Shape Optimization Process Research Articles

Hybrid topology/shape optimization under uncertainty for actively-cooled nature-inspired microvascular composites

We present a computational framework for Hybrid Topology/Shape (HyTopS) optimization of actively-cooled microvascular composites under uncertainty. We developed a novel HyTopS optimization scheme for microvascular composites which can perform the topological change during the shape optimization process. This task has been done by introducing a new set of design parameters to add/remove microchannels and change the topology of the network. The profound advantage of the HyTopS method in contrast to a solely shape optimization approach is that the design space in HyTopS is not limited to the topology of the initial configuration. We integrate the HyTopS optimization method with the non-intrusive polynomial chaos expansion (PCE) approach to conduct a robust and reliable design. The non-intrusive nature of the proposed method allows for almost any source of uncertainty to be incorporated virtually in the design optimization process. The PCE representations of the response metrics enable the optimizer to efficiently and precisely approximate the statistical moments, failure probabilities, and their sensitivities with respect to the design variables. We solve several numerical examples to demonstrate the advantages of using the suggested optimization scheme over deterministic optimization method for microvascular composites. The results reveal that the optimized designs obtained from performing optimization process under uncertainty outperform the optimized configurations of the deterministic optimization approach in terms of reducing the sensitivity of the design performance to the various random variables.

Combined interface shape and material stiffness optimization for uniform distribution of contact stress

Contact stress distribution is sensitive to the interface shape and the material stiffness. A combination of the interface shape and the material stiffness is investigated to obtain a uniform contact stress distribution. A combined interface shape and material stiffness optimization method is developed based on the evolutionary optimization techniques. The uniformity of the contact stress is represented by its variance and is defined as the objective function. The gap distance and the Young’s modulus are treated as the design variables of the shape optimization and the material stiffness optimization, respectively. A fixed optimization sequence, which is the material stiffness optimization followed by the interface shape optimization, is adopted. The effectiveness of the combined interface shape and material stiffness design on improving the uniformity of the contact stress is verified based on two cases. Additionally, the undesired nonsmooth transition of the contact stress after only the material stiffness optimization could be alleviated during the followed interface shape optimization process, thus a smoother and more uniform distribution of the contact stress could be achieved.

Automated Mesh Deformation for Computer-Aided Design Models That Exhibit Boundary Topology Changes

This paper describes an automated approach to deform an analysis mesh applied to a computer-aided design (CAD) model in response to a change in the shape of the CAD model. The process has been used to overcome a key issue that occurs during the shape optimization process. It is an advance over existing techniques because it works even when the boundary topology of the CAD model changes during the shape change, which other approaches cannot generally accommodate. To achieve the deformation, an analysis topology representation is created for each model, such that there is a one-to-one mapping between the analysis topology representing the models before and after the shape change. Strategies are presented to parameterize the superset entities in the analysis topology that have no global mathematical definition. The surface mesh is deformed in a way that allows the underlying surface/edge parameterizations to change, and it is shown how the domain mesh can be updated based on the surface mesh deformation. If the models before and after shape change have the same genus, then a small dimensional change in the shape of the model will result in meshes with the same topology, even if the topology of the underlying CAD representation has changed. This is highly desirable when evaluating sensitivities to a design change during an optimization.

Shape optimization of braced frames for tall timber buildings: Influence of semi-rigid connections on design and optimization process

With the recent development of timber as a viable structural material for high-rise structures, glulam braced frames have been recently introduced in lateral load-resisting systems of timber buildings. Based on a simple shape optimization problem of a braced frame, this paper explores one of the specificities of timber structures: the influence of semi-rigid connections on their overall structural behavior and design. Dowel-type connections are first studied to obtain a simplified relation between joint stiffness and axial load-carrying capacity. Then, the established local behavior law is introduced in the shape optimization process and design of a discrete braced frame subject to lateral drift constraint under wind load. The problem is solved by a COBYLA optimization method, combined with Optimality Criteria (OC) member sizing techniques. Solutions are then evaluated and compared with classical steel/concrete design. The semi-rigid behavior of connections finally leads to a significant increase in the volume of timber but also affects the optimal shape and topology of the X-braced frame compared with classical results.

Bending behavior of optimally graded 3D printed cellular beams

Periodic cellular materials can substantially improve the stiffness-to-weight ratio of structures. This improvement depends on the geometry of periodic cells. This article presents the idea of enhancing the bending stiffness of an architected cellular beam by an optimum distribution of relative density through its length and/or across its thickness. We employ a hybrid-homogenized model to expedite the computational evaluation of the bending performance of the graded cellular beams. Detailed finite element analysis (FEA) and experimental bending tests on specimens 3D printed by stereolithography validate the hybrid-homogenized modeling approach. The hybrid-homogenized model facilitates transforming the general optimization problem into a shape optimization process with the relative density of unit cells as design variables. The teaching-learning-based optimization (TLBO) algorithm is used to obtain the optimum relative density distribution, which maximizes the bending stiffness. The optimization results show a substantial increase in bending stiffness; as high as 43%, 155%, and 182% for a cellular beam graded through the length, across the thickness, and in both directions, respectively. It is found that varying the relative density of cells across the beam thickness is more effective than variation through the length. Detailed FEA and experimental bending tests corroborate the optimization findings and confirm the practicality of such graded designs for developing advanced lightweight structures. Investigating the effect of cell architecture also reveals that optimally graded cellular beams have a potential to outperform uniform cellular beams made out of ideal unit cells (Voigt bound for elastic properties) by reaching bending stiffness-to-density ratios greater than one. The relatively simple graded cellular designs are beneficial in applications where high bending stiffness and low density are essential. Recent advances in additive manufacturing promise extending the presented grading strategy for polymeric, composite, and metallic 3D printed cellular materials to fabricate high performance lightweight structural elements at a relatively low cost.

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.

Stress mirror polishing for future large lightweight mirrors: design using shape optimization.

This study proposes a new way to manufacture large lightweight aspherics for space telescopes using stress mirror polishing (SMP). This technique is well known to allow reaching high quality optical surfaces in a minimum time period, thanks to a spherical full-size polishing tool. To obtain the correct surface's aspheric shape, it is necessary to define precisely the thickness distribution of the mirror to be deformed, according to the manufacturing parameters. We first introduce active optics and stress mirror polishing techniques, and then we describe the process to obtain the appropriate thickness mirror distribution, allowing to generate the required aspheric shape during polishing phase. Shape optimization procedure using PYTHON programing and NASTRAN optimization solver using finite element model (FEM) is developed and discussed in order to assist this process. The main result of this paper is the ability of the shape optimization process to support SMP technique to generate a peculiar aspherical shape from a spherical optical surface thanks to a thickness distribution reshaping. This paper is primarily focused on a theoretical framework with numerical simulations as the first step before the manufacturing of a demonstrator. This two-step approach was successfully used for previous projects.

Progressive optimization of complex shells with cutouts using a smart design domain method

Thin-walled shells with cutouts are widely used as primary structures in the aerospace field. The mechanical analysis of shell structures using the Finite Element Analysis (FEA) method often needs very fine meshes to achieve a high-fidelity simulation. The advent of isogeometric analysis (IGA) method provides an alternative, yet efficient and accurate means to model complex shell structures with cutouts. In this paper, the trimmed surface analysis (TSA) method is applied in IGA to perform the trimmed elements integration of linear buckling analysis of shell structures. In the shape optimization process, the analytical formulae for the design sensitivities of IGA based shell buckling model are derived, and the sensitivity propagation from the design model to the analysis model is computed using a h-refinement method. Moreover, a novel smart design domain (SDD) method is proposed and implemented to reduce the number of design variables and further enhance the shape optimization efficiency, substantially. SDD can provide useful reference information to guide the selection of control points as design variables. Later, the comparison of optimization examples approve that the SDD method can provide a very efficient means to perform the local shape optimization for the complex shells. Finally, a multi-level progressive optimization framework based on the SDD method is developed, in which the design model and the analysis model are separated and implemented independently. In doing so, the design model is able to provide enough parameterization for optimization, and, in the meantime, the analysis model ensures efficiency and accuracy for simulation. Finally, the numerical examples on the buckling analysis and optimization of complex shells with cutouts demonstrate the greatly improved capacity and efficiency of the design framework proposed in this paper.

Integration of Dimension Reduction and Uncertainty Quantification in Designing Stretchable Strain Gauge Sensor

Interests in strain gauge sensors employing stretchable patch antenna have escalated in the area of structural health monitoring, because the malleable sensor is sensitive to capturing strain variation in any shape of structure. However, owing to the narrow frequency bandwidth of the patch antenna, the operation quality of the strain sensor is not often assured under structural deformation, which creates unpredictable frequency shifts. Geometric properties of the stretchable antenna also severely regulate the performance of the sensor. Especially rugged substrate created by printing procedure and manual fabrication derives multivariate design variables. Such design variables intensify the computational burden and uncertainties that impede reliable analysis of the strain sensor. In this research, therefore, a framework is proposed not only to comprehensively capture the sensor’s geometric design variables, but also to effectively reduce the multivariate dimensions. The geometric uncertainties are characterized based on the measurements from real specimens and a Gaussian copula is used to represent them with the correlations. A dimension reduction process with a clear decision criterion by entropy-based correlation coefficient dwindles uncertainties that inhibit precise system reliability assessment. After handling the uncertainties, an artificial neural network-based surrogate model predicts the system responses, and a probabilistic neural network derives a precise estimation of the variability of complicated system behavior. To elicit better performance of the stretchable antenna-based strain sensor, a shape optimization process is then executed by developing an optimal design of the strain sensor, which can resolve the issue of the frequency shift in the narrow bandwidth. Compared with the conventional rigid antenna-based strain sensors, the proposed design brings flexible shape adjustment that enables the resonance frequency to be maintained in reliable frequency bandwidth and antenna performance to be maximized under deformation. Hence, the efficacy of the proposed design framework that employs uncertainty characterization, dimension reduction, and machine learning-based behavior prediction is epitomized by the stretchable antenna-based strain sensor.

Simultaneous Shape and Topology Optimization of Planar Linkage Mechanisms Based on the Spring-Connected Rigid Block Model

Abstract Using the topology optimization can be an effective means of synthesizing planar rigid-body linkage mechanisms to generate desired motion, as it does not require a baseline mechanism for a specific topology. While most earlier studies were mainly concerned with the formulation and implementation of topology optimization-based synthesis in a fixed grid, this study aims to realize the simultaneous shape and topology optimization of planar linkage mechanisms using a low-resolution spring-connected rigid block model. Here, we demonstrate the effectiveness of simultaneous optimization over a higher-resolution fixed-grid rigid block-based topology optimization process. When shape optimization to change the block shapes is combined with topology optimization to synthesize the mechanism, the use of low-resolution discretized models improves the computation efficiency considerably and helps to yield compact mechanisms with less complexity, making them more amenable to fabrication. After verifying the effectiveness of the simultaneous shape and topology optimization process with several benchmark problems, we apply the method to synthesize a mechanism which guides a planar version of a human's gait trajectory.

Gradient-based hybrid topology/shape optimization of bioinspired microvascular composites

Construction of bioinspired vasculature in synthetic materials enables multi-functional performance via mass transport through internal fluidic networks. However, exact reproduction of intricate, natural microvascular architectures is nearly impossible and thus there is a need to create practical, manufacturable designs guided by multi-physics principles. Here we present a Hybrid Topology/Shape (HyTopS) optimization scheme for microvascular materials using the Interface-enriched Generalized Finite Element Method (IGFEM). This new approach, which can simultaneously perform topological changes as well as shape optimization of microvascular materials, is demonstrated in the context of thermal regulation. In the current study, we present a new feature that enables the optimizer to augment network topology by creating/removing microchannels during the shape optimization process. This task has been accomplished by introducing a new set of design parameters, which act analogous to the penalization factor in the Solid Isotropic Material with Penalization (SIMP) method. The analytical sensitivity for the HyTopS optimization scheme has been derived and the sensitivity accuracy is verified against the finite difference method. We impose a set of geometrical constraints to account for manufacturing limitations and produce a design which is suitable for large-scale production without the need to perform post-processing on the obtained optimum. The method is validated by active-cooling experiments on vascularized carbon-fiber composites. Finally, we compare various application examples to demonstrate the advantages of the newly introduced HyTopS optimization scheme over solely shape optimization for microvascular materials.

Improving the shear test to determine shear fracture limits for thin stainless steel sheet by shape optimisation

Bipolar plates are essential components in proton exchange membrane (PEM) fuel cells for electric vehicles. Micro-formed thin (thickness <0.15 mm) stainless steel bipolar plates have advantages over conventional CNC-machined graphite plates in terms of weight and cost, but the geometry of the final part shape is limited due to the fracture limits of the material. The further optimisation of micro-forming processes, therefore, requires the development of reliable fracture models, with fracture locus covering a wide range of stress states, including shear fracture at zero (or close to zero) stress triaxiality conditions. Few studies have investigated shear tests on thin specimens, as it is challenging to maintain a stable shear condition during the test. There exists a twofold problem that: (a) the sheet buckles prematurely before the failure strain is reached; (b) plane strain fracture initiates prematurely from the edge of the gauge area. In this study, in-planar shear tests are performed on 0.1 mm-thick stainless steel sheet. To achieve fracture initiation in shear, an anti-buckling device was developed and the shear test sample shape was optimised. The shape optimization process is based on finite element analysis (FEA) and verification with experimental shear test trials. The optimised shape has a minimum value of a goal function that encompasses the Lode parameter, stress triaxiality and the ratio of the centre to edge equivalent plastic strain. The experimental trials indicate that the optimised sample shape, in combination with the clamping device, enables testing thin stainless steel for fracture limits in shear. The test routine can be used to calibrate ductile fracture models.

Stochastic Natural Vibration Analyses of Free-Form Shells

This work investigates the natural vibration characteristics of free-form shells when considering the influence of uncertainties, including initial geometric imperfection, shell thickness deviation, and elastic modulus deviation. Herein, free-form shell models are generated while using a self-coded optimization algorithm. The Latin hypercube sampling (LHS) method is used to draw the samplings of uncertainties with respect to their stochastic probability models. ANSYS finite element (FE) software is adopted to analyze the natural vibration characteristics and compute the natural frequencies. The mean values, standard deviations, and cumulative distributions functions (CDFs) of the first three natural frequencies are obtained. The partial correlation coefficient is adopted to rank the significances of uncertainty factors. The study reveals that, for the free-form shells that were investigated in this study, the natural frequencies is a random quantity with a normal distribution; elastic modulus deviation imposes the greatest effect on natural frequencies; shell thickness ranks the second; geometrical imperfection ranks the last, with a much lower weight than the other two factors, which illustrates that the shape of the studied free-form shells is robust in term of natural vibration characteristics; when the supported edges are fixed during the shape optimization, the stochastic characteristics do not significantly change during the shape optimization process.

Optimal design of J-groove shape on the suppression of unsteady flow in the Francis turbine draft tube

The flow in the draft tube of a Francis turbine is highly complex and unstable when the turbine operates far from its design point. To extend the Francis turbine operation range to meet the variable energy demand, a J-groove technology is introduced on the turbine draft tube wall to stabilize the unsteady flow in the draft tube. However, J-groove contributes to energy losses in the draft tube. In this study, the J-groove shape optimization process is proposed based on steady flow analysis using the response surface method. The energy loss and swirl intensity in the draft tube are selected as objectives for the optimization design. The flow characteristics in the draft tube without J-groove and with initial J-groove and optimized J-groove shape are compared. Results show that the swirl intensity in the draft tube with the optimized J-groove is suppressed effectively with the low energy loss in the draft tube. Moreover, the pressure fluctuation in the draft tube with the optimized J-groove is mitigated significantly to stabilize the Francis turbine operation under the off-design point condition.

A Semi-Supervised Meta-Heuristic Shell Form-Finding Approach

This paper proposes a general semi-supervised form-finding approach for optimization of shells, where an initial geometry referred to as the “Directive Form (DF)” influences and directs the shape optimization process. The degree of the influence of the DF over the optimization process is controlled by the optimizer algorithm. A normal option involves stronger influence of DF within the early stages of the optimization process and a vanishing effect towards the later stages of the process. The method is illustrated through an example using the artificial intelligence for the optimization process. The selected problem is optimized with particle swarm optimization algorithm in MATLAB. The required structural analysis of the optimization steps is carried out by the finite element software SAP2000 which is linked up with the optimization algorithm via the application programming interface software. The coordinates of the nodes over the surface of the shell are selected as the optimization variables, and the semi-supervised algorithm finds the desired optimal form by minimizing the eccentricity of the forces as the target function. Finally, the paper presents a simple and workable stress calculation method for the optimized shells under gravitational loads.

On the Design of Innovative Heterogeneous Sheet Metal Tests Using a Shape Optimization Approach

The development of full-field measurement methods has enabled a new trend of heterogeneous mechanical tests. The inhomogeneous strain fields retrieved from these tests are being widely used in the calibration of constitutive models for sheet metals. However, today, there is no mechanical test able to characterize the material in a large range of strain states. The aim of this work is to present a heterogeneous mechanical test with an innovative tool/specimen shape, capable of producing rich heterogeneous strain paths and thus providing extensive information on material behavior. The proposed specimen is found using a shape optimization process where an index that evaluates the richness of strain information is used. In this work, the methodology and results are extended to non-specimen geometry dependence and to the non-dependence of the geometry parametrization through the use of the Ritz method for boundary value problems. Different curve models, such as splines, B-splines, and NURBS, are used, and C1 continuity throughout the specimen is guaranteed. Moreover, several deterministic and stochastic optimization methods are used in order to find the method or the combination of methods able to minimize the cost function effectively. Results demonstrated that the solution is dependent on the geometry definition, as well as on the optimization methodology. Nevertheless, the obtained solutions provided a wider spectrum of strain states than standard tests.

Assessment of Deterministic Shape Optimizations Within a Stochastic Framework for Supersonic Organic Rankine Cycle Nozzle Cascades

The design of converging–diverging blades for organic Rankine cycle (ORC) applications widely relies on automated shape-optimization processes. As a result, the optimization produces an adapted-nozzle cascade at the design conditions. However, only few works account for the uncertainties in those conditions and their consequences on cascade performance. The proposed solution, i.e., including uncertainties within the optimization routine, demands an overall huge computational cost to estimate the target output statistic at each iteration of the optimization algorithm. With the aim of understanding if this computational cost is avoidable, we study the impact of uncertainties in the design conditions on the robustness of deterministically optimized profiles. Several optimized blades, obtained with different objective functions, constraints, and design variables, are considered in the present numerical analysis, which features a turbulent compressible flow solver and a state-of-the-art uncertainty-quantification (UQ) method. By including measured field variations in the formulation of the UQ problem, we show that a deterministic shape optimization already improves the robustness of the profile with respect to the baseline configuration. Guidelines about objective functions and blade parametrizations for deterministic optimizations are also provided. Finally, a novel methodology to estimate the mass-flow-rate probability density function (PDF) for choked supersonic turbines is proposed, along with a robust reformulation of the constraint problem without increasing the computational cost.

NERONE: An Open-Source Based Tool for Aerodynamic Transonic Optimization of Nonplanar Wings

Transonic shape optimization of nonplanar wings is a relevant topic in current aeronautical world. Although widely studied, an appropriate level of robustness and automation is still lacking for the inclusion of aerodynamic shape optimization in a Multi-Disciplinary Optimization (MDO) process. In this paper, a new framework, opeNsource-mEsh-geneRation-fOr-aerodyNamic-Evaluations (NERONE), is presented that allows for an automatic transonic shape optimization process within an MDO loop design using open-source libraries, namely OpenCasCade (OCC) for the geometric description, GMSH for grid generation and SU2 multi-physics for aerodynamic analysis and optimization. Wing geometry is defined in the CPACS standard, converted to a mathematical continuous description (NURBS) and passed to the mesher. Grid generation process is driven by user-defined mesh dimensions defined on wing regions and on support domain boundary surfaces. The aerodynamic analysis and optimization are then launched on the obtained grid. SU2 software has been augmented to overcome robustness issues regarding point-inversion algorithm and to control geometric quality during optimization of nonplanar wings. Capability of NERONE is first demonstrated for 2D Euler and RANS simulations (on the RAE2822 airfoil) and on a 3D Euler case (ONERA M6). With very few user-specified parameters, high-quality grids are obtained providing results that correlate well with the literature data. Finally, NERONE is applied to the local shape optimization of a wing–winglet configuration in transonic flight conditions.

Topology and Size–Shape Optimization of an Adaptive Compliant Gripper with High Mechanical Advantage for Grasping Irregular Objects

SummaryThis study presents an optimal design procedure including topology optimization and size–shape optimization methods to maximize mechanical advantage (which is defined as the ratio of output force to input force) of the synthesized compliant mechanism. The formulation of the topology optimization method to design compliant mechanisms with multiple output ports is presented. The topology-optimized result is used as the initial design domain for subsequent size–shape optimization process. The proposed optimal design procedure is used to synthesize an adaptive compliant gripper with high mechanical advantage. The proposed gripper is a monolithic two-finger design and is prototyped using silicon rubber. Experimental studies including mechanical advantage test, object grasping test, and payload test are carried out to evaluate the design. The results show that the proposed adaptive complaint gripper assembly can effectively grasp irregular objects up to 2.7 kg.

Adjoint shape optimization for fluid‐structure interaction

AbstractA partitioned approach for the solution of the adjoint associated to the coupled system of time‐dependent fluid‐structure interaction is presented. Allowing for an efficient computation of both sensitivity and gradient distributions relying on the adjoint method, an optimization strategy based on the steepest descent algorithm is applied. The performance of the developed shape optimization process improving flow conditions and related cost functionals is demonstrated by application to ducted flow situations and common fluid dynamic design tasks considering the interaction between fluid loads and structural deformations. Both physical capability and feasibility are discussed in terms of theoretical and numerical aspects in order to evaluate the efficiency of the realized optimal control process.