would be needed to create a new approximation with 
a different set of inputs.</p>

<p>&nbsp;</p>

<p>The Isight model configured with Latin Hypercube sampling is
available <a href="models/Adapter-DOE-LH50.zmf">here</a>.</p>

<h4>Creating the approximation</h4>

<p>The results of executing the model are available in the
Runtime Gateway (<a href="#figure5">Figure 5</a>). We shall create RBF, RSM and Kriging approximations
of the Abaqus model by navigating to the "Visual Design" tab, clicking the <img src="images/image005a.jpg"> icon
and using the default values for technique parameters.</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure5"></a></a>
<a href="images/image006.jpg"><img width=700 src="images/image006.jpg">
Figure 5: Runtime Gateway showing the history of execution using tables and charts</a></div>
<p>&nbsp;</p>

<p>The average, maximum and RMS error values for these
approximations are listed in <a href="#table4">Table 4</a> below.</p>

<p>&nbsp;</p>
<a name="table4"></a>
<table>
<th colspan=7>Table 4: Error values for approximations</th>
 <tr>
  <td>
  </td>
  <td colspan=3>
   <b>Step_1_MISESMAX_max</b>
  </td>
  <td colspan=3>
   <b>Step_1_Mass</b>
  </td>
 </tr>
 <tr>
  <td>
  </td>
  <td>
  Average
  </td>
  <td>
  Maximum
  </td>
  <td>
  RMS
  </td>
  <td>
  Average
  </td>
  <td>
   Maximum
  </td>
  <td>
   RMS
  </td>
 </tr>
 <tr>
  <td>
  <b>RBF</b>
  </td>
  <td>
  0.02859
  </td>
  <td>0.06886
  </td>
  <td>
   0.03591
  </td>
  <td>
   0.00136
  </td>
  <td>
   0.00493
  </td>
  <td>
   0.00195
  </td>
 </tr>
 <tr>
  <td>
   <b>Quadratic RSM</b>
  </td>
  <td>
   0.02486
  </td>
  <td>0.04965
  </td>
  <td>0.02991
  </td>
  <td>6.81132R-6
  </td>
  <td>2.40225E-5
  </td>
  <td>9.28041E-6
  </td>
 </tr>
 <tr>
  <td>
  <b>Kriging</b>
  </td>
  <td>0.03334
  </td>
  <td>0.05331
  </td>
  <td>0.0375
  </td>
  <td>0.33124
  </td>
  <td>0.51563
  </td>
  <td>0.37365
  </td>
 </tr>
</table>

<p>&nbsp;</p>

<p>It can be seen that the approximation obtained with RSM is
consistently better than others for both the maximum von Mises stress and mass.
Evaluation of approximations can be done by navigating to the "Error Analysis"
tab and viewing the "Response Fit" charts. <a href="#figure6">Figure 6</a> below shows the response
fit chart for Quadratic RSM approximation. Notice that the error analysis points
lie very close to the line indicating a very good fit.</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure6"></a></a>
<a href="images/image007.jpg"><img width=700 src="images/image007.jpg">
Figure 6: Errors in the approximation can be visualized using the "Actual vs Predicted" plots; these plots show the accuracy of RSM for our problem. </a></div>


<p>The Runtime Gateway provides several ways to visualize the
design space (<a href="#figure7">Figure 7</a>). Different regions of the design space can be explored by moving
the slider for each of the inputs. In this example, it appears that the outputs
are uni-modal, meaning there are no numerous peaks and valleys(<a href="#figure8">Figure 8</a>). </p>


<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure7"></a></a>
<a href="images/image008.jpg"> <img width=700 src="images/image008.jpg">
Figure 7: The approximation can be visualized in the Runtime Gateway as 2D, 3D and contour graphs; importance of input parameters can be discerned from local and global effect plots.
</a></div>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure8"></a></a>
<a href="images/image009.jpg"> <img width=700 src="images/image009.jpg">
Figure 8: Individual graphs can be examined in greater detain; this plot shows that our problem is not multi modal.
</a></div>

<p>&nbsp;</p>
<table>
<th> Note 2: Selecting an approximation technique</th>
<tr><td>
Isight has the following approximation techniques that can be used:</p>
<ul>
<li><b>Radial Basis Functions</b>  Radial Basis Functions are a type of neural network employing a hidden 
layer of radial units and an output layer of linear units.</li>
<li><b>Elliptical Basis Functions</b> are similar to Radial Basis Function but use 
elliptical units in place of radial units. Compared to RBF, where all inputs are handled equally, 
EBF networks treat each input separately using individual weights.</li>
<li> Polynomial <b>Response Surface models</b> use polynomials of low order (from 1 to 4)
to approximate response of an actual analysis code.</li>
<li> <b>Orthogonal polynomial</b> approximation is a type of regression technique that minimizes the 
autocorrelation between the response values induced by the choice of sampling locations.
Chebyshev polynomials are a set of orthogonal polynomials that are solutions of a special 
kind of Sturm-Liouville differential equations called Chebyshev differential equations. 
</li>
<li> <b>Kriging</b> is a Gaussian process estimator that interpolates between the data points
using a correlation function. </li>
</ul>
</td></tr>
<tr><td>
Kriging is particularly effective when a multi-modal functions are to be approximated using a small 
number of sample points that are uniformly distributed. As the number of sample points increases, 
the accuracy of RBF and EBF models is higher than Kriging. Polynomial approximations are accurate for 
smooth functions and the accuracy does not improve much with additional points; Chebyshev polynomials 
are more accurate and robust compared to ordinary polynomial regression. When the sample points are 
not uniformly distributed, the accuracy of ordinary Kriging model decreases with additional points; 
EBF and RBF may be more suitable because of their low initialization times.
</td></tr>
</table>
<h3>Step 3  Find an optimization method that works for the problem</h3>

<p>With the creation of approximations in the previous step, we
can proceed to identify a better duct design  one that has lower mass ant yet
has a safety factor greater than 2. Isight provides two ways for optimizing
models that have approximations: using "design search" from within the "Visual
Design" tab of the Runtime Gateway, or, by creating a separate model in the
Design Gateway using the approximation. <i>We will create a new model for optimization 
(see next section) but the steps to use the design search capability within Runtime Gateway is provided 
for the reader to explore on their own.</i></p>

<p>In order to create a separate model, one can use the "Save
Model As" menu item under "File" menu of the Runtime Gateway. This creates a
new model that has the same components as before and in addition includes the
approximations created in the previous step. Such a model can be found <a href="models/Adapter-with-approximations.zmf">here</a>. </p>

<p>If the user wishes to remain in the Runtime Gateway and continue
with the optimization using approximations created in Step 1, navigate to the "Design
Search" tab and select the duct dimensions as design variables, maximum von Mises
stress as constraint (with upper bound = yield strength/FOS = 82000/2) and mass
as objective (minimize). NLPQL is the technique that is used by default but can
be changed by clicking the "Options" button (<a href="#figure9">Figure 9</a>). Optimization can be
executed by pressing the "search" button . </p>

<p>&nbsp;</p>
<div style="width:375; font-size:80%; text-align:center;"><a name="figure9"></a></a><a href="images/image010.jpg">
<img width=375 src="images/image010.jpg"></p>
Figure 9: Default options for the optimization technique "NLPQL".
</a></div>

<p>In this tutorial we shall now switch to the Design Gateway
and open the <a href="models/Adapter-with-approximations.zmf">model</a> that was saved along with
the approximation of the Abaqus component (<a href="#figure10">Figure 10</a> below). Now, we will use the
Optimization component in place of the DOE component. To do so, drag the
Optimization component icon over the existing DOE component. Select "Copy
existing parameters to new components" in the "Specify Action" dialog (<a href="#figure11">Figure 11</a>). Copying the parameters also copies the bounds that were defined on those
parameters and we will not have to specify those manually while configuring the
optimization.</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure10"></a></a>
<a href="images/image011.jpg"> <img width=700 src="images/image011.jpg">
Figure 10: Model with available approximations (but not activated yet).
</a></div>

<p>&nbsp;</p>
<div style="width:375; font-size:80%; text-align:center;"><a name="figure11"></a></a><a href="images/image012.jpg">
<img width=369 height=275 src="images/image012.jpg">
Figure 11: Options available when substituting one design driver with another.
</a></div>

<p>&nbsp;</p>

<p>Double click the Optimization component to open its editor. Select
the duct dimensions as design variables (<a href="#figure12">Figure 12</a>). Notice that the bounds
are automatically filled when the variable is selected. Go to the Constraints
tab and select FOS as a constraint with a lower bound of 2 (<a href="#figure13">Figure 13</a>). In the
"Objectives" tab, select Step_1__Mass as an objective to minimize (<a href="#figure14">Figure 14</a>).</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure12"></a></a><a href="images/image013.jpg">
<img width=700 src="images/image013.jpg">
Figure 12: Design variables for optimization.
</a></div>
<p>Check if the values for parameters other than design variables are correctly set to their nominal value; if not, set them.
In <a href="#figure12">Figure 12</a> above, we had to set the values of nozzle weight and thrust reverser weight.</p>  
<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure13"></a></a><a href="images/image014.jpg">
<img width=700 src="images/image014.jpg">
Figure 13: Constraints for optimization
</a></div>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure13"></a></a><a href="images/image014a.jpg">
<img width=700 src="images/image014a.jpg">
Figure 14: Objective for optimization
</a></div>

<p>&nbsp;</p>
<p>Notice that no approximation on the Abaqus component has
been activated yet. Right click on the Abaqus component; select "Approximations"
(<a href="#figure15">Figure 15</a>) to bring up the Approximations dialog. Enable the Quadratic RSM
approximation. The user may note that clicking the "Visualize" button brings up
the Approximation viewer that shows the graphs that were seen earlier in the "Visual
Design" tab of the Runtime Gateway (<a href="#figure16">Figure 16</a>).</p>

<div style="width:615; font-size:80%; text-align:center;"><a name="figure15"></a></a><a href="images/image015.jpg">
<img width=150 src="images/image015.jpg"><a><a  href="images/image016.jpg">
<img width=450 src="images/image016.jpg">
Figure 15: Approximations on the Abaqus component.
</a></div>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure16"></a></a><a href="images/image017.jpg">
<img width=700 src="images/image017.jpg"></p>
Figure 16: Approximations can also be visualized in the Design Gateway.
</a></div>

<p>&nbsp;</p>
<table>
<th>Note 3: Selecting an optimization technique </th>
<tr><td><p>Isight has the following primary optimization techniques that can be used for finding an optimum design point:</p>
<ol>

<li><b>NLPQL</b> - One of the most powerful numerical optimization algorithms. Performs gradient based search in the
direction of the local minimum of the objective function. Constraints are handled directly. Can get trapped in
local minima and performance suffers significantly for noisy output functions. Most of the described characteristics 
are applicable to other gradient based algorithms: MOST, LSGRG2, MISQP. NLPQL is strongly recommended when the objective
function is smooth; it may be necessary to repeat the optimization with several starting points to ensure that the 
result is not a local minimum.
</li>

<li><b>Hooke-Jeeves</b> - A simple optimization algorithms that replaces the need to calculate gradients by 
using a special pattern for search. Search is performed by moving each parameter one at a time in order to minimize the 
objective function. Hooke-Jeeves can be successfully applied to noisy output functions when the noise scale is smaller 
than the step size of the algorithm. Hooke-Jeeves is especially useful on non-continuous objective functions. However,
Hooke-Jeeves can get trapped in local minima like NLPQL and several starting points may be necessary.
</li>

<li><b>Downhill Simplex</b> - One of the simplest procedures that uses primitive geometrical shapes (polytopes) for stepping 
through the design space. At the cost of increased number of subflow evaluations, this technique is widely applicable due to  
its simplicity: noisy functions, non-continuous functions and multi-modal functions. However, Downhill Simplex can converge 
prematurely (need not even be a local minimum) and therefore requires re-runs with more starting points than Hooke-Jeeves.  
Like Hooke-Jeeves, constraints are not handled directly and are included into a penalty function which is added
to the objective function.
</li>  

<li><b>Particle Swarm</b> - A robust population-based search procedure where individuals (called "particles") continuously change 
position within the search area. In other words, these particles "fly" around in the design space looking for the best 
position. This algorithm does not require gradient information and can be successfully applied to noisy functions, 
non-continuous functions, and especially multi-modal functions. This technique is a "global search" algorithm and does not 
get stuck in a local optimum. Here too, constraints are not handled directly but are included into a penalty function which 
is added to the objective function. Particle swarm is also suitable for problems with multiple objectives, in which case 
a "Pareto set" is obtained. Note that Particle Swarm requires numerous subflow evaluations compared to other techniques.
</li>

<li><b>Pointer</b> - A universal optimizer which consists of a complementary set of optimization algorithms:
linear simplex, sequential quadratic programming (SQP), downhill simplex, and genetic algorithm. The Pointer 
technique can efficiently solve a wide range of problems in a fully automatic manner because of a special 
automatic control algorithm which employs suitable optimization techniques in succession. If you do not have
any knowledge of your output function behavior, Pointer may be your best choice. The Pointer possesses qualities
of a "global search" technique since it is capable of re-starting the optimization from multiple starting points
in the design space, therefore it is much less likely to get trapped in a local minimum.
</li>

</ol>
</td></tr>
<tr><td><p>The choice of optimization technique depends
on several factors including:</p>
<ol>
<li> The complexity of the objective function and constraints: For functions that are complex (multiple optima/noisy/non-continuous), 
it is better to avoid techniques that rely on gradient information and instead choose techniques such as Hooke-Jeeves or Downhill
Simplex. Pointer is recommended in case the complexity of the function is unknown.</li>

<li> Number of design variables: Most of the numerical procedures are extremely efficient for small number of design variables. </li>

<li>The time to evaluate the objective function and constraints: This determines the maximum-number of subflow evaluations that can 
be reasonably used for optimization. For problems that use long-running codes, it is recommended that the complexity of the function
be ascertained first before selecting an optimization technique to improve the time for optimization. It is also recommended that
approximations of the subflow be used instead of long-running codes.</li>
</ol>
Please see <a href="#reference2">[2]</a> for a more theoretical discussion on selecting and using an appropriate optimization technique. 
</td></tr></table>

<p>In this example, since the objective function and constraint
appear to be unimodal, NLPQL is a good starting point for optimization. Without
further insight into the problem, we shall use default values for the technique
options. The model that is obtained when these tasks are completed is available
<a href="models/Adapter-Optimization-with-Approximations.zmf">here</a>. </p>

<p>Running this model executes the NLPQL technique on the
subflow with the approximation enabled. The "History" tab of the Runtime
Gateway highlights the best point in green color (<a href="#figure17">Figure 17</a>). In this figure, we have
selected to display just the FOS, Step_1__Mass and Step_i___MISESMAX__max
variables by clicking the "Configure" button below the table (<a href="#figure18">Figure 18</a>). 
The optimum design is given in <a href="#table5">Table 5</a>.</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure17"></a></a><a href="images/image018.jpg">
<img width=700 src="images/image018.jpg">
Figure 17: History produced during optimization using NLPQL.
</a></div>

<p>&nbsp;</p>
<div style="width:270; font-size:80%; text-align:center;"><a name="figure18"></a></a><a href="images/image019.jpg">
<img width=270 src="images/image019.jpg">
Figure 18: The parameters to display in the history table can be selected from the list of all parameters.
</a></div>

<p>&nbsp;</p>
<a name="table5"></a>
<table>
<th colspan=2>Table 5: Optimum design obtained using NLPQL</th>
<tr>
<td>
<b>Shell_Section__PICKEDSET11_INCONEL625_1_Thickness</b></td><td>0.12</td></tr>
<tr>
<td>
<b>Shell_Section__PICKEDSET12_INCONEL625_1_Thickness</b></td><td>0.13690529183730882</td></tr>
<tr>
<td>
<b>Shell_Section__PICKEDSET9_INCONEL625_1_Thickness </b></td><td> 0.0504</td></tr>
<tr>
<td>
<b>FOS</b></td><td>2.000000091709937</td></tr>
<tr>
<td>
<b>Step_1__Mass</b></td><td> <b>560.2472506285862</b></td></tr>
<tr><td><b>Step_1__MISESMAX__max</b></td><td>24036.5679</td></tr>
</table>

<p>We can try to execute the same optimization problem from
several different starting points. To do so, either in the Runtime Gateway or
in the Design Gateway, open the Optimization component editor. Navigate to the "Variables"
tab and change the values of the design variables. In <a href="#figure19">Figure 19</a>, we have set
them all to their upper bound. <a href="#figure20">Figure 20</a> shows the history obtained when the
new model is executed.</p>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure19"></a></a><a href="images/image020.jpg">
<img width=700 src="images/image020.jpg">
Figure 19: Initial values for a new optimization run.
</a></div>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure20"></a></a><a href="images/image021.jpg">
<img width=700 src="images/image021.jpg">
Figure 20: History for the optimization starting at the upper bound.
</a></div>

<p>The optimum design is the same as in the previous case. The
user is encouraged to try few other starting points. At this point, the user
may also wish to change the approximation used or even the optimization
technique and repeat the optimization. </p>

<p>Since the optimum design point was obtained using an
approximation, we verify the mass and FOS at the optimum point by executing the
Abaqus model at the optimum. Isight provides couple of ways to do that:</p>
<ol>
<li>In
the "Visual Design" tab, we move the sliders of the input parameters to match
the optimum value or enter the values directly in the "Table View" (<a href="#figure21">Figure 21</a>).
Click the "Compare with actual" button at the bottom of the screen.</li>

<li>In the history table of the Abaqus component, right click on the optimum run and select 
"Init model from this run...". In the new model, disable the associated approximation. Execute 
the submodel to calculate the actual value.</li>
</ol>

<p>&nbsp;</p>
<div style="width:700; font-size:80%; text-align:center;"><a name="figure21"></a></a><a href="images/image022.jpg">
<img width=700 src="images/image022.jpg">
Figure 21: Using the "Visual Design" tab to compare the approximation with the actual Abaqus model.
</a></div>

<p>The actual and the predicted value using the approximation
are shown in a new window (<a href="#figure22">Figure 22</a>). Notice that the predicted value of the
mass is very close to the actual value; however the von Mises stress predicted
is lower than the actual value but by less than 3% - this is acceptable for
our purposes. </p>

<p>In other cases, the error between the actual and approximate
values may be significantly different. In such cases, the user can either
select a different approximation or recreate the approximation by adding
additional sample points. To add additional points, navigate to the "Visual
Design" tab and edit the approximation by clicking the <img width=24 height=22
src="images/image023.jpg">button. Isight
provides two ways to add additional points:</p>
<ol>
<li>If the user wishes to use a known sampling strategy to add new points, then in the
user should proceed to the "Sampling Data Set" page in the wizard. Here they
should select the "Preserve existing sampling data and generate additional
points" option (<a href="#figure23">Figure 23</a>) and proceed with configuring the sampling options.</li>

<li>If the user wishes to let Isight choose points that minimize the approximation
error, then they can proceed to the "Approximation Improvement Options" page (<a href="#figure24">Figure 24</a>).
Here the user can specify the target error and maximum number of points they
wish to add. Isight will use sequential sampling to add additional sample
points at locations where the error is maximum in order to improve the
approximation.</li>
</ol>

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure22"></a></a><a href="images/image024.jpg">
<img width=500 src="images/image024.jpg">
Figure 22: Comparison of actual and approximate values.
</a></div>
<p>&nbsp;</p>

<table><tr><td style="border-style:none;">
<div style="width:500; font-size:80%; text-align:center;"><a name="figure23"></a></a><a href="images/image025.jpg">
<img width=500 src="images/image025.jpg">
Figure 23: Adding additional points to the approximation.
</a></div>
</td><td style="border-style:none;">
<div style="width:500; font-size:80%; text-align:center;"><a name="figure24"></a></a><a href="images/image026.jpg">
<img width=500 src="images/image026.jpg">
Figure 24: Approximations can also be improved by letting Isight add additional points using sequential sampling.
</a></div>
</td></tr></table>

<p>&nbsp;</p>

<h2>Step 4  Perform reliability analysis</h2>

<p>In the previous step, we have obtained an optimum design
that has the minimum mass while satisfying the minimum safety factor
constraint. Although this particular design is optimum, not all manufactured
ducts will have identical dimensions due to variations in manufacturing. We
need to verify that a substantial number of ducts manufactured will satisfy the
safety factor requirement. In order to estimate the fraction of products that
satisfy this constraint, we will setup and perform a reliability study. </p>

<p>In the History tab of the Runtime Gateway right click on the
optimum design point and select "Init model from this run". This will create a
new model that has the optimum values as the nominal values for all the
parameters. We saved this model <a href="models/Adapter-with-Optimum-point.zmf">here</a>. Open the
model in the Design Gateway. Verify that the optimum values are set to the
parameters by navigating to the "Parameters" tab (<a href="#figure25">Figure 25</a>). </p>

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure25"></a></a><a href="images/image027.jpg">
<img width=700 src="images/image027.jpg">
Figure 25: Model initialized with optimum values from optimization.
</a></div>

<p>Drag the Six Sigma component over the Optimization component
and as before copy existing parameters to new component. Open the editor for
the Six Sigma component and select the following random variables:</p>

<p>&nbsp;</p>
<a name="table6"></a>
<table>
<th colspan=5>Table 6: Random Variables for Six Sigma analysis</th>
 <tr>
  <td>
  <b>Random Variable</b>
  </td>
  <td>
  <b>Distribution</b>
  </td>
  <td>
  <b>Mean</b>
  </td>
  <td>
  <b>Coefficient of Variation</b>
  </td>
  <td>
  <b>Standard Deviation</b>
  </td>
 </tr>
 <tr>
  <td>Density_1_MassDensity
  </td>
  <td>Normal</td>
  <td>
  0.304
  </td>
  <td>
  <b>0.01</b> (1%)
  </td>
  <td>3.04E-3
  </td>
 </tr>
 <tr>
  <td>Mass_&quot;_M7__PICKEDSET18_NOZZLE_WGHT__&quot;_1_MassMag
  </td>
  <td>Normal
  </td>
  <td>250
  </td>
  <td><b>0.05</b> (5%)
  </td>
  <td>12.5
  </td>
 </tr>
<tr>
  <td>Mass_&quot;_M8__PICKEDSET15_THRUST_REVERSER_WGHT__&quot;_1_MassMag
  </td>
  <td>Normal
  </td>
  <td>250
  </td>
  <td><b>0.05</b> (5%)
  </td>
  <td>12.5
  </td>
 </tr>
 <tr>
  <td>Shell_Section__PICKEDSET11_INCONEL625_1_Thickness</p>
  </td>
  <td>Normal
  </td>
  <td>0.12
  </td>
  <td>0.016667
  </td>
  <td><b>0.002</b>
  </td>
 </tr>
</table>

<p>The numbers in bold are the quantities that are fixed in the
editor. When the coefficient of variation is fixed, Isight computes the standard
deviation automatically from the mean. </p>

<p>Note that the coefficient of variation for the Nozzle weight
is high (5%)  this is because, as mentioned earlier, the design of the nozzle
has not been completed and therefore a large variation can be expected in its
mass. Once the design of the nozzle is complete, the Six Sigma analysis can be
repeated with the updated mean value and standard deviation of the nozzle (<a href="#figure26">Figure 26</a>).</p>

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure25"></a></a><a href="images/image028.jpg">
<img width=500 src="images/image028.jpg">
Figure 26: Providing the distribution information for the random variable "Shell_Section__PICKEDSET11_INCONEL625_1_Thickness".
</a></div>

<p>&nbsp;</p>

<p>Select FOS as the response with a lower limit of 2.0 (<a href="#figure27">Figure 27</a>). </p>

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure27"></a></a><a href="images/image029.jpg">
<img width=500 src="images/image029.jpg">
Figure 27: Select FOS as a response with a lower limit of 2.0.
</a></div>

<p>&nbsp;</p>

<table>
<th> Note 4: Selecting a Six Sigma technique</th>
<tr><td>

Isight provides the following techniques for performing Six
Sigma analysis:
<ol>
<li>Monte Carlo Method using <ul>
	<li> Random Sampling</li>
	<li> Descriptive Sampling</li>
	<li> Sobol Sampling</li> 
	</ul>
</li>
<li>Reliability methods such as <ul>
	<li> Mean Value Method (MVM)</li>
	<li> First Order Reliability Method (FORM) </li>
	<li> Second Order Reliability Method (SORM), and</li>
	<li>Importance sampling: <ul> 
		<li>Simple importance sampling including the option to enhance the FORM estimate</li>
		<li>Adaptive importance sampling.</li>
		</ul>
	</ul>
</ol>
</td>
</tr>
<tr><td>
<p>Among these, MVM requires the least number of subflow
evaluations to estimate the probability of satisfying the constraints, i.e.
probability of success (<i>pS</i>). Therefore, it is recommended that MVM be tried as the first step
to estimate the probabilities. Once estimates of the probability of success (and consequently 
the probability of failure (<i>pF</i>) and Sigma Level) are obtained, an appropriate Six Sigma 
technique can be chosen depending upon the accuracy needed.</p>
<p>Monte Carlo methods provide the true estimates of probabilities by simulating the 
actual underlying probability distributions at the cost of 
a very large number of samples. To calculate the number of samples required to compute 
the probabilities, the confidence interval calculation can be used:
</p>
<p>
<img height=30 src="images/MonteCarloCI.jpg">
</p>
<p>
where <i>&alpha;</i> is the confidence level (e.g., <i>&alpha;</i> = 0.05 for 95% confidence level), <i>pF</i> 
is the probability of failure and <i>z(x);</i> is the standard normal inverse cumulative distribution 
function evaluated at <i>x</i>, <i>&tau;</i> is the standard deviation of that response, 
and <i>n</i> is the number of samples.
</p>
<p>
If the required number of samples required is very high, then Reliability methods are a good alternative. 
Depending upon the estimated sigma level and the nature of the constraint, the following reliability 
techniques may be appropriate:
<ol>
<li><b>FORM</b> - either when Sigma Level > 2 and constraint almost linear, or, Sigma Level > 3 </li>
<li><b>SORM</b> - when Sigma Level > 2 and constraint close to parabolic</li>
<li><b>Simple Importance Sampling</b> - when <i>pF</i> is very small but needs to be estimated accurately</li>
<li>Simple Importance Sampling with <b>FORM correction</b> - same as simple importance sampling but constraint is known to be almost linear</li>
<li><b>Adaptive Importance Sampling </b> - Same as simple importance sampling but number of samples not known apriori</li>
</ol> 
</td>
</tr>
</table>
<p>&nbsp;</p>

<p>Since some users may not have access to all the techniques, we calculate the probability using both Monte Carlo 
as well as Reliability methods. Specifically, we will use simple random sampling for Monte Carlo and 
Mean Value method with second order Taylor series expansion for reliability calculations.
</p>
<p>
In order to estimate the probability of success, we use Mean Value method (model is available 
<a href="models/Adapter-SS-MVM.zmf">here</a>) and the results are shown in <a href="#figure28">Figure 28</a>. 
From this table, FOS has a probability of success of 0.5, i.e. 50% of designs will not satisfy the FAA safety 
requirement! This result illustrates the inadequacy of relying on the "best" design obtained with a simple 
optimization. When we look at the optimal design, we notice that the FOS constraint is active - this means
that, in the space of random variables, the FOS constraint divides the neighborhood into two halves resulting 
in 50% success rate.</p>

<p>&nbsp;</p>
<table> <tr><td style="border-style:none;">
<div style="width:500; font-size:80%; text-align:center;"><a name="figure28"></a></a><a href="images/image030.jpg">
<img width=500 src="images/image030.jpg">
Figure 28: Six Sigma Table showing the results calculated using Mean Value Method.
</a></div>
</td><td style="border-style:none;">
<div style="width:300; font-size:80%; text-align:center;"><a name="figure29"></a></a><a href="images/image031.jpg">
<img width=300 src="images/image031.jpg">
Figure 29: Six Sigma Graph showing the mean and standard deviation of response FOS calculated using Mean Value Method.
</a></div>
</td></tr></table>
<p>&nbsp;</p>

<p>
The important note here is that whenever a constraint is active at a design point, the probability of success
will be around 50% (less if more constraints are active). Therefore in order to improve the probability of 
success, the optimization problem needs to be reformulated. We will use Isight's Six-Sigma optimization 
capability in the next step to obtain a good design that also has a high probability of success.
</p>
<p>
But before we proceed to Six-Sigma optimization, we illustrate the method to calculate the number of samples 
required for Monte Carlo. Suppose we want to estimate the probability of success with &plusmn;1% error at 
95% confidence, we set <i>&alpha;</i>=0.05 and <i>&tau;</i>=0.088. Therefore, to obtain a 95% two-sided confidence 
interval of (0.49, 0.51), we need <i>n</i>=(0.088*1.96/0.005)^2=1190 samples. We will use 1200 samples (after rounding)
for simple random sampling (model available <a href="models/Adapter-SS-MC.zmf">here</a>). The results are shown 
in <a href="#figure30">Figure 30</a>. Note that the probability of success computed is 0.5142 which lies outside 
(0.49, 0.51), illustrating the fact that the estimate need not actually lie within the confidence interval. 
</p>
<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure30"></a></a><a href="images/image032.jpg">
<img width=500 src="images/image032.jpg">
Figure 30: Six Sigma Table showing the results calculated using Monte Carlo sampling.
</a></div>
<p>&nbsp;</p>


<h2>Step 5  Improve quality level with Six-Sigma optimization</h2>
<p>
In the previous step, we estimated the probability of success to be around 50%. In this step, we will improve
this probability to 99.9%. In order to do that, we open the Six Sigma editor in the Design Gateway and switch
the "Run Mode" to "Six Sigma Optimization" as shown in <a href="#figure31">Figure 31</a>. Notice that a new tab
"Optimization" is available (<a href="#figure32">Figure 32</a>).

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure31"></a></a><a href="images/image033.jpg">
<img width=500 src="images/image033.jpg">
Figure 31: Six Sigma Editor with "Six Sigma optimization" selected.
</a></div>
<p>&nbsp;</p>

<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure32"></a></a><a href="images/image034.jpg">
<img width=500 src="images/image034.jpg">
Figure 32: Six Sigma Optimization editor.
</a></div>
<p>&nbsp;</p>

<p>Similar to Step 3, we choose the adapter dimensions as design variables and mass as objective. We will, however, 
add a new constraint to the "Probability of Success" for the response "FOS" as shown in <a href="#figure33">Figure 33</a>.
We will continue to use NLPQL with default settings for optimization. The saved model is available 
<a href="models/Adapter-SS-Optimization.zmf">here</a>. We have added history plots of the 
adapter mass and FOS probability of success to the model.
<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure33"></a></a><a href="images/image035.jpg">
<img width=500 src="images/image035.jpg">
Figure 32: Constraints for Six Sigma Optimization.
</a></div>
<p>&nbsp;</p>
<p>
The results of Six Sigma optimization are shown in Figures <a href="#figure34">34</a> and <a href="#figure35">35</a> 
indicating that the mass has increased by less than 0.75% to improve the quality level.<p>

<p>&nbsp;</p>
<table> <tr><td style="border-style:none;">
<div style="width:300; font-size:80%; text-align:center;"><a name="figure34"></a></a><a href="images/image036.jpg">
<img width=300 src="images/image036.jpg">
Figure 34: History of the adapter mass during Six Sigma optimization.
</a></div>
</td><td style="border-style:none;">
<div style="width:300; font-size:80%; text-align:center;"><a name="figure33"></a></a><a href="images/image037.jpg">
<img width=300 src="images/image037.jpg">
Figure 35: History of the FOS probability of success during Six Sigma optimization.
</a></div>
</td></tr></table>
<p>&nbsp;</p>


<h2>Step 6  Validation</h2>
<p>
In the preceding steps, we have used an RSM approximation to obtain an adapter design that has 
the minimum mass and satisfies the FAA requirement with a probability of 99.9%. In doing so, we
used an approximation that was created using samples obtained using a DOE. We evaluated the quality
of the approximation across the entire design space using cross-validation. Although this approximation
had an acceptable average error across the design space, the error near the design point may be high 
resulting in a poor optimum design. In order to validate our design, we will calculate the probability 
of success directly using the subflow. 
</p>
<p>
In the Runtime Gateway, right click on the Six Sigma component and select "Save As...". At the prompt
"Do you want to set the parameter values to the values from the last run for each component?", select 
"Yes". This will create a new Six Sigma model with the last run, i.e. the best point, as the nominal values
for the adapter dimensions.</p>
</p> <p>
Open the model in the Design Gateway and navigate to the approximations of the Abaqus component. 
Deactivate all approximations on the Abaqus component. Open the Six Sigma editor and change the 
"Run Mode" to "Six Sigma Analysis". The model obtained after performing these 
steps is available  <a href="models/Adapter-SS-Validation.zmf">here</a>. The results show that the 
probability of success is merely 63%(<a href="#figure36">Figure 36</a>). This is because the approximation we created was not accurate in 
the region near the optimum design.
</p>
<p>&nbsp;</p>
<div style="width:500; font-size:80%; text-align:center;"><a name="figure36"></a></a><a href="images/image038.jpg">
<img width=500 src="images/image038.jpg">
Figure 36: Results of Six Sigma analysis without using approximations for validation.
</a></div>
<p>&nbsp;</p>

<p>In order to obtain a design that has a higher probability of success, the above procedure has to be
repeated using an approximation that is centered around the optimum point. In general, these steps 
are iterated until the error in the estimate of quality level at the optimum design is acceptable.
</p>

<h2>References</h2>
<ol>
<li> <a name="reference1">Devanathan, S., Koch, P. N., 2011, "Comparison of meta-modeling approaches for optimization," proceedings of the ASME 2011 International Mechanical Engineering Congress and Exposition, IMECE 2011, Nov 11-17, Denver Colorado, IMECE2011-65541.
</li> 
<li> <a name="reference2">Gill, P. E., Murray, W., and Wright, M. H., 2008, "Chapter 8: Practicalities," in <i>Practical Optimization</i>, Emerald Group Publishing Limited, Bingley, UK, pp. 285-355.
</li> 
</ol>
</body>

</html>
