Small Number Of Design Variables Research Articles

Computational Design of 2D Lattice Structures Based on Crystallographic Symmetries

Abstract The design representations of lattice structures are fundamental to the development of computational design approaches. Current applications of lattice structures are characterized by ever-growing demand on computational resources to solve difficult optimization problems or generate large datasets, opting for the development of efficient design representations which offer a high range of possible design variants, while at the same time generating design spaces with attributes suitable for computational methods to explore. In response, the focus of this work is to propose a parametric design representation based on crystallographic symmetries and investigate its implications for the computational design of lattice structures. The work defines design rules to support the design of functionally graded structures using crystallographic symmetries such that the connectivity between individual members in a structure with varying geometry is guaranteed and investigates how to use the parametrization in the context of optimization. The results show that the proposed parametrization achieves a compact design representation to benefit the computational design process by employing a small number of design variables to control a broad range of complex geometries. The results also show that the design spaces based on the proposed parametrization can be successfully explored using a direct search-based method.

FAST OPTIMAL DESIGN OF SHELL-GRADED-INFILL STRUCTURES WITH EXPLICIT BOUNDARY BY A HYBRID MMC-AABH PLUS APPROACH

Abstract In this study, a hybrid MMC-AABH plus approach is developed for the fast optimal design of shell-graded-infill structures. The key idea is to use a proper description about the graded microstructural infill and the coating shell. To this end, a set of moving morphable components is adopted to represent the boundary of the coating shell, while the graded infill is embodied by spatially varying orthotropic porous configurations. Under such a treatment, with a small number of design variables, both the boundary of the coating shell and the graded microstructure infill can be optimized simultaneously. Other attractive features of the present study are summarized as follows. Firstly, the smooth variation across the microstructural infill can be automatically satisfied based on the proposed approach compared with other similar method. Secondly, with the use of the extreme value principle of Laplace equation, the minimum feature size can be explicit controlled during the optimization. Thirdly, compared with other methods in the frontier, the approach proposed in the present study enjoys considerable reduction in the computation cost and can obtain near-optimal design of coating structure. The effectiveness of the proposed approach is further demonstrated with numerical examples.

Manufacturing constrained shape optimisation of variable width flat web formed channels

The trend in automotive manufacturing towards lower volumes and an increased number of car variants combined with the need for forming higher strength metals to reduce weight has led to the implementation of alternative and flexible manufacturing methods. These have new manufacturing constraints compared to conventional stamping that change the part shapes that can be formed. This requires new methods for part shape optimisation. This study proposes a novel parametrisation for shape design that allows: 1) implementation of a gradient-based optimisation approach; and 2) taking manufacturing constraints into account. Our novel parameterisation can describe most long automotive structural parts using only a small number of design variables. The parts are described using multiple series of straight and curved connected profiles. We have uniquely conducted a detailed sensitivity analysis on the profiles to determine analytical solutions for the first order derivatives of the design variables with respect to the surface area/mass of a generic part. The profiles are also used to determine the final manufacturing strains in a part based on ideal forming. These ideal manufacturing strains can be compared to manufacturing process strain limits to determine the potential manufacturability of the part. The proposed parametrisation is applied to optimise a variable width channel formed by flexible roll forming. The channel is optimised to maximise the stiffness while maintaining both mass and manufacturability. In detail, the effectiveness and the general applicability of the established parametrisation technique and shape optimisation platform are demonstrated using three case studies of a flexible roll formed automotive S-rail channel part subjected to compression and bending loads. Furthermore, the manufacturability of the optimised structure is demonstrated by a forming model of the flexible roll forming process, where the model has been previously validated against experimental data. These examples show that the presented parametrisation and the associated shape optimiser can be successfully applied to increase part stiffness while reducing weight and maintaining manufacturability. The range of problems analysed demonstrates the flexibility and capability of the newly developed optimisation platform.

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.

Comparison of Different Optimization Methods for the Microstructure of Proton Exchange Membrane Fuel Cells

Proton exchange membrane fuel cells (PEMFCs) have attracted much attention in the research and commercial sectors because they are environmental friendly and have relatively high power density compared to most other energy storage devices based on electrochemical processes. A significant part of the research on PEMFCs is related to decreasing the cost and improving the structure of PEMFCs by optimizing numerically the microstructure of these devices including the non-uniform porosity, carbon, nafion, and catalyst (i.e. platinum) distributions and the geometrical characteristics. The existing optimization techniques can generally be divided into heuristic and non-heuristic techniques. Heuristic techniques usually use stochastic approaches that search the optimization space using trial-and-error methods. Examples of such techniques include the genetic algorithm, particle swarm optimization, differential evolution, simulating annealing, harmony search, memetic algorithms, etc. These techniques can be applied to optimize a small number of design variables and often need to be used with simplified (often purely mathematic and not based on fundamental physics) models, such as compact, reduced-order models, or equivalent circuit models. On the other hand, non-heuristic techniques are deterministic and often require evaluating the gradient of the objective function. In general, these techniques have local convergence and their rate of convergence can be increased, using second-order derivatives such as the Hessian matrix, or taking advantage of the particular functional form of the objective function. Non-heuristic techniques usually require between a few tens of iterations to optimize the objective function compared to heuristic techniques which require at least tens of thousands of iterations.In this presentation we will make a thorough critical review of the different optimization methods used in the literature for PEMFCs. In additional, we will present the results of our efforts to develop a full-cell optimization algorithm for PEMFCs using the adjoint method [1-3]. The advantages and disadvantages of different methods, including the adjoint method, will be discussed in detail and examples of sample simulations results will be presented.

Comparison of structural parameterization methods for the multidisciplinary optimization of active morphing wing sections

Morphing wings promise to increase the aerodynamic efficiency of future aircraft. A challenging task is to find structures that enable the active deformation of morphing wings leading to an increased aerodynamic efficiency and being lightweight at the same time. For the development of suitable optimization methods, choosing an efficient topology parameterization method is a crucial point. Contrary demands are a small number of design variables and a wide description of valid structures. The present work compares three different parameterization methods for their applicability in the multidisciplinary multi-objective optimization of active morphing wing sections. Namely, these parameterization methods are Lindenmayer cellular systems, Voronoi diagrams and the bijective graph representation. It is shown that all three parameterization methods are basically suitable for the optimization problem addressed but have individual characteristics and benefits.

A moving morphable component‐based topology optimization approach considering transient structural dynamic responses

AbstractA novel approach is developed under the moving morphable component (MMC) framework to solve topology optimization problems considering transient structural dynamic responses, where the structure for optimization is constructed with a collection of structural components, and the structural layout can be described explicitly via only a small number of design variables characterizing the positions and geometries of the components. The transient dynamic response of the structure and corresponding sensitivities are computed via step‐by‐step numerical integration and mode reduction technique is also employed to enhance the computational efficiency. Compared with existing works, the numerical instabilities induced by localized vibration modes can be substantially alleviated and the computational effort for direct sensitivity analysis is significantly reduced in the proposed approach. Two and three‐dimensional numerical examples considering the cases of nonproportional damping mechanism and time‐varying mass are also provided to manifest the validity of the proposed approach.

Photonic crystal topological design for polarized and polarization-independent band gaps by gradient-free topology optimization.

Photonic crystals can be adopted to control light propagation due to their superior band gap feature. It is well known the band gap feature of photonic crystals depends significantly on the topological design of the lattices, which is rather challenging due to the highly nonlinear objective function and multiple local minima feature of such design problems. To this end, this paper proposed a new band-gap topology optimization framework for photonic crystals considering different electromagnetic wave polarization modes. Based on the material-field series-expansion (MFSE) model and the dielectric permittivity interpolation scheme, the lattice topologies are represented by using a small number of design variables. Then, a sequential Kriging-based optimization algorithm, which shows strong global search capability and requires no sensitivity information, is employed to solve the band gap design problem as a series of sub-optimization problems with adaptive-adjusting design spaces. Numerical examples demonstrated the effectiveness of the proposed gradient-free method to maximize the band gap for transverse magnetic field (TM), transverse electric field (TE), and complete modes. Compared with previously reported designs, the present results exhibit less dependency on the guess of the initial design, larger band gaps and some interesting topology configurations.

An Efficient Method for Complex Antenna Design Based on a Self Adaptive Surrogate Model-Assisted Optimization Technique

Surrogate models are widely used in antenna design for optimization efficiency improvement. Currently, the targeted antennas often have a small number of design variables and specifications, and the surrogate model training time is short. However, modern antennas become increasingly complex, which needs much more design variables and specifications, making the training time become a new bottleneck, i.e., in some cases, even longer than electromagnetic (EM) simulation time. Therefore, a new method, called training cost reduced surrogate model-assisted hybrid differential evolution for complex antenna optimization (TR-SADEA), is presented in this article. The key innovations include: 1) a self-adaptive Gaussian process surrogate modeling method with a significantly reduced training time while mostly maintaining the antenna performance prediction accuracy and 2) a new hybrid surrogate model-assisted antenna optimization framework that reduces the training time and increases the convergence speed. An indoor base station antenna with 2G to 5G cellular bands (45 design variables and 12 specifications) and a 5G outdoor base station antenna (23 design variables and 18 specifications) are used to demonstrate TR-SADEA. Experimental results show that more than 90% of the training time and about 20% iterations (simulations and surrogate modeling) are reduced compared to a state-of-the-art method while obtaining high antenna performance.

Optimal design of shell-graded-infill structures by a hybrid MMC-MMV approach

In the present work, a hybrid MMC-MMV approach is developed for designing additive manufacturing-oriented shell-graded-infill structures. The key idea is to describe the geometry of a shell-graded-infill structure explicitly using some geometry parameters. To this end, a set of morphable voids is adopted to describe the boundary of the coating shell, while a set of morphable components combing with a coordinate perturbation technique are introduced to represent the graded infill distribution. Under such treatment, both the crisp boundary of the coating shell and the graded infill can be optimized simultaneously, with a small number of design variables. Numerical examples demonstrate the effectiveness of the proposed approach.

Stress-related topology optimization of shell structures using IGA/TSA-based Moving Morphable Void (MMV) approach

There have been quite a few research works on the topology optimization under stress constraints. However, most of them only treated two dimensional (2D) cases. In the present work, a new approach for topology optimization of three dimensional (3D) shell structures under stress constraints is presented. This approach is constructed by combining the Moving Morphable Void (MMV) approach with trimming technique-based Isogeometric analysis (IGA), where the shell geometry is described using Non-Uniform Rational B-Spline (NURBS) and a set of trimming curves. The proposed approach not only shares the same advantages of MMV approach for topology optimization (e.g., involving small number of design variables, linking with CAD system seamlessly, etc.), but also has the capability of providing high accuracy stress analysis results using much smaller number of degree of freedoms with the help of trimming surface analysis (TSA) technique. Several numerical examples are also presented to demonstrate the effectiveness and advantages of the proposed approach.

Data-driven streamline stiffener path optimization (SSPO) for sparse stiffener layout design of non-uniform curved grid-stiffened composite (NCGC) structures

The homogenization-based streamline stiffener path optimization (SSPO) method was previously proposed by the authors for stiffener layout design of non-uniform curved grid-stiffened composite (NCGC) structures. The applications are limited to densely distributed stiffeners with no significant scale effect in the homogenization process, which is revealed in this work. A data-driven SSPO method, which is inspired by the popular principal component analysis (PCA) in data processing, is proposed for sparse stiffener layout design. With several referential layouts as input, the PCA generates a sequence of principal layouts. Together with weighting factors and distortion techniques, using only a small number of dominant principal layouts is enough to represent most possible synthesized layouts, which significantly reduces the number of design variables while preserving and extending a large design space. This strategy is further enhanced with a detailed modeling technique based on multiple point constraints (MPCs) to automate the modeling process. The data-driven SSPO method can be integrated with either gradient-based or artificial intelligence (AI) algorithms due to the small number of design variables. A flexible objective target, to maximize the critical buckling load or minimize the structural weight, can be used since the structural responses are evaluated on the detailed model. Numerical examples demonstrate the effectiveness of the proposed method in optimizing the layout of sparsely distributed stiffeners.

Data-Driven Structural Design Optimization for Petal-Shaped Auxetics Using Isogeometric Analysis

Focusing on the structural optimization of auxetic materials using data-driven methods, a back-propagation neural network (BPNN) based design framework is developed for petal-shaped auxetics using isogeometric analysis. Adopting a NURBS-based parametric modelling scheme with a small number of design variables, the highly nonlinear relation between the input geometry variables and the effective material properties is obtained using BPNN-based fitting method, and demonstrated in this work to give high accuracy and efficiency. Such BPNN-based fitting functions also enable an easy analytical sensitivity analysis, in contrast to the generally complex procedures of typical shape and size sensitivity approaches.

Indoor lighting optimization: a comparative study between grid search optimization and particle swarm optimization

Design of an indoor lighting system involves, in general, a number of design parameters, viz. maintained average illuminance, uniformity of illuminance, unified glare rating, lighting power density. These design parameters are mutually conflicting and at the same time constrained because recommended value of each of the design parameter is to be satisfied simultaneously by adjusting design variables which are also bounded by upper and lower limits. An indoor lighting design problem involves a small number of design variables, such as spacing between adjacent luminaires, mounting height of luminaires, and the objective function is non-differentiable and thus suitable to apply both the grid search optimization and particle swarm optimization methods. This study compares the performance of these two methods to find out optimized design solutions for a set of indoor lighting design strategies, usually come across by lighting designers. An objective function is formulated by assigning weights to the design parameters, where a specific design strategy can be selected by adjusting the weights. This study reveals that the performance of PSO is significantly better to meet the qualitative aspect of lighting ambiance; however, the improvement is not significant than the grid search method when a design strategy puts stress upon the quantitative aspect of lighting.

Design of sandwich panels with truss cores using explicit topology optimization

Sandwich panels with truss cores (SPTCs) are a class of novel structures with superior load bearing performance. In recent years, many efforts have been made in designing the SPTCs. However, the majority of related researches are conducted under the prespecified structural topologies of the truss cores according to subjective judgment and experience. In this paper, a novel approach under moving morphable components (MMC) based topology optimization framework is proposed for the design of SPTCs. To describe the SPTCs in an explicit and geometrical way, a multi-component topology description function (MCTDF) is developed in both 2D and 3D cases. By optimizing the configurations and layouts of the multi-components, the topological design of SPTCs with better structural performance can be generated. In this regard, the original topology optimization problem is transformed into a parameter optimization problem with only a small number of design variables. To determine the number of beams in truss core, a deletion strategy is proposed to remove the redundant multi-components. The minimum compliance problem subject to structural volume fraction constraint is investigated, and the relevant sensitivity analysis is shown. Both 2D and 3D numerical examples are presented to test the validity and effectiveness of the proposed method.

Optimal form and size characterization of planar isotropic petal-shaped auxetics with tunable effective properties using IGA

Focusing on planar isotropic petal-shaped auxetics, an isogeometric design framework is presented to achieve tunable effective properties. Specifically, the design framework includes (i) a NURBS-based parametric modelling scheme that characterizes petal-shaped auxetics with a small number of design variables; (ii) a systematic consideration of petal form, component widths and base material properties; (iii) a semi-analytical sensitivity analysis method based on material derivatives; and (iv) constraints for effective stiffness and target Poisson ratio. Three cases are considered: Case A with the same component width, Case B with different component widths, and Case C for composite designs with multiple base materials. For each case, a design limit curve is obtained for the effective Poisson ratio over a range of effective stiffness constraints, to give a quick overview on the properties attainable for each design setting. The optimization framework is next demonstrated for designing composite petal-shaped auxetics with target effective properties.

Multilevel Subdivision Parameterization Scheme for Aerodynamic Shape Optimization

Subdivision curves are defined as the limit of a recursive application of a subdivision rule to an initial set of control points. This intrinsically provides a hierarchical set of control polygons that can be used to provide surface control at varying levels of fidelity. This work presents a shape parameterization method based on this principle and investigates its application to aerodynamic optimization. The subdivision curves are used to construct a multilevel aerofoil parameterization that allows an optimization to be initialized with a small number of design variables, and then be periodically increased in resolution throughout. This brings the benefits of a low-fidelity optimization (high convergence rate, increased robustness, low-cost finite difference gradients) while still allowing the final results to be from a high-dimensional design space. In this work, the multilevel subdivision parameterization is tested on a variety of optimization problems and compared with a control group of single-level subdivision schemes. For all the optimization cases, the multilevel schemes provided robust and reliable results in contrast to the single-level methods that often experienced difficulties with large numbers of design variables. As a result of this, the multilevel methods exploited the high-dimensional design spaces better and consequently produced better overall results.

Additive Manufacturing-Oriented Design of Graded Lattice Structures Through Explicit Topology Optimization

In the present work, a new approach for designing graded lattice structures is developed under the moving morphable components/voids (MMC/MMV) topology optimization framework. The essential idea is to make a coordinate perturbation to the topology description functions (TDF) that are employed for the description of component/void geometries in the design domain. Then, the optimal graded structure design can be obtained by optimizing the coefficients in the perturbed basis functions. Our numerical examples show that the proposed approach enables a concurrent optimization of both the primitive cell and the graded material distribution in a straightforward and computationally effective way. Moreover, the proposed approach also shows its potential in finding the optimal configuration of complex graded lattice structures with a very small number of design variables employed under various loading conditions and coordinate systems.

A benchmark study on the efficiency of unconstrained optimization algorithms in 2D-aerodynamic shape design

Optimization algorithms are used in various engineering applications to identify optimal shapes. In this work, we benchmark several unconstrained optimization algorithms (Nelder–Mead, Quasi-Newton, steepest descent) under variation of gradient estimation schemes (adjoint equations, finite differences). Flow fields are computed by solving the Reynolds-Averaged Navier–Stokes equations using the open source computational fluid dynamics code OpenFOAM. Design variables vary from N=2 to N=364. The efficiency of the optimization algorithms are benchmarked in terms of: (a) computation time, and (b) applicability and ease of use. Results for lift optimizations are presented for airfoils at a Reynolds number of Re=50,000. As a result, we find for a small number of design variables N≈5 or less, the computational efficiency of all optimization algorithms to be similar, while the ease of use of the Nelder–Mead algorithm makes it a perfect choice for a low number of design variables. For intermediate and large number o...

Efficient Parameterization of Waverider Geometries

This paper summarizes the results of investigations into the development of parametric waverider geometry models, with emphasis on their efficiency, in terms of their ability to cover a large feasible design space with a sufficiently small number of design variables to avoid the “curse of dimensionality.” The work presented here is focused on the parameterization of idealized waverider forebody geometries that provide the baseline shapes upon which more sophisticated and realistic hypersonic aircraft geometries can be built. Three different aspects of rationalizing the decisions behind the parametric geometry models developed using the osculating cones method are considered. Initially, three different approaches to the design method itself are discussed. Each approach provides direct control over different aspects of the geometry for which very specific shapes would be more complex to obtain indirectly, thus enabling the geometry to more efficiently meet any related design constraints. Then, a number of requirements and limitations are investigated that affect the available options for the parametric design-driving curves of the inverse design method. Finally, the performance advantages that open up with increasing flexibility of the design-driving curves in the context of a design optimization study are estimated. This allows one to reduce the risk of overparameterizing the geometry model, while still enabling a variety of meaningful shapes. Although the osculating cones method has mainly been used here, most of the findings also apply to other similar inverse design algorithms.