Shape optimization methods
A same academic case is selected in order to compare the different methods implemented to optimize the heat transfer by conduction through solid structures . It relates to the efficient cooling of a finite-size volume generating heat, by means of a small quantity of material characterised by a high thermal conductivity. This case belongs to the generic class of Volume-to-Points (VP) problems, which are typified by the research of an optimal topology between two subdomains with dissimilar conductivities. The optimization problem linked to the physical problem is infinite-dimensional and is particularly uneasy to solve: therefore, it makes an interesting case study to benchmark the different optimization solvers.
The figure below shows a possible formulation of the VP problem in the frame of the steady state energy equation. The Ω0 domain is characterised by a heat production rate q0 and a thermal conductivity k0. The Ωp domain has a thermal conductivity kp greatly higher than k0. The boundary conditions are all adiabatic, with the exception of a small heat sink located in the middle of one of the edges, which is at a constant temperature Ts. The optimization problem is the following: what is the optimal layout of Ωp domain, in terms of shape and connexity, which minimizes one or several thermal objectives? These goals can take various shapes such as minimizing the temperature for a specific point for example, or minimizing the temperature average or variance of the whole domain Ω0 ∪ Ωp, …
Constructal theory is a philosophy of design which has been stated by A. Bejan in 1996 as follows :
for a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it. Another definition is possible in opposition to the fractal approach that describes the multi-scale systems whereas constructal theory aims explaining their formation and underlines the laws between the different scales. In particular, the constructal approach lies on an indivisible elemental volume, justified by physical or industrial reasons, which are assembled from the smallest scales to the largest ones. The animation below shows the assembly of the network made of high conductivity material kp for 6 different levels established thanks to constructal laws. The starting point is the scale 1 that is characterized by the distribution of several elemental volumes around a high conductivity axis.
With the help of J.-L. Harion, M. Nemer, S. Russeil et D. Bougeard and relying on the researches of A. Bejan, L. Kuddusi, N. Egrican et J. Denton [1-3], we have underlined three specific points regarding the application of constructal theory to the VP problems [4-6]:
- We developed an analytical solution from the slenderness hypothesis introduced by A. Bejan and based on the same physical assumptions that the ones used by L. Kuddusi and J. Denton. Thus, we contributed to the unification of two approaches introduced as opposite in the scientific literature.
- We also proposed an analytical formulation of the calculation error introduced by evaluating the area of Ωp domain: this value is one of the main optimization parameter for the constructal process.
- Finally, we examined the influence of the multi-scale system complexity on the thermal performances. In order to compare structures having different sizes, since these are a function of the constructal level (and consequently a function of the complexity), we proposed an algorithm able to reverse the constructal process and to give limits to the multi-scale structures inside a finite-size volume with fixed dimensions.
These points allowed us identifying the limits of constructal theory when it is applied to VP problems, on the basis of the formulation described in scientific literature. Moreover, our study underlines than a pure analytical approach lacks of flexibility, mainly regarding the configuration of structures, and engaged us to look more closely at other numerical methods.
In order to increase the degree of freedom of the high conductivity domain, we have been interested in the algorithms based on cellular automata implemented by Boichot et al . The main idea of this method is to fractionate the calculation field into a finite number of elements, called cellular automata, that are characterized by their own behaviour, in the same way as biological cells. Each automaton has the ability to switch its thermal conductivity from one domain to another: this swapping is carried out iteratively on the basis of their individual thermal stress, evaluated thanks to the heat flow going through them. During the convergence process, the number of cellular automata subject to this swapping is decreased, freezing gradually the structure in its final shape. The video below shows this convergence process from an initial random kp field.
Because of several numerical choices, the initial algorithm introduced by R. Boichot et al.  was subject to the creation of checkerboards: those are characterized by non-realistic structures where k0 and kp thermal conductivities are successively alternated following a staggered scheme. This observation leads us to propose a few modifications in order to solve this problem, mainly by separating the cellular automata and the control volumes used to discretize the energy equation. However, even though the emergence of global structures on the basis of a single local criterion is remarkable, the method itself does not allow the structures to evolve in order to satisfy a specific objective. In other words, the evolution of physical parameters can only be noticed during the convergence process, without that the algorithm gives priority to a configuration over another one in order to optimize a particular physical parameter.
Evolutionary Structural Optimization (ESO)
By keeping the same parameterization as the one introduced for the cellular automata, we implemented a new strategy based on the work of Q. Li et al., called Evolutionary Structural Optimization (ESO) [8,9]. We proposed to reverse this algorithm in order to model the growth of the high conductivity material inside the domain generating heat . This growth of kp material explicitly aims to minimize or maximize an objective function, also called cost function. Its main role is to identify the best position to extend the Ωp domain along its own the boundary in the course of an iterative process, overcoming the limitations of the cellular automaton method underlined in the previous section. This process is similar to a descent method based on a topological gradient evaluated for each iteration: its difficulty mainly lies on the sensitivity analysis step that aims evaluating this gradient.
The use of an objective function guiding the construction process allows reaching different structures, meeting different requirements. In other words, in contrast with the two previous methods, ESO algorithm does control the optimal topology of Ωp domain regarding the optimized objective function. However, the method presupposes the solution connexity and does not have the ability to modify its configuration in other ways than adding kp material, due to the growth assumption. Moreover, the idea of growing the solution does not seem suitable to configurations other than the ones related to VP problems. These statements lead us to consider new methods that do not have the restrictions related to the connexity hypothesis, but that aim to explicitly minimize or maximize an objective function.
Solid Isotropic Material with Penalization (SIMP)
The last method we studied lies on the works of A. Gersborg-Hansen et al.  and takes inspiration from an approach by penalization, or homogenization, initially implemented in the frame of structural optimization in solid mechanics . The originality of this method is based on the transformation of the initial discrete problem to a continuous one: the local thermal conductivity, which previously had only two possible values k0 and kp, can be continuously adjusted between these two limits from now on. The main advantage of this parameterization is to allow evaluating the sensitivity of the objective function with respect to the design parameters, since they are continuous as well. This evaluation is carried out by means of a discrete adjoint solver, which has the same order of magnitude as the direct solver. The objective function and sensitivity values are used as inputs for a gradient-based numerical optimization method, the so-called Method of Moving Asymptotes (MMA): it approximates the whole problem thanks to a local convex subproblem that is solved with the interior-point method . By gradually penalizing the problem in the course of the iterative process, an optimal structure made of high and low conductivity domains clearly arises, giving form to a discrete solution with two values from an initial homogeneous conductivity field.
The next two videos show the convergence process of the SIMP method in the frame of the minimization of two different objective functions: the average and the variance temperature of Ω0 ∪ Ωp domain. It is worth noting that objectives of different nature lead to different solutions that are topologically non-trivial. However, not any objective functions are approachable with the current algorithm implementation, they are required to be at least C1: for the time being, this particularity excludes tackling problematic such as min max problems.
In addition to the SIMP method, we developed a multi-objective approach allowing solving optimization problems with different objectives . This is based on an aggregated objective function built from a linear combination of the different objective functions, involving they have been previously rescaled in order to have the same order of magnitude. Within this context, we established that solutions minimizing both average and variance temperature have interesting structural trade-offs, as shown on the figures below. Moreover, these results point out the existence of a slip for the Pareto front, which is a function of the quantity of high conductivity material kp. Consequently, for a given average/variance minimization objective, an optimal high conductivity structure associated with a minimal Ωp area exists.
The performances of the four optimization methods introduced above can be compared on condition that their input parameters and their objective functions are similar. Indeed, it is worth noting that the structures coming from constructal theory and cellular automaton are built thanks to an implicit optimization criterion, whereas the ESO and SIMP methods explicitly use an objective function to drive their optimization process. Therefore, the predominance of one method over the others is difficult to establish, since the objectives that they pursue are not necessarily similar.
The comparison of the performances, quantified by the thermal resistance and the average temperature of the solutions, shows that the SIMP method outdistances the three other approaches, for both criteria. Even though this method is difficult to implement, numerically speaking, it has the ability of explicitly tackling the multi-objective optimization problems in various configurations. To be exhaustive, the comparison made here should include the well-known level set method, such as presented in  and which is currently under investigation.
By quantitatively and qualitatively comparing the results, the capacity of the approaches to address different optimization problems, as well as their algorithmic complexity, the SIMP method appears to have several advantages. It allows tackling a large variety of geometrical configurations, as well as solving one or more optimization problems if these ones have been previously stated in mathematical terms. Consequently, I oriented my researches towards this specific method for other shape optimization problems, always in the frame of the intensification of heat transfers.
|||Constructal-theory network of conducting paths for cooling a heat generating volume. International Journal of Heat and Mass Transfer, 40(4):799–811, 1997.|
|||Exact solution for cooling of electronics using constructal theory. Journal of Applied Physics, 93(8):4922–4929, 2003.|
|||Analytical solution for heat conduction problem in composite slab and its implementation in constructal solution for cooling of electronics. Energy Conversion and Management, 48(4):1089–1105, 2007.|
|||A new perspective of constructal networks cooling a finite-size volume generating heat. Energy Conversion and Management, 52(2):1033–1046, 2011.|
|||Refroidissement d’un volume fini générant de la chaleur : analyse du processus constructal. Acte de congrès de la Société Française de Thermique, No. 154, Le Touquet (France), 2010.|
|||Refroidissement d’un volume fini générant de la chaleur : limites de l’approche constructale. Acte de congrès de la Société Française de Thermique, No. 155, Le Touquet (France), 2010.|
|||Tree-network structure generation for heat conduction by cellular automaton. Energy Conversion and Management, 50(2):376–386, 2009.|
|||Evolutionary topology optimization for temperature reduction of heat conducting fields. International Journal of Heat and Mass Transfer, 47(23):5071–5083, 2004.|
|||Shape and topology design for heat conduction by Evolutionary Structural Optimization. International Journal of Heat and Mass Transfer, 42(17):3361–3371, 1999.|
|||Evolutionary structural optimization by extension to cool a finite-size volume generating heat. 7th International Conference on Computational Heat and Mass Transfer, No. 152, Istanbul (Turkey), 2011.|
|||Topology optimization of heat conduction problems using the finite volume method. Structural and multidisciplinary optimization, 31(4):251–259, 2006.|
|||Topology Optimization: Theory, Methods and Applications. Springer, 2nd edition, 2003.|
|||The Method of Moving Asymptotes – A new method for structural optimization. International Journal for Numerical Methods in Engineering, 24:359–373, 1987.|
|||Topology optimization using the SIMP method for multiobjective conductive problems. Numerical Heat Transfer, Part B: Fundamentals, 61(6):439–470, 2012.|
|||Topology optimization of multi-material for the heat conduction problem based on the level set method. Engineering Optimization, 42(9):811–831, 2010.|