Abstract: Topology optimization has evolved rapidly since the late 1980s. The optimization of the geometry and topology of structures has a great impact on its performance, and the last two decades have seen an exponential increase in publications on structural optimization. This has mainly been due to the success of material distribution methods, originating in 1988, for generating optimal topologies of structural elements. Previous methods suffered from mathematical complexity and a limited scope for applicability, however with the advent of increased computational power and new techniques topology optimization has grown into a design tool used by industry. There are two main fields in structural topology optimization, gradient based, where mathematical models are derived to calculate the sensitivities of the design variables, and non gradient based, where material is removed or included using a sensitivity function. Both fields have been researched in great detail over the last two decades, to the point where structural topology optimization has been applied to real world structures. It is the objective of this review paper to present an overview of the developments in non gradient based structural topology and shape optimization, with a focus on evolutionary algorithms, which began as a non gradient method, but have developed to incorporate gradient based techniques. Starting with the early work and development of the popular algorithms and focusing on the various applications. The sensitivity functions for various optimization tasks are presented and real world applications are analyzed. The article concludes with new applications of topology optimization and applications in various engineering fields.

Abstract: The automated design of synchronous reluctance (SyR) motors based on multiobjective genetic optimization and finite-element analysis is considered in this paper. Three types of barrier shapes are considered, all described by an effective limited set of input variables. The three solutions are investigated to establish which of the geometries can give the best torque output and also which one represents the best compromise between output performance and computational time. The analysis presented in this paper shows that SyR motors designed automatically can give a good performance and can be designed in a reasonable time, and it is also shown that not all design degrees of freedom are useful in terms of motor performance. Two prototypes of automatically designed machines have been fabricated and experimentally compared with a third prototype designed according to state-of-the-art design principles.

Abstract: In layer-based additive manufacturing (AM), supporting structures need to be inserted to support the overhanging regions. The adding of supporting structures slows down the speed of fabrication and introduces artifacts onto the finished surface. We present an orientation-driven shape optimizer to slim down the supporting structures used in single material-based AM. The optimizer can be employed as a tool to help designers to optimize the original model to achieve a more self-supported shape, which can be used as a reference for their further design. The model to be optimized is first enclosed in a volumetric mesh, which is employed as the domain of computation. The optimizer is driven by the operations of reorientation taken on tetrahedra with ‘facing-down’ surface facets. We formulate the demand on minimizing shape variation as global rigidity energy. The local optimization problem for determining a minimal rotation is analyzed on the Gauss sphere, which leads to a closed-form solution. Moreover, we also extend our approach to create the functions of controlling the deformation and searching for optimal printing directions.

Abstract: This paper introduces an approach to level-set topology optimization that can handle multiple constraints and simultaneously optimize non-level-set design variables. The key features of the new method are discretized boundary integrals to estimate function changes and the formulation of an optimization sub-problem to attain the velocity function. The sub-problem is solved using sequential linear programming (SLP) and the new method is called the SLP level-set method. The new approach is developed in the context of the Hamilton-Jacobi type level-set method, where shape derivatives are employed to optimize a structure represented by an implicit level-set function. This approach is sometimes referred to as the conventional level-set method. The SLP level-set method is demonstrated via a range of problems that include volume, compliance, eigenvalue and displacement constraints and simultaneous optimization of non-level-set design variables.

Abstract: An automotive engine cradle supports many crucial components and systems, such as an engine, transmission, and suspension. Important performance measures for the design of an engine cradle include stiffness, natural frequency, and durability, while minimizing weight is of primary concern. This paper presents an effective and efficient methodology for engine cradle design from conceptual design to detailed design using design optimization. First, topology optimization was applied on a solid model which only contains the possible engine cradle design space, and an optimum conceptual design was determined which minimizes weight while satisfying all stiffness constraints. Based on topology optimization results, a design review was conducted, and a revised model was created which addresses all structural and manufacturability concerns. Shape and size optimization was then performed in the detailed design stage to further minimize the mass while meeting the stiffness and natural frequency targets. Lastly, the final design was validated for durability. The initial design domain had the mass of 82.6 kg; topology optimization in conceptual design reduced the mass to 26.7 kg; and the detailed design task involving shape and size optimization further reduced the mass to 21.4 kg.

Abstract: The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Abstract: A self-folding structure fabricated by additive manufacturing (AM) can be automatically folded into a demanding three-dimensional (3D) shape by actuation mechanisms such as heating. However, 3D surfaces can only be fabricated by self-folding structures when they are flattenable. Most generally, designed parts are not flattenable. To address the problem, we develop a shape optimization method to modify a nonflattenable surface into flattenable. The shape optimization framework is equipped with topological operators for adding interior/boundary cuts to further improve the flattenability. When inserting cuts, self-intersection is locally prevented on the flattened two-dimensional (2D) pieces. The total length of inserted cuts is also minimized to reduce artifacts on the finally folded 3D shape. [DOI: 10.1115/1.4031023]

Abstract: Parallel micro or mini-channels are widely used in various devices of process and energy engineering including micro-reactors, compact heat exchangers and fuel cells. Nevertheless, the flow maldistribution due to the improper design of distributor/collector is usually observed, leading to globally poor performances of these devices. The objective of this study is to optimize the shape of the distributor/collector pipes so as to achieve a uniform flow distribution among an array of parallel mini-channels. A Z-type ladder fluid network with 10 mini-channels in parallel having square section is introduced and investigated. Two methods are used to optimize the shape of distributor/collector pipes: an optimized discrete stairway shape and a continuous tapered shape with an inclined angle varying from 0° to 30°. 3D-CFD simulations are carried out using the ANSYS FLUENT code. Numerical results obtained show that a relatively uniform flow distribution may be reached by the discrete stairway shape or by the continuous tapered shape distributor/collector under very low flow-rate conditions. Larger inclined angle or fewer channels in parallel are favorable for more uniform flow distribution under higher flow-rate conditions. Nevertheless the distributor and the collector pipes occupy a large volume so that the entire device is less compact.

Abstract: TLBO has shown better performance with less computational effort.TLBO can be effectively used for size and shape optimization of structures.TLBO can be easily extended for the optimization of other structural designs. In this study, a new meta-heuristic algorithm called teaching-learning-based optimization (TLBO) is used for the size and shape optimization of structures. The TLBO algorithm is based on the effect of the influence of a teacher on the output of learners in a class. The cross-sectional areas of the bar element and the nodal coordinates of the structural system are the design variables for size and shape optimization, respectively. Displacement, allowable stress and the Euler buckling stress are taken as the constraint for the problem considered. Some truss structures are designed by using this new algorithm to show the efficiency of the TLBO algorithm. The results obtained from this study are compared with those reported in the literature. It is concluded that the TLBO algorithm presented in this study can be effectively used in combined size and shape optimization of the structures.

Abstract: Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincar\'e type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

Abstract: We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace-Beltrami type based metrics are compared with Steklov-Poincare type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.

Abstract: We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization problem that approximates its unreduced counterpart and can be solved in milliseconds. To achieve this, we restrict the optimization to a low-dimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylov-type sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced optimization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approximation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20--100 interpolated shapes per second, even for the largest examples which have 500k vertices per example shape.

Abstract: This paper presents an explicit feature-based level-set topology optimization method involving polyline-arc profiling and 2.5D machining processes. This method relies on a feature fitting algorithm incorporated into the boundary evolvement process in order to regulate the noisy velocity fields and thus introduce new explicit feature primitives; once inserted, the feature-based shape optimization algorithm is implemented to determine the optimum part shape and topology. The research novelty lies in that, the best-fit feature primitives are inserted during the topology optimization process while other researchers so far have reported only manipulating some existing features with the conventional level-set methods. Therefore, feature-based design can be realized without special requirement of initial input or any post-processing. From the perspective of potential applications, the engineering information embedded in those feature primitives can be extracted and integrated into the optimization formulation. Such potential integration can make the topology optimization even more useful and practical. This effort is an extension of level-set topology optimization into a domain of structural optimization for manufacturing (OFM).

Abstract: An intuitive shape parameterization and control technique suitable for high-fidelity aerodynamic shape optimization is presented. It relies on the principles of free-form and axial deformation, enabling thorough exploration of the design space while keeping the number of design variables manageable. Surface sensitivities to the design variables are readily available; their inclusion in a highly efficient and robust adjoint-based optimization methodology involving linearly elastic volume mesh deformation and a Newton–Krylov solver for the Euler equations is described. The flexibility of the proposed approach is demonstrated through the exploratory shape optimization of a three-pronged feathered winglet, leading to a span efficiency of 1.19 under a height-to-span ratio constraint of 0.1, and an optimization of a regional jet wing at transonic speed where a winglet is allowed to develop starting from a planar wingtip extension, leading to an 18.8% reduction in drag.

Abstract: The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Abstract: Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore, quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.

Abstract: We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.

Abstract: The detailed outer mold line shaping of a Mach 1.6, demonstrator-sized low-boom concept is presented. Cruise trim is incorporated a priori as part of the shaping objective, using an equivalent-area-based approach. Design work is performed using a gradient-driven optimization framework that incorporates a three-dimensional, nonlinear flow solver, a parametric geometry modeler, and sensitivities derived using the adjoint method. The shaping effort is focused on reducing the under-track sonic boom level using an inverse design approach, while simultaneously satisfying the trim requirement. Conceptual-level geometric constraints are incorporated in the optimization process, including the internal layout of fuel tanks, landing gear, engine, and crew station. Details of the model parameterization and design process are documented for both flow-through and powered states, and the performance of these optimized vehicles presented in terms of inviscid L/D, trim state, pressures in the near-field and at the ground, and predicted sonic boom loudness.