Multiscale Topology Optimization Method Research Articles

Multiscale topology optimization for designing locally resonant acoustic metamaterials with specified low-frequency bandgaps

Locally resonant acoustic metamaterials (LRAMs) have significant potential to isolate low-frequency vibrations and noise, whereas designing them with specified bandgaps is challenging. Compared to the monoscale method, the multiscale method can design microstructures that form equivalent materials to act as scatterers, coatings, and matrices. This may aid in obtaining LRAMs that are sometimes difficult to achieve with the monoscale method. Therefore, a multiscale topology optimization method is presented to design LRAMs with bandgaps that meet specified frequency constraints. Since bandgap optimizations are complicated, many local minimum solutions may exist when using sensitivity-based methods. To mitigate this issue, a macro-parameter optimization stage using a genetic algorithm (GA) is added between the macro and micro optimizations in this top-down multiscale framework. As a result, the method comprises three optimization stages: macro-bandgap optimization, macro-parameter optimization, and micro-optimization. Since the macrostructures consist of multiple microstructures and the microstructures consist of multiple materials, a parameterized level-set-based multi-material topology method is adopted to obtain topologies on both scales. The effectiveness of the proposed method is demonstrated through numerical examples of two optimization problems. The first is to obtain LRAMs with the maximum bandgap width while ensuring their bandgaps encompass specified frequency ranges. The other is to obtain LRAMs with specified bandgaps. Successfully obtained LRAMs with specified bandgaps demonstrate the application prospects of this method in designing structures and devices to address issues related to vibrations and noise.

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.

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 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.

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.

Multiscale topology optimization of cellular structures using Nitsche-type isogeometric analysis

This paper proposes a multiscale topology optimization method using Nitsche-type isogeometric analysis (IGA) for design of structures described by multiple non-uniform rational B-spline (NURBS) patches. In this method, at macroscale, unknown structural responses are calculated by Nitsche-type IGA, which not only ensures the consistency between geometric and analytical models, but also offers an effective way to enforce kinematic compatibility and mechanical equilibrium on interfaces between adjacent NURBS patches. At microscale, mechanical properties of microstructures are predicted by a Kriging metamodel at a low computational cost, which is constructed via some sample lattice unit cells (LUCs). In addition, analytical computation of sensitivities for coupled elements is developed. Finally, the structures are infilled by graded LUCs according to the density distribution within the design space. Some 2D and 3D numerical examples with conforming and non-conforming NURBS patches are presented to test the proposed method. Results indicate that the proposed method is effective for design of structures described by multiple NURBS patches.

Multiscale design and experimental verification of Voronoi graded stochastic lattice structures for the natural frequency maximization problem

Additive manufacturing (AM) has gained popularity for its capacity to produce geometrically complicated structures, such as lattice structures. Lattice structures have great advantages in the lightweight design of the aerospace and automotive field, in which frequent vibration is one of the most concerning problems during the structure design process. Consequently, it is necessary to research structural vibration frequency to avoid dynamic failure, especially the natural vibration frequency of the structure. In this work, a multiscale topology optimization method is proposed to design the Voronoi graded stochastic lattice structures for the first-order frequency maximization problem. Firstly, the generation and analysis of the Voronoi stochastic lattice microstructure are carried out on the microscale. Then, the macroscale structural optimization is conducted with a penalty-free density method. Finally, the full-scale Voronoi graded stochastic lattice structure is reconstructed based on the obtained relative density distribution and mapping relationship. Numerical examples are performed to demonstrate the correctness and validity of the proposed method for designing the Voronoi graded stochastic lattice structure. Several dynamic experiments also verify the effectiveness of the developed multiscale method and the advantage of the optimized graded lattice structure in structural dynamic response.

Level set-based multiscale topology optimization for a thermal cloak design problem using the homogenization method

Artificially designed composite materials consist of microstructures, that exhibit various thermal properties depending on their shapes, such as anisotropic thermal conductivity. One of the representative applications of such composite materials for thermal control is the thermal cloak. This study proposed a topology optimization method based on a level set method for a heat conduction problem to optimally design composite materials that achieve a thermal cloak. The homogenization method was introduced to evaluate its effective thermal conductivity coefficient. Then, we formulated a multiscale topology optimization method for the composite materials in the framework of the homogenization method, where the microstructures were optimized to minimize objective functions defined using the macroscopic temperature field. We presented examples of optimal structures in a two-dimensional problem and discussed the validity of the obtained structures.

Multi-Scale Topology Optimization Method Considering Multiple Structural Performances Performances

Multi-Scale Topology Optimization Method Considering Multiple Structural Performances Performances

Open-Source Codes of Topology Optimization: A Summary for Beginners to Start Their Research

Topology optimization (TO), a numerical technique to find the optimal material layout with a given design domain, has attracted interest from researchers in the field of structural optimization in recent years. For beginners, opensource codes are undoubtedly the best alternative to learning TO, which can elaborate the implementation of a method in detail and easily engage more people to employ and extend the method. In this paper, we present a summary of various open-source codes and related literature on TO methods, including solid isotropic material with penalization (SIMP), evolutionary method, level set method (LSM), moving morphable components/voids (MMC/MMV) methods, multiscale topology optimization method, etc. Simultaneously, we classify the codes into five levels, from easy to dicult, depending on their diculty, so that beginners can get started and understand the form of code implementation more quickly.

Multiscale topology optimization for solid–lattice–void hybrid structures through an ordered multi-phase interpolation

In this paper, we propose a new multiscale topology optimization method to design the solid-lattice-void hybrid structures. A novel ordered solid-lattice-void interpolation model has been developed that realizes smooth transition across the solid, lattice, and void phases, and very importantly, this interpolation model coordinates well between the multiple variables describing the lattice configuration and the relative density variable indicating the solid, lattice, or void status. Specifically, a multi-variable lattice resulted from topology optimization is accepted as the candidate lattice and its effective elastic properties are established through regressive fitting. Then, the ordered solid-lattice-void interpolation formula is developed, which unifies the multi-phase interpolation that allows exceeding the lattice upper/lower limit to reach the solid/void status for the elements. Finally, the multiscale topology optimization algorithm is formulated and implemented for solid-lattice-void hybrid structure design. Numerical examples are studied which demonstrate significant structural performance improvement compared with pure lattice-infilled structures. Finally, additive manufacturing and mechanical tests are conducted to validate the effectiveness of the proposed method. • Propose a new multiscale topology optimization method to design the solid-lattice-void hybrid structures. • Develop a novel ordered solid-lattice-void interpolation model that realizes smooth transition across the solid, lattice, and void phases. • Develop a multi-variable lattice resulted from topology optimization as the candidate lattice. • Demonstrate significant performance improvement by involving the solid and void phases in lattice structure optimization.

Multiscale topology optimization of electromagnetic metamaterials using a high-contrast homogenization method

This study proposes a multiscale topology optimization method for electromagnetic metamaterials using a level set-based topology optimization method that incorporates a high-contrast homogenization method. The high-contrast homogenization method can express wave propagation behavior in metamaterials for various frequencies. It can also capture unusual properties caused by local resonances, which cannot be estimated by conventional homogenization approaches. We formulated multiscale topology optimization problems where objective functions are defined by the macroscopic wave propagation behavior, and microstructures forming a metamaterial are set as design variables. Sensitivity analysis was conducted based on the concepts of shape and topological derivatives. As numerical examples, we offer optimized designs of metamaterials composed of multiple unit cell structures working as a demultiplexer based on negative permeability. The mechanism of the obtained metamaterials is discussed based on homogenized coefficients.

A phase field-based systematic multiscale topology optimization method for porous structures design

In this paper, we propose a novel and systematic phase field-based multiscale topology optimization method for porous structures design. The multi-regional microstructural composite and fixed microstructural shapes are adopted to balance the objective and other constraints. The connectivity issue between different microstructures is handled. In the multiscale topology optimization, the macro and micro design variables are updated by the Allen-Cahn type equations which include a reaction-diffusion term, a sensitivity analysis term, a volume constraint term, and a correction term. The correction term is used to keep topologies of macro and micro structures closed to the prescribed structures. We use an efficient merging algorithm to handle the connectivity issue between different microstructures. Based on an interpolation technique of the phase field function and a modified Allen-Cahn equation, this algorithm can smooth the connecting boundaries and satisfy the minimum surface theorem. The final distribution of microstructures keeps the extraordinary physical and mechanical properties for the macrostructure. Some typical cantilever beam, Michell-type structure and Messerschmitt-Bölkow-Blohm beam are performed to verify the effectiveness of our method.

Comprehensive clustering-based topology optimization for connectable multi-scale additive manufacturing structures

This paper develops a multi-scale topology optimization method that realizes optimized structural stiffness design while achieves inter-connectivity among the heterogeneous unit cells. Specifically, about the technical details, lattice structure topology optimization (LSTO) is conducted by optimizing the parameter field of the specially-designed multi-variable lattices, through which the optimized lattice parameters reflect the density and stress states of the associated macro-element. Then, the macro-elements with close lattice parameters are gathered into clusters, providing the initial guess for the next-step freeform optimization. Finally, multiscale topology optimization (MTO) through the inverse homogenization approach is performed to further design the unit cell structures. The unit cell structures for each cluster are forced to be identical to save homogenization-related computational resources and the interconnectivity is ensured due to the optimized and perfectly connected initial guess from LSTO. Using the proposed method, three classical numerical examples are studied that prove the effects of improved mechanical performance, ensured micro-structure inter-connectivity, and the affordable computing scale. Finally, mechanical tests are conducted to verify the design performance benefits of the proposed method.

A multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing dynamic compliance

Compared with conventional sandwich structures with two identical face sheets, geometrically asymmetric porous sandwich structures (GAPSSs) can achieve better dynamic performances under certain load and boundary condition, since they provide more choices for geometric design. However, current studies regarding the GAPSSs are mainly analytical- and experimental-based methods with predefined face sheet thicknesses and core configurations, few works investigate the GAPSSs by optimization method. This paper presents a multiscale topology optimization method of the GAPSSs, which is capable of designing both thicknesses of two solid face sheets at macroscale and configurations of porous core at microscale for minimizing dynamic compliance. Specifically, at macroscale, a variable thickness sheet method is employed to optimize the thicknesses of two solid face sheets. Then, at microscale, a parametric level-set method integrated with a numerical homogenization approach is applied to topologically optimize the configuration of the periodic unit cell. The method of moving asymptotes is adopted to update the design variables at both scales. Several 2D and 3D numerical examples are illustrated to show the effectiveness and advantages of the proposed method for designing the GAPSSs. The results indicate that the dynamic compliance of the optimized GAPSSs show superior advantages over the conventional sandwich structures with identical face sheet thicknesses and predefined lattice core.

Multiscale topology optimization of coated structures with layer-wise graded lattice infill for maximum fundamental eigenfrequency

Coated structures with lattice infill have attracted great interest from academia to industry. Functionally, the outer coating can protect its inner substrate and lattice infill structures exhibit great ability to improve certain structural performance. In this research, we propose a novel concurrent multiscale topology optimization method to design structures with layer-wise graded lattice infill covered by an outer coating of uniform thickness. The fundamental eigenfrequency is selected as the design objective to be maximized. At the macroscale, the velocity field-based level set (VFLS) method is used to achieve optimal macroscale structures with clear and smooth boundaries. Besides, making use of the signed distance property, the thickness of the outer coating can be well-maintained. At the microscale, the substrate area of coated structures is uniformly (or near-uniformly) divided into several layers to have different microstructural configurations and the popular solid isotropic material with penalization (SIMP) method is used to perform microscale topology optimization. Two different scales are bridged using the asymptotic homogenization method. Numerical results show that coated structures with layer-wise graded lattice infill we design own higher fundamental eigenfrequency than those with periodic and uniform lattice infill microstructures, due to the enlargement of the design freedom.

Development of a multi-scale topology optimization method for thermal-fluid problems

In this presentation, we discuss how to optimize both microstructure and macrostructure in the topology optimization of 3D thermal fluid. First, the 3D thermal fluid model is approximated by a 2D thermal fluid field. Next, a proxy model of microstructural properties is constructed by learning the relationship between microstructure geometry, permeability, thermal conductivity, and heat transfer coefficient through off-line numerical analysis. The macrostructure optimization is then formulated as a microstructure selection problem from the surrogate model. Finally, a simple numerical example confirms the validity of the method. In this study, as a specific example, topology optimization is performed with two design variables on a two-dimensional model of a 3D forced air-cooled heat sink with prismatic fins. The properties of the 3D prismatic fins are analyzed individually in advance and learned offline using an RBF network so that they can be used in the two-dimensional topology optimization, which requires differentiation.

A single variable-based method for concurrent multiscale topology optimization with multiple materials

A concurrent multiscale topology optimization method which exploits a single type design variable to characterize the material properties for any number of microstructures or materials is proposed by using the Smolyak method merged with the Non-Uniform Rational B-Spline(NURBS) method. Two kinds of design variables are assigned to describe the macro and micro scale topology density fields, respectively A multiscale nested form interpolation scheme is constructed from a sequence of discrete terms coupled with the penalty principle which contributes to the selection of candidate microstructures or base materials. The proposed method eliminates the dependency of the dimension of design variables on the number of material types, and significantly reduces the number of design variables. The characteristic function which indicates the existence of each microstructure or base material is introduced to enforce an arbitrary number of volume constraints at the macro and micro scales. Sensitivity analysis of the proposed optimization formulation is also conducted. Finally, the proposed method has been verified and illustrated against multiple examples found in the literature.

Multiscale Topology Optimization Combining Density-Based Optimization and Lattice Enhancement for Additive Manufacturing

Topology optimization (TO) is a shape optimization method based on finite element (FE) analysis, and has recently been used in lightweight design on the basis of the rapid advances of additive manufacturing (AM). While the conventional TO has been applied to obtain the optimal pseudo density in a macroscale domain, microscale TO involving optimization of the strut diameters of a lattice structure has also been studied. In this study, a multiscale TO method was developed by performing the conventional macroscale TO with additional enhancement of microscale lattices. To compare the structural efficiency of the proposed multiscale TO with that of the macroscale and microscale TOs, three optimization methods were applied to a meta-sandwich beam under a three-point bending load condition. Structural FE analyses were then conducted for the three optimized beams, and their deformation behaviors were compared in terms of the structural stiffness and safety. Three optimized beams were then fabricated by the photo-polymerization type AM process using an acrylic photopolymer, and bending experiments were conducted to investigate their deformation behaviors. From the results, the multiscale TO showed the highest structural stiffness and strength owing to the enhancement of microscale lattices. The energy absorption capability was also improved compared to the result of the macroscale TO. These results demonstrate that the multiscale TO is advantageous in the design of efficient lightweight structures with enhanced structural stiffness and safety compared to the conventional macroscale and microscale TO methods.