Recent zbMATH articles in MSC 49M https://zbmath.org/atom/cc/49M 2023-11-13T18:48:18.785376Z Werkzeug Optimal control of lake eutrophication https://zbmath.org/1521.35173 2023-11-13T18:48:18.785376Z "Choquet, Catherine" https://zbmath.org/authors/?q=ai:choquet.catherine "Comte, Eloïse" https://zbmath.org/authors/?q=ai:comte.eloise The authors consider a domain $$\Omega$$ in $$\mathbb{R}^{3}$$, which represents a lake, with a $$C^{1}$$ boundary $$\partial \Omega$$ which is the union of three disjoint sets $$\partial \Omega =\Gamma _{\mathrm{in}}\cup \Gamma _{\mathrm{out}}\cup \Gamma$$, where $$\Gamma _{\mathrm{in}}$$ is the lake entrance and $$\Gamma _{\mathrm{out}}$$ is the lake exit ($$\Gamma _{\mathrm{in}}$$ and $$\Gamma _{\mathrm{out}}$$ being not necessarily connected). They consider the space-time dynamics of the phosphorus stock $$\overline{S}$$ and of the cyanobacteria concentration $$c$$ in the lake written in $$\Omega \times (0,T)$$ as: $$\partial _{t}\overline{S} +\operatorname{div}(v\overline{S})-d_{s}\Delta \overline{S}+b(\overline{S})\overline{S}-h( \overline{S})+f(\overline{S},c)c=0$$, $$\partial _{t}c+\operatorname{div}(vc)-d_{c}\Delta c+m(c)c-f(\overline{S},c)c=0$$, where $$v$$ is the velocity which governs the convection, $$d_{s}$$ and $$d_{c}$$ diffusion coefficients, $$b$$ and $$m$$ nonlinear functions which represent the rates of loss per unit stock and per cyanobacteria concentration, $$h(\overline{S})=\overline{S}^{2}/(k+\overline{S }^{2})$$ models the internal discharge of the phosphorus trapped in the sediments, and $$f$$ the Monod term, often chosen in the form $$f(\overline{S} )=u_{max}\overline{S}/(k_{s}+\overline{S})$$, which relates the phosphorus stock with the cyanobacterial dynamics, $$u_{\max}$$ representing the maximal increasing rate and $$k_{s}$$ the half-saturation value. The initial conditions $$\overline{S}\mid _{t=0}=\overline{S}_{0}$$, $$c\mid _{t=0}=c_{0}$$ are imposed in $$\Omega$$, together with the boundary conditions: $$(\overline{ S}v-d_{s}\nabla \overline{S})\cdot n=(cv-d_{c}\nabla c)\cdot n=0$$, on $$(0,T)\times \Gamma$$, $$r_{S}\overline{S}+(\overline{S}v-d_{s}\nabla \overline{S})\cdot n=R_{S}(\overline{S})$$, with $$r_{S}\leq 0$$, $$r_{c}c+(cv-d_{c}\nabla c)\cdot n=R_{c}(c)$$, with $$r_{c}\leq 0$$, on $$(0,T)\times \Gamma _{\mathrm{out}}$$, $$\overline{S}=\overline{P}$$, $$(cv-d_{c}\nabla c)\cdot n=0$$, on $$(0,T)\times \Gamma _{\mathrm{in}}$$. Assuming further hypotheses on the different terms of the parabolic problem, the authors prove the existence of a unique weak solution $$(\overline{S},c)\in (L^{2}(0,T;H^{1}(\Omega ))\cap L^{\infty }(0,T;L^{2}(\Omega ))^{2}$$ to this problem. For the proof, the authors propose a variational formulation of the above parabolic problem and they use the Faedo-Galerkin method, maximum principles and a uniqueness result. They then propose an optimal control associated to the above problem written as: Find $$(\overline{P}^{\ast }, \overline{S}^{\ast },c^{\ast })$$ such that $$J(\overline{P}^{\ast },c^{\ast })=\max\{J(\overline{P},c)$$; $$\overline{P}\in E$$ with $$(\overline{S},c)$$ satisfying the above parabolic problem$$\}$$. Here $$E$$ is the set of admissible controls defined as $$E=\{\overline{P}\in L^{\infty }((0,T)\times \Gamma _{in})$$; $$0\leq \overline{P}(t,x)\leq \overline{P}_{\max}$$ a.e. in $$(0,T)\times \Gamma _{in}\}$$, where $$\overline{P}_{\max}$$ is any given real number, and $$J$$ the objective function defined as: $$J(\overline{P} ,c)=\int_{0}^{T}\left( \int_{\Gamma _{\mathrm{in}}}B(t,\sigma ,\overline{P}(t,\sigma ))e^{-\rho t}d\sigma -\int_{\Omega }D(t,x,c(t,x))e^{-\rho t}dx\right) dt$$, where $$\rho \in ]0,1[$$ is the social discount rate and $$(\overline{S},c)$$ satisfies the above parabolic problem. Setting $$S=\overline{S}-P$$, where $$P$$ is the unique solution to: $$\partial _{t}P-d_{s}\Delta P=0$$ in $$\Omega \times (0,T)$$, $$P\mid _{t=0}=P_{0}$$ in $$\Omega$$, $$P\mid _{\Gamma _{in}}= \overline{P}$$, $$P\mid _{\partial \Omega \setminus \Gamma _{\mathrm{in}}}=0$$ on $$\partial \Omega \times (0,T)$$, $$P_{0}$$ being the solution to: $$-\Delta P_{0}=0$$ in $$\Omega$$, $$P\mid _{\Gamma _{in}}=\overline{P}\mid _{t=0}$$, $$P_{0}\mid _{\partial \Omega \setminus \Gamma _{in}}=0$$ on $$\partial \Omega \times (0,T)$$, the authors rewrite the above problem and the optimal control problem. Under the same hypotheses as in the existence and uniqueness result for a weak solution to the parabolic problem, they prove the existence of a global solution $$(\overline{P}^{\ast },\overline{S}^{\ast },c^{\ast })$$ to the optimal control problem. The proof is based on some boundedness property of a maximizing sequence for the objective function $$J$$ and on the analysis of the convergence of this sequence. Reviewer: Alain Brillard (Riedisheim) Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions https://zbmath.org/1521.47102 2023-11-13T18:48:18.785376Z "Briceño-Arias, Luis" https://zbmath.org/authors/?q=ai:briceno-arias.luis-m "Chen, Jinjian" https://zbmath.org/authors/?q=ai:chen.jinjian "Roldán, Fernando" https://zbmath.org/authors/?q=ai:roldan.fernando "Tang, Yuchao" https://zbmath.org/authors/?q=ai:tang.yuchao Summary: In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product space techniques, we derive a method for solving a composite monotone primal-dual inclusions including linear operators and we apply it to solve constrained composite convex optimization problems. Finally, we apply our algorithm to a constrained total variation least-squares problem and we compare its performance with efficient methods in the literature. Numerical analysis of the model of optimal savings and borrowing https://zbmath.org/1521.49022 2023-11-13T18:48:18.785376Z "Chernov, Alexey" https://zbmath.org/authors/?q=ai:chernov.alexey|chernov.alexey-a|chernov.alexey.1|chernov.aleksei-vyacheslavovich "Zhukova, Aleksandra" https://zbmath.org/authors/?q=ai:zhukova.aleksandra Summary: This work presents the results of the numerical analysis of the model of optimal savings and borrowing. The model is formulated as the stochastic optimal control problem where the moments of time when the state changes are random. The target (performance) functional of the problem contains the penalty for the final values of the balance account in the bank. This penalty has the form of the parameterized softplus function. The sufficient optimality conditions contain the system of functional and partial differential equations. The functional equation is solved using the functional Newton method. The paper presents the approach to numerical computation and the results of the numerical solution to the optimal control problem. For the entire collection see [Zbl 1516.90004]. A parallel algorithm for generalized multiple-set split feasibility with application to optimal control problems https://zbmath.org/1521.49023 2023-11-13T18:48:18.785376Z "Thuy, Nguyen Thi Thu" https://zbmath.org/authors/?q=ai:thuy.nguyen-thi-thu "Nghia, Nguyen Trung" https://zbmath.org/authors/?q=ai:nghia.nguyen-trung Summary: In this paper, we concentrate on the generalized multiple-set split feasibility problems in Hilbert spaces and propose a new iterative method for this problem. One of the most important of this method is using dynamic step-sizes, in which the information of the previous step is the only requirement to compute the next approximation. The strong convergence result of the suggested algorithm is proven theoretically under some feasible assumptions. When considering the main results in some special cases, we also obtain some applications regarding the solution of the multiple-set split feasibility problem, the split feasibility problem with multiple output sets, and the split feasibility problem as well as the linear optimal control problem. Some numerical experiments on infinite-dimensional spaces and applications in optimal control problems are conducted to demonstrate the advantages and computational efficiency of the proposed algorithms over some existing results. An optimal control approach to a fluid-structure interaction parameter estimation problem with inequality constraints https://zbmath.org/1521.49024 2023-11-13T18:48:18.785376Z "Chirco, Leonardo" https://zbmath.org/authors/?q=ai:chirco.leonardo "Manservisi, Sandro" https://zbmath.org/authors/?q=ai:manservisi.sandro Summary: In this work, we present a new optimal control approach to fluid-structure interaction parameter estimation problems. The goal is to obtain the desired deformation by controlling the solid material properties, such as the Young modulus. We consider a stationary monolithic FSI problem where solid and liquid forces at the interface are automatically balanced. We consider inequality constraints in order to bound the Young modulus control admissible set. For the optimization, we adopt the Lagrange multiplier method with adjoint variables and obtain the optimality system which minimizes the augmented Lagrangian functional. We implement a projected gradient-based algorithm in a multigrid finite element code suitable for the study of large solid displacements. In order to support the proposed approach, we perform numerical tests with different objectives and control constraints. Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomials https://zbmath.org/1521.49025 2023-11-13T18:48:18.785376Z "Heydari, Mohammad Hossein" https://zbmath.org/authors/?q=ai:heydari.mohammadhossein "Razzaghi, Mohsen" https://zbmath.org/authors/?q=ai:razzaghi.mohsen "Zhagharian, Shabnam" https://zbmath.org/authors/?q=ai:zhagharian.shabnam Summary: In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples. Full-space approach to aerodynamic shape optimization https://zbmath.org/1521.49031 2023-11-13T18:48:18.785376Z "Shi-Dong, Doug" https://zbmath.org/authors/?q=ai:shi-dong.doug "Nadarajah, Siva" https://zbmath.org/authors/?q=ai:nadarajah.siva-k Summary: Aerodynamic shape optimization (ASO) involves finding an optimal surface while constraining a set of nonlinear partial differential equations (PDE). The conventional approaches use quasi-Newton methods operating in the reduced-space, where the PDE constraints are eliminated at each design step by decoupling the flow solver from the optimizer. Conversely, the full-space Lagrange-Newton-Krylov-Schur (LNKS) approach couples the design and flow iteration by simultaneously minimizing the objective function and improving feasibility of the PDE constraints, which requires fewer iterations of the forward problem. Additionally, the use of second-order information leads to a number of design cycles independent of the number of control variables. We discuss the necessary ingredients to build an efficient LNKS ASO framework as well as the intricacies of their implementation. The LNKS approach is then compared to reduced-space approaches on a benchmark two-dimensional test case using a high-order discontinuous Galerkin method to discretize the PDE constraint. Multilevel quasi-Monte Carlo for optimization under uncertainty https://zbmath.org/1521.65007 2023-11-13T18:48:18.785376Z "Guth, Philipp A." https://zbmath.org/authors/?q=ai:guth.philipp-a "Van Barel, Andreas" https://zbmath.org/authors/?q=ai:van-barel.andreas Summary: This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the circulant embedding method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and single level quasi-Monte Carlo method. On the local convergence of a third-order iterative scheme in Banach spaces https://zbmath.org/1521.65043 2023-11-13T18:48:18.785376Z "Sharma, Debasis" https://zbmath.org/authors/?q=ai:sharma.debasis "Parhi, Sanjaya Kumar" https://zbmath.org/authors/?q=ai:parhi.sanjaya-kumar Summary: In a Banach space setting, we provide the local convergence analysis of a cubically convergent nonlinear system solver assuming that the first-order Fréchet derivative belongs to the Lipschitz class. The significance of this study is that it avoids the standard practice of Taylor expansion in the analysis of convergence and extends the applicability of the algorithm by using the theory based on the first-order derivative only. Also, our analysis provides the radii of convergence balls and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is studied. Different numerical tests confirm that the new technique produces better results, and it is useful in solving such problems where previous studies fail to solve. Riemannian Hamiltonian methods for min-max optimization on manifolds https://zbmath.org/1521.65051 2023-11-13T18:48:18.785376Z "Han, Andi" https://zbmath.org/authors/?q=ai:han.andi "Mishra, Bamdev" https://zbmath.org/authors/?q=ai:mishra.bamdev "Jawanpuria, Pratik" https://zbmath.org/authors/?q=ai:jawanpuria.pratik "Kumar, Pawan" https://zbmath.org/authors/?q=ai:kumar.pawan "Gao, Junbin" https://zbmath.org/authors/?q=ai:gao.junbin Summary: In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak-Łojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHMs) and present their convergence analyses. We extend RHMs to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHMs in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks. Least-squares finite elements for distributed optimal control problems https://zbmath.org/1521.65125 2023-11-13T18:48:18.785376Z "Führer, Thomas" https://zbmath.org/authors/?q=ai:fuhrer.thomas "Karkulik, Michael" https://zbmath.org/authors/?q=ai:karkulik.michael Summary: We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $$\inf$$-$$\sup$$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented. Robust topology optimization under material and loading uncertainties using an evolutionary structural extended finite element method https://zbmath.org/1521.74180 2023-11-13T18:48:18.785376Z "Latifi Rostami, Seyyed Ali" https://zbmath.org/authors/?q=ai:latifi-rostami.seyyed-ali "Kolahdooz, Amin" https://zbmath.org/authors/?q=ai:kolahdooz.amin "Zhang, Jian" https://zbmath.org/authors/?q=ai:zhang.jian.39 Summary: This research presents a novel algorithm for robust topology optimization of continuous structures under material and loading uncertainties by combining an evolutionary structural optimization (ESO) method with an extended finite element method (XFEM). Conventional topology optimization approaches (e.g. ESO) often require additional post-processing to generate a manufacturable topology with smooth boundaries. By adopting the XFEM for boundary representation in the finite element (FE) framework, the proposed method eliminates this time-consuming post-processing stage and produces more accurate evaluation of the elements along the design boundary for ESO-based topology optimization methods. A truncated Gaussian random field (without negative values) using a memory-less translation process is utilized for the random uncertainty analysis of the material property and load angle distribution. The superiority of the proposed method over Monte Carlo, solid isotropic material with penalization (SIMP) and polynomial chaos expansion (PCE) using classical finite element method (FEM) is demonstrated via two practical examples with compliances in material uncertainty and loading uncertainty improved by approximately 11\% and 10\%, respectively. The novelty of the present method lies in the following two aspects: (1) this paper is among the first to use the XFEM in studying the robust topology optimization under uncertainty; (2) due to the adopted XFEM for boundary elements in the FE framework, there is no need for any post-processing techniques. The effectiveness of this method is justified by the clear and smooth boundaries obtained. Topology optimization of heat transfer and elastic problems based on element differential method https://zbmath.org/1521.74184 2023-11-13T18:48:18.785376Z "Zhang, Si-Qi" https://zbmath.org/authors/?q=ai:zhang.siqi "Xu, Bing-Bing" https://zbmath.org/authors/?q=ai:xu.bingbing "Gao, Zhong-Hao" https://zbmath.org/authors/?q=ai:gao.zhonghao "Jiang, Geng-Hui" https://zbmath.org/authors/?q=ai:jiang.genghui "Zheng, Yong-Tong" https://zbmath.org/authors/?q=ai:zheng.yong-tong "Liu, Hua-Yu" https://zbmath.org/authors/?q=ai:liu.huayu "Jiang, Wen-Wei" https://zbmath.org/authors/?q=ai:jiang.wen-wei "Yang, Kai" https://zbmath.org/authors/?q=ai:yang.kai "Gao, Xiao-Wei" https://zbmath.org/authors/?q=ai:gao.xiaowei (no abstract) Bi-material topology optimization for fully coupled structural-acoustic systems with isogeometric FEM-BEM https://zbmath.org/1521.74209 2023-11-13T18:48:18.785376Z "Chen, L. L." https://zbmath.org/authors/?q=ai:chen.leilei "Lian, H." https://zbmath.org/authors/?q=ai:lian.haojie "Liu, Z." https://zbmath.org/authors/?q=ai:liu.zhaowei "Gong, Y." https://zbmath.org/authors/?q=ai:gong.yuhang|gong.yulin|gong.yinjiao|gong.yafang|gong.yangqing|gong.yingyao|gong.yishu|gong.yunjie|gong.yuezheng|gong.yong|gong.yanlei|gong.yiming|gong.yaling|gong.yafeng|gong.yongzhe|gong.yuexin|gong.yannian|gong.yanpeng|gong.yanglin|gong.yue|gong.yongyi|gong.youhong|gong.yuxuan|gong.yinglan|gong.yanbing|gong.yunbo|gong.yongwang|gong.yankun|gong.yaohua|gong.yanfang|gong.yihe|gong.yancheng|gong.yuyan|gong.yunlei|gong.yuhui|gong.yinghui|gong.yunfan|gong.yun|gong.yuanhao|gong.yuanjiu|gong.yuejin|gong.yan|gong.yuchen|gong.yan-xiao|gong.yonghua|gong.yonghong|gong.yudong|gong.yujie|gong.yuan|gong.yubing|gong.ye|gong.yande|gong.yongxi|gong.yungui|gong.yufei|gong.yanping|gong.yunchao|gong.yishan|gong.yaxin|gong.yunzhan|gong.yanjun|gong.yaoqing|gong.yunye|gong.yihong|gong.yubin|gong.yaonan|gong.yifan|gong.ying|gong.yeming|gong.yijun|gong.yongmin|gong.yuchang|gong.yajun|gong.yucai|gong.yuxin|gong.yuting|gong.yi|gong.yu|gong.youmin|gong.yili|gong.yicheng|gong.yongli "Zheng, C. J." https://zbmath.org/authors/?q=ai:zheng.changjun "Bordas, S. P. A." https://zbmath.org/authors/?q=ai:bordas.stephane-pierre-alain (no abstract) Deep learning driven real time topology optimization based on improved convolutional block attention (Cba-U-Net) model https://zbmath.org/1521.74440 2023-11-13T18:48:18.785376Z "Wang, Lifu" https://zbmath.org/authors/?q=ai:wang.lifu "Shi, Dongyan" https://zbmath.org/authors/?q=ai:shi.dongyang "Zhang, Boyang" https://zbmath.org/authors/?q=ai:zhang.boyang "Li, Guangliang" https://zbmath.org/authors/?q=ai:li.guangliang "Helal, Wasim M. K." https://zbmath.org/authors/?q=ai:helal.wasim-m-k "Qi, Mei" https://zbmath.org/authors/?q=ai:qi.mei (no abstract) Robust preconditioned one-shot methods and direct-adjoint-looping for optimizing Reynolds-averaged turbulent flows https://zbmath.org/1521.76124 2023-11-13T18:48:18.785376Z "Nabi, S." https://zbmath.org/authors/?q=ai:nabi.saleh "Grover, P." https://zbmath.org/authors/?q=ai:grover.piyush "Caulfield, C. P." https://zbmath.org/authors/?q=ai:caulfield.c-p Summary: We compare the performance of direct-adjoint-looping (DAL) and one-shot methods in a design optimization task involving turbulent flow modeled using Reynolds-Averaged-Navier-Stokes equations. Two preconditioned variants of the one-shot algorithm are proposed and tested. The role of an approximate Hessian as a preconditioner for the one-shot method iterations is highlighted. We find that the preconditioned one-shot methods can solve the PDE-constrained optimization problem with the cost of computation comparable (about fourfold) to that of the simulation run alone. This cost is substantially less than that of DAL, which requires $$\mathcal{O}(10)$$ direct-adjoint loops to converge. The optimization results arising from the one-shot method can be used for optimal sensor/actuator placement tasks, or to provide a reference trajectory to be used for online feedback control applications. Adjoint shape optimization coupled with LES-adapted RANS of a U-bend duct for pressure loss reduction https://zbmath.org/1521.76184 2023-11-13T18:48:18.785376Z "Alessi, G." https://zbmath.org/authors/?q=ai:alessi.g "Verstraete, T." https://zbmath.org/authors/?q=ai:verstraete.tom "Koloszar, L." https://zbmath.org/authors/?q=ai:koloszar.lilla "Blocken, B." https://zbmath.org/authors/?q=ai:blocken.bert "van Beeck, J. P. A. J." https://zbmath.org/authors/?q=ai:van-beeck.j-p-a-j Summary: Nowadays, as industrial designs are close to their optimal configurations, the challenge lies in the extraction of the last percentages of improvement. This necessitates accurate evaluations of the performance and represents a significant higher computational cost. The present work aims at integrating Large Eddy Simulations in the optimization framework for an accurate evaluation of the flow field. The number of expensive evaluations is kept to a minimum by using the adjoint method for the evaluation of the gradient of the objective function. Divergence of the gradients due to the chaotic flow motion is avoided by an additional step which decouples the Large Eddy Simulations from the gradient calculations. An adaptation process based on a Reynolds Averaged Navier-Stokes simulation is therefore sought to mimic the more accurate Large Eddy Simulation results. The obtained field is then used in combination with an adjoint shape optimization routine. The method is tested on the design of a U-bend for internal cooling channels by minimizing its pressure loss. Starting from an optimized geometry obtained through a classical approach based on RANS evaluations, further improvements of the design are achieved with the application of the proposed strategy when performances are evaluated by means of LES.