A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

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1 A Method for Geometr Optimization in a Simple Model of Two-Dimensional Heat Transfer X. Peng, K. Niakhai B. Protas Jul, 3 arxiv:37.48v [math.na] 4 Jul 3 Abstract This investigation is motivated b the problem of optimal design of cooling elements in modern batter sstems. We consider a simple model of two-dimensional stead-state heat conduction described b elliptic partial differential equations and involving a onedimensional cooling element represented b a contour on which interface boundar conditions are specified. The problem consists in finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locall optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shapedifferential calculus. The main novelt of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundar-integral formulation which eploits certain analtical properties of the solution and does not require grids adapted to the contour. This approach is thoroughl validated and optimization results obtained in different test problems ehibit nontrivial shapes of the computed optimal contours. Kewords: heat transfer, adjoint-based optimization, shape calculus, Sobolev gradients, boundar integral equations AMS subject classifications: 8M5, 35Q93, 49Q, 49Q, 65N38 Introduction. Motivation The goal of this investigation is to develop and validate a computational method for optimization of the shape of cooling elements in general stead heat transfer problems. The motivation for this work comes from problems encountered in the design of batter sstems for hbridelectric (HEV) and electric vehicles (EV) [] in which a central role is plaed b methods of the thermal batter management (TMB) ensuring that the batter operates in a suitable thermal environment []. A tpical batter sstem used in automotive applications is shown in Figure a, whereas in Figure b we present a possible design of the channels with the coolant fluid acting as Department of Mathematics and Statistics, McMaster Universit, Hamilton, ON, Canada

2 Optimal Geometr in Two-Dimensional Heat Transfer (a) (b) Figure : (a) Batter sstem used in hbrid-electric vehicles and (b) possible design of the cooling elements (courtes of General Motors of Canada). the heat-echange elements. In these applications a ke issue is optimization of the shape of the cooling elements, so that the temperature distribution is as close as possible to prescribed profiles in some selected regions of the batter sstem. Assuming a known distribution of the heat sources representing the heat generation in the batter, mathematical models of such problems lead to sstems of elliptic boundar-value problems defined on irregular domains and subject to some rather complicated boundar conditions. Optimization of geometr of the cooling elements thus leads to shape-optimization problems for such sstems of equations, and in this stud we propose an approach based on the continuous (i.e., infinite-dimensional, or optimizethen-differentiate, [3]) formulation and the methods of the shape-differential calculus. The main novel contribution is the development and validation of an accurate and efficient technique based on the boundar-integral formulation for the evaluation of the shape gradients which is a ke enabler of the proposed optimization strateg. In the literature devoted to heat transfer and the related field of fluid mechanics most of the works concerning shape optimization, or equivalentl shape identification, concern problems formulated in the discretize-then-differentiate setting, where a finite-dimensional optimization problem is set up based on a discrete version of the governing equations, somewhat limiting the fleibilit in dealing with different geometries. Such approaches were pursued, for eample, in [4, 5, 6, 7, 8, 9], and we also mention the monograph []. Approaches based on continuous adjoint formulations usuall rel on the shape-differential calculus to determine the shape sensitivities. The shape calculus, reviewed in the monographs [,, 3], is a general suite of techniques derived from differential geometr which allow one to differentiate solutions of partial differential equations (PDEs) and functionals defined on these solutions with respect to variations of the domains on which these PDEs are defined. Applications of various continuous shape-optimization approaches to problems involving heat transfer, fluid flow and phase transformations were investigated in [4, 5, 6, 7, 8, 9,, ]. We add that, as regards the numerical representation of free boundaries in PDE problems, there are two main computational approaches, namel, the interface capturing methods based on the use of suitable implicit functions, such as the level set formulation [], and the interface tracking methods

3 Optimal Geometr in Two-Dimensional Heat Transfer 3 which rel on eplicit representations of the boundar. Since the model problem considered here is described b elliptic PDEs, we surve below the state-of-the-art numerical techniques used for the solution of optimization problems for such sstem based on the continuous formulation.. Review of Computational Methods for Shape Optimization of Elliptic PDEs In both paradigms, i.e., in the approaches reling on the level sets to capture the interface and in the methods based on eplicit interface tracking, optimization problems are tpicall solved using discrete (with respect to some pseudo-time) forms of gradient flows in which suitabl defined shape gradients are used as the descent directions. Starting with the seminal work [3], most attention has recentl been focused on level-set-based techniques in which the levelset function is evolved using the Hamilton-Jacobi equation with the velocit field given as an etension of the shape gradient awa from the interface. Their advantage is that the do not require interface-fitted domain discretizations and perform well on simple Cartesian grids. The governing and adjoint problems can be solved using the immersed interface method [4], as was done for eample in [5, 6], or with a penalization technique [7]. Regularization aspects of such approaches were investigated in [8], whereas the stud [9] eplored formulations resulting from different definitions of the inner products for the shape variations. Limitations of such methods arise when the boundar conditions and/or the shape gradients defined on the interface have a more complicated form (e.g., include derivatives), as then the tend to be difficult to evaluate accuratel on Cartesian grids. On the other hand, shape optimization techniques based on eplicit interface tracking tpicall require interface-fitted discretization of the domains on which the governing and adjoint sstems are solved. This discretization then needs to be updated during iterations which can be a complicated process. Such approaches were reviewed in [3], whereas some applications to image processing are discussed in [3, 3] The approach proposed here is based on eplicit interface tracking combined with suitabl chosen Sobolev gradients. While both the shape-differentiation and Sobolev gradients are wellknown techniques, the main novelt of the proposed approach is a method for the evaluation of shape gradients which is based on a boundar-integral formulation coupled with an elliptic solver constructed using a Cartesian grid. In comparison to the approaches described above, it offers the following advantages it is characterized b a high (in principle spectral) accurac in approimating comple interface boundar conditions and epressions for the shape gradients, so that onl modest resolution is required to discretize the contour, as boundar-fitted grids need not be constructed, it can deal with fairl complicated contour shapes at a low computational cost. The proposed implementation takes advantage of the analtic structure of the governing equations. While boundar-integral techniques have been used for shape optimization of elliptic PDEs, this was tpicall done in the discrete setting (i.e., discretize-then-differentiate ) with or without the adjoint equations used to evaluate the shape sensitivities as in [33, 34, 35, 36, 37, 38]. In [39] the optimized shape was described in terms of a graph of a function, so that determination of the gradients did not require methods of the shape-differential calculus. A boundar-integral formulation for a time-dependent (parabolic) shape optimization problem was devised in [4].

4 Optimal Geometr in Two-Dimensional Heat Transfer 4 We add that all of these approaches relied on the standard techniques of the boundar-element method (BEM) to evaluate the resulting integral epressions. Finall, we also mention [4] and some references cited therein where the shape sensitivities were epressed in terms of hpersingular integral equations (obtained via shape-differentiation of the standard boundar-integral formulations). We will comment on this interesting alternative approach at the end of the paper. The structure of the paper is as follows: in the net Section we introduce the mathematical model of the sstem and state the optimization problem, in the following Section we briefl describe a gradient-based descent algorithm based on shape-differentiation and smoothed (Sobolev) gradients; the proposed computational method for the solution of the governing and adjoint sstem and evaluation of the sensitivities is presented in detail in Section 4, whereas validation tests and results demonstrating application of the method to some selected shape optimization problems are presented in Section 5; discussion and conclusions are deferred to Section 6. Mathematical Model and Optimization Problem We will consider a simplified model of the problem based on the following set of assumptions Assumptions i. heat transfer is independent of time and occurs via conduction onl with k > representing the constant thermal conductivit, ii. the batter pack is treated as a D square region Ω R with isolated boundar Ω (i.e., the heat flu vanishes on Ω), iii. the distribution of the heat sources in the batter is given b the function q : Ω R which we will assume to be square-integrable, i.e., q L (Ω); the corresponding temperature distribution will be denoted b u : Ω R, iv. the cooling element is represented b a C curve C of total length L = ds and characterized b the reference temperature u ; the densit w of the heat flu absorbed b the cooling C element at a point C C is modelled using Newton s law of cooling as w = γ(u C u ), where γ > is a constant heat transfer coefficient and the temperature field u is continuous across the contour C, v. given an arbitrar subdomain A Ω, the target temperature distribution is given b u : A R. We restrict our attention to contours which are Lipschitz-continuous and will assume that the are parameterized in terms of the arc-length coordinate s [, L]. Two versions of the problem will be considered: P: C s= = C s=l / Ω, u = Const P: C s=, C s=l Ω, C s= C s=l u = u (s) () corresponding, respectivel, to a closed contour C with a constant reference temperature and to an open contour C with the reference temperature u = u (s) varing with the arc length. All validation tests and a number of optimizations will be performed for the simper problem P.

5 Optimal Geometr in Two-Dimensional Heat Transfer 5 A Ω δω δω A n n c Ω n n c Ω s Ω (a) (b) Figure : Sketch of the domain Ω with the target region A for (a) Problem P with a closed contour C and (b) Problem P with an open contour C representing the cooling element. In addition, some optimization problems will be solved for the more realistic configuration P in which q will be taken to be the distribution of the heat sources in an actual batter (Figure b). To fi attention, in Problem P we will assume that u increases linearl with the length corresponding to the coolant liquid heating up as it absorbs heat, i.e., u (s) = T a + T b T a L s, s [, L], () where T a and T b are the prescribed temperatures at the inlet and outlet. Sketches of the domain Ω with its different attributes are shown for both cases in Figure. In Problem P with an open contour C the endpoints are assumed to attach to the domain boundar at the right angles. We will denote Ω the part of the domain Ω inside, or above, contour C, and Ω Ω\Ω its complement, cf. Figure ( means equal to b definition ). Denoting u u Ω and u u Ω the restrictions of the temperature field to the subdomains on the two sides of contour C, we have the following mathematical model of the problem k u = q in Ω, (3a) k u = q in Ω, (3b) ( ) u = u u C on C, (3c) ( u k n u ) = γ (u u ) n on C, (3d) k u = n on Ω, (3e) where n is the unit vector normal to the contour C, or the boundar Ω, and oriented as shown in Figure. The corresponding unit tangent vector will be denoted t. We add that boundar conditions (3c) and (3d) represent Newton s law of cooling mentioned in Assumption.(iv). Tpicall used to model the heat transfer in the presence of convection, this law stipulates that the heat flu k ( u ) u C absorbed into the cooling element C at a given point n n C C is proportional to the difference between the local temperature u C and the reference temperature

6 Optimal Geometr in Two-Dimensional Heat Transfer 6 u. We remark that u ma be therefore thought of as the temperature of some hpothetical coolant liquid circulating in the cooling element, although details of this process are neglected in the present model (the contour C has in fact zero thickness). Clearl, the solution u will depend on the shape of the contour, i.e., u = u(c). We also remark that, while the differential equations and boundar conditions in sstem (3) are linear in the dependent variables u and u, problem (3) is in fact geometricall nonlinear with respect to the shape of the contour C. For a discussion of the eistence and regularit of solutions to elliptic boundar-value problems in complicated domains we refer the reader to monograph [4]. The optimization problem, motivated b the industrial applications discussed in Introduction, is to find an optimal contour C such that the corresponding solution ũ u( C) of sstem (3) evaluated over A is as close as possible to the prescribed target distribution u. Defining the reduced least-squares cost functional as J (C) (u u) dω, (4) we obtain the following optimization problem A min J (C) C subject to Sstem (3). (5) Since in actual applications the length of the contour representing the cooling element ma not be arbitrar, we will also consider a second optimization problem with the additional constraint on the contour length, namel, min J (C) C subject to: Sstem (3) ds = L, C (6) where L > is the prescribed length of the contour C. Clearl, problems (5) and (6) represent PDE-constrained shape optimization problems. PDE optimization problems involving shapes of the domains as the control variables require special treatment [,, 3], and our computational approach will be based on methods of the shape-differential calculus recalled in the net Section. Finall, we add that, in principle, in the statement of optimization problems (5) and (6) we should also include the condition C Ω which is equivalent to a suitable set of inequalit constraints. However, in the interest of simplifing the formulation, this condition is omitted here, although as discussed in Section 3 below, it will be incorporated in the final computational algorithm. 3 Gradient-Based Minimization Approach In this Section we review the formulation of the optimalit conditions for problems (5) and (6) and a gradient-based descent approach for the computational solution of these problems. Since these elements of our approach are rather standard, their presentation will be brief. We consider the first-order optimalit conditions which require the vanishing of a suitabl-defined Gâteau

7 Optimal Geometr in Two-Dimensional Heat Transfer 7 (directional) differential evaluated at the optimal contour C. We remark that defining such differential and the related epression for the gradient requires differentiation of governing sstem (3) with respect to the shape of the domains Ω and Ω on which the PDEs are defined. This is properl done based on the methods of the shape-differential calculus [, ] which rel on a special parametrization of the domain geometr and provide formulas for shape-differentiation of general functionals, PDEs and the associated boundar conditions. Below we briefl present this construction and recall the main results we will need, referring the reader to monographs [, 3] for further details. As a first step, we define the velocit field V : Ω R which will parametrize the deformations of the contour C and of the domains Ω and Ω, so that for ever point C C we have C(ɛ) = C + ɛv, (7) where < ɛ is a parameter and C(ɛ) is the position of a point on the deformed contour C(ɛ). Relations analogous to (7) can also be written for points in the deformed domains Ω (ɛ) and Ω (ɛ). Given a sufficientl regular function ψ : Ω R and the functionals j (Ω (ɛ)) ψ(; Ω Ω (ɛ) (ɛ)) dω and j (C(ɛ)) ψ(; C(ɛ)) ds defined on the perturbed domain and C(ɛ) contour, the corresponding shape differentials are defined as j (Ω ; V) lim ɛ ɛ [j (Ω (ɛ)) j (Ω ())] and j (C; V) lim ɛ ɛ [j (C(ɛ)) j (C())]. One of the central results of the shape-differential calculus is summarized in the following Lemma The shape differentials of j (Ω ) and j (C) with respect to parametrization (7) are given b epressions j (Ω ; V) = ψ dω + ψ (V n) ds, (8a) Ω C ( ) ψ j (C; V) = ψ ds + n + κ ψ (V n) ds, (8b) C C where ψ is the shape derivative of the integrand function ψ defined for Ω as ψ () lim ɛ ɛ [ψ(; Ω (ɛ)) ψ(; Ω ())] and κ denotes the signed curvature of the contour C. A detailed proof of Lemma can be found, for eample, in []. We remark that, in general, when differentiating with respect to the shape of open contours, epressions (8a) and (8b) will have additional terms proportional to (V t) and localized via Dirac delta distributions at the contour endpoints [43, 8]. However, in our Problem P, owing to () and the assumption that contour C meets the domain boundar Ω at the right angle (cf. Figure b), these terms vanish identicall. Therefore, for both the closed and open contours onl the normal component ζ (V n) C of the perturbation velocit field on the contour C plas a role in epressions for shape differentials (8). The normal perturbations ζ = ζ(s), considered as functions of the arclength coordinate, must satisf certain regularit conditions. It is sufficient for the perturbation ζ to belong to the Sobolev space H (, L) of periodic functions with square-integrable derivatives on [, L] (precise definition of the corresponding inner product will be given in (7) below). We add that contour parametrization allows us to recast line integrals, such as appearing in (8a), (8b) and below, as definite integrals. The optimalit condition for problem (5) is given b ζ H (,L) J ( C; ζn) = (u u) u dω =, (9) A

8 Optimal Geometr in Two-Dimensional Heat Transfer 8 where we note that the subdomain A is fied and does not depend on the perturbation ζ, and u = u ( C, ζn) is the shape derivative of the solution of governing problem (3) evaluated for the optimal contour shape C. The sensitivit (perturbation) equation satisfied b u is obtained b considering a suitable weak form of sstem (3) and shape-differentiating the resulting integrals using formulas (8), see [3], k k u = in Ω, (a) k u = in Ω, (b) ( u u u = n u ) ζ on C, (c) n ( ) [ ] u n u γ u u = γ n n + κ (u u ) ζ γ u on C, (d) k u n = on Ω, (e) where u u Ω and u u Ω, and u is the shape-derivative of () u = u (s; ζn) = T b T a L L [H(s s ) s/l] κζ ds () in which H( ) is the Heaviside function and whose structure is a consequence of the dependence of the arc length s = s(c) on the contour shape. As a matter of course, in problem P, u, cf (). As regards the second optimization problem (6), we will incorporate the additional constraint on the length of the contour C b defining an augmented cost functional J α (C) J (C) + α ( C ds L ), () where α > is a numerical parameter. The optimalit condition for this second optimization problem is thus ( ) ζ H (,L) J α( C; ζn) = (u u)u dω + α ds L κζ ds =, (3) A where we used relationship (8b) to differentiate the second term in (). We note that, although it arises from rather different mathematical principles, the more sstematic formulation of the constrained problem using Lagrange multipliers would result in an optimalit condition quite similar to (3). More precisel, the onl difference is that the factor α ( C ds L ) in (3) would be replaced b the Lagrange multiplier λ. As a result, the modification of the descent direction would have the same form (but with a different magnitude) in the two cases. On the other hand, given the geometric nonlinearit of the constraint C ds = L, the Lagrange multiplier λ can be rather hard to compute accuratel, so for simplicit in this stud we chose formulation () (3). We emphasize that optimalit conditions (9) and (3) onl characterize local, rather than global, minimizers and due to the non-conveit of cost functional (4), resulting from the geometric nonlinearit of sstem (3) and the length constraint, eistence of multiple local minima can be C C

9 Optimal Geometr in Two-Dimensional Heat Transfer 9 epected. The (locall) optimal shape C can be found computationall as C = lim n C (n) using the following gradient-descent algorithm C (n+) = C (n) τ n J ( C (n)), n =,,..., C () = C, (4) where the points C represent the contour C used as the initial guess and τ n is the length of the step along the descent direction at the n-th iteration computed b solving a line-minimization problem τ n = argmin τ> {J (C (n) τ J (C (n) )}. (5) There are man different approaches to solving problems of this tpe and in our stud we use Brent s iterative method combining the golden section search with inverse parabolic interpolation in the neighbourhood of the minimum. This approach does not require an derivatives with respect to τ and an efficient implementation is discussed in [44]. If τ n found b solving problem (5) results in the deformed contour C (n+) intersecting the domain boundar Ω, the value of τ n is suitabl reduced to ensure the condition C (n+) Ω is alwas satisfied. We add that, while for the sake of brevit of notation formula (4) represents the steepest descent approach, more advanced optimization methods such as the Polak-Ribiére version of the nonlinear conjugate gradients method [45] were used to obtain the results presented in Section 5.. At least for the problems we investigated, these approaches were found to sstematicall outperform the steepest descent method. Clearl, a critical element of minimization algorithm (4) is evaluation at ever iteration of the cost functional gradient J (C (n) ). The Riesz representation theorem [46] guarantees that it can be etracted from the Gâteau shape differential according to the formula J (C; ζn) = H J (C), ζ, (6) where z, z H (,L) = L z z + l z s H (,L) z s ds, z,z H (,L) (7) denotes an inner product in the Sobolev space H (, L) in which l R is a parameter (which will be shown below to have the meaning of a length scale). We observe that epressions for the Gâteau differentials J (C; ζn) and J α(c; ζn) appearing in (9) and (3) are not et in the form consistent with (6), because the perturbation ζ rather than appear as a factor is hidden in boundar conditions (c) (d) of the sensitivit sstem defining u. In order to transform the differential J (C; ζn) to Riesz form (6) we will emplo the adjoint variable u : Ω R which is the solution of the following adjoint sstem k k u = (u u)χ A in Ω, (8a) k u = (u u)χ A in Ω, (8b) u u = on C, (8c) ( ) u n u = γ u on C, (8d) n u = on Ω, (8e) n

10 Optimal Geometr in Two-Dimensional Heat Transfer where u u Ω and u u Ω, whereas χ Ai is the characteristic function of the region A i Ω i A, i =,. Following the standard procedure, see e.g. [3], we obtain where J (C; ζn) = L ( ) L J = γ (u u ) κ u + u n γ κ T b T a L L L J ζ ds, (9) γ u u n [H(s s) s /L] u (s ) ds on C. The last term in () stems from the arc length dependence of the reference temperature u in Problem P, cf. (), and vanishes identicall in Problem P. Derivation details are presented in [47, 48]; in [47] we also discuss a smbolic algebra algorithm for automated determination of the adjoint boundar conditions in PDE optimization problems characterized b complicated interface conditions such as the problem considered here. While this is not the gradient we use in the actual computations, for simplicit in (9) () the gradient L J was obtained as the Riesz representer in the space L (, L) of square-integrable functions. We also add that the part of Gâteau differential (3) associated with the length constraint is alread in the Riesz form, so that the L gradient of cost functional J α (C) is ( ) L J α = L J + α ds L κ on C. () The gradients actuall used in minimization algorithm (4), namel the Sobolev gradients H J and H J α, can be obtained from () and () b identifing (6) (7) with (9), and noting the arbitrariness of the shape perturbations ζ H (, L). Then, after integrating b parts and using the boundar conditions, we arrive at ) ( l H J = L J on (, L), s Periodic boundar conditions s H J = s=,l C (P), (P). Thus, the Sobolev gradient H J is obtained b first computing the gradient L J from () or (), and then b solving elliptic boundar-value problem () defined on the contour C, a step which is known to be equivalent to low-pass filtering (smoothing) the L gradient with l acting as the cut-off length scale [49]. In Problem P the homogeneous Neumann boundar conditions ensure that the Sobolev gradient H J does not change the angle at which the contour C meets the domain boundar Ω (which therefore alwas remains π/). For some other applications of Sobolev gradients to solution of minimization problems involving PDEs we refer the reader to monograph [5], articles [3, 5] and to articles [8, 9, 9, 3] for studies concerned specificall with shape optimization. The different elements discussed in the present Section combine into Algorithm. An elegant and accurate numerical solution technique for the direct and adjoint sstems (3) and (8) and evaluation of gradient epression () is described in the net Section. () ()

11 Optimal Geometr in Two-Dimensional Heat Transfer Algorithm Iterative minimization algorithm for finding optimal contour shapes C. Input: ε J and ε τ (adjustable tolerances), C (initial contour shape) Output: C (optimal contour shape) n C () initial guess C repeat solve direct problem (3) solve adjoint problem (8) evaluate () () and solve () to determine [ H J (C (k) ) ] compute the Polak-Ribiére conjugate direction g H J (C (k) ) perform line minimization min τ> {J ( (n) C τ g [ J (C (n) ) ] ( } to find the step size τ n, ensuring that (n) C τ n g [ J (C (n) ) ]) / Ω [ ] obtain C (n+) b deforming C (n) along the conjugate direction g H J (C (n) ) with the step size τ n, n n + until τ n < ε τ or J (C (n+) ) J (C (n) ) < ε J J (C (n) ) 4 Numerical Implementation In this Section we present in detail a novel numerical approach we devised to solve the governing and adjoint sstems (3) and (8) at ever iteration of Algorithm. Since these sstems have in fact essentiall identical structure, we will focus our discussion on the solution of the first one. The methods to tackle Problems P and P are based on the same concept, but differ in regard to some technical details, and to fi attention, below we describe the approach applicable to Problem P. Modifications required to solve Problem P are summarized further below with all details available in [5]. We observe that both sstems (3) and (8) can be regarded as combinations of two Poisson problems (defined in Ω and in Ω ) which are coupled via some complicated (mied) boundar conditions on the contour C separating the two domains. It should be emphasized that this contour can have an arbitrar, though non-intersecting, shape. Given the linearit (with respect to u and u ) of equations (3a) (3b), we split problem (3) into two subproblems: a potential problem associated with the comple interface boundar condition (3d) and another elliptic problem arising from the presence of the source term q, which are then coupled using a suitable interpolation scheme. Since the solution methods for these subproblems are adapted to their analtic structure, we achieve for each of them the highest possible (spectral) numerical accurac. While similar techniques have alread been used for the solution of certain direct problems [53], to the best of our knowledge, this direction has not been eplored in applications to optimization or inverse problems. As a starting point, we consider the following ansatz for the solution u of problem (3) u = u p + u h in Ω, (3) where the fields u p and u h satisf the following sstem of PDEs and boundar conditions equiv-

12 Optimal Geometr in Two-Dimensional Heat Transfer alent to (3) k k u p = q in Ω, (4a) u h = in Ω \ C, (4b) u h = u h on C, (4c) ( uh n u ) h n = γ (u p + u h u ) on C, (4d) u p n = u h on Ω. (4e) n We note that the fields u p and u h are coupled onl through boundar conditions (4d) and (4e). Since the field u h is harmonic in Ω\C, it admits a representation in terms of the single-laer potential densit µ : C R Ω\C u h () = π C ln C µ( C ) ds. (5) Taking the limit C in (5), using boundar conditions (4c) and (4d), and taking into account the limiting properties of integrals of tpe (5) known from the potential theor [54, 55], we arrive at a singular boundar integral equation of Fredholm tpe II satisfied b the densit µ. Thus, sstem (4) can be equivalentl rewritten as µ() + γ π k C k u p = q in Ω, (6a) ln C µ( C ) ds = γ k (u p u ) on C, (6b) u p n = u h n on Ω. (6c) The new dependent variables are {u p (), Ω; µ( C ), C C} and the advantage of this formulation is that the second variable (potential densit) needs to be found on the contour C onl and, unlike in original sstem (3), there are no differential operators defined on the contour C. For the purpose of discretizing Poisson equation (6a) we cover the domain Ω with a N N dadic Chebshev grid [56], where N > is the number of grid points in each direction. Contour C is represented with M points equispaced in the arc-length coordinate s (M is taken to be an even number). These discretizations are shown in Figure 3 (in Problem P the discretization of contour C needs to be a bit different, cf. [5]). We let u N p;i,j u p ( i, j ), i, j =,..., N and µ M l µ(s l ), l =,..., M denote the discrete nodal values of the unknowns, where ( i, j ) are the coordinates of the collocation points on the dadic Chebshev grid covering Ω, whereas s l are the arc-length coordinates of the points discretizing contour C in Problem P, i.e., s l (l ) L, M l =,..., M. We then construct the vectors U and m [U] (i )N+j = u N p;i,j, i, j =,..., N, (7a) [m] l = µ M l, l =,..., M, (7b) and will use the smbol N to denote the discretization of the Laplace operator based on the Chebshev spectral collocation approach [56] and corresponding to the Neumann boundar conditions. Thus, discretization of (6a) takes the algebraic form N U = f + q, (8)

13 Optimal Geometr in Two-Dimensional Heat Transfer Figure 3: Discretization of the domain Ω and contour C. For clarit, the discretization shown is much coarser than used in the actual computations reported in Section 5. where q R N contains the values of the right-hand side (RHS) function q evaluated at the interior collocation points (and zeros in the entries corresponding to the boundar nodes) and f R N is a vector containing the values of u h at the boundar nodes, cf. (6c). It can be n epressed as f = B m, (9) in which B is a N M matri operator representing the discretization via the trapezoidal rule of the relation u h n = (b i C ) n µ( bi π b i C C ) ds, i =,..., 4N 4, (3) S where b i Ω (the rows of B corresponding to the interior grid points are zero). As regards integral equation (6b), we observe that the logarithmic kernel it contains is in fact singular and, assuming the potential densit is a Lipschitz-continuous function of s, the integral is defined as an improper one. As a standard approach to deal with this issue [54, 55], we rewrite the kernel as { ln C (t) C (t ) = ln C (t) C (t ) 4 sin t t } + ( ln 4 sin t ) t, (3) where t, t [, π] are the variables parameterizing contour C. Therefore, rewriting the line

14 Optimal Geometr in Two-Dimensional Heat Transfer 4 integral in (6b) as a definite integral, the boundar integral equation becomes {}}{ γ π µ( C (t)) + µ( C (t C (t) C (t ) )) ln π k sin( t t ) r(t ) dt (I) (II) {}}{ [ γ π ( ) ] t t + µ( C (t )) ln 4 sin r(t ) dt = γ 4π k k [u p( C (t)) u ], t [, π] where, assuming that the contour parameterization is uniform in the arc length s, we have r(t) = d(t) dt = L. We note that integral (I) has now a regular kernel (with a removable π singularit to be more precise) and can be evaluated with spectral accurac in a straightforward manner using the trapezoidal quadrature. The singularit is now contained in the improper integral (II) which can be evaluated analticall as follows. We approimate the potential densit µ(t) using the spectrall-accurate trigonometric interpolation [55] (3) µ(t) M µ M j L j (t), (33) j= in which L j (t), j =,..., M, are the trigonometric cardinal functions L j (t) ( ) ( ) M(t M sin tj ) t tj cot, t [, π], t t j, j =,..., M, where t j (j ) π. Defining now M Rj M (t) M/ M m cos [ m(t t j)] + [ ] M(t M cos tj ), j =,..., M, (34) m= the improper integral (II) in (3) is approimated as γ π ( )] t t µ(t ) ln [4 sin 4π k r(t ) dt γ L 4π k M µ M j Rj M (t), t [, π]. (35) Therefore, collocating integral equation (3) on the grid points t,..., t M ields the following discrete problem ( I + γ k K + γ ) k K m + γ k PU = γ k u, (36) where is a column vector of dimension M with all entries equal to one and the matrices K and K are defined as, cf. (35), [K ] ij = L π M ln C (t i ) C (t j ) ) sin, [K ] ij = L 4π RM j (t i ), i, j =,..., M, (37) ( ti t j j=

15 Optimal Geometr in Two-Dimensional Heat Transfer 5 whereas P is an M N matri representing interpolation of the field u p from the Chebshev grid onto the points { C (t ),..., C (t M )} discretizing the contour C. Thus, combining (8) and (36), the final discrete form of sstem (6) is [ N B γ k I + γ K k + γ K k ] [ ] U = m k [ ] q. (38) γ u The accurac of approimation represented b sstem (38) is ultimatel determined b the accurac of the interpolation operator P, and in principle can be spectral, although for reasons of the numerical stabilit we have used spline interpolation in the present stud. Sstem (38) is readil solved using standard methods of numerical linear algebra, and we refer the reader to thesis [48] for numerical validation and tests of accurac. Discretization of adjoint sstem (8) leads to a discrete problem with the same matri as in (38), but with a different right-hand side. The L gradient L J is obtained from the solution [ (U ) T (m ) ] T T of the discrete adjoint problem using relation (), where the different terms are computed using boundar conditions (8c) (8d) and the following identities, known from the potential theor [54, 55], u h u h = µ, (39a) [ u h n n n ] + u h n = π C n(c ), C C µ ( ) ds, (39b) valid for all points C C, where, denotes the inner product in R, u = u p + u h and µ is the single-laer potential densit associated with u h. We add that the kernel of the n( integral on the RHS in (39b) is in fact bounded, as we have C lim C ), C C C = κ( C) [54]. Accurac of the cost functional gradients computed in this wa is assessed in the net Section. Finall, we remark that after each step of gradient algorithm (4), the points C (t i ), i =,..., M, are no longer distributed uniforml in the arc length s. In order to retain the spectral accurac of the solution of equation (3), at ever iteration we therefore construct, using spectral Fourier interpolation [56], a new set of collocation points { C (t ),..., C (t M )} which are equispaced in the arc-length coordinate. The main modification required to adapt the method described above to Problem P concerns the solution of boundar-integral equation (6b). Since the integration domain is no longer periodic, identit (3) must be replaced with a different one and contour C must be discretized using a different set of points [5]. Moreover, the trapezoidal quadratures need to be replaced with the Clenshaw-Curtis quadratures whereas the trigonometric interpolation with a suitable polnomial technique. 5 Computational Results In this Section we first perform tests to thoroughl validate the computational algorithm introduced in Section 4 for the evaluation of cost functional gradients J. We do this here for Problem P and refer the reader to [5] for the corresponding validation tests for Problem P. Net, we appl this method in the framework of Algorithm to perform shape optimization in a number of test cases concerning Problems P and P. Throughout this Section we take the domain to be Ω = [, ] [, ].

16 Optimal Geometr in Two-Dimensional Heat Transfer 6 5. Validation of Gradients A standard computational test emploed to ascertain the accurac of the cost functional gradients in PDE optimization problems is to calculate the Gâteau differential J (C; ζn) for a given contour C and its perturbations ζ in two different was: using an approimate finite-difference formula and Riesz identit (6) [57]. Thus, the ratio of these two epressions, denoted κ(ε) J (C(ɛ)) J (C()) ɛ L J (C(), ζ L (,L), (4) should be approimatel equal to unit for a range of values of ɛ. Plotting κ(ɛ) using the logarithmic scale allows one to see the number of significant digits of accurac captured in the computation. We remark that, since different Riesz representations (L vs. H ) give the same differential J (C; ζn), for simplicit in (4) we can use the L inner product together with the corresponding gradient. To focus attention, we present our validation results for the functional J (C), i.e., without the length constraint, as the gradient of the latter part does not involve the adjoint variable u. We analze two sets of results: one in which we fi the contour C and consider different perturbations ζ and vice versa. For ever pair of the contour and the perturbation we stud the effect of different resolutions N and M. Details concerning the two test cases are collected in Table, where the different contours are specified in Table, whereas the perturbations tested are given b ζ j (t) = sin(j t), t = [, π], j =,, 3, 4. (4) In both validation tests we assume that A = Ω and use the following distribution of the heat sources and the target temperature profile q(, ) = 5 5 5(.5), (4) u(, ) = 5 + sin(4 ) cos(4 ), (43) where,. The results of TEST # and TEST # are shown in Figures 4 and 5, respectivel. In both cases we note that κ(ɛ) is fairl close to the unit for values of ɛ spanning several orders of magnitude. The quantit κ(ɛ) deviates from the unit for ver small values of ɛ which is due to the subtractive cancellation (round-off) errors, and for large values of ɛ which is due to the truncation errors, both of which are well-known effects [58]. Since we use the differentiate-then-discretize formulation, one should not epect κ(ɛ) to be at the level of the machine precision, although this quantit approaches zero as the resolution is refined. We also tested cases in which A Ω and the length constraint was included obtaining similar results as in Figures 4 and 5. Having thus validated the cost functional gradients, we now move on to discuss solution of the actual optimization problems. 5. Solution of Optimization Problems We will stud in detail solution of the following three optimization problems with and without the length constraint, as indicated below: in CASE # for Problem P we eamine the convergence of Algorithm without the length constraint for several different initial guesses C () for the contour and using A = Ω, in CASE # for Problem P we consider a configuration in which A Ω and also stud the effect of the length constraint, and in CASE #3 for Problem P we

17 Optimal Geometr in Two-Dimensional Heat Transfer 7 Table : Settings for the validation tests of the cost functional gradient J (Problem P). The contours and perturbations used are defined in Table and equation (4), respectivel. TEST Contour C Perturbations ζ Resolution (N, M) Target Domain A # C ζ, ζ, ζ 3, ζ 4 (, ), Ω (5, 5), (8, 3) # C, C 3, C 4, C 5 ζ (5, 5), (8, ), (8, ), (8, 3), (8, 4) Ω Table : Definitions of contours C,..., C 7 used in the different cases studied in Section 5. Contour Parametrization ( t π) Plot C (t) =.4 cos(t) +., (t) =.4 sin(t). C (t) =. cos(t) +.4, (t) =. sin(t) +.4 C 3 (t) =.3 cos(t), (t) =. sin(t) (t) =.4( +. cos(3t)) cos(t) +., C 4 (t) =.4( +. cos(3t)) sin(t) (t) =.4( +. cos(4t)) cos(t) +., C 5 (t) =.4( +. cos(4t)) sin(t) C 6 (t) = 3 3 cos(t).4, (t) = sin(t) +.3 π π C 7 (t) = t π, (t) =.78 π

18 Optimal Geometr in Two-Dimensional Heat Transfer log κ log κ ε ε (a) (b) log κ log κ ε ε (c) (d) Figure 4: TEST # (Table ): Dependence of log κ(ɛ) on the step size ɛ in (4) for different perturbations (a) ζ, (b) ζ, (c) ζ 3 and (d) ζ 4, cf. Equation (4), and different resolutions (asterisks) N = 5, M = 5, (circles) N =, M =, and (squares) N = 8, M = log κ log κ ε ε (a) (b) log κ log κ ε ε (c) (d) Figure 5: TEST # (Table ): Dependence of log κ(ɛ) on the step size ɛ in (4) for different contours (a) C, (b) C 3, (c) C 4 and (d) C 5, cf. Table, and different resolutions (asterisks) N = 5, M = 5, (circles) N = 8, M =, (squares) N = 8, M =, (crosses) N = 8, M = 3 and (triangles) N = 8, M = 4.

19 Optimal Geometr in Two-Dimensional Heat Transfer 9 Table 3: Parameters used in the solution of the three optimization problems in Section 5.. The contours used as the initial guesses C () are defined in Table. CASE q u (N, M) α L l C () A # (P) Eq. (44) Eq. (45a) (5,).,.3 C, C 3, C 4, C 5 Ω # (P) Eq. (44) Eq. (45a) (5,) 3.. C 6 [.5, ] [.5, ] #3 (P) Fig. 9a Eq. (45b) (5,),,, 3 6..,.5 C 7 Ω investigate a sstem in which the heat source distribution q corresponds to the temperature field in an actual batter cell, also in the presence of the length constraint. Parameters characterizing the three cases are collected in Table 3. As concerns the heat source distribution q, in CASES # and # it is given b the following epression (Figure 6a) ( q(, ) = ) (, ) Ω, (44) whereas in CASE #3 it is obtained (b appling the Laplace operator and suitable smoothing) to the temperature distribution determined eperimentall in an actual batter cell [59], see Figure 9a. In the different cases the target temperature field is given b the following epressions (see also Figure 6b) CASE #, : ( u(, ) = 5 + sin(π + π) cos π + π ), (, ) A, (45a) CASE #3 : u(, ) = 3 (, ) Ω. (45b) In CASE # and # the distribution of heat sources (44) and the target temperature field (45a) have been chosen to test the algorithm in the situation when the source field varies slowl, whereas the target field ehibits a significant variabilit, cf. Figures 6a and 6b. On the other hand, in CASE #3 the constant target temperature field (45b) represents a tpical engineering objective. The specific values assumed b the fields q and u do not have a phsical significance and were selected to make the optimization problem sufficientl challenging. The tolerances in Algorithm are set to ε J = 3 and ε τ = 8. The results characterizing the performance of Algorithm in CASE # are collected in Figure 7. First of all, we note that, depending on the choice of the initial guess C () for the contour in (4), cf. Figure 7a, the iterations in fact converge to quite distinct locall optimal shapes, cf. Figure 7b, providing evidence for the eistence of multiple local minima in the optimization problem, as discussed in Section 3. We also note that the decrease of cost functional J (C (n) ) with iterations n is quite different in these different cases, cf. Figure 7c. In Figure 7d we show the intermediate shapes found at the consecutive iterations of the algorithm starting from the initial guess C for which the largest decrease was obtained in the cost functional. We observe that simpler initial guesses (i.e., a circle or an ellipse) tend to lead to better local minimizers. However, the final temperature distributions u( C) obtained from the different initial guesses

20 Optimal Geometr in Two-Dimensional Heat Transfer (a) (b) Figure 6: (a) Distribution of heat sources q, cf. (44), and (b) target temperature field u, cf. (45a), used in CASE # and #. all capture features of the cellular pattern characterizing the target distribution u, see Figures 7f,h,j,l vs. Figure 6b. The data illustrating the performance of Algorithm in CASE # is shown in Figure 8. Since in this case we include the length constraint with a rather large value of the penalt parameter (α = ), the optimal contours are not allowed to deform much (Figure 8c). However, we remark that the algorithm is able to shift the contour so that the optimal shape C is enclosed within the target domain A in which the temperature field u is defined (Figure 8d). The data concerning CASE #3 is collected in Figure 9. Using the contour shown in Figure 9b as the initial guess (cf. Table ), we first solve Problem P assuming u = Const and with the length constraint not enforced (α = ). Then, using thus obtained optimal shape (marked with the solid line in Figure 9e) as the initial guess, we solve Problem P again, now allowing u to var with the arc length s. Mimicking changes in the inflow/outflow temperature of the coolant liquid, this is achieved b decreasing T a or increasing T b in () and corresponds to, respectivel, dash-dotted and dashed contours in Figure 9e. In Figure 9c we observe that in the initial optimization the cost functional drops b over three orders of magnitude during less than iterations (in the subsequent problems which have better initial guesses this decrease is smaller). Finall, we consider the case with T a = and T b = 9, and solve the optimization problem with the length constraint using L =.3 and increasing values of α. The resulting optimal contour shapes are shown in Figure 9f, whereas Figure 9d presents the evolution of the contour length L(C (n) ) with iterations for different values of α. As epected, we see that for increasing values of α the contour length approaches the prescribed value L while the contours themselves become less deformed. The temperature fields u(, ) obtained in the cases with u = = Const, T a = and T b =, and T a = and T b = 9, cf. (), and without the length constraint are shown in Figures 9g,h,i. This last case with the length constraint and α = is shown in Figure 9j. We see that, as compared to the temperature corresponding to the initial guess for the contour (Figure 9b), the optimal distributions in Figures 9g,h,i,j have the temperature ranges much closer to target field (43). We also observe that, with the eception of the case in which the inflow temperature T a is quite low (Figure 9h), the optimal contour shapes tend to weave around the two hot spots in the heat source distribution (Figure 9a) in a complicated manner.

21 Optimal Geometr in Two-Dimensional Heat Transfer (a) Initial contour shapes C, C 3, C 4 and C 5 (for clarit the are represented using fewer points than used in the actual computations) (b) Optimal contour shapes C corresponding to the initial contours shown in Figure (a) J n (c) Evolution of functionals J (C (n) ) for iterations starting with the initial contours shown in Figure (a) (d) Evolution of contours C (n) with iterations in the problem with initial shape C (dashed line) (e) Initial temperature field u(c ) (f) Optimal temperature field u( C) in the case with initial contour C Figure 7: Results illustrating solution of the optimization problems in CASE #, cf. Table 3, using different initial contours (asterisks) C, (circles) C 3, (squares) C 4, (pluses) C 5, cf. Table.

22 Optimal Geometr in Two-Dimensional Heat Transfer (g) Initial temperature field u(c 3 ) (h) Optimal temperature field u( C) in the case with initial contour C (i) Initial temperature field u(c 4 ) (j) Optimal temperature field u( C) in the case with initial contour C (k) Initial temperature field u(c 5 ) (l) Optimal temperature field u( C) in the case with initial contour C 5 Figure 7: (continued) Results illustrating solution of the optimization problems in CASE #, cf. Table 3. The grid shown in the Figures in the right column corresponds to the cellular pattern of the target field u, cf. Figure 6(b).

23 Optimal Geometr in Two-Dimensional Heat Transfer J α L(C (n) ) n n (a) Evolution of cost functional J α (C (n) ) with iterations (b) Evolution of contour length L(C (n) ) with iterations (c) Evolution of contours C (n) with iterations; the initial contour C 7 is marked with thin dashed line and the optimal shape C appears in red (d) Optimal temperature distribution u( C) Figure 8: Results illustrating solution of the optimization problem in CASE #, cf. Table 3. The rectangles marked with thick dashed lines in Figures (c) and (d) indicate the subregion A where the target temperature u is specified.