Multi-objective Topology Optimization Research Articles

Multi-objective topology optimization for thermo-elastic systems based on periodic constraints

Thermo-elastic systems have a wide range of applications in various engineering fields. Traditional single-objective topology optimization (TO) design of thermos-elastic structures is difficult to achieve the combined optimum of multiple properties, and the non-periodic TO design of results are complex and difficult to fabricate. This paper introduces a multi-objective TO formulation based on density for designing thermo-elastic structures with periodic constraints, aiming to overcome the aforementioned issues. This paper assesses two competing weighted objective functions, the first function corresponds to structural and thermal compliance, while the second pertains to regional temperature and global stress. To solve the optimization issue, we utilize the p-norm function with a modified coefficient for evaluating the maximum temperature and stress values. Additionally, the adjoint variable method is employed to evaluate the sensitivity of various objectives, while the method of moving asymptotes (MMA) is used to update the design variables. Then the influence of different weight coefficients and subregion numbers regarding thermos-elastic coupled analysis is demonstrated through numerical examples. The results show that the complexity of the subregion features decreases as the number of subregions increases, and that changing both the number of subregions and the weighting coefficients results in changes in the overall structural performance. Finally, the geometric model is reconstructed using the TO results, and the structural performance is validated through the simulation of the reconstructed model. The results indicate that the TO results obtained by the method of this paper have predefined performance.

Soft Robotic Joints With Anisotropic Stiffness by Multiobjective Topology Optimization

Soft Robotic Joints With Anisotropic Stiffness by Multiobjective Topology Optimization

Multi-objective topology optimization of heat transfer surface using level-set method and adaptive mesh refinement in OpenFOAM

The present study proposes a new efficient and robust algorithm for multi-objectives topology optimization of heat transfer surfaces to achieve heat transfer enhancement with a less pressure drop penalty based on a continuous adjoint approach. It is achieved with a customized OpenFOAM solver, which is based on a volume penalization method for solving a steady and laminar flow around iso-thermal solid objects with arbitrary geometries. The fluid-solid interface is captured by a level-set function combined with a newly proposed robust reinitialization scheme ensuring that the interface diffusion is always kept within a single local grid spacing. Adaptive mesh refinement is applied in near-wall regions automatically detected by the level-set function to keep high resolution locally, thereby reduces the overall computational cost for the forward and adjoint analyses. The developed solver is first validated in a drag reduction problem of a flow around a two-dimensional cylinder at the Reynolds numbers of 10 and 40 by comparing reference data. Then, the proposed scheme is extended to heat transfer problems in a two-dimensional flow at the Prandtl number of 0.7 and 6.9. Finally, three-dimensional topology optimization for multi-objective problems is considered for cost functionals with different weights for the total drag and heat transfer. Among various solutions obtained on the Pareto front, 4.0% of heat transfer enhancement with 12.6% drag reduction is achieved at the Reynolds number of 10 and the Prandtl number of 6.9. Moreover, the optimization of a staggered pin-fin array demonstrates that the optimal shapes and arrangement of the fins strongly depend on the number of rows from the inlet. Specifically, the pin-fins in the first and third rows extended in the upstream direction further enhance heat transfer, while the fins in the second row vanish to reduce pressure loss.

Multi-objective Topology Optimization of Sandwich Lattice Structures for Vibration Suppression: Numerical and Experimental Investigations

Multi-objective Topology Optimization of Sandwich Lattice Structures for Vibration Suppression: Numerical and Experimental Investigations

Augmented Lagrangian approach for multi-objective topology optimization of energy storage flywheels with local stress constraints

Augmented Lagrangian approach for multi-objective topology optimization of energy storage flywheels with local stress constraints

Structure Improvement and Optimization of Gantry Milling System for Complex Boring and Milling Machining Center

To enhance the efficiency and machining precision of the TX1600G complex boring and milling machining center, a study was conducted on the structure of its gantry milling system. This study aimed to mitigate the influence of factors such as structural quality, natural frequency, and stiffness. The approach employed for this investigation involved mechanism topology optimization. To initiate this process, a finite element model of the gantry milling system structure was established. Subsequently, an objective function, comprising strain energy and modal eigenvalues, was synthesized. This objective function was optimized through multi-objective topology optimization, taking into account certain mass fraction constraints and considering various factors, including processing technology. The ultimate goal of this optimization was to create a gantry milling structure that exhibited high levels of dynamic and static stiffness, a superior natural frequency, and reduced mass. To validate the effectiveness of these topology optimization results, a comparison was made between the new and previous structures. The findings of this study serve as a valuable reference for optimizing the structure of other components within the machining center.

Topology and Size Optimized Design and Laser Welding of the U-Frame for Free-Space Laser Communication Telescopes

The success of laser communications heavily relies on the stiffness, dynamic properties, static performance, and manufacturability of the U-frame. The U-frame is a fundamental element in satellite-to-ground laser communication telescopes. However, there is currently a lack of research on the optimal design of U-frames, leading to a significant gap between ideal construction and practical manufacturability. To address these concerns, this study proposes a comprehensive approach that combines multi-objective topology optimization and multi-start size optimization techniques. This approach considers the multidisciplinary constraints imposed by mechanical, control, and optical systems. The objective is to achieve both the conceptual and detailed design of a novel U-frame, while also ensuring thorough consideration of the structure’s manufacturability during the optimization process. The prototype of the optimized U-frame was successfully fabricated using laser welding processes. The tensile test conducted on the prototype supported the idea that laser welding can enhance the micro-grain size of the joint, leading to improved overall mechanical properties. In particular, the joint strength achieved through laser welding was found to be 1.5 times greater than that achieved through TIG (Tungsten Inert Gas) welding. Additionally, the results obtained from the free vibration experiment closely aligned with the simulation, confirming the feasibility of manufacturing the optimized structure. The optimized structure demonstrated an improvement of 7.13% in dynamic performance and 29.61% in static performance compared to the first-generation structure. Additionally, there was a reduction of 29.89% in mass without affecting the remaining performance aspects. The successful fabrication of the prototype validates the feasibility of the proposed welding process and highlights the superiority of the new U-frame.

Lightweight design method of orthotropic steel bridge deck with U-ribs based on multi-objective optimization

This paper proposed a lightweight design method for orthotropic steel bridge deck U-ribs. Firstly, the U-rib is analyzed in static structural properties. An analytic hierarchy model was constructed to maximize the stiffness for multi-objective topology optimization. Secondly, a parametric reconfiguration model of the topologically optimized U-rib was established. Finally, static tests are carried out and the performance of the 3D printing scale model was tested. The test results show that the method is effective. Weight of the original model with U-rib is 99.8 kg, and the weight of its optimized model is 76.8 kg. The method achieved a weight reduction of 23% for the U-ribs, and the stiffness still met the permitted conditions. The orthotropic steel bridge deck composed of the optimized U-ribs effectively reduced the weight by 150 kg on the basis of basically unchanged stiffness. The method provides an important reference for the lightweight transformation of assembled bridge structures.

Multi-objective topology optimization design of truss structures based on evidence theory under limited information

Uncertainty information is often limited and has great discreteness, how to ensure the structure having good robustness under epistemic uncertainty becomes a technical problem for multi-objective topology optimization design. In this paper, the evidence theory is used to quantify the epistemic uncertainty; the plausibility measurement expressing upper limit of failure probability is applied to be a new evaluation of reliability. The total weight and the reliability of structures are taken as the optimization objectives, and a multi-objective robust topology optimization design model based on evidence theory is proposed. Parallelization technique based differential evolution for multi-objective optimization (DEMO) is preferable to search above robust Pareto front due to its merits of non-requirement of any gradient and superior mechanisms of non-dominate strategy. In order to verify the effectiveness of the proposed method, the multi-objective reliability optimization of a wood truss structure is implemented with considering elastic modulus and structural load as uncertain variables. According to the optimization results, six same truss specimens are made for the static random loading test. The failure probability of the truss structure is judged by the stress and node displacement obtained from the test, so as to verify the feasibility of the reliability optimization method based on evidence theory. The experimental results also indicate that the proposed method can avoid the deviation of the optimization results caused by the fluctuation of epistemic uncertainty, and provide a new method for designers to make the optimization results robust even when the data information is insufficient and the cognitive level is limited.

Multi-objective optimization of elastic metaplates for lightweight and ultrawide bandgaps

There is inevitably a conflict between multiple objectives when designing elastic metaplates (EMPs) with desirable functionalities from the perspective of practical applications. The non-dominated sorting genetic algorithm-II (NSGA-II) and the improved fast plane wave expansion method (IFPWEM) are combined to develop a multi-objective topological optimization method for EMPs with ideal efficiency and accuracy regarding lightweight and bandgap characteristics. The results indicate that initial designs featuring concentrated scatterers can yield Pareto front solutions with greater structural diversity. Additionally, the appropriate mesh resolution and the number of iterations are determined based on convergence and computational costs. Note that the post-processing method is proposed to improve the manufacturability by utilizing the connected component labeling algorithm while achieving convergence at least 15 generations earlier than the results without post-processing. The bandgap characteristics can be effectively improved by 2.34 and 2.19 at the same relative density compared to the conventional unit cell embedded with square or circular scatterers, demonstrating the superiority of the method. Additionally, the optimization objectives are further strengthened by reducing the symmetries of the unit cells. The wave propagation within finite periodic lattices is also investigated numerically and experimentally. The results of this study would provide useful guidance for developing and optimizing EMPs in the future.

Multi-objective topology optimization of cold plates featuring branched and streamlined mini-channels for thermal management system of lithium-ion battery module

A trade-off between increasing the thermal performance of the cold plate and reducing the energy requirement of the pumping system is a big problem so far and needs to be dealt with. This study implements a novel design for aluminum cold plates using multi-objective topology optimization to effectively cool a battery thermal management system composed of twenty prismatic lithium-ion batteries (LIBs). The velocity, temperature, and pressure fields are calculated with a finite element method. Numerical results obtained from computational fluid dynamics show that the optimized cold plate with branched and streamlined mini-channels has the advantage of reducing pressure drop and improving comprehensive heat transfer performance. Using the optimized cold plate, the average temperature of the hottest LIB, the maximum average temperature deviation between LIBs, and the pressure drop can be decreased by 10.3 %, 59.4 %, and 23.9 %, respectively, compared to the conventional one under an inlet velocity of 0.4 m s−1. The optimized cold plate is an effective heat exchanger to cool the LIBs, which exhibits a 63.0 % growth in heat transfer coefficient and Nusselt number compared to the conventional one using identical inlet velocity of water coolant.

Free-mounted cooling plate multi-objective topology optimization method towards precision machine tool heat dissipation: An experimental and numerical study

Machining accuracy is reduced by precision machine tool thermal deformation. Currently, machine tool temperature and thermal deformation are usually controlled by optimizing original machine tool structure or using a cooling system. However, these methods are poor in generalization ability and convenience, and traditional cooling channels have shortcomings of a large flow resistance and low heat transfer efficiency. In this study, a cooling plate multi-objective topology optimization method towards precision machine tool heat dissipation is proposed. Firstly, multi-objective topology optimization model is established with maximum heat exchange and minimum fluid dissipation energy as objective function. Then effect of various design parameters on topology optimization results is investigated. Penalty factor and design Reynolds number are taken as design variables, and heat dissipation and flow performances of different topology optimization results are analyzed. Response surface model is constructed, and then an analytical model of average temperature in design domain and pressure drop generated by Pareto fronts under different working conditions are obtained. Furthermore, a full-size three-dimensional thermal-fluid interaction verification model is constructed, and cooling and flow performances of topology optimization cooling plate are compared with those of traditional spiral cooling plate. Numerical results show that cooling performance of topology optimization cooling plate is stronger than that of spiral cooling plate at high Reynolds number, and pressure drop of topology optimization cooling plate is reduced by 85.38% compared with that of spiral cooling plate. Finally, topology optimization and spiral cooling plates are manufactured, and experimental platform is built to evaluate heat dissipation and flow performances of cooling plates. For topology optimization plate, maximum deviations between simulation results and experimental data for temperature and pressure drop are 4.8% and 6.4%, respectively. The effectiveness and feasibility of the topology optimization design for cooling plate are verified.

Multi-objective topology optimization and thermal performance of liquid-cooled microchannel heat sinks with pin fins

The optimized structures for the liquid-cooled microchannel heat sinks with different pin-fin arrays were designed via using the topology optimization method for the better performance, including minimization of the flow energy dissipation and the average temperature of the bottom surface, aiming to facilitate the more efficient design of microchannel heat sink for electronic chips’ cooling. The heat transfer and flowing performance was simulated with steady-state incompressible Navier-Stokes equation and the energy equation. Based on the rational approximation of material properties (RAMP) method, the mathematical optimization model for heat sink was established according to the multi-objective function. The Galerkin finite element method (GFEM) was employed to calculate the flow and heat transfer system. Furthermore, the globally convergent method of moving asymptotes (GCMMA) was adopted to deal with the mathematical optimization problem as the optimization method. The entropy generation analysis and comprehensive heat transfer factor were employed to evaluate the performance of the conjugate heat transfer problem of 3D microchannel heat sinks with various pin-fin structures. The calculation results reveal that the multi-objective optimal structure has good heat transfer performance, and the temperature distribution of the optimized structure is more uniform. As the same pumping power, the comprehensive heat transfer efficiency of the optimized inline and staggered pin-fin arrays increases by 31.20% and 32.18% compared to the smooth rectangular channel. Compared with the cylindrical pin-fin configuration with the same heat transfer area, the entropy generation rates of the optimized inline and staggered pin-fin configurations decrease by 4.67% and 4.25%, respectively. The results can provide theoretical basis and application support for the electronic chip cooling technology.

Enhancing heat dissipation and mass transfer of oxygen gas flow channel in a proton exchange membrane fuel cell using multiobjective topology optimization

To enhance the heat dissipation and mass transfer from a proton exchange membrane fuel cell (PEMFC) stack, a multi-objective topology optimization model considering structural deformation, heat transfer, mass diffusion, and gas flow process is proposed herein. The average molar concentration of oxygen and average temperature in the cathode catalyst layer (CL) are considered as the optimization objectives, and the mechanical, mass transfer, and gas flow properties are treated as constraints. The results reveal that compared with a straight gas channel, the topology-optimized configuration leads to an improvement in oxygen molar concentration by 10.36% and reduction in average temperature by 2.26 K. A higher power-dissipation constraint leads to a more complex three-dimensional channel structure with more branches. A larger structural displacement constraint results in a wider and flatter flow channel topology. The average oxygen molar concentration of topologically optimized configurations increases ranging from 5.18 to 14.96%, and the average temperature in the CL decreases about 2.2 K, compared with that of straight gas channel configurations, when the velocity at inlet varies from 0.05 to 0.3 m/s. These findings can aid in the design of PEMFCs.

Multi-Objective Automatic Design of Permanent Magnet Motor Using Monte Carlo Tree Search

This study proposes a novel multi-objective design method based on Monte Carlo tree search (MCTS) for the design of permanent magnet motors. The global configurations that define the entire structure are represented by nodes in a tree structure. After MCTS is performed to select a route extending from the root to leaf node, multi-objective topology optimization (TO) is performed at the leaf node to determine the detailed shape, considering a trade-off relationship among objective functions. The novelty of this work lies in the integration of MCTS with multi-objective TO where the score of a node is provided by the number of Pareto solutions obtained by selecting that node. This enables the scoring of nodes and the determination of the node selection criterion. The proposed method is applied to the multi-objective optimization of a permanent magnet motor with respect to average torque, and either torque ripple or iron loss. The proposed method successfully obtained Pareto solutions, which comprise various global configurations and shapes.

Multi-objective topology optimization of periodic pile barriers with ambient body waves isolation performance

The attenuation zones (AZs) of periodic pile barriers can be used to mitigate ambient vibration. To identify pile structures with optimal vibration isolation performance and minimal concrete consumption, this study adopts a multi-objective topology optimization (MOTO) procedure to optimize the periodic pile units for the isolation of plane waves. The optimized pile structures are presented, and the most cost-effective one is suggested. The effects of the AZ orders, the elastic modulus and the density of soil on the optimization results are investigated. By establishing numerical models, the vibration reduction performance of the optimized pile barriers in the frequency domain and time domain is verified.

Topology optimization of the periodic pile barrier with initial stresses arranged in rectangular and equilateral triangular lattices

The attenuation zones (AZs) of periodic pile barriers can be used to suppress ambient vibration in specific frequency ranges. Topology optimization of periodic pile units can search for efficient pile structures. However, in the process of optimization, the influence of initial stresses on the AZs is usually ignored, and the periodic piles are often arranged in a square lattice. In this paper, to maximize the relative width of AZs (RWAZs), the genetic algorithm (GA) and nondominated sorting genetic algorithm-II (NSGA-II) are adopted for single- and multiobjective topology optimization of the periodic pile unit with different initial stresses. Periodic piles arranged in rectangular lattices with different aspect ratios and those in an equilateral triangular lattice are considered, and the corresponding optimized pile structures are obtained. It is found that with the same unit area, the periodic piles with the equilateral triangular layout correspond to the largest RWAZ. Due to the existence of the initial stress, the position and width of the AZs of the optimized pile structure decrease. The RWAZ of the optimized pile structure decreases with increasing initial stresses. Finally, numerical results from both frequency domain and time domain analyses demonstrate that the optimized pile barrier can effectively isolate elastic waves, which shows extensive application prospects in engineering practice.

Multi-objective Topology Optimization of the Temperature Field Considering the Material Nonlinearities

This paper presents a multi-objective topology optimization method for the temperature field considering the material nonlinearities. Although a large number of papers have studied the topology optimizations of the heat transfer problems, none of them considered the influence of the material nonlinearity properties. In this paper, the material nonlinearity is considered. Based on the nonlinear assumption, the mathematical model of the corresponding topology optimization is presented first. Then, the sensitivities of the objectives and the constraints respect to the design variables are derived. And the optimization routines are discussed. After establishing the theories and solving scheme, a software is developed by MATLAB for the topology design of the nonlinear heat transfer. Several nonlinear numerical examples are presented. And the corresponding optimal results generated under the linear assuming are also collected for comparisons. Furthermore, this paper also simply discusses the preventing approaches of instabilities and the implementation methods of some other unusual engineering constraints. These true engineering considerations are very helpful for the real industry design experiences.

Topological optimization of flow-shifting microchannel heat sinks

Efficient thermal management is required for electronic devices that generate multiple different spatially nonuniform thermal workloads during operation. Conventional heat sinks route coolant to all potential heat-generating regions of an electronic component which may result in unnecessarily broad dispersion of the coolant to regions that are sometimes inactive. To fully utilize flow for multiple potential thermal workloads, a “flow-shifting” design approach is proposed. A flow-shifting heat sink has multiple inlets, where the flow path from each inlet is optimized for cooling of a specific workload or heat map. Only the inlet corresponding to the active operating workload is open at a given time, while the rest of the inlets are kept closed until the operating workload changes, allowing a majority of the flow to be properly utilized for cooling the active heat map. This has the potential to allow many different optimal heat sinks to be simultaneously encoded into a single structure but brings forth a complex design problem to optimize the internal fluid flow pathways. A multi-objective topology optimization algorithm is implemented in this work for flow-shifting heat sink design generation. Microchannel heat sink designs are generated for a demonstration case involving two workloads. Designs optimized for flow-shifting between two inlets are compared to a benchmark that is optimized with only a single inlet. Pareto optimality curves, as well as the associated heat sink designs, are created that weigh between the thermal resistances of the two workloads. The flow-shifting designs are predicted to have lower thermal resistance under both workloads compared with every benchmark heat sink design generated along the Pareto optimality curve. Investigation of two designs selected from the Pareto curve showed that the flow-shifting heat sink achieved 10.7% and 6.8% lower thermal resistance at the two workloads relative to the benchmark. The flow-shifting approach is superior as it utilizes more flow rate for cooling the active heated region. The flow-shifting approach thereby allows for fixed heat sink structures that can be topologically optimized for many different possible heat maps and actively respond to changes in workload.

Digital design and manufacturing of spherical joint base on multi-objective topology optimization and 3D printing

Topology optimization techniques have been widely applied for the generation of highly efficient material structures in various fields. In addition, 3D printing has the ability to fabricate geometrically complex optimized results which are difficult to realize for conventional means. The research presented in here aims to implement a digital design and manufacture high-performance joints in space truss by combining topology optimization and 3D printing. Two multi-condition optimization models, i.e., minimum volume model and minimum compliance model, are employed to optimize the spherical joints under multiple conditions. The conditions are three most dangerous load conditions of joints in the space truss. Since single-objective topology optimizations only seek the optimal solution for a single objective, the mechanical properties of the optimized result may be limited. As a result, the compromise programming model, a multi-objective topology optimization model, is designed with the goals of minimum volume and compliance under each condition. In addition, multi-condition and multi-objective topology optimization design of the spherical joints are carried out based on density approach. The advantages of the multi-objective topology optimization are verified by comparing the mechanical performance of the optimized results obtained by the minimum volume model, the minimum compliance model and the compromise programming model. Finally, the additive manufacturing of the optimized results is completed by using 3D printing technology, which verifies the feasibility of digital design and manufacturing of joints.