Multi-objective Topology Research Articles

Thermomechanical coupling multi-objective topology optimization of anisotropic structures based on the element-free Galerkin method

The mathematical model of thermomechanical coupling multi-objective topology optimization for anisotropic structures was established using the element-free Galerkin (EFG) method, and the weighted function of compliance and heat dissipation was defined as the objective function. The proposed model and procedure were verified by the topological results based on the finite element method. The multi-objective EFG optimal topological structures have clearer boundary profiles even without using the sensitivity filtering technique. The effects of the weight coefficient, thermal conductivity factor, Poisson's ratio factor, off-angle and volume fraction on the multi-objective EFG optimal topological structure and multi-objective function were evaluated in detail, and reasonable ranges of the above parameters were recommended to improve the heat dissipation and mechanical performance. The multi-objective optimal results of the anisotropic structure were 3D printed and compared with the isotropic material, and their temperature, displacement and stress were improved, which reflects the advantages of orthotropic structures.

Multiobjective topology optimization with stress and strain energy criteria using the SESO method and a Multicriteria Tournament Decision

This article aims to explore the application of an evolutionary optimization technique for multiobjective optimization problem using as criteria the minimization of the Von Mises maximum stress and minimization of the maximum growth of the structure’s internal strain energy. To evaluate the global effect on the optimal design setting by the removal from the structure of inefficient material, regions in which the magnitude of stress or strain energy is relatively small, a goal weighting scheme is adopted, whereby the weight factors emphasize and balance the stress and strain energy criteria. The global criterion method for multiobjective optimization and the Pareto optimal concept were also considered in this study. Thus, contribution is made to the study of these two methods in the structural optimization procedure using linear static analysis via Finite Element Method. In addition, with the discrete approximation of the Pareto front, the choice of the preferred Pareto-optimal solution is performed using the Multicriteria Tournament Decision. A number of examples demonstrates the capabilities of the proposed method to solve structural design problems using multiobjective optimization.

Multi-objective optimization of heat transfer in microchannel for non-Newtonian fluid

This paper presents a multi-objective topology optimization method for convective heat transfer problems based on Navier-Stokes and heat transport equations. In this research, the pressure drop (or energy dissipation) function and the recoverable thermal power function are combined and optimized as a multi-objective function. A Pareto algorithm is constructed, based on a weighted-sum method using different weight coefficients. The Pareto solutions composed of a set of optimal solutions are obtained to reveal the trade-off relationship between the objective functions. The effects of non-Newtonian fluid on the flow-channel arrangement and heat transfer performance are numerically studied. In the numerical method, the interpolation term in the density-based topology optimization method is modified, while the filtering method is used to solve the problems of grayscale and the flow channel discontinuity. The results show that the differences between the optimal structures for non-Newtonian fluid and Newtonian fluid are more obvious at Re = 0.01 in the dual-terminal devices. Under the same situation, blood (non-Newtonian fluid) has greater energy dissipation and better heat transfer performance than water (Newtonian fluid).

Integrated design optimization of structural topology and heat source layout

Heat source devices embedded in a multi-component structural system are often used as integrated load-bearing components to fulfill multi-functionality and lightweight requirements. This paper proposes a multi-objective topology optimization method for integrated layout design of such multi-component structures. The layout of heat sources and the topology of host structure are optimized simultaneously to improve the structural stiffness and heat transfer performance. The velocity field level set method, which offers a convenient way to handle multiple constraints and general design variables in the implicit boundary description-based topology optimization framework, is employed in the formulation of the optimization problem. Therein, the level set model provides a relatively smooth representation of the material boundary and structure/component interfaces, and also facilitates implicitly tracking the layouts of the embedded components. An adjoint-variable sensitivity analysis scheme is derived by considering the continuity conditions on the structure/component interfaces. Several numerical examples are presented to demonstrate the validity and efficiency of the proposed framework.

A Two-Scale Multi-Physics Deep Learning Model for Smart MEMS Sensors

Smart materials and structures, especially those bio-inspired, are often characterized by a hierarchy of length- and time-scales. Smart Micro Electro-Mechanical Systems (MEMS) are also characterized by different physical phenomena affecting their properties at different scales. Data-driven formulations can then be helpful to deal with the complexity of the multi-physics governing their response to the external stimuli, and optimize their performances. As an example, Lorentz force micro-magnetometers working principle rests on the interaction of a magnetic field with a current flowing inside a semiconducting, micro-structured medium. If an alternating current with a properly set frequency is let to flow longitudinally in a slender beam, the system is driven into resonance and the sensitivity to the magnetic field may result largely enhanced. In our former activity, a reduced-order physical model of the movable structure of a single-axis Lorentz force MEMS magnetometer was developed, to feed a multi-objective topology optimization procedure. That model-based approach did not account for stochastic effects, which lead to the scattering in the experimental data at the micrometric length-scale. The formulation is here improved to allow for stochastic effects through a two-scale deep learning model designed as follows: at the material scale, a neural network is adopted to learn the scattering in the mechanical properties of polysilicon induced by its polycrystalline morphology; at the device scale, a further neural network is adopted to learn the most important geometric features of the movable parts that affect the overall performance of the magnetometer. Some preliminary results are discussed, and an extension to allow for size effects is finally foreseen.

Derivative-free level-set-based multi-objective topology optimization of flow channel designs using lattice Boltzmann method

A derivative-free level-set-based topology optimization method (DFLS-TO) was proposed and was used for channel structure design. Unlike in a conventional level-set method, quantum-behaved particle swarm optimization (QPSO) was used in DFLS-TO to optimize the level-set values at the knot points. The values at the non-knot points were interpolated by solving the Laplace equation. QPSO was combined with the Tchebycheff decomposition method to search for optimal level-set values in multi-objective problems. The channel boundary was represented by an iso-contour of the level-set function, and the B-spline method was used to convert the zigzag-like boundary into a smooth boundary. The lattice Boltzmann method (LBM) was integrated with the immersed boundary method to simulate the fluid flow. The Spalart–Allmaras model was incorporated with the LBM during the simulation of turbulent flows. To demonstrate the applicability of DFLS-TO, optimization of a pipe bend and a fluid distributor were performed.

Multi-objective topology optimization of additive manufactured alternator bracket

Topology optimization is the technique utilized in the early design phase, where these algorithms result in a complex shape, which cannot be fabricated in traditional manufacturing processes, hence the Additive manufacturing process is being the promising tool to manufacture such complex shape. Alternator mount of a passenger vehicle was taken for the redesign using OptiStruct, with objective of minimizing the weight and constrain of first natural frequency higher than 450 Hz. Topology optimized structure was smoothened using 3MATIC software and fabricated using Selective laser sintering, which had a substantially weigh lesser than the old bracket and also verified the structural stability by testing of static and dynamic loading environment. The complete methodology can produce the components that are really light in weight with an effective load carrying capacity according to their applications.

Multi-objective topology optimization for the solar thermal decomposition of methane reactor enhancement

A multi-objective topology optimization method is proposed to generate local solid structure that can enhance the solar thermal decomposition of the methane (TDM) reactor. Two dominant factors can govern the performance of solar reactor, with heat transfer entropy generation and viscous dissipation representing heat exchange capability and flow resistance, respectively. Therefore, local solid structure and flow pattern can be generated under the compromise between the extreme value of the two factors. This multi-objective topology optimization problem is solved by the calculus of variations method and globally convergent version of the method of moving asymptotes (GCMMA) with the governing equations of the solar TDM reactor as constraints. A novel local structure of the solar TDM reactor is generated, which can improve the heat exchange effect and the conversion rate of the reaction.

Generating minimal Pareto sets in multi-objective topology optimisation: an application to the wing box structural layout

Multi-objective topology optimisation problems are often tackled by compromising the cost functions according to the designer’s knowledge. Such an approach however has clear limitations and usually requires information which especially at the preliminary design stage could be unavailable. This paper proposes an alternative multi-objective approach for the generation of minimal Pareto sets in combination with density-based topology optimisation. Optimised solutions are generated integrating a recently revised method for a posteriori articulation of preferences with the Method of Moving Asymptotes. The methodology is first tested on an academic two-dimensional structure and eventually employed to optimise a full-scale aerospace structure with the support of the commercial software Altair OptiStructⓇ. For the academic benchmark, the optimised layouts with respect to static and dynamic objectives are visualised on the Pareto frontier and reported with the corresponding density distribution. Results show a progressive and consistent transition between the two extreme single-objective layouts and confirm that the minimum number of evaluations was required to fill the smart Pareto front. The multi-objective strategy is then coupled with Altair OptiStruct and used to optimise a full-scale wing box, with the clear purpose to fill a gap in multi-objective topology optimisation applied to the wing primary structure. The proposed methodology proved that it can generate efficiently non-dominated optimised configurations, at a computational cost that is mainly driven by the model complexity. This strategy is particularly indicated for the preliminary design phase, as it releases the designer from the burden to assign preferences. Furthermore, the ease of integration into a commercial design tool makes it available for industrial applications.

Designing 2D auxetic structures using multi-objective topology optimization

The behaviour of the auxetic structure under external load was regarded as the behaviour of the compliant (flexible) mechanism. The multi-objective topological optimization, based on genetic algorithms and the finite element method, was used to find the optimal shape of such two-dimensional compliant mechanism in this study. The optimization was performed on a quarter of a double-symmetric representative unit cell, which is a building block of the symmetrical auxetic structure. Static linear computational simulations were performed to determine the mechanical response of cell topologies. The proposed method leads to a set of best solutions positioned on the Pareto front with different topologies and gives a broad overview of possible designs of new auxetic structures. The method is highly effective and can be easily extended to large deformation formulations, nonlinear elasticity or elasto-plasticity.

An improved multi-objective topology optimization model based on SIMP method for continuum structures including self-weight

This work proposes an improved topology optimization model for optimizing continuum structures with self-weight loading conditions. A modified Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, the penalty factor of the SIMP model is increased to maintain the activeness of the prescribed volume constraint and to drive the design domain to binary distribution. The optimization objectives include minimizing the total strain energy of the design domain and minimizing the total displacement of the fixed domain. The shape optimization procedure is used to furtherly enhance structural performance. The whole optimization procedure is implemented with a two-dimensional model under the loading conditions of self-weight and external force. The classic MBB beam and self-weight arch are utilized to verify the proposed method, and conceptual layout designs of the steel structure bridge are conducted. It is proved that the proposed model is effective for topology optimization of continuum structures including self-weight. And it is found that the optimal structural topology is affected by the ratio of the external force to self-weight.

Topology optimization of compliant mechanisms considering strain variance

In this work, compliant mechanisms are designed by using multi-objective topology optimization, where maximization of the output displacement and minimization of the strain are considered simultaneously. To quantify the strain, we consider typical measures of strain, which are based on the p-norm, and a new class of strain quantifying functions, which are based on the variance of the strain. The topology optimization problem is formulated using the Solid Isotropic Material with Penalization (SIMP) method, and the sensitivities with respect to design changes are derived using the adjoint method. Since nearly void regions may be highly strained, these regions are excluded in the objective function by a projection method. In the numerical examples, compliant grippers and inverters are designed, and the tradeoff between the output displacement and the strain function is investigated. The numerical results show that distributed compliant mechanisms without lumped hinges can be obtained when including the variance of the strain in the objective function.

A Multi-Objective Topology Optimization Methodology Based on Pareto Optimal Min-Cut

A topology optimization (TO) design is inherently a bi-objective problem since the design goals are, at least, to minimize the material consumption and to maximize a performance parameter. To eliminate the extremely high computational burden and the checkerboard pattern problem of existing optimal methodologies, this article proposes a novel methodology based on Pareto optimal min-cut (POMC) to solve multi-objective TO problems with minimizing the volume as one objective. The good ability to handle an equality constraint based on POMC enables the proposed algorithm to achieve the optimality of the single objective under any equality volume constraint. According to the comparison of numerical results with other methodologies, the proposed methodology is capable of obtaining higher quality Pareto frontiers by using much reduced computational burdens.

Multi-material thermomechanical topology optimization with applications to additive manufacturing: Design of main composite part and its support structure

This paper presents a density-based topology optimization formulation for the design of multi-material thermoelastic structures. The formulation is written in the form of a multi-objective topology optimization problem that considers two competing objective functions, one related to mechanical performance (mean compliance) and one related to thermal performance (either thermal compliance or temperature variance). To solve the optimization problem, we present an efficient design variable update scheme, which we have derived in the context of the Zhang–Paulino–Ramos (ZPR) update scheme by Zhang et al. (2018). The new update scheme has the ability to solve non-self-adjoint topology optimization problems with an arbitrary number of volume constraints, which can be imposed either to a subset of the candidate materials, or to sub-regions of the design domain, or to a combination of both. We present several examples that explore the ability of the formulation to obtain candidate Pareto fronts and to design support structures for additive manufacturing. Enabled by the ZPR update scheme, we are able to control the complexity and the length scale of the support structures by means of regional volume constraints.

Multi-objective topology optimization and structural analysis of periodic spaceframe structures

Reduction of structural weight provides significant benefits in many engineering applications. While methods to optimise structural shape and topology of both continuous solids and discrete frame structures have existed for a while, the advent of additive layer manufacturing processes has enabled more complex geometries to be feasible. In this paper, a periodic spaceframe structure is designed for minimum mass and maximum effective flexural and torsional rigidities. A method of parametrising the spaceframe through its constituent unit cells is proposed, and Genetic Algorithm (GA) multi-objective optimisation is used to optimise its topology, size and geometry as a generic structure. The superior performance of the topology optimised periodic spaceframe is highlighted in terms of structural rigidity, large deformation capability, buckling and vibrational modal analysis in compare to equivalent beam structures of identical weight and comparable domain. The results show that the proposed method can effectively generate lightweight substitute structures of great mechanical performance in many beam structures applications, such as: aircraft wing spars. The periodic spaceframe is applied into a conventional aircraft wing structure to demonstrate the possibilities of promoting weight saving in the design of civil aircraft wings.

Topology optimization design of hydraulic valve blocks for additive manufacturing

The hydraulic valve block is a core component of an integrated hydraulic system. In practical usage, it exhibits problems such as material waste, long manufacturing cycle, significant energy loss, and leakage. Based on the aforementioned existing problems, this study presents the design of the hydraulic system valve block based on the valve block design principle. The internal valve channel of the hydraulic valve block is optimized for additive manufacturing technology to avoid auxiliary drilling, solve the problem of potential liquid leakage, and shorten the manufacturing cycle. Thus, it is more suitable for the production of customized complex hydraulic valve blocks. The multiobjective topology optimization method is applied to the lightweight design of the hydraulic valve block to save resources and decrease energy consumption. The results indicate that when compared with the original model, the minimum reduction rate of pressure loss in each oil circuit orifice after optimization of the hydraulic valve block corresponds to 32.02%, the maximum corresponds to 71.38%; the maximum stress of the final design corresponds to 542.9 MPa, which satisfies the material strength requirement; and the mass is decreased by 68.9%. Thus, the lightweight design of the hydraulic valve block is realized.

A Multi-Objective Topology Optimization Methodology and its Application to Electromagnetic Actuator Designs

In this article, a multi-objective topology optimization (MOTO) methodology based on the hybridization of the Non-dominated Sorting Genetic Algorithm II (NSGAII) and Differential Evolutionary (DE) algorithm is proposed. The framework of the proposed hybrid multiobjective optimization (MOO) algorithm is elaborated, and its performances and advantages over existing standard MOO methods are evaluated and demonstrated by solving typical mathematical test functions. To validate the proposed hybrid MOTO methodology, it is applied to the topology optimization of an electromagnetic actuator. Both linear and nonlinear cases are investigated. The numerical results demonstrate that a set of novel topologies with improved multiple objectives is obtained.

Multi-Objective Topological Optimization of an Electric Truck Frame Based on Orthogonal Design and Analytic Hierarchy Process

The electric truck frame as a vital load-bearing component has aroused growing attentions due to its enormous potential in lightweight. However, few systematic studies have been performed on the multi-objective topological design of the frame attributable to its complexity on loading and conflicting objectives. This paper aims to develop a multi-objective topology optimization strategy of the electric truck frame based on the hybrid decision making method combining orthogonal test design (OTD) and analytic hierarchy process (AHP). The hybrid strategy is performed to obtain a new set of weight ratio combination from objective data and subjective judgment. The topological results show that the overall performance of the optimal frame is better than any of the methods applied alone. By comparing, it is found that the strength and stiffness of the optimal frame is higher than that of the original frame from the perspective of static conditions, and the low-order natural frequency of the optimal frame is significantly improved. It demonstrates that the proposed approach could be as an effective tool for multi-objective topology optimization of the electric truck frame in seeking lightweight and comprehensive mechanical performance. The hybrid strategy might be expected to provide some guidance for more complicated engineering problems.

Multiobjective Topology Optimization for a Multi-layered Morphing Flap Considering Multiple Flight Conditions

Multiobjective Topology Optimization for a Multi-layered Morphing Flap Considering Multiple Flight Conditions

Multi-objective topology optimisation design of lattice structures with negative Poisson's ratio considering energy absorption and load-bearing characteristics

To make lattice structures have both load-bearing and energy absorption characteristics to protect the safety of personnel and equipment in the collision, this paper proposes a multi-objective topology optimisation method for the design of lattice structures with negative Poisson's ratio. The energy absorption and load-bearing characteristics of the lattice structure are characterised by negative Poisson's ratio and stiffness, respectively. A topology optimisation model is established to maximise the stiffness and negative Poisson's ratio of the lattice structure. The design optimisation of microscopic material is conducted by the energy homogenisation method. A modified optimality criteria method is employed to update design variables. The energy absorption and load-bearing characteristics of the optimised structure are tested and analysed by finite element simulation and compression experiment, respectively. The results show that the optimised lattice structure has both energy absorption and load-bearing characteristics. In general, the proposed method can provide a feasible reference for the topology optimisation design of anti-collision structures.