Shape Optimization Method Research Articles

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.

Structural Shape Optimization Based on Multi-Patch Weakly Singular IGABEM and Particle Swarm Optimization Algorithm in Two-Dimensional Elastostatics

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

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).

Advanced multimaterial shape optimization methods as applied to advanced manufacturing of wind turbine generators

AbstractCurrently, many utility‐scale wind turbine generator original equipment manufacturers are dependent on imported rare earth permanent magnets, which are susceptible to market risks from cost instability. To lower the production costs of these generators and stay competitive in the market, several small wind manufacturers are pursuing continuous improvements to both generator design and manufacturing. However, traditional design and manufacturing methods have yielded marginal improvements in wind power performance. This work presents novel methods to redesign a baseline 15‐kW wind turbine generator with reduced rare‐earth permanent magnets by leveraging cutting‐edge three‐dimensional (3D) printed polymer‐bonded permanent magnets and steel. Symmetric, asymmetric, and multimaterial‐magnet parametrization methods are introduced for shape optimization. We extend the symmetric and asymmetric methods to the back iron in the stator to further investigate the impact and opportunities for performance improvements with lesser active materials. We employ a design‐of‐experiments approach with parametric computer‐aided design for shape generation and evaluate different designs by magneto‐thermal modeling and finite‐element analysis. We use adaptive sampling technique to identify better performing designs with lesser magnet mass, higher efficiency, and lower cogging torque when compared with the baseline generator. Asymmetric pole designs resulted in a magnet mass in the range of 4.77–5.37 kg, which was 27%–35% lighter than the baseline generator, suggesting that a new design freedom exists that can be enabled by advanced manufacturing, such as 3D printing. Shaping the back iron in the stator resulted in material savings in electrical steel of up to 14.62 kg, which was 20% lighter than the baseline stator. We conducted a structural analysis to evaluate an optimized asymmetric rotor design from the point of view of mechanical integrity and air‐gap stiffness. The magnetically optimal shape profile was shown as having a positive impact on the radial stiffness, and an optimal solution was discovered to reduce the structural mass by nearly 30 kg, which was 29% lighter than the baseline.

Compact, Scalable, Fast‐Response Multimode 2 × 2 Optical Switch Based on Inverse Design

AbstractMode‐division multiplexing (MDM) introduces a new dimension to on‐chip optical interconnection, where the multimode optical switch is one of the core components. However, researchers are still struggling to reduce the chip size and response time, despite great efforts being made. Here, a compact, scalable, fast‐response multimode 2 × 2 switch supporting four modes is demonstrated based on the specific inverse design. The device consists of two multimode 2 × 2 CB couplers (including mode converters, multimode interference couplers, and crossings) and a thermo‐optic (TO) phase shifter. It achieves extinction ratios exceeding 15 dB for all modes at the central wavelength of 1.55 µm. All the fundamental components are designed by an improved general shape optimization method, realizing a compact size and a larger manufacturing tolerance (±20 nm). Besides, a hydrogen‐doped indium oxide microheater is introduced as a TO phase shifter to achieve a fast response (≈3.5 µs in average) for all modes. This solution provides an effective avenue toward large‐scale multimode optical switch matrices for the future MDM systems.

High-dimensional aerodynamic shape optimization framework using geometric domain decomposition and data-driven support strategy for wing design

Traditional aerodynamic shape optimization methods usually suffer from difficulties in optimization and modeling for high-dimensional aerodynamic optimization problems. To overcome these difficulties, a new high-dimensional aerodynamic shape optimization framework is proposed from the perspectives of geometric domain decomposition for dimensionality reduction and a data-driven model support strategy for changing the role of surrogate model and assisting the optimizer. Geometric domain decomposition uses proper orthogonal decomposition to transform the high-dimensional Class-Shape Function Transformation method into low-dimensional geometric modes to reduce design dimensions for parameterizing the geometric shape, which reduces optimization and modeling difficulties. Also, a data-driven support strategy is proposed to improve the effectiveness and efficiency of the optimizer by learning from historical aerodynamic data; this strategy modifies the role of surrogate model in the optimization framework to assist the optimizer; the surrogate model, serving as an auxiliary role, can reduce the dependence on model accuracy, which overcomes the difficulty of high-dimensional modeling to some extent. We applied the developed framework to transonic drag reduction of wings to validate its performance. The optimization results show that the developed framework converges faster and offers better designs compared to direct optimization and state-of-the-art surrogate-based optimization frameworks for high-dimensional aerodynamic shape optimization.

A T-splines-oriented isogeometric topology optimization for plate and shell structures with arbitrary geometries using Bézier extraction

Recently, the non-uniform rational B-splines (NURBS) have been considerably employed in modeling plate and shell structures, or developing the related size, shape or topology optimization methods for their design. However, the NURBS with the tensor product feature strongly hinders the effectiveness of the optimization on complex structures. The primary intention of the current research is to propose a new T-splines-oriented Isogeometric Topology Optimization (T-ITO) method and then address the critical design problem of plate and shell structures with arbitrary shapes in practical applications. Firstly, the Bézier extraction is utilized in the T-splines to map the geometrical model into a family of Bézier elements, each of which is presented by a local Density Distribution Function (DDF) for elementary topology. A global DDF is constructed by an assembly of all local DDFs with a natural connection to the present structural topology. Secondly, the T-splines-based IsoGeometric Analysis (T-IGA) formulation with the Bézier extraction for complex plate and shell structures is developed using the Kirchhoff-Love theory, where a universal numerical implementation framework is constructed, including the Rhino, MATLAB, export modules with all geometric information and import modules for the analysis. Thirdly, a mathematical formulation for plate and shell structures with arbitrary geometries is developed using the T-ITO method to improve structural loading-capability, and the sensitivity analysis with respect to design variables is rigorously derived in detail. Finally, several numerical examples of plate and shell structures with both regular and elaborate geometries are tested to demonstrate the validity, effectiveness, superiorities and indispensability of the T-ITO method.

An isogeometric shape optimization method for groundwater flow in porous media

This work introduces a novel isogeometric method for addressing steady, saturated groundwater flow as a free-boundary problem. The innovation lies in adopting a double spline parameterization technique for accurately representing both the porous medium and the fully saturated zone. To linearise the governing equation, we differentiate them with respect to the coefficients of the geometric parameterization. This process results in a quasi-Newton method. Several benchmark numerical tests demonstrate the applicability and effectiveness of the proposed technique.

Optimization of Buoy Shape for Wave Energy Converter Based on Particle Swarm Algorithm

In order to improve the wave energy capture rate of the buoy of a wave energy generation device, this paper proposes a multi-degree of freedom method to optimize the shape of the buoy with maximum wave energy capture. Firstly, a multi-degree of freedom wave energy converter was designed, and the buoy shape was defined using a B-spline curve to generate the shape vector; then, a numerical model of the multi-degree of freedom wave energy converter was established and numerical calculations were carried out using AQWA/WEC-Sim software; on this basis, the particle swarm optimization algorithm was introduced to find the buoy shape corresponding to the maximum wave energy capture. Finally, the optimization of the buoy shape was in irregular waves. The results show that as the wave energy capture increased, the buoy shape tended to be flatter, with a smaller taper, and the optimal buoy shape had a better motion response than the conventional cone buoy. Eventually, the correctness of the buoy shape optimization method was verified through experimental testing.

A low-resistance local component design method based on biomimicry and a random forest model: A pipe elbow case study

The energy consumption of building transmission and distribution systems accounts for more than 30% of building energy consumption, and the prevalence of localized components in building transmission and distribution systems leads to severe energy losses and resistance effects during system operation. To solve this problem, our research team has developed a series of resistance reduction components. This paper proposes a low-resistance local component shape optimization method based on biomimicry and machine learning, taking pipe elbows as an example, and the normalized shape optimization parameters of elbows with different pipe diameters are given. Through numerical simulation, full-size experiments, energy dissipation analysis and other methods, the optimized elbow resistance reduction effect and resistance reduction mechanism are verified and analyzed. The results showed that the resistance reduction rate of the elbow was 19%–26% for different pipe diameters and Reynolds numbers, and the optimized elbow significantly reduced energy dissipation during the flow process. Previously, the shape optimization problem of local components focused mainly on rectangular ducts that can be simplified in two dimensions; this study extends the optimization objective to circular pipes, which provides an effective method for guiding the optimal design of low-resistance local components in building transmission and distribution systems.

Geometric deep learning for statics-aware grid shells

This paper introduces a novel method for shape optimization and form-finding of free-form, triangular grid shells, based on geometric deep learning. We define an architecture which consumes a 3D mesh representing the initial design of a free-form grid shell, and outputs vertex displacements to get an optimized grid shell that minimizes structural compliance, while preserving design intent. The main ingredients of the architecture are layers that produce deep vertex embeddings from geometric input features, and a differentiable loss implementing structural analysis. We evaluate the method performance on a benchmark of eighteen free-form grid shell structures characterized by various size, geometry, and tessellation. Our results demonstrate that our approach can solve the shape optimization and form finding problem for a diverse range of structures, more effectively and efficiently than existing common tools.

A novel triple periodic minimal surface-like plate lattice and its data-driven optimization method for superior mechanical properties

Lattice structures can be designed to achieve unique mechanical properties and have attracted increasing attention for applications in high-end industrial equipment, along with the advances in additive manufacturing (AM) technologies. In this work, a novel design of plate lattice structures described by a parametric model is proposed to enrich the design space of plate lattice structures with high connectivity suitable for AM processes. The parametric model takes the basic unit of the triple periodic minimal surface (TPMS) lattice as a skeleton and adopts a set of generation parameters to determine the plate lattice structure with different topologies, which takes the advantages of both plate lattices for superior specific mechanical properties and TPMS lattices for high connectivity, and therefore is referred to as a TPMS-like plate lattice (TLPL). Furthermore, a data-driven shape optimization method is proposed to optimize the TLPL structure for maximum mechanical properties with or without the isotropic constraints. In this method, the genetic algorithm for the optimization is utilized for global search capability, and an artificial neural network (ANN) model for individual fitness estimation is integrated for high efficiency. A set of optimized TLPLs at different relative densities are experimentally validated by the selective laser melting (SLM) fabricated samples. It is confirmed that the optimized TLPLs could achieve elastic isotropy and have superior stiffness over other isotropic lattice structures.

Arbitrary ratio power splitter based on shape optimization for dual-band operation

Arbitrary ratio power splitters (ARPSs) are essential components in integrated photonics. However, most existing ARPSs are limited to operating within a single wavelength band. To address this limitation, in this paper, we propose a 1 × 2 ARPS capable of operating in multi-wavelength bands using the shape optimization method. Harnessing a compact footprint of 14 μm × 1.7 μm, a series of dual-band ARPSs (DBARPSs) with varying power splitting ratios (PSRs) are designed. Within the wavelength ranges of 1280–1340 nm and 1520–1580 nm, all of the DBARPSs have an excess loss of less than 0.55 dB, with PSR deviations falling within the range of −0.5 dB to 1 dB. The practical viability is confirmed through a fabricated DBARPS with a 3 dB PSR. Experimental results show that the PSR deviation is within −0.8 dB to 1 dB, while the excess loss remains below 0.54 dB within the bandwidths of 1280–1340 nm and 1520–1580 nm. Additionally, the same design methodology is extended to develop a high-performance 1310/1550 nm wavelength demultiplexer. The resulting device showcases remarkable performance, with insertion loss below 0.7 dB and extinction ratios higher than 17 dB across the specified wavelength ranges. Experimental outcomes align closely with the design objectives, underscoring the dual-band operational capability and robust generality of the approach outlined in this paper.

Evaluation of feasibility of commercial supersonic flight based on aeroacoustics

The drive to attain ever higher speeds, to be able to travel ever faster fuels the research and development for a commercial supersonic aircraft. This has previously led to the Concorde which travelled at more than twice the speed of sound. Now, in addition to business considerations about economic viability, supersonic aircraft must be quieter and emit less emissions. Considering the 20 years that have elapsed since Concordes retirement, this study aims to reevaluate the current challenges and limitations to achieving commercial supersonic flight again, in the context of noise. Identifying sonic booms and jet exhaust noise as two main challenges, it reviews current shape optimization methods, plasma as a sonic boom mitigator, sonic boom circumvention, chevron nozzles, variable cycle engines, engine positioning, and their corresponding limitations. Some of the methods have been refined for use and application in final stage design and manufacture of certain supersonic aircraft which indicates a certain feasibility.

Reduced order isogeometric boundary element methods for CAD-integrated shape optimization in electromagnetic scattering

This paper formulates a model order reduction method for electromagnetic boundary element analysis and extends it to computer-aided design integrated shape optimization of multi-frequency electromagnetic scattering problems. Firstly, a series expansion technique is adopted to decouple frequency-dependent terms from the integrands in boundary element formulation, and the second-order Arnoldi procedure is utilized to reduce the order of original systems. Secondly, to ensure geometric exactness and avoid re-meshing during shape optimization, the isogeometric boundary element method is employed, which uses the Non-Uniform Rational B-splines to represent the geometry and to discretize boundary integral equations. Lastly, we employ the Grey Wolf Optimization-Artificial Neural Network as a surrogate model for shape optimization in multi-frequency electromagnetic scattering problems, with the radar cross section as the objective function. Several examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Yet another parameter-free shape optimization method

Yet another parameter-free shape optimization method

Aerodynamic shape optimization of the vortex-shock integrated waverider over a wide speed range

The vortex-shock integrated waverider over a wide speed range could significantly improve the aerodynamic performance of traditional waveriders by introducing the vortex effect at subsonic speeds. Therefore, it is promising to be widely used in the design of wide-speed-range aerospace vehicles. However, the design of the vortex-shock integrated waverider ignores the three-dimensional effect, the subsonic effect, and the viscous effect in the establishment of the reference flow field, which fails to obtain the expected ideal performance. Therefore, it is necessary to further improve the wide-speed-range performance of the vortex-shock integrated waverider by employing the aerodynamic shape optimization method. In this study, the wide-speed-range multipoint optimization of the vortex-shock integrated waverider is carried out using a gradient-based optimizer combined with adjoint gradient evaluations. The results show that the optimum configuration increases the subsonic lift and lift-to-drag ratio by 6.7 % and 7.1 %, while increasing the hypersonic lift-to-drag ratio by 1.4 %. The improvement of the aerodynamic performance at subsonic speeds is attributed to the enhancement of the leeward vortex effect. Also, the leeward surface near the leading edge is optimized to an inner concave region to further enhance the vortex lift, while guaranteeing against the volume decline. Furthermore, the Detached Eddy Simulation (DES) method and the Dynamic Mode Decomposition (DMD) method are used to analyze the vortex dynamic characteristics at subsonic speeds. By analyzing the first three modes, it is observed that the aerodynamic performance of the optimum configuration is improved mainly by the frequency reduction and the increase of the pressure difference between the windward and the leeward surface, which enhance the stability and strength of the vortex.

Robust optimal ship hulls based on Michell's wave resistance

We seek the hull of a ship with a given volume which minimizes the water resistance with uncertainties on the cruising speed. The water resistance is based on Michell's wave resistance functional, and the speed is a random variable whose probability distribution is known. We first handle the case where the support of the hull is given and then we also optimize this support for a given area. In each case, an optimal hull is shown to exist. The numerical simulations are costly so we adapt to our problem Newton's method for shape optimization. The Dirichlet energy is used as a test case. The numerical results for the robust optimal hull are compared with the case where the cruising speed is known.

Modeling and Analysis of a Conical Bridge-Type Displacement Amplification Mechanism Using the Non-Uniform Rational B-Spline Curve.

This paper presents a new analytical model of a conical bridge-type displacement amplification mechanism (DAM) considering the effect of external loads and a piezostack actuator (PSA). With the merits of simple implementation and better fitting, the non-uniform rational B-spline (NURBS) is employed to parameterize conical connecting beams of the DAM, and an analytical model of the displacement amplification ratio and input stiffness is established based on Castigliano's second theorem. After that, considering the interactions with elastic loads and PSA, the actual displacement amplification ratio of the conical DAM is obtained, and the effect of the shape of connecting beams in the performance of the DAM is further analyzed. The proposed analytical model is verified by finite element analysis (FEA), and the results show a maximum error of 6.31% between the calculated value and FEA results, demonstrating the accuracy of the proposed model. A prototype of the conical DAM with optimized shape is fabricated and experimentally tested, which further validates the effectiveness and accuracy of the proposed analytical model. The proposed model offers a new method for analysis and shape optimization of the bridge-type DAM under specific elastic loads.