A robust objective function for topology optimization by mostafa.ahmadi.d


									COMPEL: The International Journal for Computation and Mathematics in
Electrical and Electronic Engineering
Emerald Article: A robust objective function for topology optimization
Min Li, David A. Lowther

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International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 30 Iss: 6 pp. 1829 - 1841
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         A robust objective function                                                                                         optimization
          for topology optimization
                            Min Li and David A. Lowther
    Department of Electrical and Computer Engineering, McGill University,                                                                  1829
                              Montreal, Canada

Purpose – Robust design is very important for manufacturers to ensure the quality of the finished
product. Therefore, a robustness measure is needed for the topological design of electromagnetic
problems which may be sensitive to parameter variations. The purpose of this paper is to propose a
robust objective function for topological design problems.
Design/methodology/approach – In this paper, a robust objective function for topology
optimization is defined on an uncertainty set using the worst case analysis. The robustness of a
topological design is defined as the worst response due to the variations of the location of the topology
change. The approach is based on the definition of a topological gradient.
Findings – The robust topology optimization (RTO) was applied to eddy current crack
reconstruction problems. The numerical applications showed that this method can provide more
reliable results for the reconstruction in the presence of significant noise in the measured signal.
Research limitations/implications – The RTO may be applied to some more complicated design
problems; however large computational costs may result.
Originality/value – This paper has defined a robustness metric for topology design and a robust
design model is proposed for topology optimization problems.
Keywords Sensitivity analysis, Topology optimization, Robust design, Electromagnetism,
Eddy currents
Paper type Research paper

I. Introduction
In recent years, computer-aided design has become the first choice of design engineers
when new electromagnetic devices are required. The physical problem is often mapped
into a parametric search space, and the final design is achieved through an automated
design process which determines the optimal values of the parameters using an efficient
search algorithm. The physical prototype can then be implemented based on the
specification. If the newly designed device is to be manufactured, the tolerance problem
must be taken into account. For example, in reality, small variations to the structure which
can occur during the manufacturing process are not negligible, and these variations can
sometimes cause serious deterioration in the performance of a device. Therefore, possible
variations of the design variables should be handled and integrated into the mathematical
formulation of the optimal design. Such a procedure is usually referred to as “robust
design” since the performance of the resulting product is more stable.
    The robust design of electromagnetic devices has been studied for more than ten
                                                                                                                    COMPEL: The International Journal
years. Several practical approaches have been proposed in the past. One statistically                              for Computation and Mathematics in
based formulation of robust design is to minimize the nominal value of the                                         Electrical and Electronic Engineering
                                                                                                                                      Vol. 30 No. 6, 2011
performance function and its variation at the same time. Unfortunately, the statistical                                                    pp. 1829-1841
information is sometimes not available and it can be computationally expensive to                                  q Emerald Group Publishing Limited
calculate the standard deviation. First-order sensitivity was used to approximate the                                   DOI 10.1108/03321641111168138
COMPEL   variance and the multi-objective cost optimization was solved using a weighting factor
30,6     in Yoon et al. (1999). Later, an alternative formulation based on the uncertainty set and
         worst performance analysis was proposed (Alotto et al., 2003; Steiner et al., 2004).
         A robust objective function was formulated as the worst performance of the function
         values in the small neighborhood representing the possible variations of the design
         variables. Examples of solving worst case tolerance design problems using
1830     evolutionary algorithms can be found in Spagnuolo (2003). On the other hand,
         statistical methods (e.g. Monte Carlo analysis) have also been employed for the robust
         design of electromagnetic devices (Cavaliere et al., 2003; Cioffi et al., 2004).
         Most recently, a novel approach which treated robust optimization as one of conflict
         design was presented (Guimaraes et al., 2006). This method combines the ideas of
         multi-objective robust design formulation and the worst performance analysis. In this
         formulation, the worst case performance and the nominal value of the objective
         function can be minimized simultaneously. Therefore, more robust design solutions
         were provided to the designer and decisions can be made as a tradeoff between
         performance and reliability.
            Topology optimization is an important tool for device synthesis. Unlike shape
         optimization methods which must be based on an existing device and where the
         parameters are only allowed to change within a relatively small range, the optimal
         material distribution (OMD) method is able to represent all the potential layouts of the
         devices in the design space (Dyck and Lowther, 1996). Thus, topology optimization
         methods can provide innovative designs starting from an empty space without any
            In practice, OMD is often used as a pre-processor for a boundary shape optimization
         which generates finer details of the body. The grey-scale picture of the material density
         can be converted to a solid structure using image processing techniques (Choi et al.,
         1998). However, if the original topology is sensitive to small parameter variations, it
         may result in large changes in the final shape obtained as well as performance
         deviations. Unfortunately, topics on managing uncertainties during topological design
         have rarely been discussed. The idea of reliability-based topology optimization (RBTO)
         was proposed and applied to the electromagnetic problems, e.g. the design of a c-core
         (Kang et al., 2004). Unfortunately, RBTO requires some probabilistic information and
         only the uncertainties of the environment are handled in the formulation. Rather than
         minimize the variation of the objective, RBTO focuses on the probability of the system
         response based on the distributions of the random parameters. Although there exist a
         variety of robust design formulations, they cannot be combined directly with
         traditional topological optimization methods which are based on homogenization or
         material distribution. Topological shape optimization, also known as the “bubble
         method”, emerged 30 years ago and has recently received intensive attention. The
         topological sensitivity analysis has been effectively applied to many different
         electromagnetic problems (Kim et al., 2008; Li and Lowther, 2010a, b). The topology of
         the design object, as well as the shape, is allowed to change during the design process.
         In addition, this formulation provides the possibility of incorporating a robustness
         measure into the design process.
            In this paper, we propose a robust design model for topology optimization. A robust
         objective function is defined for the design problem based on the uncertainty set and
         the worst performance analysis. The robustness measure for the topology optimization
is defined for the first time. The robust target function remains the same for both the            Topology
topological design and the shape optimization stages, despite the change of the design        optimization
parameters. The proposed robust topology optimization (RTO) formulation is also
compared to the RBTO methods.

II. Robust design formulation
A. Non-robust design formulation                                                                     1831
The design of an electrical machine is typically stated as an optimization problem
where an objective function is minimized under certain requirements of the
performance and physical constraints. A general formulation of the design
optimization problem can be expressed mathematically as:
                        min fðxÞ   s:t: gi ðxÞ # 0   i ¼ 1; 2; . . .m:                  ð1Þ
where f is an objective function which indicates the performance of the design, x is a
vector of the design variables; and gi are the inequality constraints on x. Usually, the
performance of the design must meet a desired specification. From the robust design
point of view, f and x are just the nominal values of a design. In the presence of
uncertainties, i.e. if there is a small variation to the nominal value of x, the value of f
may no longer indicate a performance which meets the specifications, or the non-robust
solution may become infeasible in the case of constrained problem.

B. Worst performance due to perturbation
Unlike the conventional formulation of design optimization, a robust design
formulation must take account of the variation of the design variables. An
uncertainty set U(x) is defined as a compact sub-domain of the design space containing
all the perturbations to the nominal values of the design variables x:
                      Uðx0 Þ ¼ {j [ Rn : x0 2 D # j # x0 þ D}:                          ð2Þ
where the vector D ¼ {D1 D2 . . . Dn}represents the largest variation to the nominal
value x0 of the design variables.
   Sometimes the variations to the design variables may be defined as proportional to
the nominal value of every dimension, especially when the vector of design variables
contains different physical quantities. Thus, the uncertainty set U(x) can be defined as:
                 Uðx0 Þ ¼ {j [ Rn : ð1 2 Di Þx0;i # ji # ð1 þ Di Þx0;1 }:               ð3Þ
where Di represents the largest variation to the ith component of vector x0 in per unit
terms. Figure 1 shows a rectangular shape of the uncertainty set in 2D.

                     ∆2* X0.2

                                                     X0 (X0.1, X0.2)
                     ∆2 * X0.2                                                                      Figure 1.
                                                                                                uncertainty set
                                   ∆1* X0.1               ∆1* X0.1
COMPEL   The advantage of the rectangular uncertainty set is the simplicity of its implementation
         and it does not require any statistical information about the design variables.
30,6        The worst performance of the objective function f due to the perturbation of the
         design variable x is given as:
                                              fw ¼ max fðjÞ                                   ð4Þ

1832     Nevertheless, if the solution x * is located close to the boundary of the feasible region,
         the shape of the uncertainty set may be altered by the constraints. To ensure feasibility
         robustness, a robust constraint function is defined as:
                                                max gi ðjÞ # 0                                  ð5Þ

         Overall, a robust design can be formulated as minimizing the worst performance of the
         objective function value within the uncertainty set:
                                   min max fðjÞ s:t: max gi ðjÞ # 0:
                                       j[UðxÞ               j[UðxÞ

         Unlike the non-robust function, the value of the robust objective function is determined
         by the worst performance, i.e. it is not sensitive to the perturbation of the nominal
         values of design variables.

         C. Multi-objective approach
         In the early attempts to tackle the robust design problem, Taguchi has developed a
         quality loss function (QLF) using the mean squared deviation to estimate the monetary
         loss as the performance deviation from the nominal value (Taguchi et al., 1989). A more
         general robust design formulation derived from Taguchi’s QLF is defined as a
         multi-objective problem:
                                                 "         #
                                             min                                             ð7Þ

         where E[f(x)] and V[f(x)] are the expectation (mean) and the variance of the objective
         function based on statistical models, respectively.
            In order to solve the problem without computing the statistical information, we may
         use the nominal value and the deviation of the worst performance instead of the
         expectation and the variance in equation (7) (Guimaraes et al., 2006):
                                              "               #
                                          min                   :                            ð8Þ
                                                fw ðxÞ 2 fðxÞ

         The nominal value of the objective function is used to represent the mean, and the
         deviation of the performance is considered as the second objective.
            The problem in equation (8) can be solved using a weighting factor a:
                    C ¼ ð1 2 aÞðfðxÞ þ aðfw ðxÞ 2 fðxÞÞ           ¼ ð1 2 2aÞfðxÞ þ afw ðxÞ:     ð9Þ
         If we choose a ¼ 0.5, C ¼ 0.5fw(x), and the worst case scenario is just one of the
         solutions to the multi-objective optimization.
   Using the multi-objective formulation, the robust design problem is treated as a                    Topology
classical conflict design and, therefore, a pareto front of the potential robust solutions           optimization
can be generated. The most suitable solution is then chosen by the designer.

III. Robust topology optimization
A. Topological shape optimization
In the conventional formulation of topology optimization based on the theory of                              1833
homogenization or material distribution, the design space is usually divided into cells.
And the design variables are the dimensions of the discretized elements. The possible
values of these design variables are numbers between 0 and 1 which represent the
intermediate states of the material properties. If only solid material is allowed in
the design, there cannot be any variation to the nominal value of the design variables,
since they must be either 0 or 1 at the end of the design. Therefore, it is not possible to
apply the robust design formulation to conventional topology optimization methods.
   Topological shape optimization has been successfully applied to electromagnetic
problems (Kim et al., 2008). This method has two stages: first, generate topology
changes to the domain of the design by creating a hole in the material based on the
values of the topological gradient; second, optimize the shape of boundary of the hole
using a standard shape optimizer (Figure 2).
   The objective function for the topological design is denoted as C in domain V.
Consider an infinitely small hole B(x, r) in V, where x is the center and r is the radius of
the hole. The topological gradient TG(x) is defined in (Li and Lowther 2010a) as:

                                           CðVÞ 2 CðV\Bðx; rÞÞ
                          TGðxÞ ¼ lim                                                 ð10Þ
                                     r!0          dðrÞ

where C is an objective function defined on V, V\B(x, r) is the domain excluding the
small hole B, and d(r) is the volume of B with a negative sign.
   In the topology optimization stage, the topology gradient is computed over the
problem domain V. A small hole B(x, r) is drilled in V where the value of TG at x is
greater than zero (Li and Lowther, 2010a). The performance function after this
topology change is denoted as C(V\B(x, r)). The design variable of the topology design
problem is the 3D coordinate of the center of the hole. Any variations of x may result
in a potential topology change. The radius r is usually set as a fixed length determined
by the size of the cells in the grids or the size of the elements in the mesh. And the
size and shape of the hole will be determined in the shape optimization stage for

                                               x r Γr


                                                                                                            Figure 2.
                                                                                              A topology change to the
                                                                                                      problem domain
COMPEL             which the robust formulation will be discussed later. We define a scalar function
30,6               J(x) ¼ C(V\B(x, r)), and the worst performance function is given as:
                                                      fw ¼ max JðjÞ:                           ð11Þ

                   B. Robust formulation of the topological gradient
1834               The worst performance of the objective function can be predicted with the assistance of
                   the topological gradient. Equation (10) can be re-arranged using an expansion on the
                   objective function J near the point x0:
                                     JðxÞ ¼ CðV\ Bðx; rÞÞ ¼ CðVÞ þ dðrÞ · TGðxÞ þ oðrÞ:                ð12Þ
                   Since C(V) and d(r) are constants with respect to x (note that d(r) has a negative sign),
                   J(j) has the largest value where TG(j) is the smallest. Therefore, the worst performance
                   of J due to the perturbation of the design variables is determined by the point j in the
                   uncertainty set, where TG(j) has the smallest value.
                       Hence, we can use the value of TG(j) to replace the nominal value of TG(x) and a
                   robust formulation for the TG can be obtained as:
                                                       TGR ðxÞ ¼ min TGðjÞ:                            ð13Þ

                   As a result, the robust formulation of the topological gradient TGR can be computed
                   and used as a decision criterion on where the topology changes will take place.

                   C. Worst vertex prediction for shape optimization
                   After the topological analysis stage has determined where to drill a small hole in the
                   problem domain, topological shape optimization is used to perform the second stage.
                   The boundary of the hole is discretized and the shape of the boundary is optimized.
                   As shown in Figure 3, several control points are created and the boundary is described
                   using a B-spline curve.
                      At the shape design stage, the design variables P ¼ [P1, P2, P3, P4, P5] are the
                   coordinates of the control points on the boundary of the hole. The uncertainty set can
                   be defined as the perturbation to the nominal value of P, and the worst performance
                   function can be applied in a similar fashion as in equation (4).
                      As mentioned earlier, a hyper-rectangular uncertainty set is chosen. We have the
                   assumption that the worst value of the performance function can be only found at one

                                              Control points

                                         P5                                       P2

Figure 3.
Boundary                                                                      B-Spline
parameterization                         P4                              P3
of the vertices of the uncertainty set (Steiner et al., 2004), and the worst vertex can be      Topology
predicted as:                                                                                optimization
                                 0                                   1
                                   signðfðx1þ Þ 2 fðx12 ÞÞ   · D1
                                 B            .                .    .C
                                 B            .                .    . C:
                  xpred   ¼ x0 þ B            .                .    .C               ð14Þ
                                 @                                    A                            1835
                                   signðfðxxþ Þ 2 fðxx2 ÞÞ   · Dx

Hence the worst performance is approximated by the function value at the worst vertex

                                     max fðjÞ < fðxpred Þ:                           ð15Þ

This approximation requires only k ¼ 2n þ 1 times function evaluations, where n is
the number of the design variables. It has seriously reduced the computational cost of
the robust design optimization compared to that required by statistical analysis.

D. Robust topology optimization
A simple algorithm for RTO based on topological shape optimization is described as
      Set the iteration number k ¼ 0.
      Calculate the robust topological gradient TGR at the center of each element.
      Define the new domain Vk where the topology changes take place by removing
      the material in the elements where TGR is greater than zero.
      Apply standard shape optimization method with a robust objective function to
      determine the shape of the boundary.
      Check convergence and exit if the optimality condition is satisfied.
      Set k ¼ k þ 1.

Uncertainties in both the topology and shape are being handled throughout the entire
design process.

E. Optimization of the robust objective function
Figure 4 shows a sample robust objective function determined by the worst case of the
objective value. The robust objective function is continuous if the original cost function
is continuous. However, the derivative of the robust objective function may not always
exist. As can be shown in Figure 4, the graph of the objective function forms a sharp
turn and its derivative changes sign. However, this may not be the case for a realistic
problem since the objective functions are usually highly non-linear and are never that
simple. In most of the cases, it is possible to use the derivatives at the predicted worst
vertex to approximate the gradient of the robust objective function and to employ some
special treatments at certain points. This should work for practical problems although
a rigorous mathematical proof would be complicated.
30,6                                               ∆ ∆
                                                                                       Robust objective
                                                                                       Non differentiable
                                  objective function
Figure 4.                            Continuous
A robust objective                  Differentiable

                     IV. RBTO vs RTO
                     The design formulations that take account of the uncertainties can be divided into two
                     major categories: the reliability-based approaches and the robust design methods.
                     The former focuses on the product safety with respect to probabilistic events while the
                     latter tends to maintain the performance level against the environmental variations.
                         A RBTO for an electromagnetic problem is formulated as:

                                                min fðxÞ
                                                s:t: Pf ðUÞ ¼ P½Gðxi ; Uj Þ , 0Š # Pt
                                                0 # xi # 1                                                  ð16Þ
                                                i ¼ 1. . .n and j ¼ 1; 2; 3
                                                U ¼ ½U1 ; U2 ; U3 Š ¼ ½m; Js ; Hc Š:
                     Where the design variables x ¼ [x1, x2, . . . , xn] are the density functions in each finite
                     element as opposed to x as dimensions in the previous section; G ¼ C-Cmin, C is the
                     magnetic energy; Pf is the probability of failure and Pt is a preset limit; and U is
                     the vector of the stochastic parameters. If G is greater than zero, the system fails,
                     therefore the probabilistic constraint must be satisfied at all times.
                        Although, RBTO is an effective tool for managing uncertainties in the topology
                     design, it is essentially different to the RTO based on robust design. In RBTO, the
                     variance of the performance is not being minimized during the design. Also the design
                     variables are the density functions in the cells, therefore only variations of the external
                     stochastic parameters, such as the permeability of the material, the source current
                     density and the coercive force, are reflected in the formulations. On the other hand,
                     RTO improves the quality of the product by minimizing the effect caused by the
                     perturbation of the design itself. From a statistical point of view, the RTO concentrates
                     more on the probabilistic distribution near the mean value rather than the behavior of
                     the function at the extreme events.

                     V. Numerical example
                     The robust design formulation is tested with a typical topology optimization problem:
                     eddy current imaging for flaws buried under the surface of a conductor. A simple 2D
model is used instead of the full 3D simulation for fast field calculation. Figure 5 shows              Topology
an ECT imaging system consisting of five coils and an aluminum block containing                      optimization
buried flaws. The thickness of the block is 10 mm. An excitation frequency of 60 Hz is
used to guarantee a deep enough penetration of the induced eddy currents.
   When one of the coils is excited with a current, the induced voltages of all five coils
are measured. The ECT physical problem is simulated using (MagNet User’s Manual
2011) and the fields are solved using finite-element method. The target signals are                            1837
obtained from a simulation model with a preset crack. And the goal of the RTO is to
reconstruct the flaw by matching the target signal with reconstructed signal. Thus, the
objective function F is formulated as minimizing the squared errors between the two
                                     5   5
                            F¼              Vij 2 Vij       ;                       ð17Þ
                                  2 i¼1 j¼1
where Vij denotes the voltage of coil i induced by coil j.
   The topological gradient for the eddy current problem is given as (Li and Lowther,
                                     3s1 ðs1 2 s2 Þ   È      É
                            TG ¼                    Re E · EA :                        ð18Þ
                                       2s1 þ s2
where s1 is the electrical conductivity of the material and s2 is the electrical
conductivity of air, EA is the electrical field computed from an adjoint problem.
    The robust topological gradient given in equation (13) is computed in order to
estimate the location of the potential crack in the problem domain and to provide an
initial crack for the robust shape optimization which in turn optimizes the shape of the
crack. The crack reconstruction is an inverse problem; therefore, a unique solution is
sought for. In the numerical tests, the value of the weighting factor as well as the size of
the uncertainty set are determined using the knowledge of cracks in order to find the
most robust solution in the presence of noises. The results from both non-robust and
robust formulations are compared.

A. Single-crack reconstruction in the presence of noise
The reconstruction results of a single crack are shown in Figure 6. The target crack is
indicated with solid lines and the reconstructed cracks are revealed with dashed lines.
A uniformly distributed random noise about 1 percent of the magnitude of the signal
was added to the target voltages.

                                    Coil 1   Coil 2   Coil 3   Coil 4   Coil 5

                 Coil 1   Coil 2    Coil 3   Coil 4   Coil 5

                   Aluminum block
                                              Buried flaw
                                                                                                            Figure 5.
                                                                                               An eddy current imaging
                                                                                                          system in 2D

1838                                                (a) Crack reconstruction by non-robust function

Figure 6.
Reconstruction of a single
                                                      (b) Crack reconstruction by robust function

                             The single crack was successfully reconstructed by both the non-robust and robust
                             formulations. Since there is only one flaw, the initial topology estimated by the
                             topological gradient is not sensitive to variations and the noise in the signal.

                             B. Reconstruction of two cracks close together
                             The second test uses a test specimen containing two cracks close together, shown in
                             Figure 7.
                                Because the two cracks are scanned simultaneously within the detection range of
                             the coils, the response voltage signals from the different cracks can interact with each
                             other. As a matter of fact, the signals generated by a second crack can be treated as
                             noise in the signals from the first crack.
                                Figure 8 shows the contour plot of both non-robust and robust topological gradient
                                As can be shown in Figure 8(b), there are two peaks in the values of the robust TG
                             which correspond to the location of the two cracks in the test plate. While in Figure 8(a),
                             the non-robust TG plot only showed one significant peak, which implies only one crack
                             may exist in the test plate.
                                The reconstruction results of the two cracks in Figure 7, are shown in Figure 9 with
                             the target crack shape displayed by solid lines and the reconstructed crack shapes
                             indicated by dashed lines.
                                The reconstruction result from Figure 9(a) shows that the non-robust formulation is
                             unable to differentiate the two cracks. The reconstructed shape of the shallow and wide
                             crack produces the closest matching signals for the target, based on the initial topology
                             generated by the non-robust topological gradient. On the other hand, the RTO method

Figure 7.
Two cracks at a close
  0                                                                                            Topology
 –2                                                                                         optimization
  –20       –15       –10        –5          0         5           10     15        20
                               (a) Non-robust TG contour plot                                         1839
 –6                                                                                                 Figure 8.
 –7                                                                                        Topological gradient
  –20       –15       –10        –5          0          5          10     15        20
                                                                                                 contour plots
                                 (b) Robust TG contour plot

                    (a) Cracks reconstruction by non-robust formulation

                                                                                                     Figure 9.
                                                                                                 of two cracks
                      (b) Cracks reconstruction by robust formulation

demonstrates the capability to distinguish the two cracks in the present of significant
noise. The estimated initial topology is robust and it leads to a successful
reconstruction of the shapes of the two cracks.

VI. Conclusion
This paper discussed the issues of managing uncertainties in topological design
optimization. A robustness measure of a topology was defined using the worst
response due to the variations of the design variables. The robustness of the design can
be evaluated during the entire design process of a topological shape optimization
where the topology and shape of the object are being optimized simultaneously.
   The robust objective function was formulated based on the worst case analysis over
an uncertainty set. The worst performance caused by the variations of the design
parameters can be predicted using the topological gradient without the evaluation of
the cost function. Therefore, this method has a huge computational advantage.
COMPEL      The proposed RTO was compared to the RBTO. RTO does not rely on the
30,6     probabilistic information and it has the capability of handling the uncertainties caused
         by the design variations as well as external stochastic parameters.
            Numerical applications of RTO to eddy current crack reconstruction problems
         showed that this method can provide reliable results in the presence of significant
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About the authors                                                                                      Topology
Min Li received a BEng in Computer Science from Shanghai Jiaotong University, Shanghai,
China, in 1999 and an MEng in Electrical Engineering from McGill University, Montreal, Canada,      optimization
in 2007. He is currently a PhD candidate in the Department of Electrical and Computer
Engineering at McGill University.
    David A. Lowther received his BSc (Eng.) Hons from King’s College, London in 1970 and his
PhD from Brighton Polytechnic in 1973. He currently holds a James McGill Chair in Electrical
Engineering at McGill University in Canada. His research interests include computational                  1841
electromagnetics, intelligent design systems, optimization and the design of electrical machines.
David A. Lowther is the corresponding author and can be contacted at: david.lowther@mcgill.ca

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