Multiscale Topology Optimization Research Articles

Numerical and Experimental Evaluation of Structured Material for Use in Multiscale Topology Optimization

Multiscale topology optimization is a powerful tool for engineers seeking a design with minimum weight and maximum stiffness, using a structured material in the form of a lattice structure. Furthermore, the current trend is to combine multiple lattice topologies in one component to achieve the best possible response to local loading conditions while minimizing weight. Therefore, herein, a numerical and experimental evaluation by compression tests in two directions is performed for six basic lattice topologies and two hypotheses are tested. The first hypothesis states that an additional weight saving of more than 30% can be achieved by a better choice of lattice topology. The second hypothesis is based on the manufacturing limitations of the laser powder bed fusion technology and the assumption that a favorable loading direction parallel to the building direction exists. The first hypothesis is only confirmed for loading in the direction parallel to the building direction and the second only for two lattice topologies. When both hypotheses are combined, the additional weight reduction of the multiscale topology optimization result is 44.5% according to the numerical results and 32.7% according to the experimental verification.

An efficient multiscale topology optimization method for frequency response minimization of cellular composites

An efficient multiscale topology optimization method for frequency response minimization of cellular composites

Multiscale topology optimization of an electromechanical dynamic energy harvester made of non-piezoelectric material

Multiscale topology optimization of an electromechanical dynamic energy harvester made of non-piezoelectric material

Multiscale topology optimization of structures by using isogeometrical level set approach

This study aims to optimize topology of structures at macro and micro scales, simultaneously, by using a level set method in an isogeometric analysis (IGA) framework. To achieve this, equilibrium and homogenization equations in the model are solved by IGA method. The level set functions are defined over a grid in parameter space of associating b-splines of the IGA model. Therefore, control net of the model and level set grid are separated and there is no need to refine the control net for having smooth boundaries. Sensitivity analyses for both scales are performed to calculate the velocity of boundary points and the level set functions are updated by solving reaction-diffusion equations. Finally, several 2D and 3D examples with different geometry and boundary conditions are provided to show performance and efficiency of the method. Obtained results show good agreement with examples in literature in terms of both topology and final value of objective function. Also, by using IGA level set method, smooth boundaries are achieved in the final topology of micro and macro structures.

Multiscale topology optimization of cellular structures with high thermal conductivity and large convective surface area

Cellular structures have excellent thermal performance due to their high specific surface areas and porosities. This paper proposes a multiscale topology optimization (MTO) method of cellular structures with high thermal conductivity and large convective surface area, which enables them to have rapid cooling capability. In the MTO of a cellular structure, the triply periodic minimal surface (TPMS) lattices are adopted and Kriging metamodels are constructed to predict their effective thermal conductivity tensors and surface areas. The thermal compliance of the cellular structure is firstly minimized under the given volume constraint by topology optimization. Then, maximizing the convective surface area of the cellular structure is considered as the objective to when imposing the optimized thermal compliance as the constraint. After conducting topology optimization, the cellular structure with high thermal conductivity and large convective surface area is achieved. Additionally, in the above MTO, isogeometric analysis (IGA) is employed to improve the accuracy of numerical analysis. A numerical example is provided to verify the effectiveness of the proposed method. By both simulation and experiment, the high thermal conductivity, large convective surface area and rapid cooling capability of the optimized cellular structure are verified.

Concurrent topology optimization of multiscale piezoelectric actuators

As an important component of smart structural devices, piezoelectric composites have strong macroscopic electromechanical coupling properties which often depend on their microstructural geometries and material parameters. Therefore, optimizing the microstructural design is crucial for achieving the desired macroscale effective properties. To this end, this paper conducts a concurrent multiscale topology optimization (TO) to optimize the material distribution of piezoelectric actuators, aiming to maximize the electrical and mechanical energy transmission to meet specific engineering requirements. Firstly, an energy method is used to homogenize the microstructures and obtain the effective parameters of the piezoelectric material. Then, the piezoelectric material with penalization and polarization (PEMAP-P) approach in conjunction with the adjoint method is employed to calculate the sensitivity of objective and constraint functions. The sensitivities at both the macroscale and microscale are obtained by considering the interscale coupling effects. Finally, the concurrent bi-level iteration based on the optimality criteria (OC) or the method of moving asymptotes (MMA) is conducted. Through this research, we have illuminated the relations between the optimized performance of the actuator and the material volume fractions at each scale. We have also compared in detail the differences between two-scale and single-scale designs, as well as the differences between the simplified half-symmetric geometric model and the full geometric model. We find that a larger volume fraction of material either at the macroscale or microscale generates a larger transmitted displacement magnitude under the same applied electric field, and for a given total material, the transmitted maximum increases with increasing microstructural volume fraction.

Lightweighting structures using an explicit microarchitectured material framework

In this paper, a new approach to design ultralight structures is developed based on a previous work called Efficient Multiscale Topology Optimization. A parameterized (or explicit) truss-based cell is introduced to generate intrinsically well-connected microstructures and to get clear interpretable optimal multiscale structures. The method uses a pre-computed database of optimal micro-cells to be computational efficient without losing in structural performances. The parameterization allows to generate a lightweight database just storing the set of parameters, that define the optimal cells, and the cells properties, that are obtained through inverse homogenization. The method has been successfully tested on two-dimensional compliance problems. Several examples demonstrate its versatility and give quantitative results. Moreover, it allows to obtain structures compatible with additive manufacturing processes, to naturally solve concurrent multi-scale problems, as well as controlled porosity and optimal fiber orientation problems.

Multiscale Topology Optimization for Lightweight Porous Linear Actuation Micro Mechanisms with Enhanced Heat Dissipation Under Laser Activation

Multiscale Topology Optimization for Lightweight Porous Linear Actuation Micro Mechanisms with Enhanced Heat Dissipation Under Laser Activation

Multi-scale topology optimization of structures with multi-material microstructures using stiffness and mass design criteria

Nowadays, there is a great interest on the part of the automotive and aerospace industry to design environmentally-friendly structures. To that purpose, stiffness-oriented designs are proposed here by extending previous work on multi-scale topology optimization to the multi-material setting reformulating the problem to include appropriately mass constraints and discussing different design domain parametrizations and algorithmic strategies.On the macroscale, the problem of minimizing the compliance subject to a global mass constraint is addressed. On the microstructure scale, the multi-material design is carried out by solving the problem of minimizing the local complementary strain energy density with mass density constraint. As a result, very efficient structures composed of spatially varying porous and multi-material microstructures are obtained. The optimal design of the multi-material microstructure can be done either in a pointwise manner or in larger subdomains, to promote design uniformity. These parametrizations are here compared and discussed. Moreover, two different algorithmic strategies to solve the multi-scale problem are proposed, and their pros and cons discussed. They differ in the way the macro and micro design variables are related and updated. The macro design variables consider the mass density distribution along the structure, while the micro variables define the microstructure's topology using a multi-material SIMP interpolation scheme.The results show very efficient structures with locally optimized multi-material microstructures, which can outperform their single-material counterparts with regards to stiffness while maintaining the same mass. Additionally, the maximum stress verified on multi-material structures tends to be lower than the one obtained in the single-material counterparts.

Multiscale topology optimization of gradient lattice structure based on volume parametric modeling

In this study, a volume parametric modeling method of lattice structure is proposed, and an efficient multiscale topology optimization framework is realized based on isogeometric analysis (IGA) to construct the gradient lattice structure. The skeleton model is constructed, which can accurately describe the topology structure and improve data utilization and computational efficiency. Based on the skeleton model, a uniform volume parameter lattice structure with an arbitrary topology of multiple types of unit cells is constructed, which is suitable for IGA. Moreover, multiscale topology optimization based on IGA is realized to construct the gradient lattice structure. The same data model is used in modeling, analysis, and optimization, which can accurately represent the geometric shape without discretization errors. At the same time, the multiscale topology optimization iteration is realized by adjusting the density of control points. The optimized model can be directly analyzed and re-optimized, thus realizing the integrated design of lattice structure modeling, simulation, and optimization. The effectiveness and robustness of the algorithm are verified by several mechanical parts and freeform models. These examples show that the gradient lattice structure has higher strength and better stress distribution than the uniform lattice structure under the same boundary conditions.

Multiscale Topology Optimization of modulated fluid microchannels based on asymptotic homogenization

Dehomogenization techniques are becoming increasingly popular for enhancing lattice designs of compliant mechanical systems with ultra-large resolutions. Their effectiveness hinges on computing a deformed periodic grid that enable to reconstruct fine-scale designs with modulated and oriented patterns. In this paper, we propose an approach for extending dehomogenization methods to laminar fluid systems. We initiate our methodology by asymptotically deriving Darcy’s law on a periodically porous medium deformed by a diffeomorphism. Unlike the mechanical context, we reveal that the homogenized permeability matrix depends not solely on local the orientation but also on the local dilation of the deformed periodic medium. This distinction presents one of the several challenges to be tackled when adapting dehomogenization-based topology optimization techniques to porous media. To accommodate existing methodologies, we formulate a simplified “poor man’s” homogenized model, which streamlines various aspects, yet still leans on periodic cell problems to estimate the spatially varying permeability matrix. Specifically, we overlook boundary layer effects, we presume periodic grid deformations, and we neglect local dilation, solely considering the relationship with local cell orientations. Subsequently, we present a numerical approach for designing a system that redistributes an input flow across numerous regularly spaced outlets at an output interface. Leveraging the homogenized model, we deduce optimized geometric arrangements of local channel spacing parameters and orientations. We then use established methods to reconstruct grid deformations and fine-scale designs. The fidelity of these reconstructions is then validated through fine-scale simulations. Our observations indicate that while the proposed designs yield satisfactory performance when subjected to the full-scale model, discernible deviations from the homogenized model persist, appealing to future improvements.

Multiscale topology optimization framework for natural frequency maximization of multi-morphology lattice structures

This paper proposes a multiscale topology optimization framework for maximizing the natural frequency of multi-morphology lattice structures (MMLSs). The proposed framework addresses the challenges of computational efficiency, design space, numerical convergence, and compatibility between adjacent microstructures in multiscale topology optimization for natural frequency problems. The macrostructural topology, the morphologies categories, distribution regions, volume fractions of different lattice materials (LMs), and the relative densities of lattice unit cells (LUCs) are simultaneously optimized to enhance the structural natural frequency. Specifically, level set functions are utilized to generate prototype LUCs, enabling obtaining graded LUCs by configuration interpolation. Multi-morphology LUCs with smooth characteristics are achieved using the Kriging-assisted morphological post-process and sigmoid function (SF) based hybrid transition strategy. Kriging metamodel-assisted Uniform Multiphase Materials Interpolation (KUMMI) schemes are constructed for the mechanical properties estimation of macro elements to promote the multiscale topology optimization procedure. Distinct processing methods for LUCs' elasticity tensor and density are incorporated within the KUMMI schemes, by which the local mode problem is avoided. The objective order natural frequency is accurately optimized with the modal assurance criterion (MAC) based mode-tracking technique. Numerical examples demonstrate that the developed design framework can efficiently maximize the natural frequency of MMLSs, while also ensuring microstructural connectivity.

Exploring the influence of initial design domain dependencies in concurrent multiscale topology optimization for heat conductivity maximization

Advancements in additive manufacturing have led to the widespread adoption of topology and shape optimization in the design process across various industrial applications. Topology optimization has emerged as a promising approach for creating ultra-lightweight structures with exceptional functionality. By designing porous structures and addressing the layout of pores, concurrent multiscale topology optimization schemes offer a means to achieve such outcomes. This paper focuses on the development of a multiscale topology optimization tool and presents a numerical investigation of the initial design domain's impact on maximizing heat conductivity through concurrent multiscale topology optimization. The study highlights the significant influence of the initial design domain in the microscale on the final microscale design. Moreover, modifying the initial micro design domain has a direct impact on the macro design domain. The findings underscore the importance of considering the initial design domain when aiming to maximize heat conductivity through concurrent multiscale topology optimization. To facilitate the replication of the results presented in this paper, comprehensive details regarding parameter settings and implementation aspects are provided.

Hygrally activated displacement inverter using a multiphysics multiscale topology optimization with considering evaporation

Hygrally activated displacement inverter using a multiphysics multiscale topology optimization with considering evaporation

Concurrent multiscale topology optimization of hollow structures considering geometrical nonlinearity

Concurrent multiscale topology optimization of hollow structures considering geometrical nonlinearity

Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB

In multi-scale topology optimization methods, the analysis encompasses two distinct scales: the macro-scale and the micro-scale. The macro-scale refers to the overall size and dimensions of the structural domain being studied, while the micro-scale pertains to the periodic unit cell that constitutes the macro-scale. This unit cell represents the entire structure or component targeted for optimization. The primary objective of this research is to present a simplified MATLAB code that addresses the multi-scale concurrent topology optimization challenge. This involves simultaneously optimizing both the macro-scale and micro-scale aspects, taking into account their interactions and interdependencies. To achieve this goal, the proposed approach leverages the Bi-directional Evolutionary Structural Optimization (BESO) method. The formulation introduced in this study accommodates both cellular and composite materials, dealing with both separate volume constraints and the utilization of a single volume constraint. By offering this simplified formulation and harnessing the capabilities of the multi-scale approach, the research aims to provide valuable insights into the concurrent optimization of macro- and micro-scales. This advancement contributes to the field of topology optimization and enhances its applications across various engineering disciplines.

Dynamic response-oriented multiscale topology optimization for geometrically asymmetric sandwich structures with graded cellular cores

Compared with conventional symmetric sandwich structures with two identical face-sheets and uniform core, geometrically asymmetric sandwich structures (GASSs) enable better dynamic performance due to the expanded design space provided by two unidentical face-sheets and graded cellular cores (GCCs). This paper proposes a dynamic response-oriented multiscale topology optimization method for the GASSs, which is capable of designing the thicknesses of two solid face-sheets, graded distribution of GCCs and their topological configurations to minimize dynamic compliance. Specifically, at macroscale, a variable thickness sheet method is employed to optimize the thicknesses of two solid face-sheets and then generate an overall free distribution of cellular cores within sandwich layers. At microscale, a parametric level set method combining with a numerical homogenization approach is adopted to progressively optimize multiple representative cellular cores (RCCs), achieving their similar topological configurations. Benefitting from the level set-based topology description of RCCs, a shape interpolation technology is conveniently applied to interpolate the shapes of these RCCs to generate the configurations of GCCs. Moreover, a Kriging metamodel constructed by some cellular cores as sample points is adopted to predict the effective properties of each cellular core within sandwich layers, significantly reducing the computational burden. Several 2D and 3D numerical examples are illustrated to show the validity of the proposed method. The results indicate that the optimized GASSs show superior dynamic performances over the conventional sandwich structures designed by microscale and macroscale topology optimization, as well as filled with the commonly applied lattice and honeycomb cores.

On multiphysics concurrent multiscale topology optimization for designing porous heat-activated compliant mechanism under convection for additive manufacture

In this research, we investigated the use of concurrent multiscale topology optimization to design additively manufacturable lightweight porous compliant mechanisms that enable harnessing heat energy as the energy source to perform mechanical actions. By establishing two independent dimensionless representations of the design problem, i.e., macro and microscale domains, a concurrent multiscale topology optimization framework is implemented, and the effective properties of the microscale (i.e., elastic and heat conductivity tensors, as well as the thermal expansion coefficient) are calculated and used as the thermo-elastic modeling effective properties of the macroscale. For heat transfer physics, heat transport in solids, as well as heat convection, are simultaneously considered in this study. Sensitivity analysis on the suggested concurrent optimization scheme was derived to address the macro and microstructure coupling as well as the thermo-elastic physics coupling. Several numerical cases are studied with single and multiple microstructure systems. To investigate the macrostructure dependency on the microstructure systems, a study was performed for multiple microstructure subsystems. Incorporating several microstructures into a single macro design domain increased design freedom and improved the performance-to-weight ratio. Furthermore, and for achieving good additive manufacturability we investigated the connectivity of the multi-microstructure optimization and implement boundary competitivity technique to attain fully connected design for attaining good additive manufacturability.

DL-MSTO+: A deep learning-based multi-scale topology optimization framework via positive definiteness ensured material representation network

A multi-scale approach to topology optimization has recently emerged due to its lightweight, robust, and multi-functional characteristics. Considering material diversity, an increasing number of materials leads to a computational burden due to numerous design variables and homogenization equations. This study proposes DL-MSTO+, a deep learning-based multi-scale topology optimization framework. The framework consists of two distinct deep neural networks for multi-scale topology optimization problems to reduce the dimensionality of design variables and predict homogenized material properties. First, a generator network learns the low-dimensional representation of a material microstructure in an unsupervised manner. Thus, the trained generator produces a micro-structural image from a low-dimensional vector. Second, a predictor network is trained in a supervised manner to predict the homogenized elasticity matrix for a given micro-structural image. Additionally, the predictor is specifically designed to ensure the positive definiteness of the predicted elasticity matrix. Lastly, the networks are integrated into the algorithm to improve the efficiency of multi-scale topology optimization. The numerical experiments demonstrate higher efficiency for the proposed algorithm than the conventional multi-scale approach. Moreover, the proposed method provides connectable multi-scale designs, and the low-dimensional latent representation enables semantic interpolation between solutions.

Complementary energy based meso-level homogenization for multiscale topology optimization

Complementary energy based meso-level homogenization for multiscale topology optimization