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This paper presents a multi-material topology optimisation (MMTO) process for the lightweight design of an electric bus roof structure including the self-weight. The usage of electric buses is increasing owing to environmental issues. However, it is challenging to design a lightweight structure, because the heavy battery pack mounted on the roof increases the deformation and reduces the safety. In a design including the self-weight, the appropriate distribution of multiple materials improves the performance more than that of a single material. The MMTO method has been applied to identify the optimal distribution of multiple materials. However, in the real-engineering problem, only a simple objective function such as compliance and a single constraint function such as the volume of material have been considered, whereas the mass reduction is the most important factor. In this paper, an MMTO process is proposed for the lightweight design of the bus roof structure to consider multiple displacement constraints including the self-weight. To control the complexity of the distribution of multiple materials for improving the manufacturability, the welding surface function is proposed. An optimisation process was constructed that can handle the complex finite-element model and multiple load cases, and it was validated according to the well-known compliance minimisation problem. Mass reduction was achieved via the lightweight optimisation, and the interfacial area between the different materials was reduced by employing the welding surface function.

In this article, a density-driven unified multi-material topology optimization framework is suggested for functionally graded (FG) structures under static and dynamic responses. For this, two-dimensional solid structures and plate-like structures with/without variable thickness are investigated as design domains using multiple in-plane bi-directional FG materials (IBFGMs). In the present approach, a generally refined interpolation scheme relying upon Solid Isotropic Material with Penalization is proposed to deal with equivalent properties of IBFGMs. This methodology’s topological design variables are totally independent of all material phases. Therefore, the present method can yield separate material phases at their contiguous boundaries without intermediate density materials. The assumption of mixed interpolation of tensorial components of the 4-node shell element is employed to analyze plate elements, aiming to tackle the shear-locking phenomenon encountered as the optimal plate thickness becomes thinner. The mesh-independence filter is utilized to suppress the checkerboard formation of the material distribution. The method of Moving Asymptotes is used as an optimizer to update design variables in the optimization process. Several numerical examples are presented to evaluate the efficiency and reliability of the current approach.

Multi-material topology optimization is a practical tool that allows for improved structural designs. However, most studies are presented in the context of continuum topology optimization – few studies focus on truss topology optimization. Moreover, most work in this field has been restricted to linear material behavior with limited volume constraint settings for multiple materials. To address these issues, we propose an efficient multi-material topology optimization formulation considering material nonlinearity. The proposed formulation handles an arbitrary number of candidate materials with flexible material properties, features freely specified material layers, and includes a generalized volume constraint setting. To efficiently handle such arbitrary volume constraints, we derive a design update scheme that performs robust updates of the design variables associated with each volume constraint independently. The derivation is based on the separable feature of the dual problem of the convex approximated primal subproblem with respect to the Lagrange multipliers, and thus the update of design variables in each volume constraint only depends on the corresponding Lagrange multiplier. Through examples in 2D and 3D, using combinations of Ogden-based, bilinear, and linear materials, we demonstrate that the proposed multi-material topology optimization framework with the presented update scheme leads to a design tool that not only finds the optimal topology but also selects the proper type and amount of material. The design update scheme is named ZPR (phonetically, zipper), after the initials of the authors’ last names (Zhang-Paulino-Ramos Jr.).

Graded porous structures combine robustness of porous structures and high stiffness of bulk designs. This study aims to design optimized graded porous bone-like structures through a novel multi-material topology optimization approach, which generalizes the concept of multiple materials. Namely, each material can have not only distinct material property but also a different level of local porosity, or a combination of both, thus allowing the realization of multiple levels of porosity. With separated density and material/porosity fields, we propose two types of multi-porosity local volume constraints to enable graded porosity considering linear and bi-linear material constitutive relations. Through the proposed framework, single- and multi-material structures can be obtained with a natural transition between the bulk and multiple levels of porous regions. We adopt the Bi-value Coding Parameterization (BCP) scheme combined with the Solid Isotropic Material with Penalization (SIMP) method to interpolate the stored energy functions. Through several examples with multiple porosity levels and various material properties, we demonstrate the effectiveness of the proposed framework with two novel constraints to generate optimized multi-material and multi-porosity structures. We further investigate the interactions among material properties, multiple porosity levels, structural stiffness, and robustness. Compared with conventional bulk designs, the optimized bone-like structures with multi-level graded porosity, although less stiff, are found to be more robust, i.e., their structural stiffness is less influenced by the load variations and material deficiency. The resulting graded porous composite designs showcase the capability of the proposed multi-material formulation to optimize the distributions of not only different types of materials but also multiple levels of porosity.

The aerospace industry is constantly looking to integrate advanced materials and manufacturing methods into their airframes to achieve new breakthroughs in lightweight design. Enabling these advancements are new computational methods such as multi-material topology optimization. While this field has expanded in recent years, the current state-of-the-art typically focuses on academic-level examples and is usually restricted to isotropic-only or anisotropic-only studies. To address this gap, this paper presents practical examples of multi-material topology optimization for the aerospace industry, including the first application of both isotropic and orthotropic material models simultaneously in the same 3D design space. Here, the structural legs and seatback for a passenger aircraft seat are considered at the conceptual level and focuses on the use of single- and multi-material designs to determine the optimum utilization of a new aerospace-grade composite alongside aluminum and magnesium. Designs are discussed and compared with preliminary considerations on performance and cost, with the inclusion of various alternative manufacturing-based constraints. Ultimately, this paper seeks to demonstrate the practical capability of multi-material topology optimization, and review methods and perspectives for evaluating various single- and multi-material design combinations.

This paper proposes a method for performing both multi-material topology optimization and multi-joint topology optimization. The algorithm can determine the optimum placement and selection of material while also optimizing the choice and placement of joint material between components. This method can simultaneously minimize the compliance of the structure as well as the total joint cost while subjected to a mass fraction constraint. A decomposition approach is used to break up the coupling between optimum structural design and optimum joint design. Multi-material and multi-joint topology optimization are then solved sequentially, controlled by an outer loop. By decomposing the problem, gradient-based optimization algorithms can be utilized, enabling the algorithm to solve large computational models efficiently. The proposed process is applied to three 3D standard TO problems. Through these example problems, the need for an iterative process is demonstrated. Improvements to joint manufacturability using the tooling and stress constraints are discussed. Finally, a review of computational cost is performed.

The development of strut-and-tie models (STMs) for the design of reinforced concrete (RC) deep beams considering a general multi-material and multi-volume topology optimization framework is presented. The general framework provides flexibility to control the location/inclination/length scale of the ties according to practical design requirements. Optimality conditions are applied to evaluate the performance of the optimized STM layouts. Specifically, the Michell number Z (or load path) is used as a simple and effective criterion to quantify the STMs. The experimental results confirm that the layout with the lowest load path Z achieves the highest ultimate load. Moreover, significantly reduced cracking is observed in the optimized layouts compared to the traditional layout. This observation implies that the optimized layouts may require less crack-control reinforcement, which would lower the total volume of steel required for the deep beams. Keywords: load path; Michell number; multi-material topology optimization; reinforced concrete (RC) deep beam; strut and tie.

In the physical world, it is common for Multiple Load Cases (MLC) to act on a body either simultaneously or at different points in time. While MLC has been widely addressed in the literature, it has been identified that MLC in 2D Multi-Material Topology Optimised (MMTO) examples using the Solid Isotropic Material with Penalisation (SIMP) method is understudied, with the majority of examples not evaluating their structural performance. It is also identified that there are currently no MLC-ready MMTO software tailored to Architects that can perform Finite Element Analysis (FEA). The current research investigates how MLC can be addressed within “Stag”, our newly developed MMTO plugin for Grasshopper, and how its results compare topologically to benchmark examples from the literature. Furthermore, an overlaying method (ALC) of individual load case results is compared to MLC. This study addresses the identified gap in the literature by evaluating and comparing the structural performance of Stag’s MMTO MLC and ALC results with those from the literature by performing FEA within the same platform using the Grasshopper plugin “Karamba3D”. It is found that Stag produces MMTO MLC results that have a similar topology and structural performance to the benchmark examples from the literature. While the ALC result surpasses the target volume fraction, it performs structurally better than the MLC result.

This research focuses on the design, fabrication, and structural and embodied carbon analysis of the world’s first topologically optimised multi-metal I-beam. Specifically, the beam under study is a European Parallel I-beam with a nominal height of 100 mm (commonly referred to as ‘IPE-100’), and the materials used are mild steel and tool steel. Topology Optimisation (TO) is performed using Altair’s OptiStruct software package, applying the Solid Isotropic Material with Penalty (SIMP) method. The multi-metal beam is fabricated using 3D printing, specifically Laser Metal Deposition (LMD), with a dual built-in metal wire feeder attached to a robotic arm. The beam is analysed both environmentally and structurally — the former focusing on an embodied carbon assessment of material extraction and component manufacturing, and the latter on four-point structural load testing. The fabrication method and analysis results are compared with those of the standard IPE-100 beam currently used in construction. Environmentally, the Multi-Material Topologically Optimised (MMTO) beam’s reduced mass results in lower carbon emissions compared with the standard IPE-100; however, due to the high emissions associated with its fabrication process, its overall carbon footprint is higher. Structurally, the MMTO beam can withstand a higher machine load than the standard IPE-100 before undergoing plastic deformation. This research is the result of an international, multidisciplinary collaboration between academia and industry across the United Kingdom, Germany, and Spain.

This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.