One of the easiest ways to create an even distribution of sampling point is by placing them in a n-dimensional grid within the factor space, see Figure 2.16. One disadvantage with this type of sampling is that the designer cannot freely decide which arbitrary points to be included and the segmentation of the factor space determine the total amount of sampling points. Another weakness is that “a projection of the design onto a subspace with reduced dimensionality would yield many replicated points” (Jurecka, 2007, p. 146), which is undesirable in cases where irrelevant factors has been identified (Jurecka, 2007).
Figure 2.16: Design sequence sampled
from a grid. Figure 2.17: Latin Hypercube Design with distance criterion
1 2
min d 2
1
20
Instead of sampling point from a grid a distance-based criterion can be added for better space-filling properties (Jurecka, 2007), see Figure 2.17. For example, a
minimum distance are so called maximin distance designs, modeFRONTIER™ provides the Incremental Space Filler (ISF) which is
a uniform way. The ISF algorithm is among other
Figure 2.18: Illustration of settings for Incremental Space Filler in modeFRONTIER™.
2.5.4
If a design sequence is to be considered orthogonal if the scalar product of any of the column vectors being evaluated is zero, or in other words, XTX is diagonal matrix. An important characteristic of orthogonal arrays is the resolution R, which explains the independent estimation of main and the interaction effects.
minimum criterion for the Euclidian distance d between the sampling points vk
and vl
Designs which maximizes the such a design, see Figure 2.18.
The points are added sequentially with a specified radius from an existing design point, filling the design space in
things used for extending the design database with points around the Pareto front when running Fast optimizers in modeFRONTIER™ (Rigoni, 2010), see description of fast optimizers in Section 2.6: Fast optimizers in modeFRONTIER™.
Two variants of the algorithm are available in modeFRONTIER™, the Genetic Algorithm Optimization (GOA) and Voronoi-Delaunay Tessellation (VDT).The GOA implements a genetic algorithm for internal optimization of the distance criterion. The method is approximative but works in a fast and robust way, in contrary to the VDT, which is a time-consuming method that will generate an exact solution for maximization of the minimum distance (Rigoni, 2010).
Taguchi Orthogonal Array Designs
21
The method might seem limited but the result is less computational effort. This due to the fact that interaction factors of interest are introduces as independent
f noise factors are the environment the
treatment combinations, factors, (Jurecka, 2007). As a result the number of sampling points will be reduced, see Table 2.2 and Table 2.3.
In modeFRONTIER™ it is possible to design experiments by a technique called Taguchi Orthogonal Arrays (TOA). TOA are orthogonal arrays which also considers both noise and design factors. Design factors, or Control factors, are parameters of the design that are effortless and cheap to control. Noise factors are factors that are hard or impossible to control but that might affect the performance of the product. Examples o
product will function in, called an external factor, and material defects or variation, also termed internal factor (Dean et. al, 1999).
Two types of designs exist for TOA, mixed and product arrays. Product arrays consist of one fractional factorial experiment for the design factors and one for the noise factors. All combinations of design factors are considered in coordination with every possible combination of noise factors. Robustness is said to be achieved if the product work consistently well regardless of unrestrained variation of noise factor levels. Mixed arrays consider
which are level combinations of design and noise factors. The robust settings for the design factors are obtained by investigation of the interaction between the noise and design factors (Dean et. al, 1999).
NO. OF SAMPLING POINTS Full Factorial Orthogonal Array
23 → 8 4
Full Factorial Orthogonal array
-1 -1 -1 -1 -1 1
Table 2.2: Comparison of the num f experiments generated by Full Factorial Design
ogonal Array D n.
Table 2.3: Design sequence of the va les v1. v2 and v3 by Full Factorial Design
omp d to Orth gonal Array Design.
H rent settings fo OA designs in od RO TIER can be applied is seen in Figure 2.19.
Figure 2.19: Possible settings for Taguchi Orthogonal Arrays in the modeFRONTIER™ interface.
2.5.5 Full factorial designs
ill be mn distributed evenly over a region of interest (Strömberg, 2010). Mathematically
e; a case of 2 design vari les varied over 3 levels would generate 9 test points and would be called a 9 experiment. The formation of 9-factorial design is illustrated in Figure 2.20 and Table 2.4.
A factorial design is design where a number of n design variables are varied over a predetermined number of m levels. The generated number of experiments w expressed as
Table 2.4: Set up of a 9-factorial design with the two variables n1 and n2.
Figure 2.20: A 9-factorial design – 2 variables varied over 3levels
23
L oria
estimated by 3-level als. As rule of thumb, a le sed to fit a polynomial of degree d. The drawback with full factorial experiments is the
r of experiments is to apply the method
less than 4 levels, see Figure 2.21.
Figure 2.21: In modeFRONTIER™ the user decides the level of every single variable. It is possible to select higher levels for some variables and a lower for another variable.
2
While full factorial generates m
onal factorial sequence contains mn-p efining the reduction of the sequence
exp ill be reduced (Dean et.
to be governed mainly by only a few of them and by low-order interaction. Another argument is the projection of the design onto other levels; if a number of p factors is excluded from the design sequence due to irrelevance the consistent dimensions of the design space is removed (Jurecka. 2007). The original fraction factorial design becomes a full factorial design, see Figure 2.22.
inear effects are estimated by 2-level fact
factori ls while quadratic behavior is
vel of d+1 has to be asses fact that the experimental effort increases rapidly when the number of levels or variables is increased (Jurecka, 2007).
A 3 level factorial with 9 variables would result in 19 683 experiments. Running a sequence with 39 experiments would fast yield a great amount of computational time. A way to decrease the numbe
Fractional Factorial instead.
In modeFRONTIER™ it is possible for the user to set the level of each variable.
A brief summary points out the fact that the full factorial is suitable for problems with less than 8 variables and
.5.6 Fractional Factorial Designs
n experiments a reduced factorial produces a subset of the full factorial sequence. A fracti
experiments, where p>0 is an integer d
compared to the full factorial. The expression 1/ is used to calculate the fractional of the full design (Jurecka, 2007).
Due to the fact that only that only a fraction of the behavior is observed, each main-effect and interaction contrast cannot be estimated independently. On the other hand, the computational effort for the eriment w
al, 1999).
Fractional Factorial Designs are motivated by some different arguments. One is the fact that in systems which depends on a large number of input variables are expected
24
Figure 2.22: Illustration of the projection of a 23-1 fractional factorial design into three 22 full factorial designs.
Fractional factorial is in modeFRONTIER™ referred to as “reduced factorial” and is based on a 2-level factorial sequence, see Figure 2.23. By selecting the largest number in the “Number of Designs”-window will result in a full factorial. For a problem with 8 variables the largest number available would be 256.
2.5.7 Central Composite Designs
For fitting second-order polynomials Central Composite Design (CCD) is one of the most popular techniques. It combines points from a two-level full factorial (or fractional factorial) sequence with a center point and so called star points. Star
points a from
the origin. On the axis
at , see Figure 2.24. The number of star points will be 2p for a number of p factors (Dean et. al, 1999).
Figure 2.23: Available settings for a Reduced Factorial Design in modeFRONTIER™
re points which are located on all coordinate axes with a distance
two points will be positioned; one at and another
25
Figure 2.24: Illustration of distribution of points for a factorial desi
The value of α is chosen depending on the properties required of n the expression
gn, the center point and the star points.
the design, ofte
is used for establishing a suitable value for , where
factorial points (Dean et. al, 1999). For this expression the resulting CCD will be
rota portant the predictio cy. The
prediction discrepancy e
de
contribute largely to tion the linear terms and they are the only points which contribute to of the teraction terms. The center point provides information ab of the s m and estimation of
α is given by are the orthogonal table, which is an im condition for n discrepan
will be equal on a sphere around the center point if th sign is rotatable (Jurecka, 2007).
The three types of points play different roles in providing information. The
factorial points the estima
computat
out curvatuion in
re yste
quadratic terms. It also provides the “pure error” of the design. If curvature is found in the system the star point will estimate the quadratic terms in an efficient way (Meyers & Montgomery, 2002)
Two types of regions can be taken in consideration for CCD, spherical and cuboidal regions. If the region interest is spherical a suitable value for
√
where n is the number of factors. This way of choosing α will not result in an exact rotatable design instead the value will create better solution from a prediction point of view. A spherical CCD will put all the star points and factorial points on a sphere with radius √ (Montgomery, 2005).
For cuboidal regions of interest it is suitable to use face-centered CCD, where
is of no greater importance when t of interest is obviously cuboidal 1. In this case the star points will be located at the centers of the faces of the factorial cube, see Figure 2.25. This type of design is not rotatable, but rotatability
he region (Meyers & Montgomery, 2002).
For sequential experimentation CCD is an efficient design which not involve a too large number of design points while still providing a reasonable amount of data for testing lack of fit. It is also a flexible design in its use of star points; it can
Full Factorial Points Center Point Star Points α
26
accommodate a spherical region by √ and with five levels for each factor, and a cuboidal region for 1 and a three-level design. The CCD can also provide for situation where √ or 1 cannot be used, the value of α can be adopted depending on the circumstances (Meyers & Montgomery, 2002). For central composite design for three variables, see Table 2.5.
In modeFRONTIER™ it is possible to choose between two es C D, see figure 2.26. The first type, Cubic Face Centered, places points at all vertexes of the cube, as in Figure 2.25, the second type, Inscribed Composite D n, p
the vertexes of a scaled hypercube within the cube
Point No.
esig laces points at .
Table 2.5: Central Comp te Design sequence e factors.
Figure 2.25: Illustration of a face-centered CCD sequence for three factors and 1, the 8 corner points are full factorial points, the 6 points on the axes are star points and the middle point is the center point.
osi for thre
27
Figure 2.26: Parameter settings for Central Composite Designs in modeFRONTIER™.
2.5.8 Box-Behnken Designs
For Box-Behnken Designs (BBD) is a special methodology which constructs balanced incomplete block designs, see Table 2.6. The variables are treated in pairs, for an example of three variables: in the first block variable 1 and 2 is paired together in a 22 factorial while variable 3 is fixed at the center. In the second block variable 2 is fixed at the center and instead variable 1 and 3 is paired together. This proceeding holds for a number of variables 2 < n < 6 (Meyers & Montgomery, 2002).
Treatment
Block 1 2 3
1 X X
2 X X
3 X X
Table 2.6: Demonstration of the order in which the variables are paired together.
For problems with 6 or more variables the procedure is slightly different, for a closer description see Meyers & Montgomery, 2002.
The resulting design sequence of Box-Behnken with three variables is shown in Table 2.7 and Figure 2.27.
28
BBD is closely related to Central Composite Designs but has the advantage of avoiding extreme values of the factors by not taking the corners of the factor space in consideration. On the other hand, the same characteristic of avoiding extreme factor settings result in BBD not being suitable for prediction of behavior in the corners of the factor space. As a conclusion this type of design should be used for cases where there is no interest of predictions of the corner points. Since BBD is a spherical design it is rotatable, or near rotatable, and all sampling points will have same distance to the center point (Jurecka, 2007).
Table 2.7: Design sequence for three variables generated by Box-Behnken. The last
row is the center point Point No.
Figure 2.27: Illustration of Box-Behnken Design for three variables.
Box-Behnken Design is available in modeFRONTIER™ without any settings for the user to configure, see Figure 2.28.
Figure 2.28: Box-Behnken Designs in modeFRONTIER™.
29
2.5.9 Latin Square Design
The Latin Square Designs (LSD) are n x n arrays with Latin letters arranged in such way that each of the letters only occurs once in each column and in each row, see Figure 2.29. If the letters in the first row and the first column of the Latin Square are in alphabetical order the square is known as a standard Latin square (Dean et. al, 1999).
LSD is used to eliminate two sources of problems concerning the changeability, Latin squares make it possible to block in two directions in a systematic way.
Small Latin Squares has the disadvantage that they have a fairly small number of error degrees of freedom, the error degree of freedom for a 3x3 Latin Square is equal to 2 . This disadvantage is compensated by replication of the squares; thus, increasing the error degrees. A common procedure for selecting a Latin square is to take a Latin square from a table providing standard Latin squares and randomly rearrange the order of rows, columns and letters of the square (Montgomery, 2005).
Figure 2.29: Examples of 4x4 and a 5x5 standard Latin Square.
In modeFRONTIER™ it is possible to create a 20x20 Latin Square, generating 400 design experiments, see Figure 2.29.
Figure 2.29: Latin Square Designs in modeFRONTIER™.
30
2.5.10 Plackett-Burman Designs
Plackett-Burman Designs (PBD) are fractional designs for a number of variables for N runs. N is a multiple of four and when N is power of 2 the designs are the same as the respective fractional design. For some special cases (for N=12, 20, 24, 28, 36), called nongeometric PBD, this type of Design of Experiment can be of interest (Myers & Montgomery, 2002).
In modeFRONTIER™ it is possible to determine the order of the variables, see Figure 2.30. The order of the variables is of significance due to the messy alias structure of PBD. The limited area of interest and the fact that the method is recommended to be used carefully by Myers & Montgomery, 2002, no closer explanation is provided for this method.
Figure 2.30: Settings for the Plackett-Burman Designs in modeFRONTIER™.
2.6 Fast optimizers in modeFRONTIER™
The fast optimizers in modeFRONTIER™ uses metamodeling to speed up the optimization process, a detailed description of metamodeling is found in Section 2.4: Metamodeling. In the following sections the working process of the fast optimizers is described in detail, as well as a closer explanation of the algorithms implemented in modeFRONTIER™; SIMPLEX and MOGA-II.
2.6.1 Working process
The working process of the fast optimizers is an iterative procedure where metamodels are trained and evaluated. Figure 2.31 illustrates the general steps of the work process.
31
Figure 2.31: The iterative process of fast optimizers.
Initially the algorithm takes in consideration a number of design experiments;
FSIMPLEX will use a number of N+1 (for a number of N variables) from the DoE as an initial population, while FMOGA-II will use the entire DoE table. The initial population m will be evaluated by means of the real solver and used for RSM training. If a set exists of previously tested points the RSM will be trained using them as a validation set instead of the initial population (Rigoni, 2010).
In the first iteration RSM by default, depending on the scale of the problem, is engaged in the virtual exploration and optimization. For the subsequent iterations RSM is selected depending on the performance throughout the previous iteration.
The training in this step only focuses on training RSM for the exploitation and optimization. The training set can be limited by the user in modeFRONTIER™ by specifying the maximum size, this reduces the computational time (Rigoni, 2010).
In the step virtual exploration the database is improved by using the Incremental Space Filler (ISF), see description of ISF in section 2.5.3 Incremental Space Fillers.
ISF is used for sampling new points around the current Pareto front and to increase robustness of the optimizer (Rigoni, 2010).
During the virtual optimization process the FMOGA-II, or FSIMPLEX, is run over the best available meta-models. As a population for next iteration the database is built up of 50 % random points and 50 % of points from the current Pareto front. The Pareto front points contribute to faster convergence and the random points increases the robustness of the optimizer. A number of 2m designs are extracted from virtual design database for the validation step. Any duplication of previous evaluated points is removed from the database, when a number of m designs remain convergence is reached (Rigoni, 2010).
In the validation process a number of m designs are randomly extracted from 2m designs from the virtual optimization step. The designs are then validated in terms of a real solver and constitute the validation set for evaluation of the different metamodels applied (Rigoni, 2010).
Metamodel
The performance of the metamodels is evaluated and compared to the predicted result in the training step. The divergence of the predicted value and the observed value becomes a measurement of the metamodels performance. The best performing metamodel is used as the primary model for the next iteration and no other metamodels are trained. If the performance would impair, all metamodels will be available for training and evaluation once again (Rigoni, 2010).
Fast optimizers are slow in terms of net computational time since they train metamodels and run ISF algorithm. Fast optimizers are primarily useful for optimizations involving an expensive CAE solver where the time for training and ISF negligible compared to the computational time of the solver (Rigoni, 2010).
2.6.2 The SIMPLEX algorithm
The original SIMPLEX is a robust algorithm for non-linear optimization problems. Due to the name it gets easily mixed up with the simplex method used for linear programming but should not be associated with each other.
For a number of N dimensions the simplex will be a polyhedron with N+1 corner points, see Figure 2.32.
Figure 2.32: A simplex, ABC, in a 2-dimension (Poles, 2003 A).
The simplex will expand and contract searching for the minimum of the objective function. Through the iterative process the values at the corner points of the simplex are evaluated and replaced by lower function values. The operation procedure is the described by the three steps: reflection, expansion and contraction (Poles, 2003 A).
In reflection, the worst vertex value is projected, or reflected, on the opposite surface to obtain a new value. As a consequence the movement will always be in a favorable direction (Poles, 2003 A). Figure 2.33 illustrates the reflection movement.
33
Figure 2.33: Reflection movement of the simplex ABC (Poles, 2003 A).
Formulated mathematically this becomes
where A is the worst vertex, D is the new point, H is the centroid of all points except for the reflected and α>0 is a reflection coefficient.
When reflection produces a new point that produces a better value than the previous the simplex expands. In expansion movement the simplex expands in the
When reflection produces a new point that produces a better value than the previous the simplex expands. In expansion movement the simplex expands in the