Shape Optimization Research Articles

Effect of 3D printed skeleton shapes on strength behavior, stress evolution and microstructural response of cement-based tailings backfills

The inherent brittle behavior of cement-based tail backfill (CTB) poses some challenges that can significantly impact the safety and effectiveness of underground mining operations. Due to its propensity for abrupt and catastrophic failure without considerable deformation, CTB can endanger both the structural solidity of mined-out cavities and workers’ security. Hence, implementing appropriate strategies is critical to mitigate the risks associated with the brittle nature of CTB. An experiment investigation was commenced to consider the influence of diverse 3D printed polymer reinforcement geometries - specifically, hexagon, staggered quadrilateral, and rhomboid shapes - on strength behavior, stress evolution, and microstructural response of CTB via peak compression strength test plus scanning electron microscope observations. Lab findings specified a decrease in fill strength upon adding these polymer reinforcements. Specifically, the N-3D-PP-reinforced CTB exhibited a strength of 0.61 MPa. In comparison, the reinforced samples showed lower strengths of 0.39 MPa for the hexagon shape, 0.44 MPa for the staggered quadrilateral shape, and 0.45 MPa for the rhomboid shape. The toughness of CTB incorporating 3D-PP was higher than that of N-3D-PP ones. The optimal shape for an ideal toughness was the rhomboid shape, with a value of 58.06 KJ.m−3. N-3D-PP-reinforced CTB specimens exhibited a semi-brittle behavior, while 3D-PP-reinforced CTBs displayed a more ductile response with high total and dissipative energies. Cracks in N-3D-PP reinforced CTB were tensile, and mainly a combination of tensile and shear in 3D-PP reinforced CTB. Despite enhanced toughness, the presence of pores, the poor interfacial bonding between the CTB matrix and the 3D-PP, the low volume of polymer reinforcement, and the potential disruption of cement particle packing compromised the CTB’s overall strength. This research highlights the potential of 3D-printed reinforcements in transforming the landscape of mining backfill technology. The findings pave the way for the widespread adoption of 3D-printed polymer-reinforced CTB in metalliferous mines, promoting safer, more productive, and, ultimately, more profitable underground mining operations.

Review of Vibration Analysis and Structural Optimization Research for Rotating Blades

AbstractBlades are important parts of rotating machinery such as marine gas turbines and wind turbines, which are exposed to harsh environments during mechanical operations, including centrifugal loads, aerodynamic forces, or high temperatures. These demanding working conditions considerably influence the dynamic performance of blades. Therefore, because of the challenges posed by blades in complex working environments, in-depth research and optimization are necessary to ensure that blades can operate safely and efficiently, thus guaranteeing the reliability and performance of mechanical systems. Focusing on the vibration analysis of blades in rotating machinery, this paper conducts a comprehensive literature review on the research advancements in vibration modeling and structural optimization of blades under complex operational conditions. First, the paper outlines the development of several modeling theories for rotating blades, including one-dimensional beam theory, two-dimensional plate–shell theory, and three-dimensional solid theory. Second, the research progress in the vibrational analysis of blades under aerodynamic loads, thermal environments, and crack factors is separately discussed. Finally, the developments in rotating blade structural optimization are presented from material optimization and shape optimization perspectives. The methodology and theory of analyzing and optimizing blade vibration characteristics under multifactorial operating conditions are comprehensively outlined, aiming to assist future researchers in proposing more effective and practical approaches for the vibration analysis and optimization of blades.

Analysing the shape memory behaviour of MWCNT-enhanced nanocomposites: a comparative study between experimental and finite element analysis

Shape memory polymers (SMPs) are known for their unique ability to withstand large deformations and revert to their original shape under specific external stimuli. However, their broader application in biomedical and structural applications is restricted by limited mechanical and thermal properties. Introducing multi-walled carbon nanotubes (MWCNTs) into SMPs has proven to significantly enhance these characteristics without affecting their inherent shape memory features. This study investigates shape memory nanocomposites (SMNCs) through dynamic and thermogravimetric analyses, along with tensile, flexural, and shape memory testing, and explores fracture interfaces using scanning electron microscopy. Findings indicate optimal shape memory, thermal, and mechanical properties with 0.6 wt% MWCNT content, showcasing a shape recovery ratio of 93.11%, storage modulus of 4127.63 MPa, tensile strength of 55 MPa, and flexural strength of 107.94 MPa. Moreover, incorporating MWCNTs into epoxy demonstrated a reduction in recovery times by up to 50% at 0.6 wt% concentration. Despite a slight decrease in shape fixity ratio from 98.77% to 92.11%, shape recoverability remained nearly consistent across all samples. The study also introduces a novel finite element (FE) method in ABAQUS for modeling the thermomechanical behavior of SMNCs, incorporating viscoelasticity, validated by matching experimental results with FE simulations, highlighting its accuracy and practical applicability in engineering.

On the Optimal Shape of a Thin Insulating Layer

On the Optimal Shape of a Thin Insulating Layer

Structural Shape Optimization Based on Multi-Patch Weakly Singular IGABEM and Particle Swarm Optimization Algorithm in Two-Dimensional Elastostatics

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

Shape Optimization of Heavy Vehicles Chassis

The main objective of this project is to modify the design to decrease the deformation of chassis.

Compact off-axis reflective optical system design combining freeform mirror and freeform detector

Minimizing system size and weight without sacrificing the desired imaging performance is a crucial and challenging goal in the design of off-axis optical systems. In this paper, we propose a design concept that combines freeform mirror and freeform detector to achieve more compact off-axis optical systems. Initially, by applying nodal aberration theory, the optimal focal surface shape for off-axis systems is demonstrated to be an off-axis paraboloid or even a freeform surface, rather than the commonly assumed spherical shape. Subsequently, we show the ability of the design concept to reduce off-axis system volume through a design example of a fast and wide-field off-axis three-mirror system. Results indicate that, under the same design parameters and imaging quality, the design with a freeform detector is 80% smaller than that with a flat detector in terms of volume. Finally, the bending capability of the freeform detector presented in this paper is ensured through finite element analysis, and its surface accuracy is achievable based on the current precision level of curved detector fabrication. The proposed design concept holds promise for application in planetary science, where the imaging performance of the instrument is highly required and the size and weight are strictly limited.

Design and shape optimization of interior permanent magnet synchronous machine based on hybrid cores

By designing stator teeth using grain-oriented sheets (GO) and other parts still with non-grain-oriented sheets (NGO), the electromagnetic performance of the interior permanent magnet synchronous machine (GO-IPMSM) can be improved greatly. As the stator core of the designed GO-IPMSM is a hybrid core that is composed of two different silicon sheets, both the electromagnetic and mechanical performances are affected by the stator joint shape between GO and NGO sheets. Meanwhile, with the adoption of GO, the torque ripple of GO-IPMSM is increased though the average torque is increased as well. For reducing the torque ripple, optimizing the rotor barrier shape is an effective way. In this paper, the piecewise linear interpolation method is employed for establishing the stator joint shape between GO and NGO sheets, and the polynomial method is proposed to establish the rotor barrier shape. Through the analysis, it can be seen that the stator joint shape plays a strong role in affecting the mechanical performance of the GO-IPMSM, while the effect on the electromagnetic performance is weak. The genetic algorithm is used to optimize the rotor barrier shape for achieving high average torque and low torque ripple. Lastly, a quick and accurate efficiency map calculation method based on the Kriging model is proposed for GO-IPMSM with less finite element method (FEM) samples are required.

Constraint‐aware optimization model for plane truss structures via single‐agent gradient descent

AbstractThis study introduces the constraint‐aware optimization model (CAOM), a novel optimization framework designed to optimize the size, shape, and topology of plane truss structures simultaneously. Unlike traditional optimization models, which rely on gradient descent and frequently struggle with managing various constraints due to their dependence on a single optimization agent, CAOM effectively addresses this challenge. It does so by incorporating a variety of assistant modules along with the Adam optimizer, a variant of the gradient descent method. Uniquely, CAOM employs the leaky rectified linear unit (ReLU) activation function beyond its conventional use in neural networks, applying it as a mechanism to integrate constraints and losses seamlessly. The model's effectiveness was validated through two numerical examples and a practical application, demonstrating that CAOM can reduce structural weight by up to 84% compared to unoptimized designs while fully adhering to structural, dimensional, and moveable constraints. Furthermore, the study found that while shape optimization plays a key role for stiffness‐governed structures, size optimization is crucial for strength‐governed structures. Optimizing size, shape, and topology together consistently leads to the most weight‐efficient designs. This emphasizes the significance of a holistic approach in the optimization processes.

Shape optimization for interface identification in nonlocal models

AbstractShape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape).

Advanced multimaterial shape optimization methods as applied to advanced manufacturing of wind turbine generators

AbstractCurrently, many utility‐scale wind turbine generator original equipment manufacturers are dependent on imported rare earth permanent magnets, which are susceptible to market risks from cost instability. To lower the production costs of these generators and stay competitive in the market, several small wind manufacturers are pursuing continuous improvements to both generator design and manufacturing. However, traditional design and manufacturing methods have yielded marginal improvements in wind power performance. This work presents novel methods to redesign a baseline 15‐kW wind turbine generator with reduced rare‐earth permanent magnets by leveraging cutting‐edge three‐dimensional (3D) printed polymer‐bonded permanent magnets and steel. Symmetric, asymmetric, and multimaterial‐magnet parametrization methods are introduced for shape optimization. We extend the symmetric and asymmetric methods to the back iron in the stator to further investigate the impact and opportunities for performance improvements with lesser active materials. We employ a design‐of‐experiments approach with parametric computer‐aided design for shape generation and evaluate different designs by magneto‐thermal modeling and finite‐element analysis. We use adaptive sampling technique to identify better performing designs with lesser magnet mass, higher efficiency, and lower cogging torque when compared with the baseline generator. Asymmetric pole designs resulted in a magnet mass in the range of 4.77–5.37 kg, which was 27%–35% lighter than the baseline generator, suggesting that a new design freedom exists that can be enabled by advanced manufacturing, such as 3D printing. Shaping the back iron in the stator resulted in material savings in electrical steel of up to 14.62 kg, which was 20% lighter than the baseline stator. We conducted a structural analysis to evaluate an optimized asymmetric rotor design from the point of view of mechanical integrity and air‐gap stiffness. The magnetically optimal shape profile was shown as having a positive impact on the radial stiffness, and an optimal solution was discovered to reduce the structural mass by nearly 30 kg, which was 29% lighter than the baseline.

Surrogate-based aerodynamic shape optimization of high-speed train heads: A review of four key technologies

With the increase in running speed, the aerodynamic characteristics of high-speed trains have a significant impact on running stability, energy consumption and passenger comfort. Since the shape of the high-speed train head can directly influence the surrounding airflow, optimizing the head shape is the primary way to improve the aerodynamic performance of the train. This paper reviews current research studies on the surrogate-based aerodynamic shape optimization of high-speed train heads, aiming to provide a comprehensive reference for designers to enhance design efficiency and optimization performance. The entire optimization process is divided into four essential steps, and the key optimization technologies in each step are discussed, including parametric modeling, computational fluid dynamics (CFD) simulation, surrogate model and optimization algorithm. By introducing the practical applications of these technologies, we summarize their advantages and disadvantages and suggest four potential research directions for the future.

Symmetry breaking in optimal transport networks

Engineering multilayer networks that efficiently connect sets of points in space is a crucial task in all practical applications that concern the transport of people or the delivery of goods. Unfortunately, our current theoretical understanding of the shape of such optimal transport networks is quite limited. Not much is known about how the topology of the optimal network changes as a function of its size, the relative efficiency of its layers, and the cost of switching between layers. Here, we show that optimal networks undergo sharp transitions from symmetric to asymmetric shapes, indicating that it is sometimes better to avoid serving a whole area to save on switching costs. Also, we analyze the real transportation networks of the cities of Atlanta, Boston, and Toronto using our theoretical framework and find that they are farther away from their optimal shapes as traffic congestion increases.

Selection of Appropriate Criteria for Optimization of Ventilation Element for Protective Clothing Using a Numerical Approach

While there are multiple methods to ventilate protective clothing, there is still room for improvement. In our research, we are using ventilation elements that are positioned at the ventilation holes in the air space between the body and clothing. These ventilation elements allow air to flow freely while preventing sun radiation, rain drops, and insects from directly accessing the body. Therefore, the shape of the ventilation element is crucial. This led us to study the shape optimization of ventilation elements through the utilization of metamodels and numerical approaches. In order to accomplish the objective, it is crucial to thoroughly evaluate and choose suitable criteria for the optimization process. We know from prior research that the toroidal cut-out shape element provides better results. In a previous study, we optimized the shape of this element based on the minimum pressure difference as a criterion. In this study, we are using different criteria for the shape optimization of ventilation elements to determine which are most effective. This study involves a metamodeling strategy that utilizes local and global approximations with different order polynomials, as well as Kriging approximations, for the purpose of optimizing the geometry of ventilation elements. The goal was achieved by a sequential process. (1) Planning the position of control points of Non-Uniform Rational B-Splines (NURBS) in order to generate elements with a smooth shape. (2) Constructing geometric CAD models based on the design of experiments. (3) Compute detailed model solutions using SolidWorks Flow Simulation. (4) Developing metamodels for responses using computer experiments. (5) Optimization of element shape using metamodels. The procedure is repeated for six criteria, and subsequently, the results are compared to determine the most efficient criteria for optimizing the design of the ventilation element.

Optimal ovoid lobe shape by an automatic calculation program and CFD using dynamic mesh technique

Optimal ovoid lobe shape by an automatic calculation program and CFD using dynamic mesh technique

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

A Non-parametric Gradient-Based Shape Optimization Approach for Solving Inverse Problems in Directed Self-Assembly of Block Copolymers

A Non-parametric Gradient-Based Shape Optimization Approach for Solving Inverse Problems in Directed Self-Assembly of Block Copolymers

Isogeometric Shape Optimization of Reissner–Mindlin Shell with Analytical Sensitivity and Application to Cellular Sandwich Structures

Structural shape optimization plays a significant role in structural design, as it can find an appropriate layout and shape to improve structural performance. Isogeometric analysis provides a promising framework for structural shape optimization, unifying the design model and analysis model in the optimization process. This paper presents an adjoint-based analytical sensitivity for isogeometric shape optimization of Reissner–Mindlin shell structures. The shell structures are modeled by multiple NURBS surfaces and design variables are associated with the position of control points. A multilevel approach is performed with a coarse mesh for the design model and a dense mesh for the analysis model. The sensitivity propagation is achieved through a transformation matrix between the design and analysis models. Structural compliance minimization problems with and without constraints are studied and the optimization history shows that the optimization can converge quickly within fewer iterations. The developed formulations are validated through several numerical examples and applied to the optimization of cellular sandwich structures, which are widely used in engineering applications. Numerical results show that optimized sandwich panels can achieve better performance in bending resistance.

Surrogate-based Shape Optimization Design for the Stable Descent of Mars Parachutes

The crucial role that the supersonic parachute plays in space exploration missions has been widely recognized, as it directly impacts the safe landing of probes. However, parachute models with optimization on different aerodynamic performances often involve design conflicts with each other. Additionally, the parachute design focusing on a single point cannot fully adapt to different speed ranges during stable descent. This complexity makes it challenging to use traditional shape design methods, which rely on empirical knowledge, to address these coupled design issues. Faced with the design challenges of Mars parachutes, this study, inspired by aircraft aerodynamic optimization principles, establishes a shape design method specifically for the stable descent phase of Mars parachutes. The method combines numerical simulation and surrogate-based optimization strategies, aiming to enhance the overall performance during stable descent and meet various demands of different exploration missions. Meanwhile, by providing a rapid estimate of the shape during the design phase, the method significantly improves computational efficiency. The optimal models effectively balance comprehensive performance in the supersonic-transonic-subsonic speed domain by conducting shape optimization research on the disk-gap-band parachute using the surrogate-based optimization strategy. Also, it exhibits better deceleration and stability across the entire speed range compared to the base model, even when deviating from the design Mach number. Importantly, the advantages of canopy-only optimization for drag performance extend to the capsule-canopy two-body system, enhancing the drag performance of the canopy in the two-body system. This strategic approach reduces the transient calculation time for the two-body system, further improving computational efficiency. The method provides a practical and forward-thinking solution for the design of Mars parachutes.

Shape optimization of annular transonic thrust nozzles via genetic algorithm and adjoint method

In this research, the idea of combining the genetic algorithm and adjoint method was proposed to leverage the benefits of both approaches to improve the accuracy of thrust nozzle shape optimization. Although the genetic algorithm can search the entire design space to approach the global optimum, its computational cost increases considerably with the number of design variables. Meanwhile, the adjoint method is unaffected by the design variables, but it cannot achieve the global optimum directly and relies on the initial guess geometry. To address these challenges, a genetic algorithm was initially employed to approximate the global optimal geometry of a thrust nozzle with a conical center body parameterized by a Bezier curve. A MATLAB code integrated the geometry parameterization, genetic algorithm, and Fluent R2 2021 flow solver. The resulting geometry was then considered as the initial guess for the adjoint method, which accurately achieved the final optimal geometry with increased control points. The genetic algorithm aimed to maximize the thrust coefficient as the objective function, while the adjoint solver aimed to minimize the axial velocity variance at the nozzle outlet as the cost function. Three distinct constraints were incorporated in the optimization process, and the influence of the center body shape on the results was thoroughly examined to consider all possible geometries throughout the optimization procedure. The findings revealed that the conical center body effectively adjusted the flow angle at the exit, significantly affecting the thrust force. Both optimization steps reduced the average velocity and increased the average pressure at the nozzle exit, enhanced the mass flow rate, and finally increased the thrust force. The genetic algorithm and adjoint method contributed to 3 % and 0.6 % increases, respectively, in the thrust force, resulting in a thrust coefficient of 96.15 % for the final optimized nozzle.