Three-dimensional Shape Optimization Research Articles

Three-dimensional aerodynamic shape optimization with high-order direct discontinuous Galerkin schemes

Three-dimensional aerodynamic shape optimization with high-order direct discontinuous Galerkin schemes

Genetic algorithm based three-dimensional shape optimization of rotor blade for a Mars multi-rotor aircraft

In the context of Mars exploration, the Mars multi-rotor aircraft plays a significant role, with its blades being the key components for achieving flight. Due to the thin Martian atmosphere, conventional Earth blades struggle to generate the required thrust for comprehensive aerial missions on the Red Planet. To solve these challenging conditions, there arises a compelling need to refine the rotor’s design, optimizing it to provide sufficient thrust while minimizing power consumption. In this research, a novel method is introduced, combining genetic algorithms and computational fluid dynamics simulations. This approach focuses on enhancing the three-dimensional structure of the rotor blade. The results of this research are verified in experiments conducted within the Martain Atmosphere Simulator, where the blade after optimization demonstrated the ability to achieve the same level of thrust while significantly reducing power consumption. Consequently, this study represents a promising avenue for improving the efficiency of Mars multi-rotor aircraft.

Three-dimensional shape optimization of fins for application in compact supercritical CO2 solar receivers

The compact solar receiver is a promising option for S-CO2 solar receivers, designed to operate safely and efficiently under high-temperature, high-pressure, and high solar flux conditions. Adding fins with outstanding thermal-hydraulic performance in the mini-channel contributes to maximizing heat transfer performance and minimizing flow resistance for compact solar receivers. However, most of the research on high-performance fins is limited to two-dimensional optimization, ignoring the effect of shape changes in the height direction of the fins. The present study employs a combination of the finite volume method and the adjoint method to optimize single-row cylindrical fin shapes in three-dimensional for compact solar receivers, and the strengthening mechanism of high efficiency and low resistance is further revealed in the view of entropy generation. The results indicate: (1) three-dimensional fins exhibit significant variation in cross-section along the height direction, and each fin is unique; (2) three-dimensional fins exhibit more excellent thermal-hydraulic performance than cylindrical fins, showing an increase of 13 % and 20 % in the performance evaluation criterion (PEC) in case 1 (qw = 160 kW·m−2, vin = 0.487 m·s−1) and in case 2 (qw = 584 kW·m−2, vin = 1.778 m·s−1), respectively; (3) the entropy generation due to heat transfer is reduced by 4.9% and 3.9%; as well as the entropy generation due to pressure drop is decreased by 8.5 % and 16.5 %, in case 1 and case 2, respectively; (4) robustness results show that three-dimensional fins can maintain higher thermal-hydraulic performance than the initial cylindrical fin even under wide working conditions.

Three-dimensional shape optimization of a submerged body under wave diffraction

This study explores the use of shape optimization to reduce wave forces on a submerged floating body subjected to wave diffraction. To this end, gradient-based shape optimization is adopted, in which dimensionless wave excitation forces are the optimization objective. A shape parameterization method based on the Fourier-series expansion is developed that permits the representation of an arbitrary three-dimensional floating body. The discrete adjoint method is utilized to calculate the gradient of the objective function with respect to the shape parameters. Using three-dimensional shape optimization, taking the initial shape to be a hemisphere, a significant reduction in surge, heave, and pitch wave forces is achieved, with a maximum reduction of 48.40%, 68.43%, and 46.22%, respectively. Furthermore, optimization effectively suppresses wave run-up, with a maximum reduction of 15.62%. A comprehensive analysis of parameters is performed to reveal the effects of wave number, incident angle, and shape parameters on the final optimized shape and wave load characteristics. This study provides a solid guide to the optimization of floating offshore platforms and the development of innovative structural systems.

Three-dimensional shape optimization of fins in a printed circuit recuperator using S-CO2 as the heat-transfer fluid

Printed circuit heat exchangers (PCHEs) are among the most promising candidates for recuperators in the S-CO2 Brayton cycle owing to their high-pressure resistance and high compactness, where the fins are usually installed in channels to improve the thermal-hydraulic performance. The fin shape optimizations have been performed in several previous studies for enhancing the heat transfer and reducing the pressure drop. However, most of them are limited to two-dimensional optimization where the shape variation along the height direction was ignored. In the present study, three-dimensional fin shape optimizations are performed for the PCHE recuperators using S-CO2 as heat transfer fluid to further improve the thermal-hydraulic performance. The heat transfer and pressure drop characteristics in channels with optimized fins are comprehensively evaluated by comparing against conventional cylindrical fins and airfoil fins. The results indicate that the optimized three-dimensional fins feature variable cross-sections along the height direction. Such three-dimensional fins can significantly diminish the impact area of fluid, which is conducive to reducing the local flow resistance caused by high-velocity gradient and consequently improving the comprehensive heat transfer performance. Under the typical working condition studied in this paper, the heat transfer capacity in hot channels of recuperators with the optimized three-dimensional fins is 21.2% greater than the cylindrical fins, and 17.3% greater than the airfoil fins when the identical pressure drop is fixed; while the heat transfer capacity in cold channels with the optimized three-dimensional fins is 24.3% greater than the cylindrical fins, and 29.3% greater than the airfoil fins. In addition, the robustness analysis indicates that the optimized three-dimensional fins can yield more excellent thermal-hydraulic performance than conventional cylindrical fins and airfoil fins over a wide range of working conditions.

Optimization of high heat flux components for DIII-D neutral beam upgrades

Upgrade of the DIII-D neutral beams leads to enhanced heat loads on many components, such as calorimeter, collimator, and pole shields which protect neutral beam magnets. Power increase from 2.6 MW to 3.2 MW per source leads to a normal heat flux loads of up to 55 MW/m2 for the calorimeter. The Princeton Plasma Physics Laboratory is responsible for the design and manufacturing of the upgrades of these components.Heat flux distribution on neutral beam components is very uneven and leads to significant thermal stresses. High heat flux density impact requires surface optimization to reduce surface heat flux projection, and avoid localized melting. Several new design features were introduced to accommodate increased heat loads, such as molybdenum inserts for the pole shields, two-dimensional shaping for the calorimeter, and three-dimensional shape optimization and replaceable copper inserts for the collimator. Additionally, all three components include an optimized cooling system design featuring peripheral cooling of copper components.The optimization process included applying analytical relations for the transient temperature distributions on the high heat flux components. These relations were confirmed by previous DIII-D experimental results. To validate the designs, numerical simulations were performed. Results of the design optimization and numerical simulations will be presented.

Investigations of Shape Definition and Aerodynamics Analysis Methods for Three-Dimensional Aerodynamic Shape Optimization

Investigations of Shape Definition and Aerodynamics Analysis Methods for Three-Dimensional Aerodynamic Shape Optimization

Three-Dimensional Shape Optimization of Claw-Pole-Motors

This paper presents three-dimensional shape optimization of claw-pole motors. The claw-pole shape is determined from the binary state of hexahedral cells. The cell states are optimized to maximize induced voltage and minimize the field current using micro-genetic algorithm. The optimization is successfully carried out without time-consuming mesh generation.

Investigation into Aerodynamic Shape Optimization of Planar and Nonplanar Wings

Problems in three-dimensional aerodynamic shape optimization can produce complex design spaces due to the nonlinear physics of the Navier–Stokes equations and the large number of design variables used. In this paper, a Newton–Krylov optimization algorithm is applied to a set of complex aerodynamic optimization problems in order to investigate its behaviour and performance. The methodology solves the Reynolds-averaged Navier–Stokes equations with a parallel Newton–Krylov algorithm. Aerodynamic geometries are meshed using structured multiblock grids, which are then fitted with B-spline control volumes for mesh deformation and geometry control. A gradient-based optimization method is used, with adjoint variables calculated using a Krylov method. The optimization of the Common Research Model (CRM) wing is revisited, with a focus on the effect of varying geometric constraints and on the possibility of multimodality. In addition, several cases are presented that involve a high degree of shape change: two planform optimizations starting from a rectangular wing, and investigation of various wingtip treatments. The results characterize the methodology, demonstrating its robustness and ability to address optimization problems with substantial geometric freedom.

Surface Mesh Movement Algorithm for Computer-Aided-Design-Based Aerodynamic Shape Optimization

This paper focuses on the development of a surface mesh movement algorithm suitable for computer-aided-design-based aerodynamic shape optimization. The algorithm interrogates the computer-aided-design system via the vendor-neutral application programming interface Computational Analysis Programming Interface and uses the Computational Analysis Programming Interface’s watertight triangulation of a modified computer-aided-design geometry to guide the movement of the structured surface mesh as the geometry changes during the optimization process. A mapping procedure is introduced that not only preserves the characteristics of the original surface mesh but also guarantees that the new mesh points are on the computer-aided-design geometry. The deformed surface mesh is then smoothed in the parametric space before it is transformed back into three-dimensional space. The procedure is efficient, in that all the processing is done in the parametric space, incurring minimal computational cost. The mesh movement tool is integrated into a three-dimensional shape-optimization framework, with a linear-elasticity volume-mesh movement algorithm, a Newton–Krylov flow solver for the Euler equations, and a discrete-adjoint gradient-based optimizer. The accuracy of the computed gradients is verified through a number of examples.

Three-dimensional shape optimization of a cemented hip stem and experimental validations.

This study proposes novel optimized stem geometry with low stress values in the cement using a finite element (FE) analysis combined with an optimization procedure and experimental measurements of cement stress in vitro. We first optimized an existing stem geometry using a three-dimensional FE analysis combined with a shape optimization technique. One of the most important factors in the cemented stem design is to reduce stress in the cement. Hence, in the optimization study, we minimized the largest tensile principal stress in the cement mantle under a physiological loading condition by changing the stem geometry. As the next step, the optimized stem and the existing stem were manufactured to validate the usefulness of the numerical models and the results of the optimization in vitro. In the experimental study, strain gauges were embedded in the cement mantle to measure the strain in the cement mantle adjacent to the stems. The overall trend of the experimental study was in good agreement with the results of the numerical study, and we were able to reduce the largest stress by more than 50% in both shape optimization and strain gauge measurements. Thus, we could validate the usefulness of the numerical models and the results of the optimization using the experimental models. The optimization employed in this study is a useful approach for developing new stem designs.

Multi-objective optimization of lean and sweep angles for stator and rotor blades of an axial turbine

The axial turbine is one of the most challenging components of gas turbines for industrial and aerospace applications. With the ever-increasing requirement for high-aerodynamic performance blades, three-dimensional aerodynamic shape optimization is of great importance. In this research, the rear part of a gas turbine consisting of a one-stage axial turbine is optimized numerically. A useful optimization algorithm is presented to improve the efficiency and/or pressure ratio of the axial turbine with two different objective functions. The three-dimensional blade-shape optimization is employed to study the effects of the turbine stator and rotor lean and sweep angles on the turbine performance. The investigation is carried out at the turbine design speed. By coupling a verified computational fluid dynamics simulation code with the genetic algorithm, an automated design procedure is prepared. Geometry candidates for the optimization algorithm are generated by re-stacking of the two-dimensional airfoil sections. Three-dimensional, turbulent, and compressible flow field is numerically investigated via a Navier–Stokes solver to calculate various objective functions. Experimental results of the gas turbine are used for specifying the boundary conditions and validation of the simulation results. The proposed method results in 1.3% and 1.5% improvements in the turbine stage efficiency in design speed and reduced mass parameter at choke condition, respectively.

Optimum nose shape of a front-rear symmetric train for the reduction of the total aerodynamic drag

For a high-speed train, the same power car is used as the first car and as the last car in a reverse direction simultaneously. Therefore, the previously optimized nose shape, considering only the first car position, is not well adopted in the last car position of a front-rear symmetric train in view of the aerodynamic drag. The three-dimensional nose shape optimization of a front-rear symmetric train is conducted to minimize the total aerodynamic drag of the entire train using CFD. The 3-D nose model is constructed by the vehicle modeling function with the optimized area distribution to minimize the micro-pressure wave. It is revealed that the total aerodynamic drag of the optimum shape for the entire train is reduced by 23.0% when compared to that of the conventionally optimized shape only for the first car of the symmetric train.

Three-Dimensional Large-Scale Aerodynamic Shape Optimization Based on Shape Calculus

Large-scale three-dimensional aerodynamic shape optimization based on the compressible Euler equations is considered. Shape calculus is used to derive an exact surface formulation of the gradients, enabling the computation of shape gradient information for each surface mesh node without having to calculate further mesh sensitivities. Special attention is paid to the applicability to large-scale three dimensional problems like the optimization of an Onera M6 wing or a complete blended-wing–body aircraft. The actual optimization is conducted in a one-shot fashion, in which the tangential Laplace operator is used as a Hessian approximation, thereby also preserving the regularity of the shape.

3-D Optimization of Ferrite Inductor Considering Hysteresis Loss

This paper presents three-dimensional shape optimization of inductors for the dc-dc converters, in which the nonconforming voxel-based finite element method (FEM) is employed to realize fast FE mesh generation during the optimization. The operating point of the inductor under the bias current condition, which is estimated from the circuit analysis, is obtained by nonlinear FE analysis. Then, the FE equation linearized around the operating point is solved being coupled with the circuit equation to obtain the magnetic fields in the inductor. The hysteresis loss is computed from the Steinmetz formula. Validity of the field computation is tested by comparing the numerical results with measured data. The multiobjective optimization of the inductor shapes is performed to minimize the winding resistance and hysteresis loss. It is shown that the present method can effectively find the Pareto solutions which can lead to improvement in the efficiency of the dc-dc converter.

Adjoint Optimization of a Wing Using the Class-Shape-Refinement-Transformation Method

This paper will demonstrate the potential of the class-shape-refinement-transformation (CSRT) method for aerodynamically optimizing three-dimensional surfaces. The CSRTmethodwas coupled to an in-house Euler solver and this combination was used in an optimization framework to optimize the ONERA M6 wing in transonic conditions. The gradients of the flow variables with respect to the design parameters were computed using an adjoint solver integrated in theEuler code.The optimizationwasperformedbya trust region reflective algorithm.A two-step approachwas used to optimize the wing. First, a general optimizationwas done using the Bernstein coefficients of the shape function. Second, a regional refinement was performed using the B-spline coefficients of the refinement function. It was shown that using this strategy a considerable improvement of the lift-to-drag ratio of 22% could be achieved. The work presented in this paper proves that the CSRT method is a very intuitive and effective way of parametrizing aircraft shapes, both in two aswell as in three dimensions. Themethod allows for a two-step approach which has the potential to significantly increase the lift-to-drag ratio of various aircraft shapes. It was also shown that using an adjoint algorithm provides the computational efficiency necessary to perform true three-dimensional shape optimization.

Optimization of Cardiovascular Stent Design Using Computational Fluid Dynamics

Coronary stent design affects the spatial distribution of wall shear stress (WSS), which can influence the progression of endothelialization, neointimal hyperplasia, and restenosis. Previous computational fluid dynamics (CFD) studies have only examined a small number of possible geometries to identify stent designs that reduce alterations in near-wall hemodynamics. Based on a previously described framework for optimizing cardiovascular geometries, we developed a methodology that couples CFD and three-dimensional shape-optimization for use in stent design. The optimization procedure was fully-automated, such that solid model construction, anisotropic mesh generation, CFD simulation, and WSS quantification did not require user intervention. We applied the method to determine the optimal number of circumferentially repeating stent cells (N(C)) for slotted-tube stents with various diameters and intrastrut areas. Optimal stent designs were defined as those minimizing the area of low intrastrut time-averaged WSS. Interestingly, we determined that the optimal value of N(C) was dependent on the intrastrut angle with respect to the primary flow direction. Further investigation indicated that stent designs with an intrastrut angle of approximately 40 deg minimized the area of low time-averaged WSS regardless of vessel size or intrastrut area. Future application of this optimization method to commercially available stent designs may lead to stents with superior hemodynamic performance and the potential for improved clinical outcomes.

Application of isogeometric analysis in structural shape optimization

In this paper, the isogeometric analysis method is utilized in the three-dimensional shape optimization of structures, instead of finite elements. The main obstacles in shape optimization problems are inherent in the currently practiced numerical analysis methods and their distinction from the geometry definition model. The newly developed isogeometrical analysis has the potential to be employed for this goal. The main advantage of this technique, besides being computationally less costly, and its ease and precision in defining boundaries, is that it does not require several remeshings in the process of optimization. In this research work, the domain of interest is defined by using Non-Uniform Rational B-Splines (NURBS). Also, a numerical analysis tool is developed, which uses the NURBS basis functions for approximation. The control points defining the design boundary surface are considered as the design variables of the shape optimization problem. To demonstrate the performance of the method, some examples of two and three dimensional problems are presented.

The effects of static, dynamic and fatigue behaviour on three-dimensional shape optimization of Kayabaşi_Ekici type hip prosthesis by finite element method and probabilistic approach

To design highly durable prostheses one has to take into account the natural processes occurring in the bone. One of the most important factors in the implant design is to reduce stress on the femur and the bone–cement. The purpose of this study is to investigate the behavior of newly designed implants under body weight load during stumbling by parametric modeling. Two different implant materials have been selected to study appropriate material and fatigue life resistant. In the parametric design, the prosthesis functional requirement is that the locking of stem to the femur head using cement should be strong enough to preclude unlocking during the life time of a patient and to prevent sliding of the implant into the bone–cement. The results of finite element simulations are compared with Charnley's implant results and appropriate material for the implant is proposed. The best stem shapes fulfilling the desired functional requirements are chosen for the design. These findings can form a base for further research such as the optimum design of bone–implant hip prosthesis. Therefore, probabilistic approach was employed to the model. In reality, uncertainties exist in the system and environment that may make the application of a deterministic design decision unreliable. That is, the values of the variables that are acting on the system cannot be predicted with certainty. For instance, probabilistic approach was applied to the model after deterministic design results. Thus, using probabilistic approach reliability of newly design cemented hip prosthesis was quantified. The new design is modeled parametrically to investigate the effects of different geometrical parameters on the relative displacement. These parameters are then optimized. Using the results of this investigation, the probability of failure was investigated for both the initial and shape-optimized prosthesis designs using several simple performance functions describing fatigue theory (Goodman, Gerber, Soderberg), static and dynamic failure of the cement–prosthesis interface. The optimum geometry and material properties are then compared with Charnley's implant results. Acknowledgement : The results presented in this study were obtained during an investigation supported by the Scientific Research Projects Commission of Marmara University.

Three-Dimensional Shape Optimization in Stokes Flow Problems

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the ...