Solid Isotropic Material with Penalization

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Comparison of constructal designs

The comparison between the diﬀerent constructal shapes meets an important issue: in function of the constructal level, the cooling tree-network does not cover the same area, as shown in Fig. 2.3. This obstacle makes diﬃcult to quantify the influence of the network complexity on the cooling eﬃciency. Bejan and Dan [107] have proposed to build an “inward” approach in opposition to the “forward” one introduced as so far, in order to constrain the constructal dendritic network to any given rect-angular shape of heat generating volume. In other words, the final area is known before beginning the constructal process and the features of the kp network are adjusted to reach an optimal con-figuration. However, this method slightly loses the philosophy of constructal theory, which assumes that the features of the final geometry are fully included into the elemental volume. Intrinsically, the “inward” approach moves away from the optimal form by aiming to fit to an external structure. This section aims to evade this problem proposing a method to create a tree-network inside a finite-size volume knowing its size, without overconstraining the constructal approach. Thereby, it is necessary to define a final shape which naturally exists in the “forward” method, meaning a shape which is involved in Fig. 2.3. Furthermore, a numerical method is introduced to solve the temperature field, which allows:

• having a reliable comparison tool which remains the same for all the tested structures;

• liberating the calculation from physical assumptions, such as the slenderness hypothesis, and obtaining the exact value of the thermal resistance;

• evaluating quickly and easily constructal tree-networks as complex as wished. This approach should allow making comparisons in order to evaluate the exact eﬃciency of constructal designs.

Presentation of the problem

The final shape of the volume depends on which constructal approach is tested. Regarding the method introduced by Bejan [83], two kinds of geometry are alternatively defined when increasing the number of constructal levels: a rectangle with an aspect-ratio of two or a square (see the left-size of Fig. 2.3). For Ghodoossi and Eğrican [102], the final volume-size is always rectangular with an √ aspect-ratio of 2 (see the right-size of Fig. 2.3). These two approaches are investigated for the even constructal levels in order to evaluate a wider range of structure. Numerical values must be fixed to apply a numerical scheme allowing calculating the temperature field. Arbitrarily, the edge size where the heat sink is located has been fixed to a = 100mm, whatever the constructal method tested. The conductivity ratio is k = 500, with kp = 500W (mK ) and k0 = 1W (mK ), and the heat generation rate is set to 10kWm 3 through k0 and kp material. Moreover, the final ratio of kp material is a result of the constructal process: as a consequence, the structures which develop a cooling network with a final porosity above 40% are excluded from this study, considering that they are not coherent from an engineering point of view. The geometries tested lie on another fundamental assumption: unlike constructal theory, it is assumed there is no technological limit concerning the elemental volume size. In other words, the constructal process can start oﬀ from any elemental volume, as small as wished. This feature allows considering cooling tree-networks which get more complex.

Equivalence between φ and n

If the fraction of kp material for the elemental volume can take any value, it is possible to link φ0 and n1 between them. In this instance, the number n1 of elemental volumes included as a part of the first construct becomes an essential parameter of the problem. This methods allows starting constructal designs by setting the geometrical parameters of the cooling network, instead of constructing the structure from the elemental volume. Eq. (2.4) introduces the optimal ratio for the elemental volume in constructal theory. The corre-spondence between the elemental volume and the first construct is based on a central assumption: n1 must be an even integer to pretend to be made up of a sum of optimized smaller volumes. Indeed, if n1 does not satisfy this requirement, it is not possible to build a first construct having the right shape. That leads to the following equation:

L1=n1(2.39) H02,Substituting Eq. (2.4) into Eq. (2.39): n1 = L1 ˆ 1 2 kφ0(2.40) L0 At this step, both constructal methods can be tackled with this approach, because the optimal features of the first construct are known in function of the ones of the elemental volume. The link between both levels is made thanks to the ratio L1 L 0: for [83], this ratio is equal to 1, whereas [102] √ proves it should be equal to 2. In such a case, Eq. (2.40) becomes respectively: n2 φ0 = ˆ1 (2.41) k n2 φ0 = 1 (2.42)ˆ 2k It is also possible to establish the width of the elemental high- conductivity path in function of n1. By definition, D0 is equal to: D0 = φ0H0Substituting Eq. (2.4) and Eq. (2.41), or Eq. (2.42), into Eq. (2.43): n1 D0=2 L0 kˆ √n1L0 D0=2 In the same way, the optimal ratio D1 D ˆ 1 2 in both cases, can be reduced 0 , which is equal to (kφ0) √, thanks to Eq. (2.41) and Eq. (2.42). It is worth to note that the width to n1, respectively n12 of the central link draining the heat is the summation of the widths of the kp links belonging to the elemental volumes for a first order construct coming from [83]. This section aims to show it is possible to substitute φ0 by n1 in a classical “forward” approach. In a standard case, φ0, k and L0 are the set of parameters allowing solving the constructal problem With this new approach, n1, k and L0 are suﬃcient to develop a constructal method.

Recurrence relations of the constructal proble

Both methods introduced in [83] and [102] have recurrence relations allowing knowing the ge-ometrical features of constructal networks. In order to constrain the constructal design inside the square area generating heat, the length L0 of the elemental volume, as well as the width D0 of the elemental conductive path, must be expressed in function of the construct order wished, called ℓ and the other geometrical parameters such as a, k and n1.

Recurrence relation of the square constructal problem

In the frame of the original constructal method and if the final assembly level is even, the shape to optimize is a square structure. In this case, L0 is expressible as: L0 = a (2.46) 2ℓ 2 The initial width of the kp link can be rewritten, thanks to Eq. (2.44) and Eq. (2.46), as: D0 = a n1 (2.47) ℓ−2ˆ 2k 2 Bejan showed in Ref. [83] that multipliers exist for describing the geometrical features of a given level in function of the previous construct order. These results, combined with this work, allow describing he dimensional parameters of each level by recurrence, only depending on a, n1, k and ℓ. The length Li of the diﬀerent assemblies are given from: L0=2ℓ 2 L1=a L0 L2=L1 for i odd and 3i < ℓ: ≤ Li=Li−1(2.48)≤≤ Li=2Li−1 for i even and 4iℓ: The same reasoning can be led about Di : D0=an1 ℓ−kˆ D1=2 n1D0 4 2=(2.49) D √3 for 3iℓ: ≤≤ Di=2Di−1 Once these parameters have been evaluated, an algorithm able to build the constructal tree-network as complex as wished is required. This algorithm takes the same input parameters as Eq. (2.48) and Eq. (2.49). This approach is fully detailed in Appendix A.

Recurrence relation of the rectangular constructal problem

The same reasoning can be led with the works of Ghodoossi and Eğrican. However, no distinction is necessary between the odd and even constructal levels, since their approach leads to a rectangular shape with the same aspect-ratio. The only diﬀerence coming from the distinction between the levels is made for the calculation of L0, due to the change of direction in the constructal network, which is expressible as: for ℓ odd:aL0= ℓ+1 for ℓ even: L0=a ℓ 2 √ The initial width of the kp link can be rewritten, thanks to Eq. (2.45) and Eq. (2.50), as: a n1 D0 = 2ℓ 2 kˆ The authors calculated in Ref. [102] the multipliers existing for describing the geometrical features of tree-networks. Combining with this study, they allow describing the geometry by recurrence, also Eq. (2.52) and Eq. (2.53) allow building the constructal designs coming from Ghodoossi and Eğrican. The algorithm designing the high-conductivity tree-network constructs it from the largest scale toward the smallest one, even though it has been calculated in the opposite direction. Fig. 2.8a shows the seven successive required steps to construct a structure including six constructal levels and four elemental volumes inside the first construct order (ℓ = 6 and n1 = 4). Appendix A provides an accurate overview of this algorithm, applying a syntax close to the one used in object-oriented programing.. One of the first observation concerns the finite number of possible structures. In certain cases, if the fraction of kp material is free to vary above 40%, the width of the high-conductivity path belonging to the last level ℓ is wider than the square side a. Under such conditions, an half square is full of the kp material spilling over the boundaries, which cannot be seen as a valid construct. It was found that the square area can only contains a small dozen of constructal designs. In Fig. 2.8b, the constructs are classified in function of their assembly level ℓ and the number of elemental volumes included in the first order construct n1. Note that an optimized elemental volume covering the whole square area has also been drawn and indexed by ℓ = 0. A two-sided description of the tree-network studied is possible:

• At isolevel ℓ, the length L0 of the elemental volumes does not vary in function of n1. On the contrary, the elemental volume becomes more and more slender and the width of the high-conductivity path increases.

• At identical number of elemental volumes included in the first element n1, the length L0 is divided into two between two consecutive even levels ℓ. On the other hand, the aspect-ratio of the elemental volumes remains identical irrespective of ℓ. These remarks are also valid for the constructal networks coming from [102], which are not pictured here because of the lack of space. There are only two minor diﬀerences with Fig. 2.8b: √

• The area of the finite-size volume is divided into 2.

• Fourteen diﬀerent tree-networks are possible, instead of eleven.

Solver and mesh

Numerical simulations have already been used in the frame of constructal theory and its optimiza-tion [99, 100, 108]. These works use computational codes such as FIDAP [99] or MATLAB [108] and take advantage of non-uniform grids to follow reasonably well the temperature gradients. It is worth to note that [108] provides numerical tests showing the achievement of grid independence. These articles have inspired the numerical methods carried out in the present study. A special care has been taken regarding the numerical solver used to solve the problem hereupon defined. The commercial software FLUENT has been used to evaluate the temperature field. A third-order scheme using a Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) reconstruction has been applied to discretize the energy equation on the calculation field. This finite volume method is particularly appropriate to evaluate constructal designs, since it provides highly accurate solution even if the solution exhibit large gradients. Indeed, massive temperature gradients take place inside the constructal design, especially at the interface between kp and k0 materials. Moreover, in order to ensure that the numerical convergence is achieved well, its criterion has been fixed at 10−15 for each calculation. Fig. 2.9 shows the mesh used through kp material and the way to refine it. An unstructured grid, made up of triangular elements, has been defined because it oﬀers more possibilities to adapt it by means of size functions, even though it makes the calculation time longer. Fig. 2.9 also displays two enlargements regarding the way to manage the grid refining for the worst locations. k0 material is meshed in the same way, using the same size functions as the ones used through kp material. The mesh refining is also a fundamental method to describe correctly the temperature gradients. So, the mesh of structures has been refined at the interface between the kp and k0 material. The main issue has been to adapt the mesh size to the constructal constraints: for instance, for a structure having eight assembly levels, the width of the kp link varies between 50µm and 14 78mm. It means that the mesh must adapt its size from a few micrometer up to a few millimeter through the kp material. This constraint leads to structures requiring between one hundred thousand and more than 5 millions of mesh to describe the systems with a good accuracy. Figure 2.9: Grid used through kp material to evaluate the temperature field. Tests have been led to make sure of the absence of correlation between the grid used and the results. As shown in Fig. 2.8b, many diﬀerent constructal structures have to be evaluated with numerical methods and carrying out a test of grid independence for each of them is not possible. As a consequence, the test has only been made for one geometry and the criterion assuring the grid independence has been transposed to the other configurations. The grid construction is mainly built from size functions which are initialized with the mesh size inside kp material. In other words, the cell number across the smallest section of kp material is the main parameter upon which lies the whole grid. The tests show that the grid independence is reached if the cell number across this section > 8. The independence is said to be achieved when the non-dimensionalized thermal resistance diﬀerence between the previous grid tested and the current one < 5 10−3 . So, all the structures have been meshed in such a way that at least 10 finite volumes are included in the width of the smallest kp path.

Table des matières

1 Introduction générale
1.1 Contexte et enjeux
1.2 Introduction aux problématiques de l’optimisation
1.2.1 Paramétrage des structures
1.2.2 Classification des algorithmes
1.2.3 Revue bibliographique
1.3 Méthodologie
1.3.1 Résolution des équations aux dérivées partielles
1.3.2 Optimisation des transferts de masse et de chaleur
1.3.3 Remarques
I Optimisation topologique des échanges de chaleur en conduction
2 Théorie constructale
2.1 Abstract
2.2 Introduction
2.3 A comparative review of two constructal methodologies
2.3.1 Optimization of the elemental volume
2.3.2 Optimization of the first order construct
2.3.3 Optimization of the higher order constructs
2.4 Analytical perspectives
2.4.1 High-conductivity material and heat generation
2.4.2 Fraction of high-conductivity material
2.5 Comparison of constructal designs
2.5.1 Presentation of the problem
2.5.2 Equivalence between φ and n1
2.5.3 Recurrence relations of the constructal problem
2.5.3.1 Recurrence relation of the square constructal problem [83]
2.5.3.2 Recurrence relation of the rectangular constructal problem [102]
2.5.4 Solver and mesh
2.5.5 Results
2.6 Conclusion
2.7 Building algorithm
2.8 Conclusion à propos de l’approche constructale
3 Automates cellulaires
3.1 Evolution structurelle par attraction locale
3.1.1 Introduction
3.1.2 Principes de fonctionnement
3.1.2.1 Algorithme
3.1.2.2 Pseudo-fonctions objectif
3.1.3 Résultats
3.1.3.1 Analyse de la convergence
3.1.3.2 Influence des paramètres numériques et physiques
3.1.4 Limitations
3.1.4.1 Discontinuité du domaine hautement conducteur
3.1.4.2 Dissociation des automates cellulaires et des volumes finis
3.1.5 Comparaison avec la théorie constructale
3.1.6 Conclusion
3.2 Evolutionary Structural Optimization by extension
3.2.1 Introduction
3.2.2 Evolutionary Structural Optimization by extension
3.2.2.1 Problem introduction
3.2.2.2 Setting up algorithm
3.2.3 Results and discussion
3.2.3.1 Shape of cooling networks
3.2.3.2 History of construction
3.2.4 Comparison between ESO and constructal methods
3.2.5 Conclusion
3.3 Conclusion à propos des automates cellulaires
4 Solid Isotropic Material with Penalization
4.1 Abstract
4.2 Introduction
4.3 SIMP presentation
4.3.1 Introduction to SIMP
4.3.1.1 Penalization process
4.3.1.2 Checkerboard problems
4.3.2 Finite Volume Method
4.3.3 Sensitivity analysis
4.3.3.1 Average temperature
4.3.3.2 Variance temperature
4.3.3.3 Additional comments on sensitivity analysis
4.3.4 Sensitivity filter
4.3.5 Method of Moving Asymptotes (MMA)
4.4 Multi-objective optimization
4.5 Results
4.5.1 Convergence process
4.5.2 Influence of numerical parameters
4.5.2.1 Mesh independence analysis
4.5.2.2 Numerical filter analysis
4.5.2.3 Sensitivity filter analysis
4.5.2.4 Influence of heat generation rate
4.5.3 Multi-objective results
4.6 Conclusion
4.7 Remarques complémentaires
4.7.1 Conditions limites du filtre numérique
4.7.2 Influence de la porosité φ
4.7.3 Conditions limites thermiques complémentaires
4.8 Comparaison entre les différentes approches
4.9 Conclusion à propos de l’approche SIMP
II Optimisation topologique des échanges de chaleur conducto-convectifs
5 Optimisation topologique des transferts de masse et de chaleur : régime laminaire
5.1 Abstract
5.2 Introduction
5.3 Topology optimization for heat and mass transfer
5.3.1 Topology optimization analysis
5.3.1.1 Problem formulation
5.3.1.2 Algorithmic scheme
5.3.2 Finite Volume Method
5.3.2.1 Modified FVM formulation
5.3.2.2 Shear force correction for solid domain
5.3.3 Sensitivity analysis
5.3.3.1 General formulation
5.3.3.2 Objective functions
5.3.3.3 Multi-objective optimization
5.3.4 Method of Moving Asymptotes
5.4 Results
5.4.1 Diffuser
5.4.2 Bend pipe
5.4.3 Double pipe
5.4.4 Single pipe with constant wall temperature
5.5 Conclusion and perspectives
5.6 Limitations de l’approche par homogénéisation
5.7 Conclusion à propos de l’optimisation des transferts de masse et de chaleur
6 Conclusion et perspectives
6.1 Conclusion
6.2 Perspectives
6.2.1 Automates cellulaires
6.2.2 Méthodes d’homogénéisation
6.2.2.1 Optimisation des transferts de chaleur par conduction
6.2.2.2 Optimisation des transferts de masse et de chaleur