Multiscale Topology Optimization Research Articles

A Two-Player Game for Multi-Scale Topology Optimization of Static and Dynamic Compliances of Triply Periodic Minimal Surface-Based Lattice Structures

A Two-Player Game for Multi-Scale Topology Optimization of Static and Dynamic Compliances of Triply Periodic Minimal Surface-Based Lattice Structures

Multi-scale topology optimization of periodic structures with orthotropic multiple materials using the element-free Galerkin method

ABSTRACT A multi-scale topology optimization model is proposed for orthotropic multi-material periodic structures using the element-free Galerkin method. The macro multi-material distribution is optimized with the alternating active-phase algorithm, and the effective elastic tensor of multi-type microstructures is determined by energy-based homogenization. The feasibility of the model is verified and the effects of the quantities of material types and design subdomains, the Poisson's ratio factor and the multi-material off-angle on topological configuration and compliance are studied. The results indicate that the quantity of material types affects the mechanical properties of the periodic structure, and different microstructure configurations can be obtained under different numbers of design subdomains. The increase in the Poisson's ratio factor and decrease in the off-angle can enhance the effective elastic properties of the microstructure and reduce the compliance. Reasonable values of the Poisson's ratio factor and multi-material off-angle are suggested to be 2–4 and 0–45°.

Multiscale topology optimization of anisotropic multilayer periodic structures based on the isogeometric analysis method

Multiscale topology optimization of anisotropic multilayer periodic structures based on the isogeometric analysis method

Multi-Scale Topology Optimization Using Multi-Class Unit Structures

Multi-Scale Topology Optimization Using Multi-Class Unit Structures

Generating a Multi-Material Lattice Structure through a Modified Relative Density Mapping Algorithm

Multi-material lattice structures demonstrate superior mechanical advantages and lightweight potential, enabling it preferable in many engineering fields, especially with the rise of additive manufacturing. However, employing multi-scale topology optimization inherently involves computational complexities of designing the macromaterial distribution and microlattice geometric size simultaneously. In this paper, a modified relative density mapping (MRDM) method is proposed, which generates a multi-material lattice structure by projecting the continuum multi-material topology optimization results. The proposed method extends the standard relative density mapping (RDM) method to multi-material problem and improves the performance. To achieve such purpose, a new mapping relation considering multi-material density and stress information is introduced to generate mapping weight of materials, which is the most novel contribution. Second, to avoid intersection of different materials, two types of material mapping strategies, namely continuous mapping strategy, and discrete mapping strategy, are introduced to generate multi-material lattice structure using the mapping weight of materials. Several numerical examples are exhibited, which demonstrate that the proposed method is capable of generating a multi-material lattice structure with clear material interface and good performance.

Non-probabilistic reliability-based multi-scale topology optimization of thermo-mechanical continuum structures with stress constraints

A reliability-based non-probabilistic multiscale topology optimization (NRBMTO) method with stress constraints is proposed for thermo-mechanical continuous structures with uncontrollable stresses. The physical parameters, external loads and temperature values at the macro scale, are regarded as non-probabilistic uncertain parameters in the optimization of structural topologies with complex physical fields at multi-scale. The homogenization-based finite element method is employed to quantify thermo-mechanical structures with multi-scale uncertain parameters in the established multi-scale topology model. The ellipsoid model is applied to describe the uncertainty of non-probabilistic random variables, and the non-probabilistic reliability index is obtained by estimating the failure probability based on the first-order reliability method (FORM). The unit stresses are aggregated to the global maximum stresses with the normalized p-norm function, taking into account the mechanical and thermal stresses. The sensitivity information of the compliance and stress constraint to the macro- and micro-design variables and uncertain variables are derived simultaneously. The macro- and micro- design variables are solved by the method of moving asymptotes (MMA), respectively. Several numerical examples are given to verify the effectiveness and feasibility of the proposed NRBMTO method. The results demonstrate that the optimized structure based on the NRBMTO method provides better security with reliability index β=3 and minimum compliance (244.39) while stress is controlled below 235 MPa compared to the classical deterministic multiscale topology optimization (DMTO) method.

Multiscale Topology Optimization Design and Additive Manufacturing of Thermal Expander Metadevices

Thermal expander metadevices can yield a large uniform temperature field powered by a linear heat source. Previous design of thermal expander metadevices can be regarded as a combination of achieved thermal cloaks and their background materials. However, these thermal-cloak-inspired expander metadevices have an inherent flaw, i.e., their thermal functionality will be lost when the background material is changed, thus limiting their practical applications. To solve this problem, the multiscale topology optimization (MTO) method is employed to design thermal expander metadevices that can maintain their expander functionality under different background materials. In MTO, transformation thermotic technology is used to determine the anisotropic thermal conductivities inside a thermal expander metadevice and topology optimization is performed to generate the topological configuration of each microstructure with the target effective thermal conductivity. Subsequently, the thermal functionalities of thermal double and triple expander metadevices with different background materials are numerically verified via simulations. Finally, the thermal double expander metadevice is fabricated via additive manufacturing and experimentally tested for its thermal functionality. The findings of this study address the challenge of designing thermal expander metadevices with background material-independent functionality.

Concurrent multiscale topology optimization for size-dependent structures incorporating multiple-phase materials

The significant progress in additive manufacturing has resulted in the development of well-engineered microstructures with innovative topological configurations, showcasing exceptional performance across diverse fields. Topology optimization is widely recognized as an advanced design tool that provides engineers with new insights for creating lightweight structures. As integration devices continue to trend towards miniaturization, the size effect in small-scale structures remains a persistent challenge. The absence of characteristic parameters to describe this size effect hinders proper explanation through traditional multiscale topology optimization based on classical mechanics. Therefore, we propose a size-dependent topology optimization framework for concurrent multiscale design involving multiple-phase materials. The modified couple stress theory is proposed to account for the size effect in small-scale structures. The numerical findings suggest that the size effect is significantly influenced by the ratio of elastic modulus, volume fraction, and width-to-length ratio of the representative volume element as its characteristic size approaches the length of the material. These results highlight the importance of considering size-dependent effects in topology optimization for multiscale design and provide valuable insights for engineers and researchers in this field.

Multiscale topology optimization for the design of spatially-varying three-dimensional lattice structure

This paper introduces three dimensional (3D) topology optimization specifically tailored for designing spatially-varying primitive-cubic (CP) type lattice structures. The developed design process consists of three steps: pre-processing, main processing, and post-processing. In the pre-processing step, a surrogate model between lattice geometry variables and effective elasticity tensor is constructed using an artificial neural network. This surrogate model is essential because it enables the quick calculation of the effective elasticity tensor of lattice microstructure. In the subsequent main processing step, multiscale topology optimization is performed. During this step, lattice microstructure size and rotation design variables are optimized together with the macrostructure density design variables. Macrostructures are designed using a well-established three-field density scheme with a SIMP formulation. Formulations for spatially-varying 3D lattice microstructure are newly developed based on a unit quaternion rotation representation scheme. In the final post-processing step, a spatially-varying microstructure is restored using a de-homogenization scheme, newly developed particularly for the primitive-cubic (CP) type 3D lattice structures. The effectiveness and robustness of the proposed formulations are validated through three design examples for the compliance minimization problem. Moreover, a designed lattice structure is fabricated using an additive manufacturing machine.

Stress-constrained optimization of multiscale structures with parameterized microarchitectures using machine learning

A multiscale topology optimization framework for stress-constrained design is presented. Spatially varying microstructures are distributed in the macroscale where their material properties are estimated using a neural network surrogate model for homogenized constitutive relations. Meanwhile, the local stress state of each microstructure is evaluated with another neural network trained to emulate second-order homogenization. This combination of two surrogate models — one for effective properties, one for local stress evaluation — is shown to accurately and efficiently predict relevant stress values in structures with spatially varying microstructures. An augmented lagrangian approach to stress-constrained optimization is then implemented to minimize the volume of multiscale structures subjected to stress constraints in each microstructure. Several examples show that the approach can produce designs with varied microarchitectures that respect local stress constraints. As expected, the distributed microstructures cannot surpass density-based topology optimization designs in canonical volume minimization problems. Despite this, the stress-constrained design of hierarchical structures remains an important component in the development of multiphysics and multifunctional design. This work presents an effective approach to multiscale optimization where a machine learning approach to local analysis has increased the information exchange between micro- and macroscales.

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.