Concurrent Topology Optimization Research Articles

Multi-scale concurrent topology optimization of cellular structures with multiple microstructures for minimizing dynamic response in the time domain

This paper proposes a concurrent topology optimization method for optimizing the structures that are periodically filled with multiple microstructures excited by dynamic response in the time domain. At the macroscale, different microstructures are considered as different materials. To generate the various distribution of different microstructures, a multi-material interpolation approach based on the Solid Isotropic Material with Penalization (SIMP) is integrated. At the microscale, the energy-based homogenization method (EBHM) is employed to determine the macroscopic effective properties of the microstructures. All macroscale elements with identical material are represented by a distinct microstructure. For the proportional damping model, the HHT-α is implemented as a time integration technique to obtain the dynamic response of multi-scale and assumed multi-material structures. The objective function of the topology optimization problem is to minimize the mean dynamic compliance. Combined differentiate-then-discretize method with the adjoint variable method, the sensitivity analysis applies the gradient-based Zhang-Paulino-Ramos Jr. (ZPR) algorithm or Method of Moving Asymptotes (MMA) to update the design variables under multiple constraints in the time and space-discretized system. The proposed approach is numerically performed through 2D and 3D examples to demonstrate its effectiveness.

Multi-scale concurrent topology optimization of frequency- and temperature-dependent viscoelastic structures for enhanced damping performance

Multi-scale concurrent topology optimization of frequency- and temperature-dependent viscoelastic structures for enhanced damping performance

Multiscale concurrent topology optimization for structures with multiple lattice materials considering interface connectivity

Multiscale concurrent topology optimization for structures with multiple lattice materials considering interface connectivity

EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples.

Concurrent topology optimization of shells with pattern-guided infills for intuitive design and additive manufacturing

This paper introduces a systematic design optimization approach for shells with pattern-guided infills, aiming to address both intuitive design requirements and considerations for additive manufacturing. The approach utilizes a density-based topology optimization method and assigns two sets of design variables to distinctly define the base, coating, and infill structures. By evolving the solid shells and enriched infills concurrently, the design optimization process ensures a cohesive integration of both components. To guide the infill design towards matching the geometrical features of a predefined pattern, a geometrical constraint is imposed on the infill field. Enhanced by an updated and approximate volume constraint, the infill materials with prescribed appearance performances are accordingly distributed in the base region. The simultaneous optimization of shell geometry and infill layout is then performed within the framework of minimum compliance topology optimization. A range of numerical results illustrate the effectiveness and robustness of the proposed approach. The topology optimized results demonstrate that a given pattern can directly control the design features of the infill in a visually explicit manner and the shell-infill composites ensure convenient manufacturability.

Multiscale concurrent topology optimization for thermoelastic structures under design-dependent varying temperature field

Multiscale concurrent topology optimization for thermoelastic structures under design-dependent varying temperature field

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.

MFSE-based two-scale concurrent topology optimization with connectable multiple micro materials

The concurrent design of different lattice material microstructures and their corresponding macro-scale distributions has great potential in achieving both lightweight and desired multiphysical performances. In such design problems, the lattice microstructures are usually separately optimized on the basis of the homogenization method, and the possibly poor connectivity between them is a key factor that severely hinders the fabrication and application of optimized two-scale structures. To handle the microstructure connectivity issue, this paper proposes a novel microstructure connectable strategy to bridge the gap between the two-scale and the full-scale model using the material-field series expansion (MFSE) method. Assuming different lattice material types in several pre-defined macro-scale regions, describe all types of microstructural topology with different portions of one material field function and update them simultaneously during the optimization process. Benefits from the material field definition with spatial correlation, the microstructures are well-connected without requiring additional constraints in the topology optimization model. The energy-based homogenization method is utilized for bridging the two-scale with different microstructures, while a decoupled sensitivities analysis for the microscale is employed to enhance the computation efficiency. Additionally, the proposed method significantly reduces the dimension of design variables, resulting in lower optimizer spending. The effectiveness and efficiency of the proposed method are demonstrated by several benchmark two-scale problems. Compared to density-based connectable methods, the proposed framework is easy to implement and reduces computational time by an order of magnitude in the 2D case.

Concurrent topology optimization of parts and their supports for additive manufacturing with thermal deformation constraint

A typical manufacturing failure in powder bed additive manufacturing (AM) is caused by the collision between the already printed structure and the powder recoater due to large thermal deformation. This paper presents a density-based topology optimization (TO) approach to concurrently design parts and their supports that are not susceptible to such a manufacturing failure. The thermal deformation during the layer-by-layer AM process is predicted based on an inherent strain method, and the top surfaces' upward deformations are constrained below the powder layer thickness to avoid recoater collision. To increase the design freedom of TO, the parts and their supports are composed of lattice unit cells with different volume fractions, whose effective stiffness matrixes are predicted by the numerical homogenization and surrogate models. Besides, an overhang constraint and a two-field-based length scale control formulation are also utilized, ensuring overall manufacturability. With the proposed approach, lightweight structures with optimized mechanical properties and controllable manufacturing qualities can be obtained. Numerical examples are given to demonstrate the effectiveness of the proposed approach.

Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted.

Multiscale concurrent topology optimization of hierarchal multi-morphology lattice structures

This paper proposes a multiscale concurrent topology optimization method for design of hierarchal multi-morphology lattice structures (HMMLSs), which features in the Kriging metamodel assisted Uniform Multiphase Materials Interpolation (KUMMI) model and the sigmoid function (SF) based hybrid transition strategy. Specifically, level set functions are adopted to model hierarchal lattice unit cells (LUCs). Then LUCs with different morphology are treated as distinct lattice materials (LMs) and the SF-based hybrid transition strategy is employed to realize smooth transition between multi-morphology LUCs. Besides, Kriging metamodels are constructed to predict the equivalent properties of LUCs efficiently so that the topology optimization of HMMLSs can be achieved with an affordable computational cost. Two kinds of design variables are involved in the topology optimization of HMMLSs, where discrete variables indicate the categories of lattice materials and continuous variables determine the relative densities of lattice microstructures. The proposed KUMMI model can couple the two kinds of design variables dexterously to realize concurrent optimization of relative densities of LUCs, and distribution regions and percentages of different LMs within HMMLSs. Several numerical examples are presented to demonstrate the effectiveness and applicability of the proposed method. Results indicate the optimized HMMLSs exhibit rational distribution of hierarchal multi-morphology LUCs, hence achieving superior structural performance.

Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures

In this paper, the Proportional Topology Optimization (PTO) algorithm is extended for the two-scale concurrent topology optimization, in which both the structure and material cellular micro-structure are subject to design. PTO was originally developed on the concept that the amount of material being distributed to an element would be proportional to the contribution of that element in the objective function. Sensitivity analysis is not required. In a two-scale concurrent topology optimization problem, two sets of design variables are defined, one for macro-structure and one for micro-structure. Here, the objective function is reformulated such that the contribution of each micro-scale design variable can be determined, facilitating the employment of PTO. The macroscopic effective elastic tensor is evaluated by the energy-based homogenization method (EBHM), providing a link between micro-structure and macro-structure. Feasibility and efficiency of the proposed PTO approach are demonstrated via several benchmark examples of both two and three dimensional structures.

Dynamic concurrent topology optimization and design for layer-wise graded structures

There are many requirements in engineering fields to improve the structural performance. Thus, the hierarchical structures have attracted lots of attention because of the appealing material efficiency and functionality, e.g., the bearing capacity and interfacial strength from large aerospace structures to small device facilities. However, the dynamic topology optimization design of these structures composed of multiple microstructures remains challenge by the connectivity issue between different microstructures and the extensive computational cost when handling the interval-frequency problem during multiscale concurrent design. In this work, the connectivity model is proposed for improving the manufacturability of hierarchical structure, which includes the connectable layer scheme between different microstructures and the enlarged filter domain. To break the bottleneck of computation cost induced by the simultaneous optimization of macro-microscopic design variables, the combined method of modal superposition and model order reduction is incorporated to efficiently compute the frequency response. Two response vectors are presupposed before decoupling sensitivity analysis. In addition, in order to implicitly ensure the bearing capacity of the structure, the stiffness maximization design is integrated into the response minimization problem when the frequency interval range is high. The layer-wise graded cantilever beam is used as an example to demonstrate the effectiveness and accuracy of proposed method in handling dynamic concurrent topology optimization problem. The manufacturability and systematic performance of hierarchical structures are significantly improved when they simultaneously experience the interval-frequency harmonic and static loads.

Concurrent topology optimization design for CNT orientation and CNTRC layout

In this study, the topology optimization (TO) design of carbon nanotube (CNT) orientation and material layout for carbon nanotube-reinforced composites (CNTRCs) is performed for the first time. The equivalent performance of CNTRCs is calculated by the extended easy-to-implement rule of mixtures. Then, by considering two independent angle variables, a 3-dimensional (3D) concurrent optimization model for CNT orientation and material layout is constructed. To decrease the possibility of falling into local optimal solutions, a discrete interval method is proposed to divide the search space into specific subintervals, and thus the original optimization problem is changed into material selection and subinterval angle optimization problems. In addition, different from conventional fiber-reinforced composites, CNTRCs employ nanoscale filler as reinforcement, which makes it more beneficial to achieve local material strengthening. Therefore, to decrease the possibility of local optimal solutions and achieve local structural strengthening, a concurrent TO model involving CNTRCs and isotropic materials under the volume constraint is proposed to achieve the optimal balance between structural stiffness and economy. Subsequently, the sensitivities of structural compliance with respect to design, selection, and angle variables are derived. Also, the effectiveness of the proposed method is verified by 2D and 3D examples.

Gradient-based concurrent topology and anisotropy optimization for mechanical structures

The present paper introduces a novel gradient-based optimization framework to obtain discrete topologies consisting of either none or complete presence of anisotropic material, modeled by means of the polar formalism. Current topology optimizations with the polar formalism are based on optimality criteria, and limited to performing compliance minimization for thermodynamically feasible materials. The proposed optimization approach uses a sequential approximations approach, based on the Methods of Moving Asymptotes. The material density, orientation and anisotropic modules are updated separately at each iteration, in parallel sub-problems constructed with different types of approximations and settings. The proposed optimization approach is successfully validated for compliance minimization against the Alternate Directions method for general orthotropic materials, defined by the thermodynamic bounds. The importance of the anisotropy initialization in the gradient-based approach is highlighted to obtain stiffer solutions. The gradient-based strategy is also extended to incorporate geometric bounds on the polar parameters, defining the domain of existence of composite laminates. Obtained results for compliance minimization with laminates are compared to published results using lamination parameters. Finally, new solutions are presented showing the improvement of the compliance with the expansion of the anisotropy design domains. This paper paves the way for strength-based topology optimization of parts made of orthotropic materials or composite laminates.

Two-scale concurrent topology optimization method of constrained layer damping structure subjected to non-uniform blast load

Two-scale concurrent topology optimization method of constrained layer damping structure subjected to non-uniform blast load

Concurrent topology optimization of multiple components sharing partial design domain based on distance regularized parameterized level set method

AbstractIn this article, the distance regularized parameterized level set method (DRPLSM) is proposed, which introduces a specified energy functional into the level set equation as the distance regularization term. Geometric verification shows that the distance regularization scheme keeps the level set function near the structural boundary as an approximate signed distance function. Based on benchmark example of the cantilever beam, it is verified that the DRPLSM can avoid the re‐initialization of level set function in the optimization process and obtain smooth structural boundary. The DRPLSM is applied to multicomponent concurrent topology optimization with partially shared design domain for the first time. It makes the optimization results have smooth structural boundaries. The optimization objective of this algorithm is to minimize the structural compliance of all components. The sharing domain is divided into different design domains according to the strain energy of elements. The optimality of the proposed algorithm for partitioning the sharing domain is verified using the literature example with known optimal solutions. A symmetric structure example, a main and auxiliary optimization structure example and an example with design‐dependent boundary loads show that the proposed concurrent optimization algorithm has wide applicability.

On concurrent multiscale topology optimization for porous structures under hygro‐thermo‐elastic multiphysics with considering evaporation

AbstractLightweight polymeric and natural composite materials are extensively used in modern structures, especially with the demands for environmentally friendly products as well as lowering energy consumption. Furthermore, a high performance‐to‐weight ratio can be attained by utilizing porous composites. However, hygral and thermally induced loads are limiting the robustness of polymeric and natural composite materials, therefore; in this research, concurrent multiscale multiphysics topology optimization is used to design lightweight porous composite structures that have resilience toward mechanical as well as hygral and thermal loads. By establishing two independent representations of the design problem, that is, macro and microscale domains, a concurrent topology optimization framework is implemented, and the effective properties of the microscale (i.e., elastic, thermal conductivity, moisture diffusivity tensors, and hygral as well as thermal expansion coefficients) are calculated and used as the hygro‐thermo‐elastic properties of the macroscale using in‐house MATLAB codes. For hygral physics, moisture transport, as well as evaporation, are simultaneously considered in this study. A sensitivity analysis was conducted on the multiphysics concurrent optimization scheme in order to account for the coupling of macro and microstructure, as well as hygro‐thermo‐elastic physics. Multiple numerical cases were examined, which included different loading and boundary conditions, as well as various spatial configurations. The results showed attaining a high stiffness‐to‐weight ratio for the multiscale optimized porous structure compared to the single‐scale solid structure. Furthermore, a study was conducted on multiple microstructure subsystems to examine the impact of microstructure systems on macrostructure dependence. By combining several microstructures into a single macro design domain, design flexibility was enhanced and the performance‐to‐weight ratio was improved. The study was expanded to include the evaluation of hygro‐thermo‐elastic multiscale multiphysics with an evaporation problem, which was demonstrated through several numerical examples. The introduced formulations showed a successful application of the concurrent multiscale optimization formulations and good coupling on the macro and microscale. Also, the formulations demonstrated a strong influence between the macro and the microscale of the design problem for the topology optimization methods. The successful application of the concurrent multiscale optimization method in this research highlights its potential for designing more efficient and effective structures in the future.

Concurrent topology optimization of multiscale structure under uncertain dynamic loads

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

Concurrent Topology and Fiber Path Optimization for Continuous Fiber Composite Under Thermo-Mechanical Loadings

Concurrent Topology and Fiber Path Optimization for Continuous Fiber Composite Under Thermo-Mechanical Loadings