Concurrent Topology Optimization Research Articles

Tailored Functionally Graded Materials design and concurrent topology optimization with implicit fields

Tailored Functionally Graded Materials (FGMs) offer the ability to design and engineer materials with specific properties at a changing volume fraction and are widely used in various fields such as aerospace, biomedical engineering, etc. The precise control of physical properties and the connectivity of microstructural sequences are two main challenges in multiscale problems. This paper constructs a novel optimization model for generating FGMs under customized performance, employing an implicit field representation governed by tensor product B-splines. The cross-sectional profile aligns with a microstructure, and thus varying heights correspond to a sequence of microstructures. Fine-tuning the implicit field on connectable FGMs is achieved by optimizing specific properties and addressing the 2-norm problem under connectivity constraints. Therefore, these FGMs can serve as fundamental units for bottom-up multiscale infills. Additionally, we also develop a new model to investigate a more universally applicable concurrent topology optimization method without initial input restrictions. These multiscale optimization results demonstrate excellent performance under tested working conditions.

Concurrent topology optimization of sandwich structures with multi-configuration and variable-diameter lattice infill

The superior stiffness-to-weight and strength-to-weight mechanical advantages of sandwich structures can be fully exploited through concurrent design of entire topology, infill configuration and density, where the high-performance yet complicated structure can be fabricated through additive manufacturing. However, the emerging design challenges are concurrent design updating related to sandwich topology, infill configuration and density, which is a design problem with continuous and discrete variables mathematically. In this paper, a concurrent topology optimization is proposed for sandwich structures with multi-configuration and variable-diameter lattice infill. Three design variable fields are employed to describe the fundamental topology considering sandwich structural topology, infill configuration and density simultaneously. Corresponding material interpolation model is developed by combining DSP-based shell-infill description and multi-response latent-variable surrogate model based effective material property calculation. Two-stage design model is formulated as a rough design model considering all design variables followed by a refined design model with only infill density variables, which is developed to strictly satisfy the material allowance constraint due to the mapping between discrete infill configuration variables and continuous latent configuration variables. Corresponding sensitivities of compliance and constraint with respect to the structural topology, infill configuration and density variables are derived, and the method of moving asymptotes (MMA) is employed to solve the design model efficiently. Several numerical examples are presented to systematically demonstrate the effectiveness of the proposed approach.

Concurrent Topology Optimization of Curved-Plate Structures with Double-Sided Stiffeners

Due to their high specific stiffness, particularly in bending, along with their strong design capabilities, stiffened plates have become a prevalent structural solution in aerospace and various other fields. In pursuit of optimizing such structures, a topology optimization method named Heaviside-function-based directional growth topology parameterization (H-DGTP) was proposed in our previous work. However, this approach is limited to designing planar, single-sided stiffened structures. Thus, this paper extends the scope of this method to encompass double-sided, curved, stiffened panels, presenting a topology optimization technique tailored for such configurations. Specifically, considering the position, shape of the curved panels, and the arrangement and height of the stiffeners as design variables, while prioritizing structural stiffness as the objective, a topology optimization model for double-sided curved stiffened plate structures is established, and the corresponding sensitivities of the objective with respect to the design variables are analytically derived. Numerical examples illustrate that simultaneously optimizing the position and shape of the plate, as well as the layout and height of the stiffeners on both sides of the curved plate, results in greater stiffness compared to optimizing only part of these variables, validating the necessity and effectiveness of the proposed method.

Efficient reliability-based concurrent topology optimization method under PID-driven sequential decoupling framework

The continuous improvement of advanced equipment performance has promoted the development of detailed structural design. The detailed design of the structure should meet many requirements such as high reliability, strong robustness, lightweight, and high efficiency. This paper proposes a proportional-integral-derivative-driven efficient reliability-based concurrent topology optimization strategy under the sequential framework. Considering the uncertainty of load position, load direction, load amplitude, and material properties, an efficient reliability-based concurrent topology optimization model is constructed based on the sequential strategy. Based on effectively quantifying the nonlinear characteristics brought by multi-source uncertainty, a constraint refinement movement strategy based on the proportional-integral-derivative controller is proposed, which to some extent overcomes the tedious calculation of uncertainty quantification. This paper provides sensitivity information on macroscopic and microscopic level set functions under displacement constraints. Four numerical examples demonstrate the effectiveness, efficiency, and generalization ability of the proposed method.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

Multiscale design based on non-penalization smooth-edged material distribution for optimizing topology (SEMDOT)

This paper presents a concurrent topology optimization method for macro and micro phases based on non-penalization smooth-edged material distribution for optimization topology (SEMDOT) method. Although there is existing research on the multiscale design method, grayscale elements are always emerged especially for penalization method for example the solid isotropic material penalization (SIMP) method, also high computational cost are required when large scale of elements are utilized for obtaining high resolution structures. The methodology proposed here aims to apply a new tech called non-penalization SEMDOT method to find the optimum layout on both scales of elements, it is assumed that the macro structure is composed of periodic materials and both element scales are optimized through their linearly interpolated grid points. The effective macroscopic properties are evaluated by the homogenization method. The approach could provide smooth and clear boundaries for multiscale system without grayscale elements or high computational cost. A series of numerical examples are presented to demonstrate the effectiveness and the efficiency of the proposed method.

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.

Concurrent Topology Optimization of Shell Structures with Multi-Configuration and Variable-Density Infill

Concurrent Topology Optimization of Shell Structures with Multi-Configuration and Variable-Density Infill

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.