Microstructural Topology Optimization Research Articles

Topology optimization of orthotropic multi-material microstructures with maximum thermal conductivity based on element-free Galerkin method

A topology optimization model for periodic heat transfer microstructure with orthotropic multi-material is established by using the element-free Galerkin method (EFGM) and alternating active-phase algorithm (AAPA). The maximum thermal conductivity is chosen as the opposite number of objective function, and the 3-D printing of the microstructure and heat transfer analysis are performed to validate the proposed model. The effects of the number of multi-material categories, the initial distributions of multi-material, the thermal conductivity factor, and the multi-material off-angle on the topological configuration with maximum thermal conductivity are studied. The results show that the multi-material topological configuration is comprised the material with a higher thermal conductivity as the primary frame and surrounded by the material with the lower thermal conductivity. The initial distribution of multi-material has a little impact on the maximum thermal conductivity. The alteration of the thermal conductivity factor will contribute to the accumulating of materials in the direction of the higher thermal conductivity, and the material stacking direction is more inclined to the direction of multi-material off-angle. The reasonable values of multi-material thermal conductivity factor is suggested to be 2–4 and multi-material off-angle is suggested to be 0°–30° and 60°–90°.

Multi-objective topology optimization of porous microstructure in die-bonding layer of a semiconductor

Multi-objective topology optimization of porous microstructure in die-bonding layer of a semiconductor

Topology optimization of controllable porous microstructure with maximum thermal conductivity

Porous structures are lightweight and thus possess tailorable thermophysical properties through topological design. A multi-constraint topology optimization scheme is developed for designing 2D periodic lattice material with controllable porosity and optimal thermal conductivity. The porosity, pore size and specific surface area are modulated by imposing local density constraints, and thus the structure-determined thermal conductivity can be systematically investigated. The isotropic porous structures after optimization show effective thermal conductivity close to the Hashin-Shtrikman theoretical bound. The specific surface area of base cell is enlarged with imposing local density constraint. In addition, microstructure with anisotropic thermal conductivity can be also obtained. The samples of lattice structure are additively manufactured via selective laser melting, and the thermal conductivities are experimentally validated with deviation within 10 %. The proposed porous structures with targeted porosity have potential application in skeleton embedded with phase change materials.

Topology optimization of differentiable microstructures

Recent years have seen a growing interest in topology optimization of functionally graded microstructures, characterized by an array of microstructures with varying volume fractions. However, microstructures optimized at slightly different volume fractions do not necessarily connect well when placed adjacently. Furthermore, optimization is commonly performed on a finite set of volume fractions, limiting the number of microstructure configurations.In this paper, we introduce the concept of differentiable microstructures, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. To construct such microstructures, we propose a novel formulation for topology optimization. In this approach, a series of 2-dimensional microstructures is represented using a height field, and the objective is to maximize the bulk modulus of the entire series. Through this optimization process, an initial microstructure with a small volume fraction undergoes non-uniform transformations, generating a series of microstructures with progressively increasing volume fractions. Notably, when compared to traditional uniform morphing methods, our proposed optimization approach yields a series of microstructures with bulk moduli that closely approach the theoretical limit.

Multiscale topological design of coated structures with layer-wise bi-material lattice infill for minimum dynamic compliance

In this paper, a novel multiscale topological design strategy for coated structures infilled with layer-wise bi-material lattice microstructures is presented, where the objective is to minimize the dynamic compliance of the macrostructure under time-harmonic excitation frequency. The optimization of bi-material microstructures is concurrently performed with the topological design of the macrostructure to fully consider the interaction of structural evolution between the two scales. At the macroscale, the material distribution for the macrostructure is optimized using the velocity field-based level set method, which inherits the implicit geometrical representation, and smooth structural boundaries can be obtained. Due to the signed distance property, the thickness of the outer coating is able to be well-maintained during the iteration. As for the microscale, the density method is chosen to conduct topology optimization of lattice microstructures. The lattice infill for coated structures is designed to be layer-wise by dividing it uniformly (or near-uniformly) into different layers, and each layer is composed of periodic and uniform bi-material microstructures, which can enlarge the design space and generate various microscale structural patterns to better resist the vibration response. The sensitivity analysis for the dynamic compliance with respect to the design variables at different scales is carefully performed and the influences of the number of divided layers, excitation frequency, coating thickness, damping coefficients, element meshes, static constraint, and division direction for the lattice infill on the optimized macroscale and microscale structures are studied by several numerical examples. Moreover, a body-fitted mesh for a Computer-Aided Design (CAD) model is constructed and the finite element analysis results further verify the effectiveness of the proposed approach.

Topology optimization of microstructures with perturbation analysis and penalty methods

Topology optimization at the continuum nano/microscale is of wide interest in designing and developing more efficient micro/nano electromechanical systems. This paper presents a new methodology for topology optimization of microstructures that is based on perturbation analysis and the penalty methods. The homogenized material coefficients are numerically computed based on perturbation analysis, and periodic boundary conditions are imposed by the penalty methods. The sensitivity analysis is implemented directly without the adjoint method. The extension of the proposed method to the design of components for multi-field analysis is straightforward. The capability and performance of the presented methodology are demonstrated through several numerical examples.

Topology optimization of connectable microstructure using enlarged connective domain

Topology optimization of connectable microstructure using enlarged connective domain

Topology optimization of periodic microstructures for prescribed relaxation moduli of viscoelastic composites

This paper presents the microstructural topology optimization for the prescribed relaxation moduli of viscoelastic composites. The microstructures of the viscoelastic composites are assumed to be composed of periodic unit cells. A numerical homogenization method is systematically developed to obtain the effective relaxation modulus of the unit cell. The density-based topology optimization is adopted to find optimal microstructures of the viscoelastic composites. The design objective is to minimize the difference between the prescribed relaxation moduli and effective relaxation moduli at specified time instants. An adjoint sensitivity analysis is developed to compute the derivatives of the effective relaxation moduli with respect to the design variable. Several numerical examples demonstrate the effectiveness of the proposed approach. Various optimized microstructures of viscoelastic composites are presented and discussed based on the volume fraction of each constituent material, relaxation time, and stiffness.

Multi-material and strength-oriented microstructural topology optimization applied to discrete phase and functionally graded materials

Multi-material and strength-oriented microstructural topology optimization applied to discrete phase and functionally graded materials

Bézier extraction based isogeometric approach to multi‐objective topology optimization of periodic microstructures

AbstractThis article presents the Bézier extraction based isogeometric approach to multi‐objective topology optimization of periodic microstructures. The approach utilizes the B‐splines based Bézier elements as the finite element representation at both macro and micro levels. The equivalent elastic properties of the representative volume element (RVE) are computed by the numerical homogenization method with the periodic boundary conditions on the control points (CPs) of the B‐spline mesh. The multi‐objective optimization problems including the material bulk modulus maximization, negative Poisson's ratio, and concurrent topology optimization of the composite structures are formulated by using the Bézier elements based isogeometric analysis. The distance‐based density distribution on the CPs of the B‐spline mesh is proposed as the initialization for designing the RVE, which is more robust than the uniform density distribution towards the optimal results. Several numerical examples are presented to illustrate the effectiveness of the proposed approach, and a variety of isotropic and anisotropic RVEs and composite structures are obtained. Meanwhile, various influences on the optimal design of the RVE and macrostructure are also discussed.

A well connected, locally-oriented and efficient multi-scale topology optimization (EMTO) strategy

Multi-scale topology optimization (a.k.a. micro-structural topology optimization, MTO) consists in optimizing macro-scale and micro-scale topology simultaneously. MTO could improve structural performance of products significantly. However, a few issues related to connectivity between micro-structures and high computational cost have to be addressed, without resulting in loss of performance. In this paper, a new efficient multi-scale topology optimization (EMTO) framework has been developed for this purpose. Connectivity is addressed through adaptive transmission zones which limit loss of performance. A pre-computed database of micro-structures is used to speed up the computing. Design variables have also been chosen carefully and include the orientation of the micro-structures to enhance performance. EMTO has been successfully tested on two-dimensional compliance optimization problems. The results show significant improvements compared to mono-scale methods (compliance value lower by up to 20% on a simplistic case or 4% on a more realistic case), and also demonstrate the versatility of EMTO.

Topology optimization of metamaterial microstructure for wireless power transfer with high power transmission efficiency

An efficient short-range magnetic coupling wireless power transferring (WPT) system by embedding metamaterials (MTM) resonator between the source coil and load coil, whose configuration was designed based on topology optimization, was proposed in this paper. The topology optimization model for designing the MTM microstructures to achieve WPT with maximum power transfer efficiency (PTE) at the required frequency was established. The different MTM microstructures were realized by different conductor patch combination in the discretized pixelated design domain. As a designing example, a short-range magnetic coupling MTM-based WPT system with high efficiency at specific frequency 1 GHz was designed. Taking two classical MTM microstructures as reference, the PTE of WPT system with the topology optimized MTM was improved considerably. The variation of the transmission characteristics of the designed WPT system caused by misalignment was analyzed. The comparison between the MTM-based WPT system and the typical 4-coil WPT system was also discussed.

Microstructural topology optimization for patch-based sandwich panel with desired in-plane thermal expansion and structural stiffness

Apart from the lightweight and excellent mechanical properties, sandwich panels can be endowed with tailorable in-plane coefficient of thermal expansion (CTE) through an elaborate design of periodic face-sheets. However, albeit that the microstructural topology of their periodic face-sheets promises unique thermal expansion behaviors, it may also bring significant influences to the structural stiffness of sandwich panels. In this study, we apply the topology optimization method to design face-sheet microstructures to enable the sandwich panels to possess desired in-plane CTEs, lightweight and benign mechanical properties, simultaneously. By introducing the patch-based cell as initial configuration, the existing thermally bending adjustment mechanism for thermal deformation control is integrated to the process of topology optimization. The entire topology optimization process including the equivalent mechanical properties prediction and the sensitivity computation is performed within an in-house program coupled with commercial finite element analysis software. To this end, a matching numerical sensitivity analysis method to extract sensitivities straightforwardly from software’s output is also developed on the basis of asymptotic homogenization method. Three types of specific optimization problems in terms of different objective functions and constraint conditions are proposed, solved, and studied, namely, in-plane zero thermal expansion combining with maximum stiffness, the other for in-plane zero thermal expansion optimal specific stiffness, and minimizing in-plane isotropic thermal expansion. Some specific resulting topologies, microstructural features, and design details are subsequently obtained. In particular, the current strategy of integrating effective mechanism and topological technology can be extended to design more microstructures for simultaneously tailorable CTE and high mechanical performance by replacing present thermal deformation control mechanism with others.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

Stress-based and robust topology optimization for thermoelastic multi-material periodic microstructures

To take advantage of multi-material additive manufacturing technology using mixtures of metal alloys, a topology optimization framework is developed to synthesize high-strength spatially periodic metamaterials possessing unique thermoelastic properties. A thermal and mechanical stress analysis formulation based on homogenization theory is developed and is used in a regional scaled aggregation stress constraint method. Since specific load cases are not always known beforehand, a method of worst-case stress minimization is also included to efficiently address load uncertainty. It is shown that the two stress-based techniques lead to thermal expansion properties that are highly sensitive to small changes in material distribution and composition. To resolve this issue, a uniform manufacturing uncertainty method is utilized which considers variations in both geometry and material mixture. Test cases of high stiffness, zero thermal expansion, and negative thermal expansion microstructures are generated, and the stress-based and manufacturing uncertainty methods are applied to demonstrate how the techniques alter the optimized designs. Large reductions in stress are achieved while maintaining robust strength and thermal expansion properties.

Microstructural topology optimization of periodic beam structures based on the relaxed Saint-Venant solution

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

Topology optimization of micro structure of porous material to maximize dissipated energy by using homogenization method

Topology optimization of micro structure of porous material to maximize dissipated energy by using homogenization method

Topological design of microstructures using periodic material-field series-expansion and gradient-free optimization algorithm

Light-weight cellular materials with periodic repetitive microstructures are widely used in various fields due to their superior mechanical/multi-physical performances. As the microstructural design problem is known to have multiple local minima, most gradient-based topology optimization methods significantly depend on the initial guess of the microstructural geometry, thus requiring the designer’s experiences. This paper presents an effective gradient-free framework for periodic microstructure design, which exhibits powerful global searching capabilities and requires no sensitivity information. The proposed framework combines the material-field series-expansion (MFSE) topology representation of periodic microstructures and the sequential Kriging-based optimization algorithm. The MFSE method decouples the topological representation from the finite element discretization, and describes a relatively complex microstructural topology with high-quality boundary description using a greatly reduced number of design variables. Based on the Kriging surrogate model, a solution scheme is suggested to solve material microstructural topology optimization successively in a sequence of sub-optimization problems with self-adaptive design spaces. With the present gradient-free optimization method, high-performance cellular materials that approach the H-S upper bound with porosities from 0.2 to 0.6, or that achieve negative Poisson’s ratios of -0.94 in the principal directions for materials with square symmetry, are obtained without prior knowledge of the optimum microstructural topology.

Microstructural Topology Optimization of Constrained Layer Damping on Plates for Maximum Modal Loss Factor of Macrostructures

This paper presents microstructural topology optimization of viscoelastic materials for the plates with constrained layer damping (CLD) treatments. The design objective is to maximize modal loss factor of macrostructures, which is obtained by using the Modal Strain Energy (MSE) method. The microstructure of the viscoelastic damping layer is composed of 3D periodic unit cells. The effective elastic properties of the unit cell are obtained through the strain energy-based method. The density-based topology optimization is adopted to find optimal microstructures of viscoelastic materials. The design sensitivities of modal loss factor with respect to the design variables are analyzed and the design variables are updated by Method of Moving Asymptotes (MMA). Numerical examples are given to demonstrate the validity of the proposed optimization method. The effectiveness of the optimal design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to the plates with CLD treatments.

Combination of BESO and harmony search for topology optimization of microstructures for materials

This paper introduces a metaheuristic algorithm for topology optimization of microstructures for cellular materials. It combines the harmony search (HS) algorithm with bi-directional evolutionary structural optimization (BESO) features. It is assumed that the cellular material comprises periodic base cells. Bulk and shear moduli are selected as the objectives of topology optimization with a volume constraint on the material solid phase. Each element in the base cell ground structure is treated as a discrete design variable and sensitivity numbers are calculated using homogenization theory. Through an iterative process, new sets of design variables are selected and the volume constraint is imposed using the BESO principals. Algorithm parameters such as filter radius, harmony memory considering rate (HMCR) and pitch adjusting rate (PAR) are selected experimentally. Numerical examples show the capability of the algorithm in designing microstructures of 2D and 3D materials with a high convergence speed. The proposed algorithm can also be easily extended to design of materials with other functional properties.