Shape Optimization Problems Research Articles

Discretization-independent node-based shape optimization with the Vertex Morphing method using design variable scaling

AbstractThe Vertex Morphing method is a node-based shape parameterization that uses an explicit filtering approach to regularize the optimization problem and generate smooth shapes. It has been successfully applied to shape optimization problems of industrial size in recent years. This work investigates in detail how irregular discretizations, design surface boundaries, and complex geometries can influence the progress of a gradient-based optimization using the standard Vertex Morphing formulation. A sensitivity weighting approach based on the available shape morphing functions is presented, which eliminates all of the aforementioned influences. Subsequently, a design variable scaling strategy is developed that transforms the optimization problem into an alternative design space and allows the use of arbitrary, even highly irregular surface discretizations in combination with black-box optimization algorithms for shape optimization with the Vertex Morphing method. Illustrative academic examples and an application case of an additively manufactured part are presented to support the work.

Adaptive Free-Form Deformation Parameterization Based on Spring Analogy Method for Aerodynamic Shape Optimization

An adaptive Free-Form Deformation parameterization method based on a spring analogy is presented for aerodynamic shape optimization problems. The proposed method effectively incorporates the gradients of the objective and constraint functions, achieving automatic control point adjustment based on variances in design variable components. To evaluate the performance of the adaptive FFD parameterization method, two 2D airfoil optimization design problems are examined. The optimization of the RAE2822 airfoil with 12, 18 and 24 design variables demonstrates superior results for the adaptive method compared to uniform parameterization. The adaptive method requires fewer iterations and achieves lower objective function values. Additionally, the optimization design from NACA0012 to RAE2822 airfoil with 18 design variables shows that the adaptive parameterization method achieves a lower drag coefficient while satisfying the optimization objective. This validates the method’s capability to finely adjust airfoil shapes and capture more optimal design points by exerting stronger control over local shapes. The proposed adaptive FFD parameterization method proves highly effective for optimizing aerodynamic shapes, offering stability and efficiency in the early stages of optimization, even with a limited number of design variables.

Framework of acoustic analysis and shape optimization for three-dimensional doubly periodic multilayered structures

In this research, a framework of acoustic analysis and shape optimization, based on isogeometric boundary element method (IGA-BEM), is proposed for three-dimensional doubly periodic multilayered structures. The study addresses a gap in the literature by focusing on the shape optimization of such structures, which has not been extensively explored previously. The interface between different acoustic media is an infinite doubly periodic surface, which can be constructed by an open non-uniform rational B-splines. A periodic IGA-BEM is developed for the sound field analysis of the doubly periodic multilayered structure, in which the Ewald method is used to accelerate the calculation of periodic Green function. Furthermore, the shape derivative of the doubly periodic multiple boundaries is derived by imposing boundary perturbation and using the adjoint variable method. The control points of the NURBS surfaces are defined as the shape design variables, and all shape sensitivities can be quickly calculated by discretizing the shape derivative formula. Finally, in according with shape sensitivities, the corresponding shape optimization problem is solved by the method of moving asymptotes, so that the optimized shape design can be obtained. A series of numerical examples validates the accuracy and applicability of the proposed approaches.

Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form −∆u β = f (x, u β) endowed with the Robin boundary conditions ∂ν u β +β(x)u β = 0. The optimisation variable is the function β, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form J (β) =\int_{Ω or ∂Ω} j(u β). We prove that, depending on the monotonicity of the function j, the optimisers may be of bang-bang type (in other words, the optimisers write 1Γ for some measurable subset Γ of ∂Ω) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann (∂ν u = 0) and constant Robin conditions (∂ν u + u = 0) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal β * .

CFD-based shape optimization of structured packings for enhancing separation efficiency in distillation

Free-form shape optimization techniques are investigated to improve the separation efficiency of structured packings in laboratory-scale distillation columns. A simplified simulation model based on computational fluid dynamics (CFD) for the mass transfer in the distillation column is used and a corresponding shape optimization problem is formulated. The goal of the optimization is to increase the mass transfer in the column by changing the packing's shape, which has been previously used as criterion for increasing the separation efficiency of the column. The computational shape optimization yields promising results, with an increased mass transfer of nearly 20%. For validation, the resulting optimized shape is additively manufactured using 3D-printing and investigated experimentally. The experimental results are in good agreement with the performance improvement predicted by the computational model, yielding an increase in separation efficiency of around 20%.

The continuous adjoint to the incompressible (D)DES Spalart-Allmaras turbulence models

This article formulates the continuous adjoint method for the gradient-based shape optimization of fluid flows governed by the incompressible Detached Eddy Simulation (DES) and Delayed-DES (DDES) models, based on the Spalart-Allmaras turbulence model. As both flow models are inherently unsteady, challenges arise regarding the availability of flow fields during the backward in time integration of the unsteady adjoint equations. To minimize both the computational cost and the memory demands, the computed flow fields are compressed using the iPGDZ+ lossy compression technique, recently developed by the authors. Its application in the context of turbulence-resolving flows, where the compression of flow fields poses an intricate challenge, is a second original contribution of this article. Everything is implemented as an extension to the publicly available adjointOptimisation library of OpenFOAM, which is used to solve the flow and adjoint equations and conduct the optimization. Using two shape optimization problems in external aerodynamics, it is demonstrated that including the adjoint to the turbulence model equation is crucial for the computation of accurate sensitivity derivatives. In contrast to sensitivities computed under the “frozen turbulence” assumption, which neglects variations in turbulent viscosity due to changes in the design variables, the proposed adjoint method yields sensitivities that align with those obtained using Finite Differences. This is due to the Think-Discrete Do-Continuous adjoint method which, inspired by hand-differentiated discrete adjoint, gives rise to consistent discretization schemes of the terms involved in the equations derived by continuous adjoint. Furthermore, it is demonstrated that the proposed adjoint method can significantly benefit from the iPGDZ+ algorithm, by reducing memory requirements by more than two orders of magnitude, eliminating the need for flow recomputations, while maintaining the accuracy of the computed derivatives. Ways to handle large integration windows of the objective function with this type of flow models are beyond the scope of this article.

Shape optimization for nonlocal anisotropic energies

In this paper we consider shape optimization problems for sets of prescribed mass, where the driving energy functional is nonlocal and anisotropic. More precisely, we deal with the case of attractive/repulsive interactions in two and three dimensions, where the attraction is quadratic and the repulsion is given by an anisotropic variant of the Coulomb potential. Under the sole assumption of strict positivity of the Fourier transform of the interaction potential, we show the existence of a threshold value for the mass above which the minimizer is an ellipsoid, and below which the minimizer does not exist. If, instead, the Fourier transform of the interaction potential is only non-negative, we show the emergence of a dichotomy: either there exists a threshold value for the mass as in the case above, or the minimizer is an ellipsoid for any positive value of the mass.

Mathematical derivation of a unified equations for adjoint lattice Boltzmann method in airfoil and wing aerodynamic shape optimization

Unified equations of the adjoint lattice Boltzmann method (ALBM) are derived for five applicable objective functions in 2D/3D aerodynamic shape optimization problems. The derived equations include the adjoint equation, boundary condition, terminal condition and gradient of the cost function. In this research, firstly, these relations are extracted for each objective in details and then the general form of ALBM equations are presented for all defined practical aerodynamic objective function. Five applicable cost functions which are the most important objectives in optimization of aerodynamic geometries include desired pressure and viscous shear stress (VSS) inverse design, drag and moment at fixed lift and finally lift to drag ratio at fixed angle of attack. The new extracted relations are based on the circular and spherical function scheme, and are valid for viscous/inviscid, compressible/incompressible and 2D/3D flows in all continuous flow regimes. Proof of new extracted general relations have been performed by authors.

Acoustic shape optimization using energy stable curvilinear finite differences

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inverting for the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance.

Nacelle optimisation through multi-fidelity neural networks

PurposeAerodynamic shape optimisation is a complex problem usually governed by transonic non-linear aerodynamics, a high dimensional design space and high computational cost. Consequently, the use of a numerical simulation approach can become prohibitive for some applications. This paper aims to propose a computationally efficient multi-fidelity method for the optimisation of two-dimensional axisymmetric aero-engine nacelles.Design/methodology/approachThe nacelle optimisation approach combines a gradient-free algorithm with a multi-fidelity surrogate model. Machine learning based on artificial neural networks (ANN) is used as the modelling technique because of its ability to handle non-linear behaviour. The multi-fidelity method combines Reynolds-averaged Navier Stokes and Euler CFD calculations as high- and low-fidelity, respectively.FindingsRatios of low- and high-fidelity training samples to degrees of freedom of nLF/nDOFs = 50 and nHF/nDOFs = 12.5 provided a surrogate model with a root mean squared error less than 5% and a similar convergence to the optimal design space when compared with the equivalent CFD-in-the-loop optimisation. Similar nacelle geometries and aerodynamic flow topologies were obtained for down-selected designs with a reduction of 92% in the computational cost. This highlights the potential benefits of this multi-fidelity approach for aerodynamic optimisation within a preliminary design stage.Originality/valueThe application of a multi-fidelity technique based on ANN to the aerodynamic shape optimisation problem of isolated nacelles is the key novelty of this work. The multi-fidelity aspect of the method advances current practices based on single-fidelity surrogate models and offers further reductions in computational cost to meet industrial design timescales. Additionally, guidelines in terms of low- and high-fidelity sample sizes relative to the number of design variables have been established.

The continuous adjoint method to the γ−R˜eθt$$ \gamma -\tilde{R}{e}_{\theta t} $$ transition model coupled with the Spalart–Allmaras model for compressible flows

AbstractThe continuous adjoint method for transitional flows of compressible fluids is developed and assessed, for the first time in the literature. The gradient of aerodynamic objective functions (aerodynamic forces) with respect to design variables, in problems governed by the compressible Navier–Stokes equations coupled with the Spalart–Allmaras turbulence model and the transition model (in three, non‐smooth and smooth, variants of it), is computed based on the continuous adjoint method. The development of the adjoint to the smooth transition model variant proved to be beneficial. The accuracy of the computed sensitivity derivatives is verified against finite differences. Programming is performed in an in‐house, vertex‐centered finite‐volume code, efficiently running on GPUs. The proposed continuous adjoint method is used in 2D and 3D aerodynamic shape optimization problems, namely the constrained optimization of the NLF(1)–0416 isolated airfoil and that of the ONERA M6 wing. The impact of “frozen transition” (assumption according to which the adjoint to the transition model equations are not solved) or “frozen turbulence” (by additionally ignoring the adjoint to the turbulence model) are evaluated; it is shown that both lead to inaccurate sensitivities.

Scalar Field Prediction on Meshes Using Interpolated Multi-Resolution Convolutional Neural Networks

Abstract Scalar fields, such as stress or temperature fields, are often calculated in shape optimization and design problems in engineering. For complex problems where shapes have varying topology and cannot be parametrized, data-driven scalar field prediction can be faster than traditional finite element methods. However, current data-driven techniques to predict scalar fields are limited to a fixed grid domain, instead of arbitrary mesh structures. In this work, we propose a method to predict scalar fields on arbitrary meshes. It uses a convolutional neural network whose feature maps at multiple resolutions are interpolated to node positions before being fed into a multilayer perceptron to predict solutions to partial differential equations at mesh nodes. The model is trained on finite element von Mises stress fields, and once trained it can estimate stress values at each node on any input mesh. Two shape datasets are investigated, and the model has strong performance on both, with a median R-squared value of 0.91. We also demonstrate the model on a temperature field in a heat conduction problem, where its predictions have a median R-squared value of 0.99. Our method provides a potential flexible alternative to finite element analysis in engineering design contexts. Code and datasets are available at: https://github.com/kevinferg/sfp-cnn.

Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics

We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape.First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in C1,1 sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the C1,α sense, for every α<1. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest.

AONN-2: An adjoint-oriented neural network method for PDE-constrained shape optimization

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.

A mid-range approximation method assisted by trust region strategy for aerodynamic shape optimization

This paper presents an efficient solution for high-fidelity large-scale aerodynamic shape optimization problems based on several developments in the mid-range approximation method within a trust region optimization framework. The mid-range approximation method is an iterative optimization technique that utilizes mid-range approximations to replace the physical experiments or simulations during the optimization based on the selected trust region strategy. It could transform the original optimization problem into a sequence of approximate sub-optimization problems. In this work, an improved trust region strategy is proposed to contain more optimization states with a flexible and controllable performance to suit different types of problems. A metamodel assembly technique and its gradient-enhanced version are developed to further relax the requirements of computational costs in the mid-range approximation method. Its performance is discussed through a detailed comparison of metamodel performance using a mathematical benchmark case named Vanderplaats scalable beam. The wing only of the Common Research Model is offered to the proposed method for aerodynamic shape optimization. The optimization has 1 design objective, 232 design variables, and 135 design constraints. With all constraints satisfied, the optimized configuration has a 4.85 % improvement in wing drag performance. The shock region is greatly reduced and the wing pressure distribution is smooth and nearly parallel. These results show that the proposed method could achieve the design goal successfully within a reasonable computational cost for large-scale problems.

Shape optimization for interface identification in nonlocal models

Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape).

Multi-fidelity deep learning for aerodynamic shape optimization using convolutional neural network

Aerodynamic shape design is essential for improving aircraft performance and efficiency. First, this study introduces a data-driven optimization framework utilizing a multi-fidelity convolutional neural network (MFCNN) for aerodynamic shape optimization. To achieve better optimization results with reduced computational cost, the framework dynamically incorporates new data in each optimization cycle. Specifically, it constantly involves the optimal solution from previous cycle as a new high-fidelity sample and employs a low-fidelity infilling strategy that maximizes the minimum Euclidean distance for selecting new low-fidelity samples. Moreover, a standard synthetic benchmark is used to elaborate the procedure of optimization and show the capability and effectiveness of the framework. Finally, the framework is applied to two aerodynamic shape optimization problems: maximizing the lift-to-drag ratio for the Royal Aircraft Establishment 2822 (RAE2822) airfoils and minimizing the cruise drag coefficient for the three-dimensional (3D) drooped and scarfed non-axisymmetric nacelles. The framework increases the lift-to-drag ratio by 51.21% over the baseline and achieves an 18.79% reduction in the cruise drag coefficient for nacelle optimization, outperforming traditional multi-fidelity deep neural network optimization framework. Sufficiently utilizing the implicit relations between different fidelity levels of data through defined local perceptual fields and convolution, our MFCNN-based optimization framework signifies a step forward in the efficiency and accuracy of aerodynamic shape optimization.

Designing gradient coils with the shape derivative and the closed B-spline curves

This study proposes a versatile and efficient optimisation method for discrete coils that induce a magnetic field by their steady currents. The prime target is gradient coils for MRI (Magnetic Resonance Imaging). The derivative (gradient) of the z-component the magnetic field, which is calculated by the Biot–Savart's law, with respect to the z-coordinate in the Cartesian xyz coordinate system is considered as the objective function. Then, the derivative of the objective function with respect to a change of coils in shape is formulated according to the concept of shape optimisation. The resulting shape derivative (as well as the Biot–Savart's law) is smoothly discretised with the closed B-spline curves. In this case, the control points (CPs) of the curves are naturally selected as the design variables. As a consequence, the shape derivative is discretised to the sensitivities of the objective function with respect to the CPs. Those sensitivities are available to solve the present shape-optimisation problem with a certain gradient-based nonlinear-programming solver. The numerical examples exhibit the mathematical reliability, computational efficiency, and engineering applicability of the proposed methodology based on the shape derivative/sensitivities and the closed B-spline curves.

Variable-Fidelity Surrogate Model-Based Airfoil Optimization at a Moderate Reynolds Number

Airfoil shape optimization for enhanced aerodynamic performance at a moderate Reynolds number is carried out using a novel technique for constrained optimization of expensive engineering functions. The proposed method is a co-kriging surrogate model using two levels of fidelity in conjunction with a gradient-free trust-region method to drive the model towards the global optimum. The methodology is first applied on a generic test function to demonstrate its efficiency. Later, the proposed optimization framework is applied on constrained airfoil shape optimization problems involving maximizing the lift-to-drag ratio, maximizing the endurance factor, and minimizing the drag coefficient of an Eppler E214 airfoil. The optimal lift-to-drag ratio and the endurance factor are found to be 13.25% and 16.05%, respectively, higher than the baseline airfoil. The optimal drag coefficient is 6.86% lower than the baseline airfoil. The flow transition is also delayed for the some of the optimal airfoils. The present results show that the proposed optimization methodology is successful in improving the aerodynamic characteristics in the sensitive low/moderate Reynolds number regime.

The boundary Harnack principle on optimal domains

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.