Shape Optimization Research Articles

Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper, we consider a minimization problem of a nonlinear functional I_{\beta,p}(D,\Omega) related to a thermal insulation problem with a convection term, where \Omega is a bounded connected open set in \R^{n} and D\subset\overline{\Omega} is a compact set. The Euler–Lagrange equation relative to I_{\beta,p} is a p -Laplace equation, 1<p<\infty , with a Robin boundary condition with parameter \beta>0 . The main aim is to study extremum problems for I_{\beta,p}(D,\Omega) , among domains D with given geometrical constraints and \Omega\setminus D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes I_{\beta,p} . In the n -dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.

Reduced order model enhanced source iteration with synthetic acceleration for parametric radiative transfer equation

Applications such as uncertainty quantification, shape optimization, and optical tomography, require solving the radiative transfer equation (RTE) many times for various parameters. Efficient solvers for RTE are highly desired.Source Iteration with Synthetic Acceleration (SISA) is a popular and successful iterative solver for RTE. Synthetic Acceleration (SA) acts as a preconditioning step to accelerate the convergence of Source Iteration (SI). After each source iteration, classical SA strategies introduce a correction to the macroscopic particle density by solving a low order approximation to a kinetic correction equation. For example, Diffusion Synthetic Acceleration (DSA) uses the diffusion limit. However, these strategies may become less effective when the underlying low order approximations are not accurate enough. Furthermore, they do not exploit low rank structures concerning the parameters of parametric problems.To address these issues, we propose enhancing SISA with data-driven ROMs for the parametric problem and the corresponding kinetic correction equation. First, the ROM for the parametric problem can be utilized to obtain an improved initial guess. Second, the ROM for the kinetic correction equation can be utilized to design a low rank approximation to it. Unlike the diffusion limit, this ROM-based approximation builds on the kinetic description of the correction equation and leverages low rank structures concerning the parameters. We further introduce a novel SA strategy called ROMSAD. ROMSAD initially adopts our ROM-based approximation to exploit its greater efficiency in the early stage of SISA, and then automatically switches to DSA to leverage its robustness in the later stage. Additionally, we propose an approach to construct the ROM for the kinetic correction equation without directly solving it.Through a series of numerical tests, we compare the proposed methods with SI-DSA and DSA preconditioned Krylov solver. Particularly, for a multiscale parametric pin-cell problem, ROMSAD achieves approximately 10 times the acceleration compared to SI-DSA and 4 times acceleration compared to DSA preconditioned GMRES.

Hysteretic behavior optimization of hollow laminated viscoelastomer-filled steel tube damper

This study proposes a shape optimization approach for designing hollow laminated viscoelastomer-filled steel tube dampers (HLVSTDs), aiming to reduce stress concentration, enhance energy dissipation capacity, and improve low-cycle fatigue performance. The shape optimization curve is obtained based on the definition of stress contours for HLVSTD bending moments and shear forces, with the assumption that points on the same contour yield simultaneously. Cyclic loading experiments were conducted on four HLVSTDs. Numerical models were established and validated to further investigate the mechanical properties, energy dissipation capacity, and seismic performance of the shape-optimized HLVSTDs. The results demonstrate that the shape-optimized HLVSTDs specimen exhibits excellent energy dissipation capacity and improved low-cycle fatigue performance. The calculated initial stiffness and yield force closely align with experimental values. Compared to non-optimized HLVSTDs, the shape-optimized HLVSTDs exhibit more uniform stress and equivalent plastic strain distributions, effectively reducing the concentration of plastic strains. Additionally, structures equipped with shape-optimized HLVSTDs demonstrate superior seismic performance.

Design optimization of cubic-shaped pressurant tank for CubeSat propulsion system

From a technical and economic point of view, cold gas propulsion for nanosatellites is one of the more rational systems. There is a problem, however: the optimal spherical shape for a high-pressure vessel only takes up half the cubic volume of the CubeSat unit. To increase the volume, it is proposed to switch to a cubic-shaped tank. The aim of this work is to synthesize the more rational structural design of a pressurant tank that fits within a cubic constraint, which provides the perfect balance between the mass of the structure and the delta-V budget available for the propulsion system. The proposed variants are as follows: a set of four cylinders with spherical bottoms; 14 equal spheres arranged in a face-centered cubic packing; a thin-walled cube supported by ribs, the location and parameters of which provide the smallest mass of the structure; a cube, the reinforcement scheme of which is calculated using topology optimization method; an original polyspheric and bypass design schemes where the filling of the volume of the cube is created by a set of spherical and cylindrical shells of different radii. All the considered vessel variants can be obtained using additive technologies, such as selective laser sintering. For each variant, the internal volume and mass of the vessels were obtained, as well as the delta-V budget, taking into account the mass of the tank and the mass of the stored gas. The analysis of the results reveals that there are two structures that are more rational. The new bypass scheme provides the largest delta-V budget, but it has a 1.6 times larger mass than the initial spherical vessel design. It should be noted that the difference in vessel mass is not crucial for CubeSat form factor satellites. Choosing a rational design may require a larger delta-V budget. The possibility of increasing the reserve delta-V of the CubeSat cold gas propulsion system by 25–35 % due to the transition from a spherical to a cube-shaped tank has been demonstrated. At the same time, the mass of the tank structure increases by 1.6–6.3 times, which can be considered an acceptable price for increasing the efficiency of the propulsion system. The new bypass scheme offers the optimal balance of mass and delta-V reserve.

Deep reinforcement learning for tuning active vibration control on a smart piezoelectric beam

Piezoelectric transducers are used within smart structures to create functions such as energy harvesting, wave propagation or vibration control to prevent human discomfort, material fatigue, and instability. The design of the structure becomes more complex with shape optimization and the integration of multiple transducers. Most active vibration control strategies require the tuning of multiple parameters. In addition, the optimization of control methods has to consider experimental uncertainties and the global effect of local actuation. This paper presents the use of a Deep Reinforcement Learning (DRL) algorithm to tune a pseudo lead-lag controller on an experimental smart cantilever beam. The algorithm is trained to maximize a reward function that represents the objective of vibration mitigation. An experimental model is estimated from measurements to accelerate the DRL’s interaction with the environment. The paper compares DRL tuning strategies with [Formula: see text] and [Formula: see text] norm minimization approaches. It demonstrates the efficiency of DRL tuning by comparing the control performance of the different tuning methods on the model and experimental setup.

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.

Introducing a comprehensive methodology for optimizing the mass of realistic roofing structural systems using cold-formed steel profiles

This study aims to introduce a comprehensive methodology for optimizing complete real structural systems for roofs involving trusses, purlins, and bracing systems jointly, taking into account realistic loads and constraints dictated by technical codes, thereby offering a more accurate representation of practical scenarios. The objective is to achieve the minimum mass through size, shape, and topology optimization of both the main truss and purlin structural subsystems. To achieve this goal, the Enhanced Particle Swarm Optimization (EPSO) algorithm is implemented. An example of a realistic case, which takes into account multiple actual constraints such as stress, displacement, buckling, and natural frequency limits, is thoroughly evaluated. After that, 144 other interactions among dimensions of the building and loads applied are simulated, and the mass of the system is obtained for each one of them. The results indicated that the graphs generated from the various simulations allow for the determination of the optimized mass for different building dimensions. Consequently, the cost and raw material consumption can be estimated for common applications. Therefore, it is concluded that this work presents a significant contribution to structural designers, as the proposed methodology enables structural optimization quickly and easily for practical engineers.

Integrated Aerodynamic Shape and Aero-Structural Optimization: Applications from Ahmed Body to NACA 0012 Airfoil and Wind Turbine Blades

Integrated Aerodynamic Shape and Aero-Structural Optimization: Applications from Ahmed Body to NACA 0012 Airfoil and Wind Turbine Blades

Experimental validation of asymmetric PZT optimal shape in the active vibration reduction of triangular plates

Experimental validation of asymmetric PZT optimal shape in the active vibration reduction of triangular plates

Two-dimensional electromagnetic scattering analysis based on the boundary element method

An effective formula for the shape-sensitivity analysis of electromagnetic scattering is presented in this paper. First, based on the boundary element method, a new electromagnetic scattering formula is derived by combining the traditional electromagnetic scattering formula with the non-uniform rational B-spline (NURBS) curve, and the geometric model is represented by NURBS, which ensures the geometric accuracy, avoids the heavy grid division in the optimization process, and realizes the fast calculation of high-fidelity numerical solutions. Second, by deducing the sensitivity variables, the electromagnetic scattering equation of shape optimization is obtained, which can provide reliable data references for shape optimization. Finally, the effectiveness and accuracy of the algorithm are demonstrated by an example, and the sensitivity data of some examples are given.

Blade shape optimization in hover and forward flight

Blade shape optimization in hover and forward flight

Shape optimization of cast steel tubular joints based on subdivision surface and genetic algorithm

Cast steel tubular (CST) joints are extensively employed in steel structures for their excellent integrity and mechanical performance. However, the diverse configurations of CST joints result in the absence of standard verification formulas, leading to inefficient trial-and-error design approaches. To tackle this issue, this paper introduces a novel shape optimization method for CST joints that integrates subdivision surface techniques with genetic algorithms. The method comprises three key components: geometric modeling, structural analysis, and optimization algorithm, with shape optimization achieved through their collaborative operation. Applied to two types of CST joints, this approach resulted in significant reductions in joint volume by 39.2 % and structural stress by 49.0 %, respectively. The shape-optimized joints were compared with topology-optimized joints from published literatures, and demonstrated enhanced mechanical performance while presenting superior manufacturability. The proposed method offers considerable advantages over traditional trial-and-error approaches and topology optimization methods. It generates designs with continuous and smooth boundaries, better suited for the conventional casting process, and overcomes manufacturability issues of topology optimization. Additionally, the genetic algorithm allows for flexible selections of optimization objectives, addressing the restrictions of topology optimization methods. Furthermore, the methodʼs high level of automation is anticipated to expedite the joint design process, significantly reducing the requirement for manual intervention.

Surrogate-based integrated design optimization for aerodynamic/stealth performance enhancements

The shape optimization considering both aerodynamic and radar stealth performances has been a considerable attention area for next-generation aircraft, with challenges involving balancing design contradictions and improving optimization efficiency. This paper develops a surrogate-based integrated design optimization method coupling Computational Fluid Dynamics (CFD) and Computational Electromagnetics (CEM) for aerodynamic/stealth performance enhancements. The topology-based 3D hexahedral mesh generation and 2D triangular mesh generation with adaptive refinement provide efficient preprocessing for CFD/CEM solvers. The surrogate-based optimization algorithm with parallel infill-sampling strategy and adaptive design space expansion is employed to search the global optimum and improve optimization efficiency while filtering out numerical noise. The Weighted Sum method is employed to address design contradictions in different disciplines. Optimizations considering aerodynamic, stealth, and aerodynamic/stealth performance of a flying wing aircraft are conducted to verify the effectiveness of the developed method. The optimization results indicate that exclusive consideration of a singular discipline leads to a significant deterioration in the performance of the other discipline. Balanced performance is achieved through coupling optimization. The numerical noise in the stealth discipline is markedly higher than in the aerodynamics discipline. The results demonstrate that the developed method is capable of addressing aerodynamic/stealth design optimization problems and significantly reducing the number of CFD/CEM evaluations while maintaining global optimization effectiveness.

Study on the Influence of Wind Fairing Parameters on the Aerodynamic Performance of Long-Span Double-Deck Steel Truss Suspension Bridge

A long-span double-deck steel truss suspension bridge is easy to produce vortex-induced vibration (VIV) at low air velocity, which affects bridge service life. Additional aerodynamic measures play a role in suppressing VIV by changing the aerodynamic shape, which is a common control method. As the main aerodynamic measure to suppress the VIV response, wind fairing is widely used in engineering practice. In order to obtain the optimal additional position and shape parameters of the fairing, Huangjuetuo Yangtze River Bridge is the research target. Through the combination of a wind tunnel test and numerical simulation, the VIV response of the original and fairing section is studied. Based on data analysis, it is revealed that these additional fairings to the upper chord can significantly reduce the VIV response. When the shape parameters of the fairing are h/D = 1/4 and l/D = 1, the VIV inhibition efficiency is the highest, which can reach 65.51%. By analyzing the flow distribution, it can be seen that VIV is caused mainly by vortex separation in the upper bridge board area. Although this wind fairing does not change the original vortex shedding forms, it changes the first separation point and movement direction of the airflow, making the vortex scale generated by the airflow smaller and the vorticity lower, thus effectively suppressing VIV.

Shape Optimization of Structures by Biological Growth Method

Shape Optimization of Structures by Biological Growth Method

A study of rare-earth magnet profiles for torque ripple improvement of axial flux permanent magnet machine

Purpose The purpose of this study is to reduce the cogging torque in axial flux permanent magnet (AFPM) machine using optimal magnet shape. Design/methodology/approach This study analyzes different magnet shapes for AFPM machine performance enhancement. Three-dimensional (3D) finite element analysis is performed to see the effects of pole shaping on the cogging torque of the AFPM machine. Findings The magnetic pole shape has a significant effect on cogging torque and overall efficiency. The conventional model has the highest torque whereas the conventional skewing affected cogging torque positively and significantly reduced the cogging torque. The combination of skewing the pole along with face curving is more effective and decreases the cogging torque from 3.88 Nm to 1.5 Nm. Originality/value Rare-earth magnets are the most expensive and important part of AFPM machines. Shape and volume optimization of rare-earth magnets is crucial for the performance of AFPM machines. The research aims to analyze the different permanent magnet designs for performance improvement of the AFPM machine. Conventional flat top trapezoidal, curved-top and skewed-magnet shapes are analyzed and the performance of the AFPM machine is compared with different magnet shapes. Curved-top shape and skewed magnet significantly reduce the cogging torque. Furthermore, a combination of curved-top shape and skew magnet shape is proposed to reduce the cogging torque further and improve the AFPM machine’s overall performance. Newly proposed magnet profile gives skewed curve magnet shapes which reduce the cogging torque further. 3D finite element analysis has been used to analyze the single-sided AFPM with all four different magnet shapes. The research focuses on single-sided AFPM machines, but the results are also valid for double-sided AFPM machines and can be extended to other topologies of AFPM machines.

Shape- and frequency-dependent self-excited forces emulation for the aero-structural design of bluff deck bridges

The shape design and optimization of bluff decks prone to aeroelastic phenomena require emulating the fluid-structure interaction parameters as a function of the body shape and the oscillation frequency. This is particularly relevant for long- and medium-span bridges equipped with single-box decks that are far from being considered streamlined and for other girder typologies such as traditional truss decks and modern twin- and multi-box decks. The success of aero-structural design frameworks, which are inherently iterative, relies on the efficient and accurate numerical evaluation of the wind-induced responses. This study proposes emulating the fluid-structure interaction parameters of bluff decks using surrogate modeling techniques to integrate them into aero-structural optimization frameworks. The surrogate is trained with data extracted from forced-vibration CFD simulations of a typical single-box girder to emulate the values of the flutter derivatives as a function of the deck shape and reduced velocity. The focus is on deck configurations ranging from streamlined to bluff cross-sections and on low reduced velocities to capture eventual aerodynamic nonlinearities. The girder cross-section geometry is tailored based on its buffeting performance. This design tool is fundamental to finding the optimum balance between the structural and aeroelastic requirements that drive the design of bluff deck bridges.

Classroom Interior Design: Wooden Furniture Prototype with Feedback from Students and Teachers

Studies indicate that natural wooden materials positively affect students’ well-being in classrooms. In addition, students spend a considerable amount of their time in classrooms predominantly seated, making school interiors a suitable place to reduce sedentary behaviors of children. A mixed team of experts in human factors, architecture, design and engineering designed a prototype wooden standing desk for indoor use and formed focus groups with students and teachers to gather feedback on the development process and to evaluate the suitability of the prototype. The prototype desk was well received by the primary school teachers and students. The students appreciated plywood as the main material for the construction of the desk; however, they criticized that the wooden tabletop should be more resistant. The height adjustability of the desk and the tiltable tabletop were the most appreciated features of the prototype. Further studies should be conducted to investigate the optimal material, shape and color of the school desk, especially the tabletop, and additional efforts should be made to design furniture that promotes a less sedentary classroom and improves students’ well-being at school.

Compact Heat Exchangers With Curved Fins for Hydrogen Turbofan Intercooling

Abstract Hydrogen is being considered as a possible path toward carbon-neutral aviation. There are additional advantages besides its main benefit of CO2-free combustion. One application is to use it for aero engine heat management due to its cryogenic temperature and high heat capacity, including intercooling and exhaust heat recuperation. The focus of this paper is on the design of a compact heat exchanger (HEX) integrated into an intermediate compressor duct (ICD), which could decrease compression work and specific fuel consumption (SFC). This compact heat exchanger features curved fins to promote flow turning and decrease pressure losses compared to more conventional straight fin heat exchangers. Conceptual design and duct shape optimization has been carried out which produced integrated ICD heat exchanger designs with significantly lower air-side total pressure losses compared to their conventional straight fin counterparts, which could improve system level integration and engine performance. A direct outcome of this study is a pressure loss correlation, which can be used in future engine system-level trade studies.

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.