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In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an Lp best approximation problem under divergence constraints to the shape tensor introduced by Laurain and Sturm [ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 1241-1267]. More precisely, the main result of this paper states that the Lp distance of the above approximation problem is equal to the dual norm of the shape derivative considered as a functional on W1,p* (where 1/p + 1/p* = 1). This implies that for any given shape, one can evaluate its distance from being stationary with respect to the shape derivative by simply solving the associated Lp-type least mean approximation problem. Moreover, the Lagrange multiplier for the divergence constraint turns out to be the shape deformation of steepest descent. This provides, as an alternative to the approach by Deckelnick, Herbert, and Hinze, a way to compute shape gradients in W1,p* for p* ∈ (2, ∞). The discretization of the least mean approximation problem is done with (lowest-order) matrix-valued Raviart--Thomas finite element spaces leading to piecewise constant approximations of the shape deformation acting as a Lagrange multiplier. Admissible deformations in W1,p* to be used in a shape gradient iteration are reconstructed locally. Our computational results confirm that the Lp distance of the best approximation does indeed measure the distance of the considered shape to optimality. Also confirmed by our computational tests is the observation of Deckelnick, Herbert, and Hinze [ESAIM Control Optim. Calc. Var., 28 (2022), 2] that choosing p* (much) larger than 2 (which means that p must be close to 1 in our best approximation problem) decreases the chance of encountering mesh degeneracy during the shape gradient iteration. [ABSTRACT FROM AUTHOR]

We consider numerical shape optimization of a fluid–structure interaction model. The constrained system involves multiscale coupling of a two-dimensional unsteady Navier–Stokes equation and a one-dimensional ordinary differential equation for fluid flows and structure, respectively. We derive shape gradients for both objective functionals of least-squares type and energy dissipation. The state and adjoint state equations are numerically solved on the time-dependent domains using the Arbitrary-Lagrangian–Eulerian method. Numerical results are presented to illustrate effectiveness of algorithms. [ABSTRACT FROM AUTHOR]

We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see R. Herzog and E. Loayza-Romero, Math. Comput. 92 (2022) 1-50. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of a globally optimal solution for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation. [ABSTRACT FROM AUTHOR]

The spontaneous directional transportation (SDT) of water and gas has functions such as efficient water collection, enhanced heat transfer, underwater drag reduction, and so on, having great application prospects in aerospace and navigation fields. Therefore, it is important to efficiently prepare spontaneous directional water droplet transportation (SDWT) surfaces and spontaneous directional gas bubble transportation (SDBT) surfaces and apply them in different fields. In recent years, researchers have used biological structures as the basis for their studies and have continued to analyze the SDT transport mechanism in depth, aiming to find more efficient transportation methods. In this review, we first summarize the important basic theories related to fluid transportation. Then, the related methods and the limitations corresponding to SDWT and SDBT are introduced and discussed. In addition, we review the applications of SDWT and SDBT. Finally, we highlight the challenges and future perspectives of SDWT and SDBT. [ABSTRACT FROM AUTHOR]

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims to explain several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered. [ABSTRACT FROM AUTHOR]

This work aims at investigating the convergence of shape gradients for shape optimizations governed by the Stokes-Brinkman equations. Two types of optimal control problems are considered, the shape inverse problem and the dissipated energy minimization problem, where the distinction between these two problems lies in the difference of the objective functionals. Lagrangian functional is introduced to obtain the adjoint equations and Eulerian derivatives at a fixed domain Ω in the direction V in volume and boundary integral forms are derived by the Piola material derivative approach and function space parametrization technique. Mixed finite element method is applied to discretize both the state and adjoint equations, as well as the corresponding Eulerian derivatives. The optimal error estimates for both forms of the shape derivatives are obtained. We infer that the volume-based shape derivative has a convergence rate of order h 2 , while the boundary-based one has a convergence rate of order h | log ⁡ h | under the MINI element. Numerical experiments are reported to demonstrate the theoretical analysis and indicate the better accuracy of volume-based expressions. [ABSTRACT FROM AUTHOR]

For shape optimization of fluid flows governed by the Navier–Stokes equation, we investigate effectiveness of shape gradient algorithms by analyzing convergence and accuracy of mixed finite element approximations to both the distributed and boundary types of shape gradients. We present convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the distributed formulation has superconvergence property. Numerical results with comparisons are presented to verify theory and show that the shape gradient algorithm based on the distributed formulation is highly effective and robust for shape optimization. [ABSTRACT FROM AUTHOR]

The spontaneous directional transportation (SDT) of water and gas has functions such as efficient water collection, enhanced heat transfer, underwater drag reduction, and so on, having great application prospects in aerospace and navigation fields. Therefore, it is important to efficiently prepare spontaneous directional water droplet transportation (SDWT) surfaces and spontaneous directional gas bubble transportation (SDBT) surfaces and apply them in different fields. In recent years, researchers have used biological structures as the basis for their studies and have continued to analyze the SDT transport mechanism in depth, aiming to find more efficient transportation methods. In this review, we first summarize the important basic theories related to fluid transportation. Then, the related methods and the limitations corresponding to SDWT and SDBT are introduced and discussed. In addition, we review the applications of SDWT and SDBT. Finally, we highlight the challenges and future perspectives of SDWT and SDBT.

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims to explain several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered.

We propose in this paper two kinds of continuity preserving discrete shape gradients of boundary type for PDE-constrained shape optimizations. First, a modified boundary shape gradient formula for shape optimization problems governed by elliptic Dirichlet problems was proposed recently based on the discrete variational outward normal derivatives. The advantages of this new formula over the previous one lie in the improved numerical accuracy and the continuity along the boundary. In the current paper we generalize this new formula to other shape optimization problems including the Laplace and Stokes eigenvalue optimization problems, the shape optimization of Stokes or Navier--Stokes flows, and the interface identification problems. We verify this new formula's numerical accuracy in different shape optimization problems and investigate its performance in several popular shape optimization algorithms. The second contribution of this paper is to propose a continuous discrete shape gradients of boundary type for Neumann problems, by using the ideas of gradient recovery techniques. The continuity property of the discrete boundary shape gradient is helpful in certain shape optimization algorithms and provides certain flexibility compared to the previous discontinuous ones, which are extensively discussed in the current paper. [ABSTRACT FROM AUTHOR]

Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be more specific, the boundary type shape gradient is more popular in practice due to its concise formulation and convenience in combining with shape optimization algorithms but has lower numerical accuracy. In this paper we provide a simple yet useful boundary correction for the normal derivatives of the state and adjoint equations, motivated by their continuous variational forms, to increase the accuracy and possible effectiveness of the boundary shape gradient in PDE-constrained shape optimization. We consider particularly the state equation with Dirichlet boundary conditions and provide a preliminary error estimate for the correction. Numerical results show that the corrected boundary type shape gradient has comparable accuracy to that of the distributed one. Moreover, we give a theoretical explanation for the comparable numerical accuracy of the boundary type shape gradient with that of the distributed shape gradient for Neumann boundary value problems. [ABSTRACT FROM AUTHOR]

We consider scalar-valued shape functionals on sets of shapes which are small perturbations of a reference shape. The shapes are described by parameterizations and their closeness is induced by a Hilbert space structure on the parameter domain. We justify a heuristic for finding the best low-dimensional parameter subspace with respect to uniformly approximating a given shape functional. We also propose an adaptive algorithm for achieving a prescribed accuracy when representing the shape functional with a small number of shape parameters. [ABSTRACT FROM AUTHOR]

This paper concerns the accuracy of Galerkin finite element approximations to two types of shape gradients for eigenvalue optimization. Under certain regularity assumptions on domains, a priori error estimates are obtained for the two approximate shape gradients. Our convergence analysis shows that the volume integral formula converges faster and offers higher accuracy than the boundary integral formula. Numerical experiments validate the theoretical results for the problem with a pure Dirichlet boundary condition. For the problem with a pure Neumann boundary condition, the boundary formulation numerically converges as fast as the distributed type. [ABSTRACT FROM AUTHOR]

In this article, we study the shape sensitivity of optimal control for the steady Stokes problem. The main goal is to obtain a robust representation for the derivatives of optimal solution with respect to smooth deformation of the flow domain. We introduce in this paper a rigorous proof of existence of the material derivative in the sense of Piola, as well as the shape derivative for the solution of the optimality system. We apply these results to derive the formulae for the shape gradient of the cost functional; under some regularity conditions the shape gradient is given according to the structure theorem by a function supported on the moving boundary, then the numerical methods for shape optimization can be applied in order to solve the associated optimization problems. [ABSTRACT FROM AUTHOR]

We study a new framework for a geometric inverse problem in linear elasticity. This problem concerns the recovery of cavities from the knowledge of partially overdetermined boundary data. The boundary data available for the reconstruction are given by the displacement field and the normal component of the normal stress, whereas there is lack of information about the shear stress. We propose an identification method based on a Kohn-Vogelius error functional combined with the shape gradient method. We put special focus on the identification of cavities and prove uniqueness for the case of monotonous cavities.

Abstract In this paper, we propose an imaging technique for the detection of porous inclusions in a stationary flow governed by Stokes–Brinkmann equations. We introduce the velocity method to perform the shape deformation, and derive the structure of shape gradient for the cost functional based on the continuous adjoint method and the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape inverse problem. The numerical results demonstrate the proposed algorithm is feasible and effective for the quite high Reynolds numbers problems. [ABSTRACT FROM AUTHOR]

Abstract Eulerian derivatives of shape functionals in shape optimization can be written in two formulations of boundary and volume integrals. The former is widely used in shape gradient descent algorithms. The latter holds more generally, although rarely being used numerically in literature. For shape functionals governed by the Stokes equation, we consider the mixed finite element approximations to the two types of shape gradients from corresponding Eulerian derivatives. The standard MINI and Taylor–Hood elements are employed to discretize the state equation, its adjoint and the resulting shape gradients. We present thorough convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the volume integral formula has superconvergence property. Numerical results are presented to verify the theory and show that the volume formulation is more accurate. Highlights • We present convergence analysis of approximate shape gradients in Stokes equation. • Both volume and boundary formulations of shape gradients are considered. • MINI and Taylor–Hood elements are used to discretize shape gradients. • The volume formulation convergences faster and offers better accuracy. [ABSTRACT FROM AUTHOR]

The identification of Lamé parameters in linear elasticity is considered. An optimum experimental design problem is formulated, which aims at minimizing the size of an associated confidence ellipsoid by optimizing the shape of the specimen. Representations of the Eulerian shape derivative and the shape gradient are derived by means of shape calculus and adjoint techniques. Numerical experiments are conducted, yielding specimens of improved shape. [ABSTRACT FROM AUTHOR]

Fog harvesting of the Namib desert beetles has inspired many researchers to design artificial fog harvesting hybrid surfaces, which commonly involve flat hydrophilic patterns on hydrophobic surfaces. However, relatively less interest has been shown in the bumpy topography of the Namib desert beetle’s dorsal surface as well as its curved body shape when designing artificial hybrid surfaces. In this work, we explore a fog harvesting flexible hybrid surface that has a superhydrophilic 3D copper oxide pattern on a hydrophobic rough elastomer background surface enabled by transferring a copper layer from a prepared donor substrate to a receiving elastomer substrate. The water collection rates of the hybrid surface and control samples are measured, and the results reveal the advantages of 3D bumpy structures on a curved shape surface to facilitate fog harvesting, particularly in more unfavorable fog stream conditions. The curved 3D bumpy hybrid surface exhibits an over 16 times higher water collection rate than the flat 2D hybrid surface in the fog stream in parallel to the hybrid surface. This work provides an improved understanding of the role of the Namib desert beetle’s bumpy dorsal surface and curved body shape, and offers an insight into the design of novel surfaces with enhanced fog harvesting performance.

We consider to solve numerically the shape optimization models with Dirichlet Laplace eigenvalues. Both volume-constrained and volume unconstrained formulations of the model problems are presented. Different from the literature using boundary-type Eulerian derivatives in shape gradient descent methods, we advocate to use the more general volume expressions of Eulerian derivatives. We present two shape gradient descent algorithms based on the volume expressions. Numerical examples are presented to show the more effectiveness of the algorithms than those based on the boundary expressions. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes-Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient-type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study a new framework for a geometric inverse problem in linear elasticity. This problem concerns the recovery of cavities from the knowledge of partially overdetermined boundary data. The boundary data available for the reconstruction are given by the displacement field and the normal component of the normal stress, whereas there is lack of information about the shear stress. We propose an identification method based on a Kohn-Vogelius error functional combined with the shape gradient method. We put special focus on the identification of cavities and prove uniqueness for the case of monotonous cavities. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain Ω from partial boundary measurements in the two dimensional case. The fluid ow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape. [ABSTRACT FROM AUTHOR]

A technique of the shape optimization of radiant enclosures with specular-diffuse surfaces is proposed. The shape optimization problem is formulated as an operator equation of the first kind with respect to a surface to be optimized. The operator equation is reduced to a minimization problem for a least-squares objective shape functional. The minimization problem is solved by a combination of the pure random (or blind) search (the simplest stochastic minimization method) and the conjugate gradient method. The random search is used to find a starting point for the gradient method. The latter needs the gradient of the objective functional. The shape gradient of the objective functional is derived by means of the shape sensitivity analysis and the adjoint problem method. Eventually, the shape gradient is obtained as a result of solving the direct and adjoint problems. If a surface to be optimized is given by a finite number of parameters, then the objective functional becomes a function in a finite-dimensional space and the shape gradient becomes an ordinary gradient. Numerical examples of the shape optimization of “two-dimensional” radiant enclosures with polyhedral specular or specular-diffuse surfaces are given. [ABSTRACT FROM AUTHOR]

In this article, we consider the shape optimization problem of a body immersed in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model. Based on the continuous adjoint method, we formulate and analyze the shape optimization problem. Then we derive the structure of shape gradient for the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape optimization problem. Finally, numerical examples demonstrate the feasibility and effectiveness of the proposed algorithm. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1700-1715, 2014 [ABSTRACT FROM AUTHOR]

Abstract: Light diffraction by biperiodic grating structures can be simulated by a boundary value problem of the equation for the electric field u. To optimize the geometry parameters of the grating, a quadratic functional of u is defined. The minimization of this functional by gradient based optimization schemes requires shape derivatives of the functional with respect to the geometry parameters. However, a simple application of classical shape calculus is not possible since the energy space for the electric fields is not invariant with respect to the transformation of geometry. In a recent paper, Hettlich (2012) [15] has proposed to replace the electric field by a simple transform which leads to a differentiable vector field in the energy space. We follow here a different approach. For constant magnetic permeability, the magnetic field is piecewise in . Applying the shape calculus to the magnetic field equation, substituting the magnetic field by the curl of the electric field, and employing some technical transformations, we derive stable formulas for the material derivatives depending on the electric field. Numerical tests confirm the formulas. [Copyright &y& Elsevier]

The flow around a rigid obstacle is governed by the compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain in two spatial dimensions with a compact obstacle in its interior. The flight of the airflow is characterized by the work shape functional, to be minimized over a family of admissible obstacles. The lift of the airfoil is a given function of temporal variable and should be maintain closed to the flight scenario. The continuity of the work functional with respect to the shape of obstacle in two spatial dimensions is shown for a wide class of admissible obstacles compact with respect to the Kuratowski-Mosco convergence. The dependence of small perturbations of approximate solutions to the governing equations with respect to the boundary variations of obstacles is analyzed for the nonstationary state equation. [ABSTRACT FROM AUTHOR]

We introduce a new class of data-fitting energies that couple image segmentation with image restoration. These functionals model the image intensity using the statistical framework of generalized linear models. By duality, we establish an information-theoretic interpretation using Bregman divergences. We demonstrate how this formulation couples in a principled way image restoration tasks such as denoising, deblurring (deconvolution), and inpainting with segmentation. We present an alternating minimization algorithm to solve the resulting composite photometric/geometric inverse problem. We use Fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates. We derive the shape gradient of our data-fitting energy and investigate convex relaxation for the geometric problem. We introduce a new alternating split-Bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real-world images. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present a technique for shape reconstruction based on the topological and shape gradients. The shape in consideration is a solution of an inverse conductivity problem. To solve such a problem numerically, we compute the topological gradient of a Kohn–Vogelius-type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole. The reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation. Then, we use a gradient-type algorithm to perform a good reconstruction. Various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm. [Copyright &y& Elsevier]

We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design. [ABSTRACT FROM AUTHOR]

This article tackles the problem of using variational methods for evolving 3D deformable surfaces. We give an overview of gradient descent flows when the shape is represented by a triangular mesh-based surface, and we detail the gradients of two generic energy functionals which embody a number of energies used in mesh processing and computer vision. In particular, we show how to rigorously account for visibility in the surface optimization process. We present different applications including 3D reconstruction from multiple views for which the visibility is fundamental. The gradient correctly takes into account the visibility changes that occur when a surface moves; this forces the contours generated by the reconstructed surface to match with the apparent contours in the input images. [ABSTRACT FROM AUTHOR]