Tool For Shape Optimization Research Articles

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

Application of an automated grid deformation tool for divertor shape optimization in SOLPS‐ITER

AbstractAn optimized divertor design is crucial to maximize the lifetime of plasma‐facing components and reduce costs of future fusion power plants. Numerical shape optimization could be a powerful tool to obtain improved designs in an automated way. However, it is not trivial to apply due to the need of a field‐aligned and boundary‐fitted grid for simulating the plasma and quantifying the heat load. This paper shows how a grid deformation tool can automate the gridding process while safeguarding the grid quality. Additionally, sensitivities of shape parameters are computed using finite differences and compared to those obtained by remeshing using standard tools such as CARRE2. The plasma and neutral behavior is simulated using the unstructured SOLPS‐ITER code with the latest advanced fluid neutrals model for an ASDEX Upgrade test case. The comparison shows that, contrary to the remeshing strategy, the grid deformation approach yields smoother sensitivities. Furthermore, it is shown that the deformed grids have better mesh quality in terms of poloidal cell size ratio compared to the grid generated with CARRE2, which improves the accuracy of the simulation. This supports the use of the grid deformation tool for automated shape design in future work.

Aerodynamic shape optimization of bluff body for galloping-based wind energy harvester by the class function/shape function transformation technique

This paper presents an aerodynamic modification scheme by using the class function/shape function transformation (CST) method to determine the optimal shape of the bluff body and hence improve the performance of the associated galloping-based wind energy harvester (WEH). The CST method, accompanying with the Latin Hypercube sampling method, is used to generate 50 representative samples of galloping bluff bodies with varying sectional geometries, by using only four parameters to describe each sectional geometry. Wind tunnel tests are conducted to measure the output power of each WEH and the transverse wind force acting on the associated bluff body. A Kriging surrogate model is constructed based on the results from wind tunnel test, to determine the optimal shape in terms of the output power. Results show that the optimal shape has the tapered leeward side and concave windward side, which can double the output power compared to a square shape. The underlying mechanism for the high performance is explained by analyzing the measured transverse force in the wind tunnel and the vorticity contours of the flow field generated by CFD analysis. On the one hand, the tapered leeward side avoids flow attachment on the two lateral sides of the shape, providing considerable transverse force that sustains the unstable steady-state galloping. On the other hand, the concave windward side promotes the development of large circulation bubbles, causing significant asymmetry of wind flow and hence remarkable galloping. The findings of the research can guide the design for the bluff body of the galloping-based WEHs, and the CST method can be used as a powerful tool for aerodynamic shape optimization.

Adjoint shape optimization of Ahmed body to improve aerodynamics

Analyses are done on the bluff body of a simple ground vehicle's wake structure. A 4 million Reynolds number depending on model length. Results for various geometric configurations of the Ahmed body's flow visualization were achieved. There was a flexible geometric element called the base slant angle. Drag research found that body drag makes up about 85% of all drag. The body's rear end is where most drag is created or developed. Horseshoe vortices build a three-deck structure in wake flow. The base slant angle affects these vortices' strength, existence, and merging. Examined is the applicability of these phenomena to actual ground vehicles. To reduce drag and enhance aerodynamics, the body's shape was modified using the adjoint approach and the adjoint shape optimization tool in ANSYS Fluent. In this study, we found that by changing the tail-end vortex structure and flow separation at the rear part of the vehicle, the optimized back shape or rear slant design of the body provides better flow separation that results in better pressure recovery and reduces drag by roughly 7 to 10 percent, respectively. The comparison of experimental CD values for the Ahmed body at different rear slant angles, including 12.5o, 25o, and 30o. Since it had less variance from empirical data, the computationally computed CD for a 30° base slant angle was chosen to do adjoint shape optimization. Mesh cells of less than three lakhs were used in the CFD process, and skewness quality was checked. 60 m/s of velocity was the boundary condition for the velocity inlet.

Tailored elastic properties of beam-based lattice unit structures

AbstractIn this paper a structural optimization framework is developed to design three-dimensional periodic lattice unit cells that meets specific mechanical requirements. The work is motivated by the high design freedom of additive manufacturing technologies, which enable complex multiscale lattice structures to be printed. An optimized lattice unit cell delivers desired orthotropic elastic material properties, providing a tailored metamaterial. The design variables are the coordinates of lattice skeleton nodes defined within the three-dimensional lattice cell space, and the connectivities between them resulting a strut-skeleton. Genetic algorithm (GA) is combined with posterior particle swarm optimization (PSO) algorithm to establish an integrated topology and shape optimization tool. For the calculation of the elastic properties of the individual lattice cells, an effective Timoshenko beam-based finite element calculation method was developed. The novelty of the work stems from its free topology optimization nature, excluding the strut diameters from the optimization variables. The method is demonstrated by four lattice cell optimization cases, where extreme orthotropic elastic properties were targeted and achieved. The tailored lattice cells represent a metamaterial, that can be used to build a structural component on the macroscopic scale, by stacking the cells periodically together, to fill the macroscopic 3D design space. This framework is a strong basis that can be extended to meet further nonlinear metamaterial requirements, such as energy absorption.

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

Designing Custom Mechanics in Running-Specific Prosthetic Feet via Shape Optimization.

Running-specific prosthetic feet are curved cantilever springs, distinguished from ordinary prosthetic feet by their size, shape, and greater ability to store and return elastic potential energy. Finite element analyses show how modifications to prosthesis shape alter the multidimensional and nonlinear mechanics, but designers seeking to prescribe custom mechanics are limited by the complex constraints and high dimensionality of possible geometries. We introduce two simple formulations for describing foot mechanics, and use them with a custom spline-based shape optimization to generate new prosthetic foot shapes given desired endpoint mechanics. We then tackle a relevant example problem using this approach, designing and characterizing three running-specific prosthetic feet with similar vertical and angular deflections but varied horizontal deflections, with the expectation that knee extension moments would increase with anterior deflection of the toe. Finally, we compare the prostheses' endpoint mechanics and resulting biomechanics in an athlete with unilateral transfemoral amputation. The shape optimization was able to derive shapes that substantially alter prosthesis mechanics along all dimensions of endpoint behavior, and benchtop testing validated the behavior of two new feet constructed from the optimization. The subject's knee moments increased with horizontal endpoint deflection as expected. We developed and validated a shape optimization tool using a simplified formulation of foot behavior to achieve desired running prosthesis mechanics. With this framework, researchers can begin to elucidate the link between prosthesis mechanics and athlete biomechanics and performance.

Organic Rankine cycle turbogenerator cooling – Optimization of the generator water jacket heat exchange surface

Discharge of excess heat from the micro-turbine electric generator is an important issue for the micro-turbine operation and should be considered during the design process. The manuscript is intended to present a new tool for rapid shape optimization of the heat exchanging surface of the ORC (Organic Rankine Cycle) micro-turbine electric generator. The optimization of the water jacket geometry of a 10 kWe ORC micro-turbine generator is performed by making use of a modified, with respect to the models presented in the literature, 0D heat transfer model of the finned heat exchange surface of the cooling water jacket, and a genetic algorithm. As a result of optimization, an apparent reduction in fin size and increased fin density at the finned wall surface is achieved, yielding a significant increase of the heat transfer coefficient and a temperature decrease of the electric generator stator. The temperature at the shell surface tangential to the electric generator stator was found to decrease by over 6°C. A considerable reduction of temperature gradients around the water jacket wall is also reported.The results of optimization from the 0D model are verified by numerical simulation using a full 3D CFD model of fluid flow and conjugate heat transfer (CHT) in Ansys CFX for the original and optimized geometries. The CFD CHT model was first validated using experiment-based correlations for heat transfer in liquid flow over a circular cylinder pipe for a wide range of Reynolds numbers.The obtained temperature decrease in the optimized ORC electric generator stator for the nominal operating point of the ORC generator is found similar for both 0D and 3D models. The temperatures calculated in the 0D model at the shell surface tangential to the electric generator stator deviate from respective CFD values by 0.5–1.5°C. Another operational case beyond the nominal operating point has been analysed making use of the 3D CFD method as well, showing that the optimized geometry is characterized by an improved heat exchange performance not only for the nominal operating point.

Inverse Design of Silicon Photonic Modulators

A novel shape optimization procedure is formulated for obtaining a higher phase change while maintaining high transmission in phase modulators. The formulation is implemented as a shape optimization tool. Subwavelength grating-based Si carrier depletion and carrier injection modulators are considered for testing the optimizer. Carrier transport simulations were performed to extract the carrier concentration profiles and were used in the optimizer. The optimized structures are then validated using simulation for transmission and phase change. The results show that the optimization tool developed can optimize both carrier injection and depletion type modulators, in terms of both transmission and phase modulation. The optimized structures can be placed in a ring or Mach-Zehnder modulator to make compact modulators with low drive voltages.

Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

In this paper, we are interested in the inverse problem of the determination of the unknown part ∂ Ω , Γ 0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝ N from the measurement of the normal derivative ∂ n v on suitable part Γ 0 of its boundary, where v is the solution of the wave equation ∂ t t v x , t − Δ v x , t + p x v x = 0 in Ω × 0 , T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂ Ω . From necessary conditions, we estimate a Lagrange multiplier k Ω which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.

Guidelines for Designing Micropillar Structures for Enhanced Evaporative Heat Transfer

Abstract Over the last several decades, cooling technologies have been developed to address the growing thermal challenges associated with high-powered electronics. However, within the next several years, the heat generated by these devices is predicted to exceed 1 kW/cm2, and traditional methods, such as air cooling, are limited in their capacities to dissipate such high heat fluxes. In contrast, two-phase cooling methods, such as microdroplet evaporation, are very promising due to the large latent heat of vaporization associated with the phase change process. Previous studies have shown that nonaxisymmetric droplets have different evaporation characteristics than spherical droplets. The solid–liquid and liquid–vapor interfacial areas, volume, contact angle, and thickness of a droplet confined atop a micropillar are the primary parameters that influence evaporative heat transport. These parameters have a strong influence on both the conduction and diffusion resistance during the evaporation process. For example, a droplet with a higher liquid–vapor interfacial area will favorably increase heat transfer. Increased droplet thickness, on the other hand, has a detrimental influence on the evaporation rate. The dimensions of these droplets will vary in response to changes in each of the aforementioned parameters. Lowering the droplet thickness can be achieved by decreasing the liquid volume while maintaining a constant solid–liquid area. However, if the solid–liquid area and volume vary simultaneously, the average droplet thickness may increase, decrease, or remain constant. Furthermore, changes in the shape of the droplet modify the local equilibrium contact angle of the droplet for different azimuthal angles. As a result, the optimal combination of these parameters must be identified to maximize the heat transfer performance of an evaporating microdroplet. These droplet parameters can be manipulated by selecting different micropillar cross sections. In this work, we develop a shape optimization tool using the particle swarm optimization algorithm to maximize evaporation from a droplet confined atop a micropillar. The tool is used to optimize the shape of a nonaxisymmetric droplet. Compared to droplets atop circular and regular equilateral triangular micropillar structures, we find that droplets confined on pseudo-triangular micropillar structures have 23.7% and 5.7% higher heat transfer coefficients, respectively. The results of this work will advance the design of microstructures that support droplets with maximum heat transfer performance.

Unsteady continuous adjoint to URANS coupled with FW-H analogy for aeroacoustic shape optimization

In this paper, an unsteady aeroacoustic shape optimization tool is presented based on the continuous adjoint method. Aeroacoustic noise radiated by bodies in unsteady flows can be computed using a hybrid acoustic prediction tool, where the near-field flow and acoustics result from an unsteady CFD simulation while the acoustic propagation to far-field relies upon an acoustic analogy. The CFD solution is performed using the in-house GPU-enabled unsteady RANS solver for which a continuous adjoint solver is also available. Next to this, the noise prediction tool is developed based on the permeable version of the Ffowcs Williams and Hawkings analogy in the frequency domain and its implementation is verified by comparison to a well-known analytical solution of the sound field from a monopole source in uniform flow as well as comparing the results computed by the FW-H analogy with that of a time-accurate CFD run. The full mathematical formulation of the continuous adjoint method for the aforementioned acoustic analogy, coupled with the adjoint to the mean-flow and turbulence model equations, is presented. The accuracy of the gradients computed by the hybrid adjoint method is verified through comparisons with finite differences in two cases, both governed mainly by the tonal component in noise generation; these include the vortex shedding cylinder in a laminar flow and the rod-airfoil canonical benchmark in a turbulent flow. Finally, the programmed software is used to optimize the shape of the airfoil in the last test case, aiming at min. noise for a receiver located at far-field.

Vortex-in-cell simulation of the flow and performance of a Savonius hydraulic turbine with S-shaped blades

The volume penalization vortex-in-cell (VP-VIC) method, which combines the vortex-in-cell and vortex penalization methods, is a kind of immersed boundary method that has the advantage of accurately calculating the vorticity field near the boundary. It is expected to be used as a design tool for shape optimization. In this study, the flow and performance of a small Savonius turbine rotor (rotor diameter DR = 245 mm) with S-shaped blades, which is used for small-scale distributed power generation with output less than 100 kW, were analyzed by numerical simulation. The results were compared with those of previous experimental studies to examine the usefulness of the VP-VIC method for fluid analysis around the Savonius turbine. The Reynolds number (Re=U0DR/ν) based on the streamwise velocity U0 is 1.32×105. The volume penalization vortex-in-cell (VP-VIC) method is applied for flow simulation, and the rotational motion of the rotor is analyzed using the calculation method previously proposed by the authors for the fluid force acting on the solid boundary (blade surface). As a result, the relationship between the rotational speed of the rotor and the output coefficient obtained in this simulation is similar in the experimental results of Golecha et al. [Appl. Energy 88, 3027–3217 (2011)] using S-shaped blades. In particular, with an increase in the load torque, the rotational speed of the rotor decreases, and the rotor stops momentarily. When the rotor stops, the positive torque of clockwise rotation acts on the advancing blade, while the negative torque of counterclockwise rotation acting on the returning blade decreases; hence, the rotor restarts immediately after stopping. In addition, it is established that the rotor output can be accurately predicted. Based on the above, it is reconfirmed that the VP-VIC method is beneficial for predicting the performance of a Savonius turbine.

Automatic generators of multi-patch B-spline meshes of blade cascades and their comparison

In this paper, we propose several approaches for automatic construction of planar multi-patch B-spline parameterizations around one or between two consecutive blade profiles in a blade cascade starting from given design parameters (blade cascade is obtained by unfolding a cylindrical cross-section of the runner), which have been implemented in C++ as fully automatic generators and are an integral part of our in-house automatic shape optimization tool for water turbines. These B-spline parameterizations have to be prepared for the definition of periodic boundary conditions on the corresponding parts of the boundary, which makes finding of analysis-suitable multi-patch B-spline parameterizations even more challenging. All approaches are compared with respect to the selected geometric quality measures and also with respect to the results from our in-house isogeometric incompressible turbulent fluid flow solver and we discuss pros and cons of these approaches.

ADflow: An Open-Source Computational Fluid Dynamics Solver for Aerodynamic and Multidisciplinary Optimization

Computational fluid dynamics through the solution of the Navier–Stokes equations with turbulence models has become commonplace. However, simply solving these equations is not sufficient to be able to perform efficient design optimization with a flow solver in the loop. This paper discusses the recommendations for developing a flow solver that is suitable for efficient aerodynamic and multidisciplinary design optimization. One of the major recommendations is to be able to load the flow solver as a library that provides direct memory access to the relevant data. Other recommendations are to use a higher-level language for scripting and to pay special attention to solution warm starting, code efficiency, flow solver robustness, and solution failure handling. As an example of a flow solver that follows these recommendations, the open-source flow solver ADflow is presented. Results from aerodynamic optimization, aerostructural analysis, and aerostructural optimization using ADflow demonstrate the performance advantages claimed in the recommendations. The publication of these recommendations and the availability of the source code open the door for other solvers to adopt the same application programming interface. ADflow is part of a wider aerodynamic shape optimization tool suite that is also available under an open-source license.

Additive manufacturing scanning paths optimization using shape optimization tools

This paper investigates path planning strategies for additive manufacturing processes such as powder bed fusion. The state of the art mainly studies trajectories based on existing patterns. Parametric optimization on these patterns and allocating them to the object areas are the main strategies. We propose in this work a more systematic optimization approach without any a priori restriction on the trajectories. The typical optimization problem is to melt the desired structure, without over-heating (to avoid thermally induced residual stresses) and possibly with a minimal path length. The state equation is the heat equation with a source term depending on the scanning path. First, in a steady-state context, shape optimization tools are applied to trajectories. Second, for time-dependent problems, an optimal control method is considered instead. In both cases, gradient-type algorithms are deduced and tested on 2-D examples. Numerical results are discussed, leading to a better understanding of the problem and thus to short- and long-term perspectives.

Flexible Formulation of Spatial Integration Constraints in Aerodynamic Shape Optimization

In aircraft design, spatial integration limits aerodynamic and structural performance. While the outer mold line shape determines aircraft aerodynamic characteristics, aircraft systems and passengers or payloads must fit inside. Emerging categories of aircraft, such as electric aircraft, are likely to experience new and critical spatial integration challenges. Existing aerodynamic shape optimization tools use a narrowly defined set of geometric constraints. However, no known aerodynamic optimization framework can handle spatial integration constraints derived directly from three-dimensional (3-D) geometry. A general geometric constraint formulation based on triangulated 3-D geometry is proposed. The constraint consists of two metrics: the length of the intersection curve(s) between any two objects, and the Kreisselmeier–Steinhauser function aggregated minimum distance. The proposed implementation of the intersection and distance computations is fast and analytically differentiable with respect to geometric design variables, making it suitable for efficient gradient-based optimization. The constraint formulation is validated using three Reynolds-averaged Navier–Stokes-based aerodynamic shape optimization problems: a 2-D fairing design problem, a 3-D fairing design problem, and the design of an aeroshell to surround a human avatar. It is shown that this approach is robust and efficient, enabling the aerodynamic optimization of bodies containing objects with arbitrary shapes.

Free surface vortices at hydropower intakes: – A state-of-the-art review

For years, the study of free surface vortices at hydropower plant intakes has been a topical and intriguing subject among engineers and researchers. This subject will continue to attract attention especially as the world strives to meet the ever-increasing demand for energy. Despite the numerous benefits associated with hydropower, the sustainability of some hydropower plants is being threatened due to low inflows often associated with climate change. Free surface vortices associated with low water levels or submergence at plant intakes can have very detrimental consequences on the operation of hydropower plants if not addressed. Notwithstanding this, free surface vortex flows have also been found to be very relevant in emerging technologies such as the water vortex hydropower plant system. This paper, therefore, presents a state-of-the-art review of the subject including summarised historical findings, but with an emphasis on current developments, findings and research gaps to guide practitioners and researchers. In response to the research gaps identified, the authors make a number of recommendations for further studies which include establishing relationships between free surface vortices formation and turbine efficiency, development of more accurate models for critical submergence and free surface vortices, assessment of free surface vortices at multiple and multi-level intakes, establishing the relationship between free surface vortices and sediment transport at intakes, application of Computational Fluid Dynamics (CFD) shape optimization tools for intake and anti-vortex device optimisation, as well as the continuing development of CFD tools to simulate air-entrained vortices at hydropower intakes.

A review of “performance analysis and optimization of car air spoilers”

Before last decade, due to surplus amount of fuel, cars are designed for high speed operation, comfort and safety. As scarcity of fuel increases due to high consumption of fuel in automobiles, many researchers started working on the idea of alternative fuels, redesign of car body and reducing aerodynamic losses. To overcome these losses spoilers are used and hence it is needed to optimize its shape. This study is primarily focused to find out optimized shape of car spoiler so that mass can be minimized which help in reducing fuel consumption without affecting its aerodynamic properties and strength. Computational Fluid Dynamics (CFD) analysis of two dimensional model of spoiler is done whose results are validated by earlier research work in this field for understanding variation of aerodynamic property of cross-section. Three dimensional CFD analysis of spoiler gives aerodynamic property and pressure data which is needed to compare result of optimized model formed by optimized cross-sectional shape. Shape optimization is done by Shape Optimization tool of ANSYS 14.0 which is further tested for design failure in ABAQUS 6.11. The result of shape optimization has saved 18.74% of material keeping all its strength and aerodynamic property intact. This study open provides a numerical tool for improvement of future model of spoiler in terms of reduction of mass.

Weight optimization of fixed landing gear for medium range UAV

Weight reduction of Unmanned Aerial Vehicle (UAV) components and systems is one of the most essential factors for aviation. Take-off gross weight can be divided into crew weight, payload, fuel, and empty weight. The empty weight is divided into the structure, engine/s, landing gear, avionics and fixed equipment. Engine/s, landing gear, avionics and fixed equipment are selective items. Weight optimization can be made for structure and landing gear to minimize aircraft total weight. The main objective of the present study is to optimize weight and reach safe landing for a medium range UAV landing gear. The reduced weight can be used to increase payload and/or fuel to increase range or endurance of the UAV. In the present study, ANSYS shape (Topology) optimization tool is used to get a proposed shape for the UAV landing gear. Then parametric size optimization of the shape proposed was done. Design of experiment (DOE) technique is used in order to get the minimum weight for the landing gear at a specified stress limit. Aluminum alloy 7075-T61 and bi-directional carbon fiber which are the most common used materials in UAVs landing gear were used, It was found that using topology weight optimization technique reduces Aluminum Alloy landing gear weight by (30.4 %) changing the material to be bi-directional carbon fiber without optimization reduce landing gear weight with additional value (27.9 %). Optimizing bi-directional carbon fiber landing gear reduces the weight with additional value (4 %) to be the total weight ratio by (62.3 %). The optimization technique for weight optimization is a guide to optimize any other type of landing / part to minimize the total weight for the Unmanned Aerial Vehicle.