Shape Optimization Method Research Articles

A variable gradient descent shape optimization method for transition tee resistance reduction

In this paper, a local component optimization method is proposed for reducing resistance and consumption in building transmission and distribution systems. The gradient descent method is used to investigate energy dissipation minimization constrained by the Navier‒Stokes equation with nonlinear boundary conditions, and the optimality conditions of the shape are derived from the shape optimization problem using the function space parameterization method. The shape of the upper wall of the transition tee is iteratively optimized until the optimal shape with the minimum energy dissipation is obtained. The derivative of the free boundary shape of the transition tee and the objective function are proposed, and the normalized wall height of the transition tee with different area ratios are provided; these results can enable the adaptation of various sizes of transition tees in engineering to achieve air supply conditions that meet the engineering requirements. In this paper, the flow field and pressure loss of the optimized and conventional tees are compared, and the optimized tee performance improvement and energy saving potential are evaluated. The optimized transition tee has a resistance reduction rate of 6∼99% for the straight duct and 2–98% for the branch duct under different area ratios and flow ratios. Our method has the advantages of being more accurate, flexible and general and provides a novel method for the resistance optimization design of building transmission and distribution systems.

Direct Method of Aerodynamic Shape Optimization For Supersonic Aircraft Design

The efficient direct optimization method for Supersonic Aircraft Design is presented. Aerodynamic drag and sonic boom signature parameters are used as the objective functions minimized under geometric and aerodynamic constraints. The method combines Newton based algorithm with Euler space-marching solver. The efficiency of the method is demonstrated on examples of two- and three-dimensional aerodynamic design. Nose shapes that ensure minimum wave drag for specified constraints on the overall size are defined. It is shown that the near-optimal axisymmetric noses have a flat forward face. The median surface of a complex plane form wing with minimal drag due to lift is constructed. The simplest shape deformation of the wing providing the main part of aerodynamic drag diminishing is established. To profile the aircraft wing and fuselage providing optimal sonic boom parameters approaches of theory for sonic boom propagation are included in the method. The initial shock intensity as the primary value of the pressure signature is minimized. The results obtained within the framework of the Eulers equations and simplified flow models are compared.

Efficient variable cell shape geometry optimization

A fast and reliable geometry optimization algorithm is presented that optimizes atomic positions and lattice vectors simultaneously. Using a series of benchmarks, it is shown that the method presented in this paper outperforms in most cases the standard optimization methods implemented in popular codes such as Quantum ESPRESSO and VASP. To motivate the variable cell shape optimization method presented in here, the eigenvalues of the lattice Hessian matrix are investigated thoroughly. It is shown that they change depending on the shape of the cell and the number of particles inside the cell. For certain cell shapes the resulting condition number of the lattice matrix can grow quadratically with respect to the number of particles. By a coordinate transformation, which can be applied to all variable cell shape optimization methods, the undesirable conditioning of the lattice Hessian matrix is eliminated.

In-plane bending performance of novel negative Poisson’s ratio structures and its enhancement through the application of shape optimization method

NPR structures possess unique mechanical characteristics rendering them attractive for various engineering applications. However, their low in-plane stiffness limits their structural applications. The present study proposes a two-fold solution to overcome this issue. Firstly, new negative Poisson’s ratio (NPR) structures were designed with improved NPR thereby yielding better performance than the conventional NPR structures. Secondly, shape optimization was applied to the structures, whereby material was redistributed in each cell according to the applied load. The bending performance of structures was evaluated through Finite Element Method, duly validated, and quantified in terms of their flexural stiffness and energy absorption capacity. According to the results, the newly proposed structures offered higher performance than the conventional structure, i.e., 37–213 % stiffness and 9–37 % specific energy absorption capacity. This improvement further increased by many folds when shape optimization was applied to the structures, i.e., 427 % stiffness and 151 % specific energy absorption as compared to unoptimized conventional structures. This study reveals that the bending performance of NPR structures can be enhanced by improving the cell design/NPR and invoking shape optimization. The results detailed herein study can serve as a guideline to realize further improvements.

Level set-based shape optimization of deformable structures for achieving desired sound transmission and reflective responses

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framework of a level set-based shape and topology optimization method. Finally, we provide two-dimensional optimization examples that demonstrate the effectiveness of our proposed method in providing optimized designs with desired functionality, while considering the structural deformation.

Effects of Static Stability Margin on Aerodynamic Design Optimization of Truss-Braced Wing Aircraft

Currently, the aviation industry is facing an oil and energy crisis and is contributing much more greenhouse gas emissions to the environment. Aircraft design approaches, such as aerodynamic shape optimization, new configuration concepts, and active control technology, have been the primary and effective means of achieving goals concerning fuel burn, noise, and emissions. For now, the design problems of relaxed static stability (RSS, an active control technique) and truss-braced wing (TBW) configurations with high-fidelity aerodynamic shape optimization methods have been investigated widely to promote aerodynamic performance. Nevertheless, they are studied almost always separately, and the combination of exploration and refined design is rarely presented. Therefore, the purposes of this work are to evaluate the benefits of RSS on a full TBW wing–body–tail configuration under various flight conditions and the effects on multi-components and to further explore the potential and analyze the aerodynamic features with the combination of shape optimization and RSS. To address these issues, on the one hand, a range of seven static stability margins are adopted to evaluate its effects with a high-fidelity Reynolds-averaged Navier–Stokes solver. On the other hand, seven cases of drag minimization multipoint aerodynamic design optimization are performed, which are with 600 shape variables and 13 twist variables, subject to lift coefficient, trim, and thickness constraints. The results indicate that with RSS only, the initial configuration has a 2.39% drag reduction under cruise conditions and a 3.01% and a 5.24% drag reduction under two off-design conditions. Additionally, the effects on the multi-components are observed and analyzed. Moreover, all of the optimized configurations with RSS have 2.13%, 2.42%, and 2.12% drag reductions under cruise conditions, drag divergence conditions, and near-buffet-onset conditions, respectively. The most promising optimized configuration has a lift-to-drag ratio of 24.48 with an aerodynamic efficiency of 17.14. The evaluations with a series of off-design points also present high-level aerodynamic efficiency.

Design for the Vent Holes of Gas Turbine Flow Guide Disks Based on the Shape Optimization Method

Two shape optimization methods, based on non-parametric and geometric parameters, were developed to address stress concentrations in the vent holes of gas turbine flow guide disks. The design optimization focused on reducing the maximum equivalent stress at the hole edge in an aero-engine gas turbine flow guide disk. The effectiveness of both methods in achieving this objective was studied. The results indicated that the non-parametric-based optimization method reduced the maximum equivalent stress at the hole edge by 24.5% compared to the initial design, while the geometric parameter-based optimization method achieved a reduction of 20.2%. Both shape optimization methods proved effective in reducing stress concentrations and improving fatigue life. However, the non-parametric shape optimization method resulted in a better design for the vent holes based on the study’s findings.

Shape Transformation Approaches for Fluid Dynamic Optimization

The contribution is devoted to combined shape- and mesh-update strategies for parameter-free (CAD-free) shape optimization methods. Three different strategies to translate the shape sensitivities computed by adjoint shape optimization procedures into simultaneous updates of both the shape and the discretized domain are employed in combination with a mesh-morphing strategy. Considered methods involve a linear Steklov–Poincaré (Hilbert space) approach, a recently suggested highly non-linear p-Laplace (Banach space) method, and a hybrid variant which updates the shape in Hilbert space. The methods are scrutinized for optimizing the power loss of a two-dimensional bent duct flow using an unstructured, locally refined grid that initially displays favorable grid properties. Optimization results are compared with respect to the optimization convergence, the computational effort, and the preservation of the mesh quality during the optimization sequence. Results indicate that all methods reach, approximately, the same converged optimal solution, which reduces the objective function by about 18% for this classical benchmark example. However, as regards the preservation of the mesh quality, more advanced Banach space methods are advantageous in comparison to Hilbert space methods even when the shape update is performed in Hilbert space to save costs. In specific, while the computational cost of the Banach space method and the hybrid method is about 3.5 and 2.5 times the cost of the pure Hilbert space method, respectively, the grid quality metrics are 2 times and 1.7 times improved for the Banach space and hybrid method, respectively.

Data-driven non-intrusive shape-topology optimization framework for curved shells

Aiming at obtaining an optimal configuration of curved shells in the compact design space, it is an effective way to perform the shape and topology optimization simultaneously. Considering the complexity in solving the coupled shape and topology optimization problem, this paper presents an easy-to-implement data-driven non-intrusive shape-topology optimization framework for curved shells, including two steps. In the off-line step, the shape equation-driven mesh deformation method is firstly proposed, contributing to using a small number of design variables for fast shape change and optimization of the skin of curved shell without re-modeling and re-meshing. Latin hypercube sampling is used to generate sample points as the input data in the design space, and the corresponding skin shape is obtained. In each design domain determined by different skin shapes, the topology optimizations are performed in parallel or by multiple optimization solvers, after that above multi-source optimization results are collected into the output dataset. The mapping relationship is trained between the input data and output data via surrogate modeling technique, and this non-intrusive concept contributes to integrating the independent shape and topology optimization into a single-loop data-driven optimization. The cross validation-Voronoi adaptive model updating method is used to improve the prediction accuracy. In the on-line step, the single-loop data-driven optimization is carried out using covariance matrix adaptive evolution strategy and adaptive optimization updating method, which can obtain optimized results more efficiently than direct topology optimization or shape optimization method. In order to verify the effectiveness of the proposed data-driven non-intrusive shape-topology optimization framework, three optimization examples are carried out. Example results show that compared with the results by the single topology optimization, the strain energy results by the proposed framework can be reduced by 20.55%, 15% and 51.6%, respectively, highlighting the outstanding optimization ability of the proposed framework.

Free form shape optimization of three dimensional beams using cross section analysis

We introduce an efficient method for shape optimization of three dimensional beams that is based on free-form design of the beam’s cross section. The cross section boundary of each beam segment is parameterized by a closed, unclamped B-spline curve, whose control points’ coordinates serve as design variables. The response of the beam is evaluated globally as a one-dimensional entity using 3D beam finite elements, while each cross section is modeled locally using cross section analysis following anisotropic beam theory. A fully consistent analytical sensitivity analysis is derived to account for the relation between the control points and the beam cross-sectional properties and design updates are performed using gradient-based optimization. The proposed method is applied to several test cases, aiming at minimization of compliance or maximization of the fundamental frequency. The numerical results and a comparison to other methods for beam optimization show that the method can generate highly competitive designs. A consistent quantitative comparison to high resolution continuum topology optimization shows that a performance which is only slightly inferior can be achieved, for only a small fraction of the computational effort.

Study on optimizing transition layer thickness to reduce tire sidewall delamination

This paper begins with a common disease of tires. The analysis of the disease of the sidewall area reveals that the thickness of the transition layer is closely related to the tire side delamination. This paper combines the phenomenon of sidewall thinning with the actual working process. The shape optimization method is used to optimize the design of the tire side transition layer while maintaining the thickness of the tire’s inner liner layer. Finally, finite element analysis was used for simulation verification. The findings demonstrated that the optimum thickness of the transition layer structure enhanced the inner von Mises stress of the tire carcass chord by 8.04% and the shear stress by 8.79%-15.08%, hence boosting the sidewall resistance to thickness decrease by 12.98%-22.48%.

Efficient multimodal-based shape optimization of acoustic horns with application to subwavelength perfect transmission

This paper presents an efficient multimodal-based shape optimization method for maximizing sound radiation from open-ended acoustic horns. The proposed method tightly integrates the improved multimodal admittance method with interpolation-based geometry parameterization to allow flexible design degrees of freedom, with computational cost for gradient evaluation nearly independent of the number of design variables. We demonstrate that our method outperforms an existing discrete-model-based approach in solving typical narrowband and wideband horn design problems, offering increased efficiency and flexibility. For the first time, we achieve novel resonator-like waveguide designs with subwavelength perfect transmission well below the cut-off frequency of classical horns by performing shape optimization. The optimal waveguides exhibit single to multiple transmission peaks or bandpass characteristics. We validate the designs numerically and experimentally and demonstrate that the perfect transmission is related to the excitation of the acoustic resonant mode. Our method has potential applications in the design of compact high-output subwoofers, anechoic duct terminations, and acoustic filters.

The implicit bulk-surface filtering method for node-based shape optimization and a comparison of explicit and implicit filtering techniques

This work studies shape filtering techniques, namely the convolution-based (explicit) and the PDE-based (implicit), and introduces the implicit bulk-surface filtering method to control the boundary smoothness and preserve the internal mesh quality simultaneously in the course of bulk (solid) shape optimization. To that end, the volumetric mesh is governed by the pseudo-solid smoothing equations, which are stiffened by the mesh-Jacobian and endowed with the Robin boundary condition, which involves the Laplace-Beltrami operator on the mesh boundaries. Its superior performance from the non-simultaneous (sequential) treatment of boundary and internal meshes is demonstrated for the shape optimization of complex solid structures. Well-established explicit filters, namely Gaussian and linear, and the Helmholtz/Sobolev-based (implicit) filter are critically examined for shell optimization in terms of consistency (rigid-body-movement production), geometric characteristics, and computational cost. It is demonstrated that implicit filtering is more numerically efficient and robustly enforces fixed boundaries compared to explicit filtering. Supported by numerical experiments, a regularized Green’s function is introduced as an equivalent explicit form of the Helmholtz/Sobolev filter. Furthermore, we give special attention to deriving mesh-independent filtered sensitivities for node-based shape optimization with non-uniform meshes. It is shown that mesh-independent filtering can be achieved by scaling discrete sensitivities with the inverse of the mesh mass matrix.

Coupled physics analysis of blended-wing-body underwater glider equipped with electromagnetic active flow control

Lift-drag ratio is an important index to determine the hydrodynamic performance, glide performance, and economy of the underwater glider(UG). Limited by many factors, the shape optimization method has been unable to meet the needs of UG hydrodynamic performance optimization. Aiming at this crucial problem, the electromagnetic active flow control method for the UG is proposed in this paper. The coupled physical analysis of the blended-wing-body underwater glider(BWB-UG) equipped with electromagnetic active flow control is carried out by numerical method. The numerical calculation model of electromagnetic active flow control is established. The coupled physical field structure of the BWB-UG equipped with electromagnetic active flow control is studied by Detached-Eddy Simulation. The mechanism of electromagnetic active flow control on the hydrodynamic performance of the BWB-UG is further revealed by coupling physics analysis. The results show that a layer of the periodically distributed electromagnetic force is formed on the surface of BWB-UG, and the boundary layer structure can be effectively controlled. By equipping with electromagnetic active flow control, the lift-drag ratio of BWB-UG can be increased by 11.659%.

Concurrent shape and build orientation optimization for FDM additive manufacturing using the principal stress lines (PSL)

Additive Manufacturing (AM) with the consisting constantly evolving technologies is a particularly popular research area. Based on the shape forming freedom, size, shape, and topology optimization techniques can be validated by AM produced parts. However, in every manufacturing process, AM also has some adverse inherent properties. One and maybe the most significant optimization problem is the mechanical anisotropy caused by the layered structure. In this paper, a simultaneous build orientation and shape optimization method is presented. Both of the approaches are intended to increase the mechanical performance of the produced parts. Shape optimization was accomplished by varying the cross-section of the beam geometries, based on the angle between a PSL section and the characteristic load direction. To test the efficiency and validate the method 2D structures (with relatively small 3rd dimension) and their tensile properties were tested. Based on the results, we can prove that the PSL method works and help to increase the mechanical performance by 19.2% with only 7.8% size increment.

An efficient method for helicopter fuselage shape optimization

PurposeThis paper aims to develop a new method of fuselage drag optimization that can obtain results faster than the conventional methods based on full computational fluid dynamics (CFD) calculations and can be used to improve the efficiency of preliminary design.Design/methodology/approachAn efficient method for helicopter fuselage shape optimization based on surrogate-based optimization is presented. Two numerical simulation methods are applied in different stages of optimization according to their relative advantages. The fast panel method is used to calculate the sample data to save calculation time for a large number of sample points. The initial solution is obtained by combining the Kriging surrogate model and the multi-island genetic algorithm. Then, the accuracy of the solution is determined by using the infill criteria based on CFD corrections. A parametric model of the fuselage is established by several characteristic sections and guiding curves.FindingsIt is demonstrated that this method can greatly reduce the calculation time while ensuring a high accuracy in the XH-59A helicopter example. The drag coefficient of the optimized fuselage is reduced by 13.3%. Because of the use of different calculation methods for samples, this novel method reduces the total calculation time by almost fourfold compared with full CFD calculations.Originality/valueTo the best of the authors’ knowledge, this is the first study to provide a novel method of fuselage drag optimization by combining different numerical simulation methods. Some suggestions on fuselage shape optimization are given for the XH-59A example.

Comparison of Shape Optimization Methods for Heat Exchanger Fins Using Computational Fluid Dynamics

Inverse design techniques are one way to leverage advances in 3D printing, artificial intelligence, and computational resources to achieve increased performance of heat exchangers. Two optimization techniques (genetic algorithm and particle swarm) and three geometry representations (binary level set, composite Bézier, and free form deformation) are used to increase the heat transfer and reduce the pressure drop of a heat exchanger fin. After running 210,810 OpenFOAM simulations, results indicate that a significant performance increase of the fin can be realized in less than 48 hrs, allowing for such a process to be integrated in traditional design processes. The best design increased the performance of the objective function, compared to the baseline rectangular geometry by 75%. A custom distributed infrastructure was built, allowing for all methods to reach 95% of the final objective values in a little over 4 hrs, handling 1674 OpenFoam simulations per hour.

Aeroacoustic shape optimization using adjoint sensitivity analysis based on lattice Boltzmann method for bluff bodies

The present study developed a shape optimization method using adjoint sensitivity analysis based on lattice Boltzmann method to suppress aerodynamically sound generation around bluff bodies. In this method, flow and acoustic fields are directly simulated using the lattice Boltzmann equation (LBE) with an athermal model under low-Mach-number conditions. The sensitivities of far-field sounds to numerous design variables that define an object shape can be evaluated by solving the adjoint equation, which is derived from the LBE. The adjoint equation can be computed easily and efficiently, similar to the LBM. The proposed method was applied to the cylinder Aeolian tone, and the cylinder shape was optimized.

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

Cathode Shape Design for Steady-State Electrochemical Machining

The inverse or cathode shape design problem of electrochemical machining (ECM) deals with the computation of the shape of the tool cathode required for producing a workpiece anode of a desired shape. This work applied the complex variable method and the continuous adjoint-based shape optimization method to solve the steady-state cathode shape design problem with anode shapes of different smoothnesses. An exact solution to the cathode shape design problem is proven to exist only in cases when the function describing the anode shape is analytic. The solution’s physical realizability is shown to depend on the aspect ratio of features on the anode surface and the width of the standard equilibrium front gap. In cases where an exact and physically realizable cathode shape exists, the continuous adjoint-based shape optimization method is shown to produce accurate numerical solutions; otherwise, the method produces cathode shapes with singularities. For the latter cases, the work demonstrates how perimeter regularization can be applied to compute smooth approximate cathode shapes suitable for producing workpieces within the range of manufacturing tolerance.