Multi-material Topology Optimization Research Articles

Energy-absorbing structures with multi-material topology optimization considering large deformation

Energy-absorbing structures with multi-material topology optimization considering large deformation

Sustainability-oriented multimaterial topology optimization: designing efficient structures incorporating environmental effects

Sustainability-oriented multimaterial topology optimization: designing efficient structures incorporating environmental effects

A geometric nonlinear multi-material topology optimization method based on univariate combination interpolation scheme

A geometric nonlinear multi-material topology optimization method based on univariate combination interpolation scheme

Multi-material topology optimization of 2D structures using the SESO and SIMP method with reliability

Topology optimization is essential for the efficient distribution of structural form within the desired design domain, ensuring that structural elements are positioned to resist the requested forces and the specific boundary conditions for which they were designed. Therefore, the main objective of the article is the multimaterial topological optimization of a variety of two-dimensional structural problems, based on literature benchmarks, such as Michell structures and the MBB beam. These problems are approached considering a multi-material perspective, to distribute different materials throughout the structure, thus seeking their optimal solution. The methodology was generated based on the use of the finite element method for structure analysis and computing techniques were implemented for multi-material topology optimization (M.T.O.), Smoothing-ESO (SESO), and Solid Isotropic Material with Penalization (SIMP) via MATLAB. Furthermore, a reliability analysis is incorporated to deal with uncertainties, using Reliability-Based Topology Optimization (RBTO) with the First-Order Reliability Method (FORM), dealing with the random variables involved, such as geometry, modulus of elasticity, volume fraction, compliance, and loading, such as normal and lognormal probability distributions. The results obtained show a satisfactory convergence between the two topological optimization methods studied, thus highlighting the potential for applying these techniques to various structures as an effective tool in the search for economic efficiency in structural design.

Multi-material topology optimization design of microstructures with extreme mechanical properties

Multi-material topology optimization design of microstructures with extreme mechanical properties

Topology optimization framework of multiple-phase materials with stress and dynamic constraints under self-weight loads

This work aims to optimize the multi-material structures subjected to self-weight loading for the first time. Ignoring the self-weight loads results in less reliable designs, and to enhance the strength of optimized designs, stress-constrained multi-material topology optimization (MMTO) is considered with body forces. Two stress constraint aggregation schemes are employed and comparable, such as the K-S and p-norm aggregation schemes. Both of these schemes are effective with respect to stress constraints due to local stress issues. Moreover, eigenvalue constraints (MMTO) are introduced in this study with self-weight loading and stress constraints. Thus, the optimized results not only ensure self-weight load criteria but also reduce stress concentration and improve stability due to free vibration of structures. The proposed approach is an alternating-active-phase algorithm (AAPA) for dealing with the problems considered in this work. Results show that the results caused by self-weight loads are totally different from optimized results when they are not considered with self-weight loads using multi-materials.

Performance enhancement of a magnetoactive elastomer actuator through a coupled magnetoelastic topology optimization scheme

Abstract We have found that considering local magnetic fields, large deformations, and magnetoelastic coupling simultaneously significantly affect the resulting shape in magnetoelastic topology optimization in a uniaxial actuator case. In contrast to the work presented here, other works incorporate magnetoelastic formulations that include simplifying assumptions on the local field, and subsequent effects on the magnetization response of the material, or the absence of large deformation mechanics, or both. These assumptions were shown to produce solutions that differ substantively from cases where local fields and large deformations are addressed concurrently. Magnetoelastic topology optimization schemes are needed to optimize magnetoactive elastomer (MAE) devices. MAE devices are magnetic particle-filled polymer matrices designed for specific actuations and controlled remotely by an external magnetic field. They garner considerable research interest as an emerging technology for actuators in soft robots or in applications requiring untethered actuation. The material properties of MAEs are dependent on the volume fraction of particles in the elastomer matrix, where a high-volume fraction increases relative permeability (for soft magnetic particles) but also increases elastic modulus. For optimal actuation, a tradeoff between low stiffness and high magnetic response must be made by adjusting volume fraction and controlling material placement. Using a topology optimization scheme that considers both the magnetic and mechanical properties of the material, the shape and material composition of the device can be tuned to best achieve the desired actuation displacement. In this work, a density-based magnetoelastic multimaterial topology optimization scheme for soft magnetic material is developed in COMSOL Multiphysics. The optimization scheme uses multiphysics coupling that considers local magnetic fields and large deformations at each iteration through a Maxwell stress tensor formulation. A simulated example is then considered to demonstrate the effectiveness and necessity of a coupled optimization. The effect of considering large deformations during optimization is also investigated. It was found that a coupled topology optimization scheme with large deformations produced shapes with modes of actuation not captured by schemes with simplifying assumptions, leading to better performance at lower material cost. Considering large deformations in the coupled scheme offered significantly better performance, with an increase of 81.3% in a side-by-side performance simulation when compared to uncoupled cases.

Uniform multiple laminates interpolation model and design method for double–double laminates based on multi-material topology optimization

Double–Double (DD) laminates, incorporating a repetition of sub-plies featuring two groups of balanced angles, offer broad design flexibility together with the ease of design and manufacturing. In this work, a novel optimization design method is proposed for DD composite laminates based on multi-material topology optimization. First, the uniform multiple laminates interpolation (UMLI) model is proposed to describe the certainty of the stacking direction in multi-layer composite structures, inspired by the interpolation model in multi-material topology optimization. Specifically, the stiffness matrices of all alternative angle combinations of laminates are interpolated to form virtual laminates. The UMLI model eliminates the need for adding interlayer constraints during the optimization process. Then, the optimization problem is defined to minimize the compliance of the composite structures and is solved using the gradient-based optimization algorithm. Finally, the proposed method is applied to the design of the composite stiffened panel, the composite Unmanned Aerial Vehicle (UAV) wing, and the rear fuselage. The results demonstrate that the UMLI model and proposed optimization method have considerable potential in the angle optimization design of multi-layer structures.

Approach for multi-valued integer programming in multi-material topology optimization: Random discrete steepest descent (RDSD) algorithm

The present study models the multi-material topology optimization problems as the multi-valued integer programming (MVIP) or named as combinatorial optimization. By extending classical convex analysis and convex programming to discrete point-set functions, the discrete convex analysis and discrete steepest descent (DSD) algorithm are introduced. To overcome combinatorial complexity of the DSD algorithm, we employ the sequential approximate integer programming (SAIP) to explicitly and linearly approximate the implicit objective and constraint functions. Considering the multiple potential changed directions for multi-valued design variables, the random discrete steepest descent (RDSD) algorithm is proposed, where a random strategy is implemented to select a definitive direction of change. To analytically calculate multi-material discrete variable sensitivities, topological derivatives with material contrast is applied. In all, the MVIP is finally transferred as the linear 0–1 programming that can be efficiently solved by the canonical relaxation algorithm (CRA). Explicit nonlinear examples demonstrate that the RDSD algorithm owns nearly three orders of magnitude improvement compared with the commercial software (GUROBI). The proposed approach, without using any continuous variable relaxation and interpolation penalization schemes, successfully solves the minimum compliance problem, strength-related problem, and frequency-related optimization problems. Given the algorithm efficiency, mathematical generality and merits over other algorithms, the proposed RDSD algorithm is meaningful for other structural and topology optimization problems involving multi-valued discrete design variables.

Incorporating interface effects into multi-material topology optimization by improving interface configuration: An energy-based approach

Interfaces between structural multi-materials generally exhibit asymmetric resistance to tension and compression. Given this interface behavior, this work suggests an energy-based approach to improve the interface configuration for multi-material topology optimization. Based on the strain spectral decomposition, we decompose the structural elastic strain energy into tensile and compressive portions. In the density-based topology optimization framework, we use the gradient-based method to track the interface between multiple materials. Then, we construct an interface-associated scalar field to penalize the tensile portion of the strain energy, causing a pseudo-degradation of the strain energy at the interface region. Finally, within limited material usages and by minimizing the linear weighted structural strain energy and its pseudo-degradation, multi-material topology optimization with improved interface configuration is achieved. Several 2D and 3D numerical examples are investigated, by which the effectiveness and robustness of the suggested approach are fairly validated.

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

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

Design of Locally Resonant Acoustic Metamaterials with Specified Band Gaps Using Multi-Material Topology Optimization.

Locally Resonant Acoustic Metamaterials (LRAMs) have significant application potential because they can form subwavelength band gaps. However, most current research does not involve obtaining LRAMs with specified band gaps, even though such LRAMs are significant for practical applications. To address this, we propose a parameterized level-set-based topology optimization method that can use multiple materials to design LRAMs that meet specified frequency constraints. In this method, a simplified band-gap calculation approach based on the homogenization framework is introduced, establishing a restricted subsystem and an unrestricted subsystem to determine band gaps without relying on the Brillouin zone. These subsystems are specifically tailored to model the phenomena involved in band gaps in LRAMs, facilitating the opening of band gaps during optimization. In the multi-material representation model used in this method, each material, except for the matrix material, is depicted using a similar combinatorial formulation of level-set functions. This model reduces direct conversion between materials other than the matrix material, thereby enhancing the band-gap optimization of LRAMs. Two problems are investigated to test the method's ability to use multiple materials to solve band-gap optimization problems with specified frequency constraints. The first involves maximizing the band-gap width while ensuring it encompasses a specified frequency range, and the second focuses on obtaining light LRAMs with a specified band gap. LRAMs with specified band gaps obtained in three-material or four-material numerical examples demonstrate the effectiveness of the proposed method. The method shows great promise for designing metamaterials to attenuate specified frequency spectra as required, such as mechanical vibrations or environmental noise.

Multimaterial filtering applied to the topology optimization of a permanent magnet synchronous machine

PurposeThis study aims to propose a general methodology to handle multimaterial filtering for density-based topology optimization containing periodic or antiperiodic boundary conditions, which are expected to reduce the simulation time of electrical machines. The optimization of the material distribution in a permanent magnet synchronous machine rotor illustrates the relevance of this approach.Design/methodology/approachThe optimization algorithm relies on an augmented Lagrangian with a projected gradient descent. The 2D finite element method computes the physical and adjoint states to evaluate the objective function and its sensitivities. Concerning regularization, a mathematical development leads to a multimaterial convolution filtering methodology that is consistent with the boundary conditions and helps eliminate artifacts.FindingsThe method behaves as expected and shows the superiority of multimaterial topology optimization over bimaterial topology optimization for the chosen test case. Unlike the standard approach that uses a cropped convolution kernel, the proposed methodology does not artificially reflect the limits of the simulation domain in the optimized material distribution.Originality/valueAlthough filtering is a standard tool in topology optimization, no attention has previously been paid to the influence of periodic or antiperiodic boundary conditions when dealing with different natures of materials. The comparison between the bimaterial and multimaterial topology optimization of a permanent magnet machine rotor without symmetry constraints constitutes another originality of this work.

Novel multi-material topology optimization method for multi-segmented permanent magnet motors

PurposeThe purpose of this paper is to develop a novel optimization method that can improve the convergence of the multi-material topology.Design/methodology/approachIn the proposed method, the optimization procedure is divided into two steps. In the first step, a global search is performed to probabilistically determine the material distribution of multi-segmented magnets. In the second step, the design area is limited and a local search is performed to determine the detailed magnet shape.FindingsBecause the first optimization step determines the arrangement of the magnetization vectors according to the rotational position, as in a d-axis flux concentration orientation, the optimal solution can be obtained with a smaller volume of magnets than the conventional method.Research limitations/implicationsBecause a few case studies are considered in this paper, additional verification is required, such as application to different types of motors, to clarify scalability.Practical implicationsThe solution obtained using the proposed method has a smaller amount of magnet than the solution obtained using the conventional method and can fully satisfy the average torque constraint.Originality/valueThe proposed method differs from the conventional method in that the material distribution is determined according to the probability function in the first optimization step.

Multi-material topology optimization considering both isotropy and anisotropy with fibre orientation optimization

Research on multi-material topology optimization (MMTO) considering both isotropic and anisotropic materials as candidate materials is limited. Previous studies required researchers to preselect the anisotropic material fibre orientations, which are not subject to change during the optimization. As the preselected orientations cannot change, better MMTO solutions may be overlooked. To address this issue, a novel MMTO algorithm incorporating the anisotropic material fibre orientation as a design variable is proposed, eliminating the need to preselect fibre orientations and leading to better MMTO solutions. A numerical approximation method named the carry-through method is also proposed. This allows the compliance sensitivity over anisotropic material fibre orientation to be calculated without the need for the strain–displacement matrix information, simplifying the MMTO method implementation. Comparative studies on three models demonstrate that the proposed MMTO method outperforms the existing MMTO method, achieving improvements of 7%, 10% and 14% on the objectives in these models.

An Incremental Interpolation Scheme With Discrete Cosine Series Expansion for Multimaterial Topology Optimization

Abstract Topology optimization is a powerful tool for structural design, while its computational cost is quite high due to the large number of design variables, especially for multilateral systems. Herein, an incremental interpolation approach with discrete cosine series expansion (DCSE) is established for multilateral topology optimization. A step function with shape coefficients (i.e., ensuring that no extra variables are required as the number of materials increases) and the use of the DCSE together reduces the number of variables (e.g., from 8400 to 120 for the optimization of the clamped–clamped beam with four materials). Remarkably, the proposed approach can effectively bypass the checkerboard problem without using any filter. The enhanced computational efficiency (e.g., a ∼89.2% reduction in computation time from 439.1 s to 47.4 s) of the proposed approach is validated via both 2D and 3D numerical cases.

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

Enhancing the degrees of freedom of topology optimization via variable-porosity metal foams: Design of heat conduction paths in a volume-to-point problem

Multi-material topology optimization determines the optimal distribution of different materials within a design domain in order to achieve specific performance goals. This work takes advantage of such technique to enhance the thermal performance of a benchmark heat conduction problem. The aim is to investigate the impact of enhancing the degrees of freedom introducing different materials, i.e., variable-porosity metal foams in addition to the high-conductivity solid, in the design of heat conduction paths under a constant weight constraint. The study leverages a well-established case study – volume-to-point problem – in the field of thermal management, ensuring a consistent basis for comparison. It consists of a square domain with a uniform heat source and a Dirichlet condition at a point on the boundary. Interpolation functions – ordered solid isotropic material with penalization (SIMP) type – allow the properties – in this case thermal conductivity – of materials to be correctly assigned. The distinction between materials is made by means of different thresholds on the projection function. By varying the foam porosity, we investigate the trade-offs between weight and heat dissipation efficiency. The objective is to find the ideal combination of materials that maximizes heat transfer – minimizing the average domain temperature – while conforming to weight constraints. Findings reveal that multi-material topology optimization – when applied to heat conduction problems – can outperform other design approaches, such as the growth-based algorithm, the evolved constructal tree, and the classical topology optimization (with one solid material), thereby paving the ground to innovative heat sink designs in weight-constrained environments.

Multi-material topology optimization based on enhanced alternating active-phase algorithm

Multi-material topology optimization based on enhanced alternating active-phase algorithm

Transient heat conduction in multi-material topology optimization of thermoelastic structures involving dynamic constraints

This work presents a modified solid isotropic material with penalization (SIMP) for conducting dynamic-constrained multi-material topology optimization (MMTO) of thermoelastic structures subjected to transient thermal conduction. The novelty of this study is the multi-material optimization of instantaneous temperature conduction under specific frequency constraints, which has never been studied before. The proposed method has been developed for general models of multi-material interpolations such as thermal stiffness, heat capacity, and thermal stress coefficient. The temperature environment is formulated by applying prescribed mechanical and thermal loads that are exposed over a period of time. In this work, we focus on the objective of minimizing compliance in a transient heat conduction structure while simultaneously imposing constraints on the dynamics. With high temperatures, the stiffness of the optimization results decreases while stability is still guaranteed. The sensitivity of the objective function is subsequently calculated using the discretize-then-differentiate approach for the dynamic problem as well as the adjoint variable method. This study examines the impact of thermal fields and different time increments in conjunction with dynamic criteria. The outcomes of heat conduction result in notable alterations under identical dynamic constraint levels.