Shape Optimization Problems Research Articles

Analysis of General Shape Optimization Problems in Nonlinear Acoustics

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem, uniformly with respect to shape variations, and prove that it does not degenerate under the hypothesis of small initial and boundary data. Additionally, we prove Hölder-continuity of the acoustic potential with respect to domain deformations. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives.

Reducing memory requirements of unsteady adjoint by synergistically using check‐pointing and compression

AbstractThe unsteady adjoint method used in gradient‐based optimization in 2D and, particularly, 3D industrial problems modeled by unsteady PDEs may have significant storage requirements and/or computational cost. The reason for this is that the backward in time integration of the adjoint equations requires the previously computed instantaneous flow fields to be available at each time‐step. This article proposes remedies to this problem, by extending/upgrading relevant techniques proposed by the group of authors as well as other researchers. Their applicability is wide, even if these remedies are herein demonstrated in shape optimization problems in unsteady fluid mechanics. Check‐pointing is in widespread use as it reduces the memory footprint and CPU cost of the optimization with a controllable computational overhead. Alternatively, flow field time‐series can be stored in a lossless or lossly compressed form. The novelty of this article is the development of a Compressed Coarse‐grained Check‐Pointing strategy for second‐order accurate schemes in time, by optimally combining check‐pointing and lossy compression. The latter includes (a) the incremental Proper Generalized Decomposition (iPGD) algorithm and (b) a hybridization of the iPGD with the ZFP and Zlib algorithms. This is implemented within OpenFOAM, which is used to solve the flow and adjoint equations and conduct the optimization, and assessed in 2D/3D aerodynamic shape optimization problems on unstructured grids. Effectiveness in data reduction, computational cost, and reconstruction accuracy are compared, vis‐à‐vis also to the “standard” binomial check‐pointing technique after adjusting it to second‐order accurate schemes in time.

Shape-supervised Dimension Reduction: Extracting Geometry and Physics Associated Features with Geometric Moments

In shape optimisation problems, subspaces generated with conventional dimension reduction approaches often fail to extract the intrinsic geometric features of the shape that would allow the exploration of diverse but valid candidate solutions. More importantly, they also lack incorporation of any notion of physics against which shape is optimised. This work proposes a shape-supervised dimension reduction approach. To simultaneously tackle these deficiencies, it uses higher-level information about the shape in terms of its geometric integral properties, such as geometric moments and their invariants. Their usage is based on the fact that moments of a shape are intrinsic features of its geometry, and they provide a unifying medium between geometry and physics. To enrich the subspace with latent features associated with shape’s geometrical features and physics, we also evaluate a set of composite geometric moments, using the divergence theorem, for appropriate shape decomposition. These moments are combined with the shape modification function to form a Shape Signature Vector (SSV) uniquely representing a shape. Afterwards, the generalised Karhunen–Loève expansion is applied to SSV, embedded in a generalised (disjoint) Hilbert space, which results in a basis of the shape-supervised subspace retaining the highest geometric and physical variance. Validation experiments are performed for a three-dimensional wing and a ship hull model. Our results demonstrate a significant reduction of the original design space’s dimensionality for both test cases while maintaining a high representation capacity and a large percentage of valid geometries that facilitate fast convergence to the optimal solution. The code developed to implement this approach is available at https://github.com/shahrozkhan66/SSDR.git . • Shape signature vector (SSV) comprises a shape modification vector and geometric moments (GM). • Karhunen–Loève expansion of SSV results in a shape-supervised space (SSS). • GM can induce the notion of physics resulting in a physics-informed SSS. • SSS retains higher geometric variance with fewer latent variables. • SSS is robust and compact, providing accelerated optimisation convergence.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Enhanced Salp Swarm Algorithm for Optimizing the Shape of Developable Bézier-Like Surfaces

The shape optimization of developable surfaces is a key and difficult technology in CAD/CAM. This paper studies the shape optimization of generalized developable Bézier-like surfaces using the improved salp swarm algorithm. First, aiming at the problems of slow convergence speed and low calculation accuracy, quickly falling into the optimal local solution for the salp swarm algorithm, and by adjusting the control parameters, the added Morlet wavelet mutation strategy to the individual position updates operation. An improved salp swarm algorithm is proposed, called FMSSA for short. Numerical results show that FMSSA performed the best among 18 (78%) comparisons with other improved SSAs in the CEC2005 test set. In addition, FMSSA serves the best among 16 (69%) comparisons with other algorithms. Therefore, the convergence speed and computational accuracy of FMSSA are improved, and the computational results are significantly better than other intelligent algorithms. Second, according to the duality principle of point and plane, the shape optimization problem of generalized developable Bézier-like surface is transformed into the minimization problem of arc length, energy, and curvature change rate of the dual curve, and three shape optimization models of the developable surface are established. Finally, shape optimization of expandable surfaces is mathematically an optimization problem that the swarm intelligence algorithm can efficiently handle, so FMSSA is used to solve these three optimization models. Numerical experiments show that FMSSA obtains optimal generalized developable Bézier-like surfaces in solving these three optimization models, demonstrating the superiority of the proposed FMSSA in effectively solving the shape optimization models in terms of precision and robustness.

Shape optimization method for wheel rim of automobile wheels based on load path analysis

The wheel rim is an annular and thin-walled structure with complex structural features and working conditions. Guaranteeing its safety and lightweight are two major concerns in wheel rim design. In this study, a shape optimization method based on load path analysis is proposed to evaluate and optimize the structure of the wheel rim. The load-transfer law of the wheel rim is identified based on the load path visualization. Two design criteria are put forward to evaluate the load-bearing performance and give the improvement suggestions. Then, the shape optimization problem is simplified to a parameter optimization problem, and the optimal size combination performance-oriented is obtained using the non-dominated sorting genetic algorithm II (NSGA-II). As a case study, a wheel rim is redesigned and the optimization result shows that the stiffness and strength are improved under bending and radial fatigue simulation, while the weight is reduced by 3.78%. Two stages of experimental studies are carried out to verify the optimization method. The static pressure test shows that the bending resistance performance of the wheel rim is improved by 4.57%, and the load transfer is more uniform. The radial impact test shows that the impact deformation at a high energy level is reduced by 13.77%, indicating that the impact resistance performance is significantly improved.

An Efficient Numerical Scheme Based on Radial Basis Functions and a Hybrid Quasi-Newton Method for a Nonlinear Shape Optimization Problem

The purpose of this work is to construct a robust numerical scheme for a class of nonlinear free boundary identification problems. First, a shape optimization problem is constructed based on a least square functional. Schauder’s fixed point theorem is manipulated to show the existence solution for the state solution. The existence of an optimal solution of the optimization problem is proved. The proposed numerical scheme is based on the Radial Basis Functions method as a discretization approach, the minimization process is a hybrid Differential Evolution heuristic method and the quasi-Newton method. At the end we establish some numerical examples to show the validity of the theoretical results and robustness of the proposed scheme.

A Phase-Field Approach to Shape and Topology Optimization of Acoustic Waves in Dissipative Media

We investigate the problem of finding the optimal shape and topology of a system of acoustic lenses in a dissipative medium. The sound propagation is governed by a general semilinear strongly damped wave equation. We introduce a phase-field formulation of this problem through diffuse interfaces between the lenses and the surrounding fluid. The resulting formulation is shown to be well-posed and we prove that the corresponding optimization problem has a minimizer. By analyzing properties of the reduced objective functional and well-posedness of the adjoint problem, we rigorously derive first-order optimality conditions for this problem. Additionally, we consider the $\Gamma$-limit of the reduced objective functional and in this way establish a relation between the diffuse interface problem and a perimeter-regularized sharp interface shape optimization problem.

Shape optimization of a thermal insulation problem

We study a shape optimization problem involving a solid Ksubset {mathbb {R}}^n that is maintained at constant temperature and is enveloped by a layer of insulating material Omega which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all (K,Omega ) with prescribed measure for K and Omega , and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on partial Omega ) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

Shape and topology optimization for maximum probability domains in quantum chemistry

This article is devoted to the mathematical and numerical treatments of a shape optimization problem emanating from the desire to reconcile quantum theories of chemistry and classical heuristic models: we aim to identify Maximum Probability Domains (MPDs), that is, domains \(\Omega \) of the 3d space where the probability \({\mathbb {P}}_\nu (\Omega )\) to find exactly \(\nu \) among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional \({\mathbb {P}}_\nu (\Omega )\) arises as the integral over \(\nu \) copies of \(\Omega \) and \((n-\nu )\) copies of the complement \({\mathbb {R}}^3 \setminus \Omega \) of an analytic function defined over the space \({\mathbb {R}}^{3n}\) of all the spatial configurations of the n electron system. Our first task is to explore the mathematical well-posedness of the shape optimization problem: under mild hypotheses, we prove that global maximizers of the probability functions \({\mathbb {P}}_\nu (\Omega )\) do exist as open subsets of \({\mathbb {R}}^3\); meanwhile, we identify the associated necessary first-order optimality condition. We then turn to the numerical calculation of MPDs, for which we resort to a level set based mesh evolution strategy: the latter allows for the robust tracking of complex evolutions of shapes, while leaving the room for accurate chemical computations, carried out on high-resolution meshes of the optimized shapes. The efficiency of this procedure is enhanced thanks to the addition of a fixed-point strategy inspired from the first-order optimality conditions resulting from our theoretical considerations. Several three-dimensional examples are presented and discussed to appraise the efficiency of our algorithms.

Shape Optimization with a Flattening-Based Morphing Method

In shape optimization problems, generating variously shaped designs is an important task. In this study, a new design method called the flattening-based morphing method, which can create various designs efficiently based on baseline objects, is proposed. In the flattening-based method, anchor points are defined for each baseline object to set correspondence among the baseline objects, and each baseline object is mapped to 2D parametric space in a way that places all corresponding anchor points of the baseline objects at the same location. Then, remeshing is carried out to make the baseline objects’ mesh topologically identical in the parametric space. After these remeshed baseline objects are parameterized back to the physical space, the morphed object is created by computing the positions of its vertices as a weighted sum of the baseline meshes’ vertices. When the flattening-based morphing method is applied to find the optimal shape of a blended-wing body aircraft using an artificial neural network (ANN), the aerodynamic performance enhanced optimal model with an appropriate loading capacity is successfully achieved using three baseline models. The simulation results of the baseline models and optimization results are also provided in the current study.

Comparative study of inner–outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

Advanced Krylov subspace methods are investigated for the solution of large sparse linear systems arising from stiff adjoint-based aerodynamic shape optimization problems. A special attention is paid to the flexible inner–outer GMRES strategy combined with most relevant preconditioning and deflation techniques. The choice of this specific class of Krylov solvers for challenging problems is based on its outstanding convergence properties. Typically in our implementation the efficiency of the preconditioner is enhanced with a domain decomposition method with overlapping. However, maintaining the performance of the preconditioner may be challenging since scalability and efficiency of a preconditioning technique are properties often antagonistic to each other. In this paper we demonstrate how flexible inner–outer Krylov methods are able to overcome this critical issue. A numerical study is performed considering either a Finite Volume (FV), or a high-order Discontinuous Galerkin (DG) discretization which affect the arithmetic intensity and memory-bandwidth of the algebraic operations. We consider test cases of transonic turbulent flows with RANS modeling over the two-dimensional supercritical OAT15A airfoil and the three-dimensional ONERA M6 wing. Benefits in terms of robustness and convergence compared to standard GMRES solvers are obtained. Strong scalability analysis shows satisfactory results. Based on these representative problems a discussion of the recommended numerical practices is proposed.

A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches

In this work, we study shape optimization problems in the Stokes flows. By phase-field approaches, the resulted total objective function consists of the dissipation energy of the fluids and the Ginzburg–Landau energy functional as a regularizing term for the generated diffusive interface, together with a Lagrangian multiplier for volume constraint. An efficient decoupled scheme is proposed to implement by the gradient flow approach to decrease the objective function. In each loop, we first update the velocity field by solving the Stokes equation with the phase field variable given in the previous iteration, which is followed by updating the phase field variable by solving an Allen–Cahn-type equation using a stabilized scheme. We then take the cut-off post-processing for the phase-field variable to constrain its value in [0,1]. In the last step of each loop, the Lagrangian parameter is updated with an appropriate artificial time step. We rigorously prove that the proposed scheme permits an unconditionally monotonic-decreasing property. To enhance the overall efficiency of the algorithm, in each loop we update the phase field variable and Lagrangian parameter several time steps but update the velocity field only one time. Numerical results for various shape optimizations are presented to validate the effectiveness of our numerical scheme.

Numerical approximation of optimal convex and rotationally symmetric shapes for an eigenvalue problem arising in optimal insulation

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R3, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

Surrogate Model-Based Parameter Tuning of Simulated Annealing Algorithm for the Shape Optimization of Automotive Rubber Bumpers

A design engineer has to deal with increasingly complex design tasks on a daily basis, for which the available design time is shrinking. Market competitiveness can be improved by using optimization if the design process can be automated. If there is limited information about the behavior of the objective function, global search methods such as simulated annealing (SA) should be used. This algorithm requires the selection of a number of parameters based on the task. A procedure for reducing the time spent on tuning the SA algorithm for computationally expensive, simulation-driven optimization tasks was developed. The applicability of the method was demonstrated by solving a shape optimization problem of a rubber bumper built into air spring structures of lorries. Due to the time-consuming objective function call, a support vector regression (SVR) surrogate model was used to test the performance of the optimization algorithm. To perform the SVR training, samples were taken using the maximin Latin hypercube design. The SA algorithm with an adaptive search space and different cooling schedules was implemented. Subsequently, the SA parameters were fine-tuned using the trained SVR surrogate model. An optimal design was found using the adapted SA algorithm with negligible error from a technical aspect.

Formulation of branched transport as geometry optimization

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.

Optimal shape of stellarators for magnetic confinement fusion

We are interested in the design of stellarators, devices for the production of controlled nuclear fusion reactions alternative to tokamaks. The confinement of the plasma is entirely achieved by a helical magnetic field created by the complex arrangement of coils fed by high currents around a toroidal domain. Such coils describe a surface called “coil winding surface” (CWS). In this paper, we model the design of the CWS as a shape optimization problem, so that the cost functional reflects both optimal plasma confinement properties, through a least square discrepancy, and also manufacturability, thanks to geometrical terms involving the lateral surface or the curvature of the CWS.We completely analyze the resulting problem: on the one hand, we establish the existence of an optimal shape, prove the shape differentiability of the criterion, and provide the expression of the differential in a workable form. On the other hand, we propose a numerical method and perform simulations of optimal stellarator shapes. We discuss the efficiency of our approach with respect to the literature in this area.

A shape design problem for the Navier–Stokes flow with a convective boundary condition

In this study, a shape optimization problem for the two-dimensional stationary Navier–Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the usual do-nothing boundary condition. The goal of the optimization problem is to maximize the vorticity of the said fluid by determining the shape of an obstacle inside the channel. Meanwhile, the shape variation is limited by a perimeter functional and a volume constraint. The perimeter functional was considered to act as a Tikhonov regularizer and the volume constraint is added to exempt us from possible topological changes. The shape derivative of the objective functional was formulated using the rearrangement method, and this derivative was later on used for gradient descent methods. Additionally, an augmented Lagrangian method and a class of solenoidal deformation fields were considered to take into account the goal of volume preservation. Lastly, numerical examples based on gradient descent and the volume preservation methods are presented.

Multi-condition multi-objective optimization using deep reinforcement learning

A novel multi-condition multi-objective optimization method that can find Pareto front over a defined condition space is developed using deep reinforcement learning. Unlike the conventional methods which perform optimization at a single condition, the present method learns correlations between conditions and optimal solutions. The exclusive capability of the developed method is examined in solutions of a modified Kursawe benchmark problem and an airfoil shape optimization problem. The solutions include nonlinear characteristics which are difficult to be resolved using conventional optimization methods. Pareto front with high resolution over a condition space is successfully determined in both problems. Compared with multiple operations of a single-condition optimization method for multiple conditions, the present multi-condition optimization method shows a greatly accelerated search of Pareto front by reducing the required number of function evaluations. An analysis of aerodynamic performance of optimally designed airfoils confirms that multi-condition optimization is indispensable to avoid significant degradation of target performance for varying flow conditions.

Design optimization of metamaterials to control waves in cylindrical rods

Metamaterials can control different modes of waves by tuning their bandgap structures. Various design strategies are implemented to obtain desired responses of metamaterials by solving multiple forward problems using parametric sweeps, topology, and shape optimization problems. This study focuses on the design of cylindrical metamaterials to control wave propagation in a cylindrical rod using a shape optimization approach. Topology optimization requires relatively more computational resources and time as well as it also gives relatively complex structures in comparison with shape optimization. Optimal widths of layered elastic materials arranged periodically along an axis, on subsurface, and an outer surface of cylindrical surface are obtained by solving multiple time-dependent shape optimization problems. Optimization problems are solved using the finite element method and non-gradient optimization algorithm. Single and multi-objective functions are defined to reduce only axial and both axial as well as radial displacements integrated over the end surface, and monochromatic Gaussian input pulse is considered in this study. Every design solution obtained during each design iteration can be easily manufactured as a weak constraint is applied to the total length of the metamaterial. Prior knowledge of possible bandgaps helps to set an effective optimization problem, but it is not a necessary condition.