Shape Optimization Framework Research Articles

An adaptive T-spline finite cell method for structural shape optimization

In this work, an adaptive T-spline finite cell method is developed for structural shape optimization based on the finite cell method (FCM). This method has the advantage of using local T-mesh refinement on the uniform B-mesh to ensure a good computing accuracy in the shape optimization process. To do this, we first carry out a pre-analysis of the structure using a uniform coarse mesh with relatively low computational effort. Then, cells for local refinements are identified by adopting the stress gradient criterion. Subsequently, an adaptive quadtree-based refinement scheme is employed to generate the adaptive T-mesh which preserves the linear independence of the corresponding T-spline shape functions. Finally, the shape optimization framework is correspondingly established, and the proposed adaptive refinement scheme is validated by numerical examples. Its efficiency is fully demonstrated by shape optimization procedures of mechanical structures.

Design of Gas gun barrel support structure by finite element model updating using optimization techniques

Abstract Impact tests in aerospace industries are usually carried out by single stage gas guns. It has three major components namely pressure chamber, valve & barrel. Pressure chamber is used to hold operating gas at high pressure which can be used to accelerate the projectile. Main factor in deciding the design parameters of the pressure chamber is the maximum operating pressure of the working medium. The barrel length plays a major role in the final achievable projectile velocity. This paper talks about designing of the support structures of the barrel for peak recoil load. For designing of the support structures a finite element formulation is made for the assembly of the barrel support and analysed for a peak recoil load. Thereafter, the problem is optimized based on the framework of shape optimization. Considering objective function as to minimize the mass of the support structure and to be able to withstand the recoil loads. The above finite element model is subjected to optimization. Design space is explored for the optimal shape of support structure for a given barrel length using the above-mentioned frame work

Shape optimization of shell structure by using Genetic Algorithm (GA) method

Shell structure has been popular for various application in building, aerospace, oil and gas, ship and automobile industry. This type of element is light weight and can be provide great stiffness for large structure. Furthermore, it is very useful for building architecture and appearance. However, currently, construction is not only concerning structural strength but also focusing on building cost and its impact to environment. Finding the balance between structural strength, cost and aesthetic appearance is essentials this nowadays. Hence, construction framework to find the leeway between those parameters is greatly important. This paper will discuss the framework for shape optimization of shell element by using Genetic Algorithm (GA) method. The shape of structure will be developed from Non-Uniform Rationale Bi-Spline (NURBS). Finite element analysis based on Reissner-Mindlin plate theory is utilized to find the element stress within the structure. Genetic algorithm method will evaluate and perform search operation inside the result domain with constraints of structural weight and element stress. The framework then was implemented in numerical example to find its effectiveness on obtaining the most optimum shape of shell structure. Based on the numerical result, it is clear that the proposed frame work is able to reach the convergence to the optimum structure without violating the given constraints.

In vivo and in vitro evaluation of a biodegradable magnesium vascular stent designed by shape optimization strategy

The performance of biodegradable magnesium alloy stents (BMgS) requires special attention to non-uniform residual stress distribution and stress concentration, which can accelerate localized degradation after implantation. We now report on a novel concept in stent shape optimization using a finite element method (FEM) toolkit. A Mg-Nd-Zn-Zr alloy with uniform degradation behavior served as the basis of our BMgS. Comprehensive in vitro evaluations drove stent optimization, based on observed crimping and balloon inflation performance, measurement of radial strength, and stress condition validation via microarea-XRD. Moreover, a Rapamycin-eluting polymer coating was sprayed on the prototypical BMgS to improve the corrosion resistance and release anti-hyperplasia drugs. In vivo evaluation of the optimized coated BMgS was conducted in the iliac artery of New Zealand white rabbit with quantitative coronary angiography (QCA), optical coherence tomography (OCT) and micro-CT observation at 1, 3, 5-month follow-ups. Neither thrombus or early restenosis was observed, and the coated BMgS supported the vessel effectively prior to degradation and allowed for arterial healing thereafter.The proposed shape optimization framework based on FEM provides an novel concept in stent design and in-depth understanding of how deformation history affects the biomechanical performance of BMgS. Computational analysis tools can indeed promote the development of biodegradable magnesium stents.

Isogeometric shape optimization of smoothed petal auxetics with prescribed nonlinear deformation

Auxetic materials with negative Poisson’s ratio have potential applications across a broad range of engineering fields. Several FE based design techniques have been developed to achieve auxetic materials with targeted effective properties, mostly in linear deformation regime. In this paper, an isogeometric shape optimization framework for designing 2D auxetic materials with prescribed deformation over large strain intervals is presented. Taking into account practical manufacturing considerations, a minimum thickness for each member is imposed via a spline-based geometric constraint. The capability of the framework is demonstrated through two examples. First, the paper considers shape optimization of smoothed hexa-petals in plane strain condition to achieve constant Poisson’s ratios ranging from a null value to -0.5 to an effective tensile strain of 50%. The second example showcases the shape optimization of smoothed tri- and hexa-petals in plane stress condition for targeted nonlinear deformation behaviour of cat’s skin up to 90% tensile strain.

Hydrodynamic shape optimization by high fidelity CFD solver and Gaussian process based response surface method

A computational framework for hydrodynamic shape optimization of complex ship hull form is proposed and applied to improve the calm water performance of the KRISO Container Ship (KCS). The framework relies on three key features: a novel shape morphing method based on a combination of subdivision surfaces and free form deformations, a robust three dimensional viscous computational fluid dynamic solver based on the openFOAM open-source libraries and a Gaussian process-response surface method (GP-RSM) based on ordinary Kriging model which has been created to speed-up the evaluation of the quantity of interest (QoI) of the design process.The accuracy of the hydrodynamic solver is proven by comparing the obtained results against available experimental measurements. A preliminary sensitivity analysis on the mesh size has been carried out aiming at reducing the computational burden required by the CFD predictions. Three GP-RSMs have been trained relying on increasing number of hull designs. Each surrogate model has been cross-validated by both leave-one-out and k-fold techniques. The behaviours of these multi-dimensional surfaces have been analyzed in details by sampling the investigated design space with 107 points according to a Full-Factorial algorithm, highlighting the regions of maximum deviation with respect to the resistance of the reference hull. The three optimum designs provided by the corresponding GP-RSM models have been verified by using high-fidelity CFD simulations with a refined mesh configuration. Calm water resistance, wave patterns and pressure distributions over the selected hull surfaces have been discussed in the light of the generated shape variations.

Design and Optimization of a Blended-Wing-Body Underwater Glider

The shape of blended-wing-body underwater glider (BWBUG) is an important factor in determining the hydrodynamic efficiency. In order to reduce the needed computation time and efforts during the search of the optimum shape, a new surrogate-based shape optimization framework is proposed to solve the complicated BWBUG shape design optimization problem in this paper. During the search, seven baseline airfoils are used to build the parametric geometric model of the BWBUG, with the planar surface being fixed. Moreover, a newly proposed ensemble of surrogates based global optimization algorithm using a hierarchical design space reduction method (ESGO-HSR) is employed to optimize each baseline airfoil. Then, the optimum shape of the BWBUG can be determined and rebuilt based on all baseline airfoils that are successful optimized. As the optimization target, the maximum lift to drag ratio of the initial design is increased by 13.72% under the given operating conditions. The results demonstrate that the presented surrogate-based shape optimization framework is efficient and capable in identifying the optimum shape of the BWBUG.

Robust aerodynamic shape optimization—From a circle to an airfoil

Aerodynamic design optimization currently lacks robustness with respect to the starting design and requires trial and error in the flow solver and optimization algorithm settings to get a converged optimal design. We address need issue by developing ways to overcome robustness issues arising from shape parametrization, mesh deformation, and flow solver convergence. Our approach is demonstrated on the Aerodynamic Design Optimization Discussion Group (ADODG) airfoil optimization benchmarks to show the factors that dominate the robustness and efficiency. In the ADODG NACA 0012 benchmark, we address the additional issue of non-unique solutions. In the ADODG RAE 2822 case, we address solver failure due to shock waves, separation, and gradient accuracies due to the frozen turbulence model. Finally, we create a new, challenging aerodynamic shape optimization case that starts with a circle to test the robustness of our aerodynamic shape optimization framework. We use both fixed and adaptive parametrization methods to tackle this problem and show how we can exploit the advantages of adaptive parametrization methods to improve both robustness and efficiency. The combination of flow solver robustness, precision of gradient information, robust mesh deformation, and adaptive parametrization brings us closer to a “push-button” solution for airfoil design.

Structural optimization under internal porosity constraints using topological derivatives

Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this article is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modeling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. Our approach is illustrated by several two- and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect.

A gradient-based shape optimization scheme via isogeometric exact reanalysis

PurposeThis paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.Design/methodology/approachThe authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.FindingsThe proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.Originality/valueThe authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.

Isogeometric sizing and shape optimization of thin structures with a solid-shell approach

This work explores the use of solid-shell elements in the the framework of isogeometric shape optimization of shells. The main difference of these elements with respect to pure shell ones is their volumetric nature which can provide recognized benefits to analyze, for example, structures with non-linear behaviors. From the design point of view, we show that this geometric representation of the thickness is also of great interest since it offers new possibilities: continuous sizing variations can be imposed by modifying the distance between the control points of the outer surfaces. In other words, shape and sizing optimization can be performed in an identical manner. Firstly, we carry out a range of numerical experiments in order to carefully compare the results with the commonly adopted technique based on the Kirchhoff-Love formulation. These studies reveal that both solid-shell and Kirchhoff-Love strategies lead to very similar optimal shapes. Then, we apply a bi-step strategy to integrate shape and sizing optimization. We highlight the potential of the proposed approach on a stiffened cylinder where the cross section along the stiffener is optimized leading to a final design with smooth thickness variations. Finally, we combine the benefits of both Kirchhoff-Love and solid-shell formulations by setting up a multi-model optimization process to efficiently design a roof.

Approximate Hessian for accelerated convergence of aerodynamic shape optimization problems in an adjoint-based framework

The current work explores the use of an approximate Hessian to accelerate the convergence of an adjoint-based aerodynamic shape optimization framework. Exact analytical formulations of the direct-direct, adjoint-direct, adjoint-adjoint, and direct-adjoint Hessian approaches are presented and the equivalence between the adjoint-adjoint and direct-adjoint formulations is demonstrated. An approximation of the Hessian is obtained from the analytical formulation by partially solving first-order sensitivities to reduce computational time, while neglecting second-order sensitivities to ease implementation. Error bounds on the resulting approximation are presented for the first-order sensitivities through perturbation analysis. The proposed method is first assessed using an inverse pressure problem for a quasi-one-dimensional Euler flow. Additionally, three-dimensional inviscid transonic test cases are used to demonstrate the effectiveness of the method.

Multi-fidelity optimization of super-cavitating hydrofoils

We present an effective multi-fidelity framework for shape optimization of super-cavitating hydrofoils using viscous solvers. We employ state-of-the-art machine learning tools such as multi-fidelity Gaussian process regression and Bayesian optimization to synthesize data obtained from multi-resolution simulations, and efficiently identify optimal configurations in the design space. We validate our simulation results against experimental data, and showcase the efficiency of the proposed work-flow in a realistic design problem involving the shape optimization of a three-dimensional super-cavitating hydrofoil parametrized by 17 design variables.

A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry

We present a novel computational framework for shape optimization problems arising in spectral geometry. The goal in such problems is to identify domains in Rd which are the global optima of certain functions of the spectrum of elliptic operators on the domains. We propose the use of a combined finite element and Bayesian optimization (FEM–BO) framework in this context, and demonstrate the key ideas on two concrete examples. We study the Pólya–Szegö conjecture on polygons, and demonstrate that our proposed framework yields the theoretically proven result for triangles and quadrilaterals, and also provides compelling numerical evidence for the case of pentagons. We next study a variant of this conjecture for the Steklov eigenvalue problem.

A multilevel framework for decomposition-based reliability shape and size optimization

A method for decoupling reliability based design optimization problem into a set of deterministic optimization and performing a reliability analysis is described. The inner reliability analysis and the outer optimization are performed separately in a sequential manner. Since the outer optimizer must perform a large number of iterations to find the optimized shape and size of structure, the computational cost is very high. Therefore, during the course of this research, new multilevel reliability optimization methods are developed that divide the design domain into two sub-spaces to be employed in an iterative procedure: one of the shape design variables, and the other of the size design variables. In each iteration, the probability constraints are converted into equivalent deterministic constraints using reliability analysis and then implemented in the deterministic optimization problem. The framework is first tested on a short column with cross-sectional properties as design variables, the applied loads and the yield stress as random variables. In addition, two cases of curvilinearly stiffened panels subjected to uniform shear and compression in-plane loads, and two cases of curvilinearly stiffened panels subjected to shear and compression loads that vary in linear and quadratic manner are presented.

Design of optimized soft soles for humanoid robots

We describe a methodology to design foot soles for a humanoid robot given walking gait parameters (i.e. given center-of-mass and zero-moment-point trajectories). In order to obtain an optimized compliant sole, we devised a shape optimization framework which takes — among other inputs, an initial rough (simplified) shape of the sole and refines it through successive optimization steps under additional constraints and a cost function. The shape is optimized based on the simulation of the sole deformation during an entire walking step, taking time dependent input of the walking pattern generator into account. Our shape optimization framework is able to minimize the impact of the foot with the ground during the heel-strike phase and to limit foot rotation in case of perturbations. Indeed, low foot rotation enforces a vertical posture and secures the balance of the humanoid robot. Moreover, weight restriction (formulated as a constraint on the sole volume) is added to our optimization problem.

Design of thermoelastic multi-material structures with graded interfaces using topology optimization

A level-set based shape and topology optimization framework is used to investigate the influence of graded interfaces in the optimization of micro-architectured multi-materials. In contrast to other studies found in the literature, interfaces are considered as smooth and graded transitions between constitutive phases instead of sharp delimitations. Case studies for extreme thermoelastic properties of 2D isotropic composites are analyzed and optimal designs are presented. It is shown that explicitly accounting for interfaces can influence the design of heterogeneous materials in composite microstructures.

A computational compensation method for fabric panels of tensioned membrane structures using a shape optimization method based on gradientless algorithms

Tensioned fabric membrane structures, in which a coated fabric material is used as both cladding and supporting structure, are classified as unconventional structures and demand a dedicated design method. In this paper, we propose a method to combine different stages in design and analysis of these structures within the shape optimization framework. The proposed method facilitates the incorporation of various sources of nonlinearities in the analysis of tensioned membrane structures. Especially, the amount of fabric panels compensation when using simple material model for the coated fabric, i.e., linear orthotropic elasticity, as is widely used in practice, is presented. The novelty of the proposed method lies on the postulation of a stress-free 3-D intermediate configuration, which is built from flat fabric panels. The goal is to find an optimum shape of this 3-D intermediate configuration such that the stress distribution in the resulting structure, which is formed by deforming the 3-D intermediate configuration, remains at desired levels. A nonlinear finite element analysis is employed to identify these stress fields in the resulting structures. In this analysis, material and geometrical nonlinearities are taken into account. Moreover, the nonlinear kinematical interaction between the boundary cables and the membrane is also considered. Hence the stress fields on the resulting structures are expected to be reliable and close to the physical ones. In this work, different gradientless optimization algorithms, viz., genetic algorithm, pattern search, and Bayesian, are considered. The obtained results show that the stress levels in the resulting structures obtained from the proposed method are close to the desired ones, and the Bayesian optimization algorithm has the best performance among the considered algorithms.

Geometric constraints for shape and topology optimization in architectural design

This work proposes a shape and topology optimization framework oriented towards conceptual architectural design. A particular emphasis is put on the possibility for the user to interfere on the optimization process by supplying information about his personal taste. More precisely, we formulate three novel constraints on the geometry of shapes; while the first two are mainly related to aesthetics, the third one may also be used to handle several fabrication issues that are of special interest in the device of civil structures. The common mathematical ingredient to all three models is the signed distance function to a domain, and its sensitivity analysis with respect to perturbations of this domain; in the present work, this material is extended to the case where the ambient space is equipped with an anisotropic metric tensor. Numerical examples are discussed in two and three space dimensions.

Safer passenger car front shapes for pedestrians: A computational approach to reduce overall pedestrian injury risk in realistic impact scenarios

Vehicle front shape has a significant influence on pedestrian injuries and the optimal design for overall pedestrian protection remains an elusive goal, especially considering the variability of vehicle-to-pedestrian accident scenarios. Therefore this study aims to develop and evaluate an efficient framework for vehicle front shape optimization for pedestrian protection accounting for the broad range of real world impact scenarios and their distributions in recent accident data. Firstly, a framework for vehicle front shape optimization for pedestrian protection was developed based on coupling of multi-body simulations and a genetic algorithm. This framework was then applied for optimizing passenger car front shape for pedestrian protection, and its predictions were evaluated using accident data and kinematic analyses. The results indicate that the optimization shows a good convergence and predictions of the optimization framework are corroborated when compared to the available accident data, and the optimization framework can distinguish ‘good’ and ‘poor’ vehicle front shapes for pedestrian safety. Thus, it is feasible and reliable to use the optimization framework for vehicle front shape optimization for reducing overall pedestrian injury risk. The results also show the importance of considering the broad range of impact scenarios in vehicle front shape optimization. A safe passenger car for overall pedestrian protection should have a wide and flat bumper (covering pedestrians’ legs from the lower leg up to the shaft of the upper leg with generally even contacts), a bonnet leading edge height around 750mm, a short bonnet (<800mm) with a shallow or steep angle (either >17° or <12°) and a shallow windscreen (≤30°). Sensitivity studies based on simulations at the population level indicate that the demands for a safe passenger car front shape for head and leg protection are generally consistent, but partially conflict with pelvis protection. In particular, both head and leg injury risk increase with increasing bumper lower height and depth, and decrease with increasing bonnet leading edge height, while pelvis injury risk increases with increasing bonnet leading edge height. However, the effects of bonnet leading edge height and windscreen design on head injury risk are complex and require further analysis.