Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models
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Abstract
In the paper we consider a new method based on the genetic algorithm for finding the location and size of small holes in the domain, in which the coupled linear and nonlinear boundary value problems are defined. The linear and nonlinear components are connected by the transmission conditions on the interface boundary. The expansion with respect to small parameter of the shape functional for nonlinear component and the expansion of SteklovPoincaré operator for linear component are derived in order to determine the form of topological derivative for shape functional defined for coupled model. Then the topological derivative is employed for evaluation of the probability density applied to generation of location of holes by the genetic algorithm.
Keywords
Shape optimisation Topological derivative Genetic algorithm SteklovPoincaré operator1 Introduction
We consider a coupled model described by the domain bounded in ℝ^{2} and decomposed into two subdomains Ω and ω in such way that the interior subdomain ω is surrounded by the exterior subdomain Ω. In the interior subdomain the physical phenomenon are described by the linear partial differential equation (PDE), and in the exterior domain the processes are governed by nonlinear PDEs subject to some external function. An example of such a system constitutes a gravity flow around an elastic obstacle. Here the NavierStokes equation, which is nonlinear, is coupled through transmission conditions with the linear static or dynamic elasticity system. Such situations have numerous physical interpretations. Let us mention here only two : the water flow around submarine or gas flow inside the jet engine. For real life models the coupling conditions are still a subject of research (Ignatowa et al. 2013).
In the paper we assume that the number and the radii of hollow voids are given, and we determine only the locations.
The topological derivative, introduced in Sokolowski and Zochowski (1999), and developed in the papers Sokolowski and żochowski 2003, 2005, 2008, was created specifically in order to asses the influence of voids or inclusions on the solutions of PDE’s. The method of topology optimization based on the asymptotic analysis works very well for the solutions strongly sensitive on the appearance of holes, otherwise the information obtained, while analytically correct, is more fuzzy, especially if contaminated by numerical errors. What’s more, particularly in case of multiple voids, there may occur several configurations constituting local minima for the associated topological optimization problem. Of course, in case of general domain optimisation from scratch, the approach based on topological derivative is not the only one, see Allaire (2002) and Bendsoe and Sigmund (2003).
Our goal in this paper consists in proposing the combination of genetic algorithm and information given by the topological derivative for solving such difficult problems or at least providing the initial approximation of the solutions. The idea of combining heuristic and strict methods for modelling physical structures was recently used for example in Patelli and Schuëller (2012) but the way we use information contained in the topological derivative is new. Namely, we construct on its basis the probability density used in generation of initial populations and supplying additional elements as needed.
Of course, in order to use the topological derivative it was necessary to compute its value. Therefore the essential part of the paper is devoted to obtaining the appropriate analytic formulas for the given nonlinear problem. We employ the domain decomposition technique in order to split the coupled model into separate parts. The interaction of these parts in both domains is modelled by the appropriate SteklovPoincaré non–local boundary operator, which can be defined for the linear part and includes the information about the design in its interior. It means that we can apply the asymptotic analysis for decoupled elliptic boundary value problems and determine topological derivatives for shape functionals of interest.
The SteklovPoincaré operators appear in boundary conditions of the fluid or gas (non–linear) models in the second subdomain. This transmission of the behaviour of the linear part to the non–linear part allows us to apply the tools of shape and topology design to the coupled models. For example, if the non–linear part represents the compressible NavierStokes equation, we may use recent results concerning its shape optimization, see Plotnikov and Sokolowski (2008).

the algorithm using probability density obtained from topological derivative,

the same algorithm using uniform density, that is lacking any knowledge about the physics of the problem,
Let us stress again that the approach proposed here applies to situation, when the topological derivative does not give clear answers (as happens quite often) but nevertheless may help in the search for them.
2 Problem formulation
3 Expansion of the SteklovPoincaré operator
4 Topological derivative
Theorem 1
Proof
5 Numerical investigations
The shape optimization problem considered in this section consists in finding locations and sizes of a finite number of ballshaped holes in the domain which minimize a certain integral functional. The standard approach would use the values of topological derivative for initial location of these holes, and then shape derivative would be applied for fine tuning their sizes and positions (Fulmanski et al. 2007; Iguernane et al. 2009). Such a method is fast, but there is a danger of landing in a local optimum, especially when the number of holes is bigger than one.

one hole of fixed size;

two holes of fixed sizes;

three holes of fixed sizes;
The approach exploiting the information supplied by topological derivative is compared to the method using exactly the same genetic algorithm, but with uniform distribution of holes locations. The performance is measured in both cases statistically over several runs.
5.1 Solution of boundary value problems
5.2 Optimization problem
In our numerical experiments we use the tracking type shape functional with a known element z _{ d }, so an optimal value is 0.
All numerical computations were performed using Matlab and its PDE toolbox. In particular, the triangulation of intact and perforated domains was done using the builtin procedures. The boundary value problems were solved using linear finite elements.
For the sake of obtaining the topological derivative we are interested only in values of functions in the subdomain ω. We assume that ω is divided into M triangles represented by the matrix T = [t _{ i j }], i = 1, …, M, j = 1, 2, 3, where t _{ i∗} denotes labels of points constituting vertices of the ith triangle. We assume also that vertices in ω have labels k = 1, …, N.
Using these values the probability density is constructed in the following way. First we approximate, using the standard procedure, the function D by the piecewise linear function represented by the vector of nodal values \(\tilde {D}=[\tilde {d}_{k}], \;k=1, \dots , N\). Then let \(\tilde {D}_{max}=\max \limits _{k=1, \dots , N} \tilde {d}_{k}\) and \(\tilde {s}_{k}=\tilde {D}_{max}\tilde {d}_{k}\). Thus the function represented by nodal values \(\tilde {S}=[\tilde {s}_{k}], \;k=1, \dots , N\) is the biggest at nodes where \(\tilde {D}\) has minimum and vanishes at nodes where \(\tilde {D}\) achieves maximum.
5.3 Genetic algorithm
Initial population
The initial population consists of circular holes represented by the coordinates of their centres (the radius of the hole is fixed and is the same for each hole). Thus, each element of population represents a point in a space for which the dimension is determined by the number of coordinates of the holes centres. In case of one hole, the element of the population constitutes a vector [x _{0}, y _{0}] of two coordinates of the center of the circular hole. In case of two holes, the individual is given by a vector of four coordinates defining centres [x _{1}, y _{1}, x _{2}, y _{2}] of two holes. For three holes, the individuals are represented by vectors of six coordinates [x _{1}, y _{1}, x _{2}, y _{2}, x _{3}, y _{3}]. The initial population is randomly initialized with a constant number \(\mathcal {S}\) of individuals. Since we compare two versions of algorithms, two kinds of initializations are used. In the first case, we apply the probability density function defined in (41). In the second case, when we do not use the information contained in topological derivative, the probability density is uniform on ω, which corresponds to d _{ i } = const in (40).
Since the values of probability density function are defined only in the nodes of the discretized domain, we need the following procedure to draw a random point, not necessarily the node, in the domain.

first we select the triangle according to its average probability;

next we select the point inside this triangle taking into account the values of probability density in the corners.
In case of two holes, selection of the initial population consists in drawing the pairs of holes. A pair of holes is retained only if the distance between their centres is greater that the fixed value 2ε, where ε is the radius of the hole. Similar condition is used for triples. If the condition is not verified, the pair or triple is eliminated and the drawing continues until the required number or elements is reached.
Evaluation
The fitness value of every individual is evaluated using the cost functional 𝓙 such that the minimization of 𝓙 is equivalent to the maximization of the fitness value. Since the population is considered as a vector of individuals sorted according to their fitness values, the lower value of the cost functional means the higher position in the population vector.
Crossover
Mutation
New generation
In the next step the population consisting of old individuals and those produced in the crossover stage is pruned by removing the elements violating the constraints (distance between centres greater then 2ε in case of two or three holes) and sorted according to fitness value.
The next generation contains \(\frac {2}{3}S\) of the best elements. In order to prevent locking in local optima, \(\frac {1}{3}S\) individuals are again drawn randomly using appropriate probabilities, i.e. the one based on topological derivative or uniform on ω.
5.4 Numerical results
In the third example we consider a case with three holes. As a target domain we take a square with three circular holes located at points (−0.1, 0.2, ), (0.2, −0.1) and (−0.2, −0.2) inside the interior subdomain ω. The goal functional consists of the sum of two parts with different reference functions z _{ d1}, z _{ d2} corresponding to right–hand sides f _{1} = x _{1} + x _{2} and f _{2} = x _{1} − x _{2}. As a result, the topological derivative was also computed twice, and the value of the final 𝓣 was the sum of these two parts.
6 Conclusions
The problems considered in the paper, especially those including several holes, are quite difficult because the voids which we want to localize are screened from the observation. They are contained in ω, while the goal functional is computed on Ω. As a result, the goal functionals are not very sensitive to the changes of the configuration and this impedes the use of algorithms based on gradients (shape derivatives).
The genetic algorithms do not use derivatives and are suitable for such situation. What’s more, as shown in the paper, they can utilize even imprecise information through modifying appropriate probabilities. The idea to use topological derivative in this context seems to be new.
References
 Allaire G (2002) Shape optimization by the homogenization method. Springer, New YorkCrossRefzbMATHGoogle Scholar
 Bendsoe MP, Sigmund O (2003) Topology optimization. Theory, methods and applications. Springer, New YorkGoogle Scholar
 Fulmanski P, Lauraine A, Scheid JF, Sokołowski J (2007) A level set method in shape and topology optimization for variational inequalities. Int J Appl Math Comput Sci 17(3):413– 430CrossRefzbMATHMathSciNetGoogle Scholar
 Garreau S, Guillaume Ph, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39(6):1756–1778CrossRefzbMATHMathSciNetGoogle Scholar
 Giusti SM, Novotny AA, Padra C (2008) Topological sensitivity analysis of inclusion in twodimensional linear elasticity. Eng Anal Bound Elem 32(11):926–935CrossRefzbMATHGoogle Scholar
 Ignatowa M, Kukavica I, Lasiecka I, Tuffaha A (2013) On wellposedness for a free boundary fluidstructure model. J Math Phys 53(11):115624, 13. doi: 10.1063/1.4766724
 Iguernane M, Nazarov SA, Roche JR, Sokolowski J, Szulc K (2009) Topological derivatives for semilinear elliptic equations. Int J Appl Math Comput Sci 19(2):191–205CrossRefzbMATHMathSciNetGoogle Scholar
 Kowalewski A, Lasiecka I, Sokolowski J (2010) Sensitivity analysis of hyperbolic optimal control problems. Comput Optim Appl 52:147–179Google Scholar
 Mazja VG, Nazarov SA, Plomenevskii BA (2000) Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, vol 1. Birkhäuser, BaselGoogle Scholar
 Michalewicz Z (1996) Genetic Algorithms + Data Structures = Evolution Programs. SpringerVerlag, BerlinCrossRefzbMATHGoogle Scholar
 Nazarov SA (1999) Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. Am Math Soc Trans 198(2):77–125Google Scholar
 Nazarov SA, Sokołowski J (2003) Asymptotic analysis of shape functionals. J Math Appl 82:125–196zbMATHGoogle Scholar
 Patelli E, Schuëller GI (2012) Computational optimization strategies for the simulation of random media and components. Comput Optim Appl 53(3):903–931CrossRefzbMATHMathSciNetGoogle Scholar
 Plotnikov PI, Sokolowski J (2008) Inhomogeneous boundary value problems for compressible NavierStokes equations, wellposedness and sensitivity analysis. SIAM J Math Anal 40:1152–1200CrossRefzbMATHMathSciNetGoogle Scholar
 Sokołowski J, żochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272CrossRefzbMATHMathSciNetGoogle Scholar
 Sokołowski J, żochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42(4):1198–1221CrossRefzbMATHMathSciNetGoogle Scholar
 Sokołowski J, żochowski A (2005) Modelling of topological derivatives for contact problems. Numerische Mathematik 102(1):145–179CrossRefzbMATHMathSciNetGoogle Scholar
 Sokołowski J, żochowski A (2008) Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions. Eng Anal Bound Elem 32:533–544CrossRefGoogle Scholar
 Sokołowski J, Zolésio JP (1992) Introduction to shape optimization. Shape sensitivity analysis. SpringerVerlag, New YorkCrossRefzbMATHGoogle Scholar