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Fine pitch microprobe arrays are microneedle-like probes for inspecting the pixels of LCD panels or IC. They are usually made of multi-layers of metallic, nonmetallic, or combination of the two. The design requirement for a contacting force is less than 2 gf and a deflection should be less than 100 µm. Many microprobe shapes satisfying the design requirements are possible. A cantilever-type microprobe having many needles was chosen and optimized in this study. Several candidate shapes were chosen using topology and shape optimization technique subjected to design requirements. Then, the microprobe arrays were fabricated using the process applied for MEMS fabrication and they were made of BeNi, BeCu, or Si. The contact probing forces and deflections were measured for checking the results from optimum design by newly developed measuring equipment in our laboratory. Numerical and experimental results were compared and both showed a good correlation.

In this paper we present a new approach in the study of Aorto–Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor-like patch is found. A feedback procedure with Navier–Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI.

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology of several engineering problems. This derivative provides the sensitivity of a problem when a small hole is created at each point of the domain under consideration. In the present work the Topological Derivative for Poisson's problem is calculated using two different approaches: the Domain Truncation Method and a new method based on Shape Sensitivity Analysis concepts. By comparing both approaches it will be shown that the novel approach, which we call Topological-Shape Sensitivity Method, leads to a simpler and more general methodology. To point out the general applicability of this new methodology, the most general set of boundary conditions for Poisson's problem, Dirichlet, Neumann (both homogeneous and nonhomogeneous) and Robin boundary conditions, is considered. Finally, a comparative analysis of these two methodologies will also show that the Topological-Shape Sensitivity Method has an additional advantage of being easily extended to other types of problems.

In this paper an optimal shape control problem dealing with heat transfer in enclosures is studied. We have considered an enclosure heated by a flame surface (taking into account radiation, conduction and convection effects), and we look for an optimal flame shape which minimizes a cost functional defined on the temperature field. This kind of problem arises in industrial furnaces optimization, as temperature uniformity is one of the most important aspects in industrial plant analysis and design. Analytical results (smoothness of the control-to-state mapping, existence of an optimal shape in a certain admissible class) as well as numerical optimization results by the boundary element method have been obtained; we have employed the gradient method to optimize the flame shape, exploiting the adjoint equation associated with our state equation and cost functional.

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

The purpose of this paper is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a "small" uncertainty exists over some of its features. The main idea of the method is to linearize the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, which can be rewritten as a more "classical" function of the design, owing to standard adjoint techniques from optimal control theory. The resulting "linearized worst-case" objective function turns out to be the sum of the initial cost function and of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations. This formal approach is very general, and can be justified in some special cases. In particular, it allows to address several problems of considerable importance in both parametric and shape optimization of elastic structures, in a unified framework.

An algorithm for aerodynamic shape optimization using a gradient-based approach is presented. A stabilized finite element method is utilized to solve the equations governing the fluid flow and adjoint variables. The effectiveness of the method in finding nontrivial designs that satisfy the posed objectives and constraints is demonstrated. It is shown that the initial guess plays an important role in the optimal solution that is realized. Shapes of bumps that are associated with a specified volume, lift and drag are determined and presented. The method is applied to design of airfoils in unsteady flows. The objective function is based on time-averaged force coefficients. The effect of enriching the design space is studied. It is shown that this being a local method, where the search direction is based on gradient of the objective function, the gradual enrichment of design space leads to superior performance. The idea is demonstrated via spontaneous appearance of corrugations on an airfoil surface during the optimization, for maximum lift, by gradually increasing the number of control points. The method is extended to three dimensions. Its application is demonstrated via optimization of the planform of a finite wing for maximum lift-to-drag ratio. A bi-parametric tensorial NURBS (non-uniform rational bi-cubic spline) surface is interpolated on a 3D control net to represent a wing surface. It is shown that for low Reynolds number, it is possible to design a planform that is more efficient than an ellipse. Unlike the elliptic planform, the optimal wing computed by the present method, is associated with a short winglet-like structure at the wing-tip and the maximum chord length at around mid-span.

This paper is concerned with aquaporins (AQPs), that are proteins playing the role of water-selective channels also called nanopores, involved in many biological systems. From a technological point of view, it is relevant to design systems enjoying as good filtration properties. Inspired by [S. Gravelle, L. Joly, C. Ybert and L. Bocquet, Large permeabilities of hourglass nanopores: From hydrodynamics to single file transport, J. Chem. Phys.141 (2014) 18C526], we investigate in a quite general framework shape optimization issues related to the improvement of hourglass-shaped aquaporins performances, in terms of energy dissipated by the fluid through the channel. After modeling this problem mathematically, we show that it is well-posed in some sense, and compute the so-called shape derivative of the cost functional in view of numerical simulations. Noting that our framework requires regularity properties of the free boundary, we introduce a dedicated numerical method, using in particular a proper shape gradient extension-regularization to adapt the mesh at each iteration, in an adequate way. Optimal shapes of aquaporins are then provided for relevant values of parameters, and we finally discuss the observed performances with respect to the existing results/literature.

An approach to optimal actuator design based on shape and topology optimization techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology.

This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure $\mu$, describing the distribution of leaves, we introduce a sunlight functional ${𝒮}(\mu )$ computing the total amount of light captured by the leaves. On the other hand, given a measure $\mu$ describing the distribution of root hair cells, we consider a harvest functional ${\mathscr{H}}(\mu )$ computing the total amount of water and nutrients gathered by the roots. In both cases, we seek to maximize these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk and from the trunk to the leaves. The main results establish various properties of these functionals, and the existence of optimal distributions. In particular, we prove the upper semicontinuity of ${𝒮}$ and ${\mathscr{H}}$, together with a priori estimates on the support of optimal distributions.

This paper is devoted to the shape optimization of the internal structure of an electric motor, and more precisely of the arrangement of air and ferromagnetic material inside the rotor part with the aim to increase the torque of the machine. The governing physical problem is the time-dependent, nonlinear magneto-quasi-static version of Maxwell’s equations. This multiphase problem can be reformulated on a 2D section of the real cylindrical 3D configuration; however, due to the rotation of the machine, the geometry of the various material phases at play (the ferromagnetic material, the permanent magnets, air, etc.) undergoes a prescribed motion over the considered time period. This original setting raises a number of issues. From the theoretical viewpoint, we prove the well-posedness of this unusual nonlinear evolution problem featuring a moving geometry. We then calculate the shape derivative of a performance criterion depending on the shape of the ferromagnetic phase via the corresponding magneto-quasi-static potential. Our numerical frameworkto address this problem is based on a shape gradient algorithm. The nonlinear time periodic evolution problems for the magneto-quasi-static potential is solved in the time domain, with a Newton–Raphson method. The discretization features a space-time finite element method, applied on a precise, meshed representation of the space-time region of interest, which encloses a body-fitted representation of the various material phases of the motor at all the considered stages of the time period. After appraising the efficiency of our numerical framework on an academic problem, we present a quite realistic example of optimal design of the ferromagnetic phase of the rotor of an electric machine.

This paper deals with shape optimization issues under vibroacoustic criteria. The aim of the conducted research is to minimize the energy density in the cavity by changing its geometry parameters. The energy density is obtained through an energy method called simplified energy method (MES). The optimization method is based on a transformation function mapping 3D cavity surface on a 2D domain. The optimization process directly relies on this function and thus avoids remeshing of the geometry. The proposed method allows to describe the geometry through Bezier, Bspline and NURBS parametrization. To illustrate the method, we process a shape optimization on a simple acoustic cavity.

Reducing the noise level in the acoustic cavities is the important problem when treating inflight conditions of commercial planes or boats. Shape optimization of the acoustic cavity that will take into account the geometrical and material uncertainties, arising during the manufacturing process, is presented in this paper. The noise level is controlled by minimizing the energy density in the cavity, obtained through an energy method called Simplified Energy Method. Such formulation is based on our previous published work where transformation function mapping 3D cavity surface on a 2D domain was proposed. The optimization process directly relies on this function and thus avoids remeshing of the geometry. Robust optimization is performed using the nondominated sorting genetic algorithm (NSGA-II) together with the Kriging surrogate model. Influence of geometrical and material characteristics on the optimal solution is identified.

In this paper, we prove a sharp upper bound for the first nontrivial eigenvalue of the $p$-Laplacian with Neumann boundary conditions. This applies to convex sets with given diameter. Some variants and extensions are investigated as well.

We consider the problem of minimizing or maximizing the quantity $\lambda (\text{Ω})T^q(\text{Ω})$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $\lambda (\text{Ω})$ denotes the first eigenvalue of the Dirichlet Laplacian on $H_0^1(\text{Ω})$, while $T(\text{Ω})$ is the torsional rigidity of $\text{Ω}$. The optimization problem above is considered in the class of all domains$\text{Ω}$, in the class of convex domains$\text{Ω}$, and in the class of thin domains. The full Blaschke–Santaló diagram for $\lambda (\text{Ω})$ and $T(\text{Ω})$ is obtained in dimension one, while for higher dimensions we provide some bounds.

In this paper, the elastic lateral-torsional buckling of tapered cantilever strip beams acted by a tip load is investigated. Analytical solutions of the buckling problem are derived for cantilevers with a linearly varying depth. It is shown that the corresponding governing differential equation is a particular case of the general confluent equation, whose solution is given in terms of confluent hypergeometric functions. The buckling load corresponds to the lowest positive root of a 4×4 characteristic determinant arising from the introduction of the general solution of the differential equation into the boundary conditions. Finally, one discusses the optimization of the cantilever geometrical shape (for constant volume), as far as its resistance to lateral-torsional buckling is concerned.

In order to improve hydrodynamic journal bearings (HJB) performances, a novel shape optimization method for HJB based on the general film thickness equation was proposed. The general film thickness equation was studied which cannot only include the existing classical bearings shapes but can also be used to develop some other new shapes. An efficient mathematical model of shape optimization for HJB based on the general film thickness equation was established. Feasibility of shape optimization for hydrodynamic bearings based on the general film thickness equation was proved. An example was given and a new HJB shape with better performances was obtained.

In this paper, we describe a new type of humanoid robot designed for made-to-measure garment industry — a shape-changing robotic mannequin. This mannequin is designed to imitate body shapes of different people. The main emphasis of this paper is on modeling and shape-optimization algorithm used to adjust mannequins shape to resemble the shape of any given person. We represent the whole procedure of adjusting the mannequin to the body shapes of real people. Finally, we provide the estimate of the mannequin's model precision and suitability of the proposed solutions for made-to-measure tailoring application. The results show that the mannequin and the optimization methods are sufficiently precise for the requirements in tailoring industry.

This paper presents a numerical analysis method for solving the shape identification problems of thermoelastic fields. The square error integral between the actual thermal deformation distributions and the prescribed thermal deformation distributions on the prescribed sub-boundaries is used as the objective functional. The shape gradient of the shape identification problems is derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping is accomplished using a traction method that was proposed as a solution to domain optimization problems. In addition, a new numerical procedure for the shape identification is proposed. The validity of the proposed method is confirmed based on the results of 2D numerical analysis.

The minimum volume of the nonlinear longitudinal fin with rectangular and triangular profiles by using the modified Newton–Raphson method is presented in this paper. The dimension of the fin profile is regarded as the optimization variable. Furthermore, a mechanism called “volume updating” is added into the modified Newton–Raphson algorithm to obtain the minimum volume of the fin. Two examples are illustrated to demonstrate the proposed method. The obtained results showed that the proposed method can be used efficiently and accurately in finding the minimum volume of the nonlinear longitudinal fin problem with rectangular and triangle profiles.