Abstract: Abstract Isogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff–Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Abstract: Abstract Interest in electric aircraft has increased due to developments in electric propulsion technology and concerns regarding aircraft carbon emissions. The emerging urban air mobility (UAM) industry aims to provide convenient short-range air travel using electric aircraft. An important factor in the design of electric aircraft is the modeling and design of electric motors. The many degrees of freedom in electric motor design makes it a complex design problem. To mitigate this complexity, we have developed a novel adjoint-based electric motor design methodology using finite-element electromagnetic analysis and PDE-based mesh warping with exact derivatives. This paper highlights the approach and details of the proposed method and presents results from its application to a representative motor design problem. The optimization results in a 35\% decrease in motor mass and a 3\% increase in efficiency in comparison to the baseline design. These results demonstrate the efficacy of an adjoint-based optimization approach with exact analytical derivatives for electric motor design.

Abstract: In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by proposing a simple yet powerful approach to combine parameter and shape optimization in a framework using Isogeometric Analysis (IGA).The optimization employs sensitivity analysis by determining the gradients of an objective function with respect to parameters and control points that represent the geometry. The gradients with respect to the control points are calculated in an analytical way using the adjoint method, which enables straightforward shape optimization by altering these control points. Given that a change in a single geometry parameter corresponds to modifications in multiple control points, the chain rule is employed to obtain the gradient with respect to the parameters in an efficient semi-analytical way.The presented method is exemplarily applied to nonlinear 2D magnetostatic simulations featuring a permanent magnet synchronous motor and compared to designs, which were optimized using parameter and shape optimization separately. It is numerically shown that the permanent magnet mass can be reduced and the torque ripple can be eliminated almost completely by simultaneously adjusting rotor parameters and shape. The approach allows for novel designs to be created with the potential to reduce the optimization time substantially.

Abstract: Abstract This study proposes a shape optimisation framework for unsteady electromagnetic scattering problems on the basis of the time-domain boundary integral equation method, focusing on the perfectly electric conductors (PECs). The boundary-only formulation is ideal for treating a shape optimisation problem in exterior domains. However, the electromagnetic shape optimisation in concern has been unrealised with the boundary integral approach regardless of the fact that the boundary-type shape derivative has been known in the literature. The first contribution of the present study is to derive a novel expression of the shape derivative in terms of the surface current densities of the primary and adjoint problems, by considering that the surface current density is handled by usual integral equations methods. The second contribution is to clarify the integral representations and equations of the adjoint electromagnetic fields in terms of the reversal time. These theoretical achievements possess a high affinity with the standard spatial discretising approach (i.e. RWG basis) whenever the temporal basis is sufficiently smooth. The numerical experiments confirmed the reliability of the proposed shape optimisation methodology and indicated the capability to deal with scientific and engineering applications.

Abstract: The purpose of this study is to demonstrate the suitability and efficacy of the adjoint method on the aerodynamic shape optimization on a simple symmetrical airfoil NACA0018 at low Reynolds number for wind turbine application. The adjoint method has been used in many pressure-based numerical simulations with various degrees of success leading to optimized geometries in their respective uses. ANSYS Fluent code was used in this simulation. Lift to drag ratio was defined as the observables for which adjoint sensitivities were formulated. The objective function of the optimization was set to maximize the lift to drag ratio of the airfoil by 20%. The optimization regime showed significant increase in lift and drag ratio from the initial baseline NACA0018 value of 0.0211 up to 3.66 for the optimal NACAOpt. The results demonstrate the potential of the adjoint solver paired together with the gradient-based optimizer to improve the geometry for shape optimization in many CFD applications.

Abstract: The purpose of this study is to demonstrate the suitability and efficacy of the adjoint method on the aerodynamic shape optimization on a simple symmetrical airfoil NACA0018 at low Reynolds number for wind turbine application. The adjoint method has been used in many pressure-based numerical simulations with various degrees of success leading to optimized geometries in their respective uses. ANSYS Fluent code was used in this simulation. Lift to drag ratio was defined as the observables for which adjoint sensitivities were formulated. The objective function of the optimization was set to maximize the lift to drag ratio of the airfoil by 20%. The optimization regime showed significant increase in lift and drag ratio from the initial baseline NACA0018 value of 0.0211 up to 3.66 for the optimal NACAOpt. The results demonstrate the potential of the adjoint solver paired together with the gradient-based optimizer to improve the geometry for shape optimization in many CFD applications.

Abstract: The present thesis is concerned with the topic of shape optimization for the first eigenvalue $d_1$ of a fourth order Steklov problem. Therefore, we first consider a smooth domain $\Omega$ and its concerning family $\Omega_t$ of perturbations are considered. We are interested in both volume preserving perturbations and perimeter preserving perturbations. First, we show the differentiability of the first eigenvalue for smooth domains. Based on this, we compute the first area variation of $d_1$ under measure preserving perturbation or perimeter preserving perturbation for a general smooth domain $\Omega$. In the second part of the thesis, we study the first and second shape variation for the unit ball. We compute the second shape variation for $d_1(B)$ in n dimensions both under the condition of volume conservation and under the condition of perimeter preserving perturbation. Finally, we succeed in showing that in two dimensions the ball maximizes the first eigenvalue under volume preserving perturbation locally. Observing perimeter preserving perturbations we obtain in two dimensions that the circle minimizes the first eigenvalue locally. In the last part of the thesis we restrict our admissible domains to bounded convex domains and show that the first eigenvalue is continuous with respect to Hausdorff convergence.

Abstract: For the numerical solution of shape optimization problems, particularly those constrained by partial differential equations (PDEs), the quality of the underlying mesh is of utmost importance. Particularly when investigating complex geometries, the mesh quality tends to deteriorate over the course of a shape optimization so that either the optimization comes to a halt or an expensive remeshing operation must be performed before the optimization can be continued. In this paper, we present a novel, semi-discrete approach for enforcing a minimum mesh quality in shape optimization. Our approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem. The proposed constraints bound the angles of triangular and solid angles of tetrahedral mesh cells and, thus, also bound the quality of these mesh cells. The method treats these constraints by projecting the search direction to the linear subspace of the currently active constraints. Additionally, only slight modifications to the usual line search procedure are required to ensure the feasibility of the method. We present our method for two- and three-dimensional simplicial meshes. We investigate the proposed approach numerically for the drag minimization of an obstacle in a two-dimensional flow and for the large-scale, three-dimensional optimization of a structured packing used in a distillation column. Our results show that the proposed method is indeed capable of guaranteeing a minimum mesh quality for both academic examples and challenging industrial applications. Particularly, our approach allows the shape optimization of extremely complex structures while ensuring that the mesh quality does not deteriorate.

Abstract: Abstract A set of numerical approaches ranging from the time-consuming unsteady full turbine RANS approach to a very fast periodic steady-stage approach were considered for CFD simulation of flow in Francis pump-turbines. Their computational efforts and accuracy of efficiency prediction were assessed through comparison with experimental data in both pump and turbine modes. It was shown that the unsteady full turbine approach is needed for accurate simulation of low discharge regimes in pump mode. From the other hand, near the optimum and in the high discharge zone, periodic steady-stage approach can be utilized. Next, an approach for multi-objective runner shape optimization of pump-turbine runners was proposed. Thirty free design parameters allowed flexible variation of runner blade shape, hub and band curves, as well as the height of the distributor, linked to the guide vane opening angle. Two or three reference operating points, indicating the turbine performance and head in both turbine and pump modes, were taken for optimization. Cavitation characteristics were accounted for through special constraints. The approach was tested for a 500 m head pump-turbine. It was shown that the optimized runners significantly outperform the initial one in terms of efficiency with comparable cavitation performance.

Abstract: In this paper, we study a shape optimization problem for the torsional energy associated with a domain contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes and show some of its geometric and topological properties. As this issue is closely related to the question of characterizing domains in cylinders that admit solutions to an overdetermined problem, our minimization result allows us to deduce interesting consequences in that direction. In particular, we find that, for some cylinders and some volumes, the ``trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. Moreover, it is not even a minimizer, which indicates that solutions with flat level sets are not always the best candidates for optimizing the torsional energy.

Abstract: The exterior Bernoulli free boundary problem is being considered. The solution to the problem is studied via shape optimization techniques. The goal is to determine a domain having a specific regularity that gives a minimum value for the Kohn-Vogelius-type cost functional while simultaneously solving two PDE constraints: a pure Dirichlet boundary value problem and a Neumann boundary value problem. This paper focuses on the rigorous computation of the first-order shape derivative of the cost functional using the Hölder continuity of the state variables and not the usual approach which uses the shape derivatives of states.

Abstract: Reducing the resistance of compressible flow around a blunt body is of great interest in engineering applications, while an efficient shape optimization method for compressible flows remains far from well established, especially for high Mach numbers. To this end, a volume penalization method for simulating compressible flows past a no-slip and isothermal solid is established by introducing an artificial body force and a heat sink into the governing equations. The level-set functions are introduced as design variables, and the cost functional is defined as the total drag acting on the solid. Then, a continuous adjoint-based shape optimization algorithm for drag reduction is developed by deriving the adjoint equations, the adjoint boundary conditions, and the shape update formula. Both the forward and adjoint simulations are verified by existing studies. The results show that the relative deviations of the drag coefficients obtained in the present study from those reported in the reference studies are around 5% at most, and also a comparable drag reduction rate and also optimal shapes can be reproduced by the present optimization scheme for benchmark problems at relatively low Mach numbers considered in previous studies. Finally, the present method is applied to shape optimization of an initially two-dimensional cylinder and also a three-dimensional sphere in the transonic regime of Ma_∞ = 1.2. The drag reduction of over 20% is achieved for both two-dimensional and three-dimensional cases.

Abstract: Abstract Reducing the resistance of compressible flow around a blunt body is of great interest in engineering applications, while an efficient shape optimization method for compressible flows remains far from well established, especially for high Mach numbers. To this end, a volume penalization method for simulating compressible flows past a no-slip and isothermal solid is established by introducing an artificial body force and a heat sink into the governing equations. The level-set functions are introduced as design variables, and the cost functional is defined as the total drag acting on the solid. Then, a continuous adjoint-based shape optimization algorithm for drag reduction is developed by deriving the adjoint equations, the adjoint boundary conditions, and the shape update formula. Both the forward and adjoint simulations are verified by existing studies. The results show that the relative deviations of the drag coefficients obtained in the present study from those reported in the reference studies are around 5% at most, and also a comparable drag reduction rate and also optimal shapes can be reproduced by the present optimization scheme for benchmark problems at relatively low Mach numbers considered in previous studies. Finally, the present method is applied to shape optimization of an initially two-dimensional cylinder and also a three-dimensional sphere in the transonic regime of Ma ∞ = 1.2. The drag reduction of over 20% is achieved for both two-dimensional and three-dimensional cases.

Abstract: The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

Abstract: Abstract Preform design refers to the intermediate geometric design between the initial tube blank and the desired product. A proper preform design can improve the formability of the tubular structure with small fillets. It is difficult to obtain a reasonable preform design based on engineering experience or trial-and-error methods. In this work, a Non-Uniform Rational B-Splines (NURBS) curve control point inversion algorithm-based approach is introduced to define and parameterize the preform cross-section profiles. The corresponding mathematical optimization model of preform cross-section shape design for tube hydroforming with low pressure is established, and an optimization procedure was accordingly proposed. In this procedure, the relationships between the design variables and responses are formulated by combining the DoE techniques with multiple surrogate models. The k-fold cross-validation strategy is adopted to quantify the performance of the candidate surrogate models on fresh datasets. The selected and verified surrogate models are integrated with the heuristic algorithm to determine the optimal preform shape. Compared to the direct hydroforming process, the optimized preform designs decreased the forming pressure of square, trapezoidal, and irregular polygonal tubes by 89.29%, 70.29%, and 32.67%, respectively. The validity of the proposed method is thus verified.

Abstract: This research focuses on the improvement of porosity distribution within the electrode of an all-vanadium redox flow battery (VRFB) and on optimizing novel cell designs. A half-cell model, coupled with topology and shape optimization framework, is introduced. The multiobjective functional in both cases aims to minimize pressure drop while maximizing reaction rate within the cell. Topology optimization results reveal dependencies on initial value, porosity constraint, and flow rate. The distribution with lower porosity is preferred downstream of the inlet manifold. This design enhances active surface area, thus facilitating more effective conversion of incoming educts and improving mass transport of products. Compared to homogeneous electrodes, two-part design demonstrates superior performance at specific porosity values. For combined porosities of 0.7 and 0.95, optimized distribution results in 81 % reduction in pressure drop, while conversion rate decreases by 7%. As regards various cell designs, optimization suggests a need to reconsider the vertical format of a rectangular cell. Horizontal cells are favored for nearly all porosities and flow rates. Trapezoidal and radial designs characterized by reduced downstream cross sections lead to higher pressure drops and are not preferred. This work provides further valuable insight into optimizing VRFB electrodes and challenges conventional cell design assumptions.

Abstract: PDE-constrained shape optimization has been playing an important role in a large variety of engineering applications. Traditional mesh-dependent shape optimization methods often encounter challenges due to mesh deformation. To address this issue, we present a new adjoint-oriented neural network method, AONN-2, for PDE-constrained shape optimization problems. This method extends the capabilities of the original AONN method [1], which is developed for efficiently solving parametric optimal control problems. AONN-2 inherits from AONN the direct-adjoint looping (DAL) framework for computing the extremum of an objective functional and the involved neural network methods for solving complicated PDEs. Furthermore, AONN-2 expands the application scope to shape optimization by taking advantage of the shape derivatives to optimize the shape represented by discrete boundary points. AONN-2 is a fully mesh-free shape optimization approach, naturally sidestepping issues related to mesh deformation, with no needs for maintaining mesh quality and additional mesh corrections. A series of experimental results are presented, highlighting the flexibility, robustness, and accuracy of AONN-2.

Abstract: The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a known heuristic algorithm. The speed method of shape optimization is applied to obtain necessary optimality conditions for representative test examples. The algorithm originates from the biomimetic approach to compliance optimization. The trabecular bone adapts its form to mechanical loads and is able to form structures that are lightweight and very stiff at the same time. In this sense, it is a problem pertaining to both the nature or living entities which is similar to structural optimization, especially topology optimization. The paper presents the biomimetic approach, based on the trabecular bone remodeling phenomenon, with the aim of minimizing the compliance in multiple load cases. The method employed aims at minimizing the energy and combines structural evolution inspired by trabecular bone remodeling and the shape gradient framework, with strict analysis based on functionals in the 3-dimensional elasticity model. The method is enhanced to handle the problem of structural optimization under multiple loads. The new biomimetic approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Structural evolution is based on shape gradient approximation by the speed method, and it is separated from the finite element method of the model solution. Numerical examples confirm that the heuristic algorithm for structural optimization is efficient.

Abstract: While shape optimization using isogeometric shells exhibits appealing features by integrating design geometries and analysis models, challenges arise when addressing computer-aided design (CAD) geometries comprised of multiple non-uniform rational B-splines (NURBS) patches, which are common in practice. The intractability stems from surface intersections within these CAD models. In this paper, we develop an approach for shape optimization of non-matching isogeometric shells incorporating intersection movement. Separately parametrized NURBS surfaces are modeled using Kirchhoff–Love shell theory and coupled using a penalty-based formulation. The optimization scheme allows shell patches to move without preserving relative location with other members during the shape optimization. This flexibility is achieved through an implicit state function, and analytical sensitivities are derived for the relative movement of shell patches. The introduction of differentiable intersections expands the design space and overcomes challenges associated with large mesh distortion, particularly when optimal shapes involve significant movement of patch intersections in physical space. Throughout optimization iterations, all members within the shell structures maintain the NURBS geometry representation, enabling efficient integration of analysis and design models. The optimization approach leverages the multilevel design concept by selecting a refined model for accurate analysis from a coarse design model while maintaining the same geometry. We adopt several example problems to verify the effectiveness of the proposed scheme and demonstrate its applicability to the optimization of the internal stiffeners of an aircraft wing.

Abstract: Abstract Flexible and compact shape representation schemes are essential for design optimization problems. Current shape representation schemes for coronary stent designs concern predominantly idealized or Independent Ring (IR) designs, which are outdated and only consider a small number of core design variables (such as strut width, height, and thickness) and ignore clinically critical characteristics such as the number of connectors. No reports exist on the geometry parameterization of the latest Helical Stents (HS) that have more complex designs than IR stents. Here, we present two new shape parameterization schemes to fully capture the 3D designs of contemporary IR and double-helix HS stents. We developed a 3D stent geometry builder based on 17 (IR) and 18 (HS) design variables, including strut width, thickness, height, number of connectors and rings, stent length, and strut centerline shape. The shape of the strut centerline was derived via a combination of NURBS, PARSEC, quarter circle, and straight line segments. Shape matching for complex 3D geometries, such as the contemporary stents within limited function evaluations, is not trivial and requires efficient parameterization and optimization algorithms. We used shape matching optimization with a limited function evaluation budget to test the proposed parameterization and two surrogate-assisted optimization algorithms relying on predictor believer and an expected improvement maximization formulation. The performance of these algorithms is objectively compared with a gradient-based optimization method to highlight their strengths. Our work paves the way for more realistic, full-fledged stent design optimization with structural and hemodynamic objectives in the future.