This paper presents an efficient MATLAB framework for large-scale density-based topology optimization and porous infill optimization in 3D. Besides showing comparable computational efficiency with existing MATLAB implementations at equivalent simulation scales, this framework supports significantly larger models with up to 128 million hexahedral simulation elements on a standard PC equipped with 64 GB RAM. Furthermore, it can handle arbitrary non-cuboid design domains and does not require powers-of-two differences in the elements’ spatial resolutions. To achieve this, the technical contribution concentrates on solving the linear system of static finite element method (FEM). A tailored element-based matrix-free computing stencil is demonstrated to circumvent the vast memory consumption in large-scale FEM. Its computational efficiency is assured by fully leveraging the efficient matrix–vector operations and indexing functionalities in MATLAB. We further improve the computational efficiency and memory consumption of the MATLAB-implemented geometric multigrid method with a non-dyadic Galerkin coarsening and a diagonal relaxation scheme.

Designs generated by topology optimization are often geometrically too complex for conventional manufacturing techniques. While additive manufacturing holds promise for producing such complex designs, several manufacturability constraints must be addressed, including overhang and overheating. Unlike the well-studied overhang constraints, which can be described geometrically, overheating lacks a straightforward and reliable geometric characterization and therefore requires thermal process simulations to identify regions prone to it. However, these simulations are computationally expensive and thus unsuitable for topology optimization, which involves numerous design evaluations. This paper proposes a computationally efficient alternative for detecting zones prone to overheating. The key idea is to estimate local thermal conductivity—and thereby potential overheating—by analyzing the local material distribution. This geometric approach provides a physically motivated approximation of thermal behavior. The method is then integrated into topology optimization, resulting in optimized structures that exhibit clear heat conduction paths to the baseplate. Comparisons with high-fidelity thermal simulations demonstrate the effectiveness and efficiency of the proposed method in mitigating overheating in topology optimization.

Residual stresses and distortions are major barriers to the broader adoption of wire arc additive manufacturing. These issues are coupled and arise due to large thermal gradients and phase transformations during the directed energy deposition process. Mitigating distortions may lead to substantial residual stresses, causing cracks in the fabricated components. In this paper, we propose a novel method to reduce both residual stresses and distortions by optimizing the fabrication sequence. This approach explores the use of non-planar layers, leveraging the increased manufacturing flexibility provided by robotic arms. Additionally, our method allows for the concurrent optimization of the structural layout and corresponding fabrication sequence. We employ the inherent strain method as a simplified process simulation model to predict residual stresses and distortions. Local residual stresses are aggregated using a p-norm function, which is integrated into distortion minimization as a constraint. Through numerical examples, we demonstrate that the optimized non-planar fabrication strategies can effectively reduce both residual stresses and distortions.

We introduce the Stress-Guided Lightweight Design Benchmark (SGLDBench), a comprehensive benchmark suite for applying and evaluating material layout strategies to generate stiff, lightweight designs in 3D domains. SGLDBench provides a seamlessly integrated simulation and analysis framework, including six reference strategies and a scalable multigrid elasticity solver to efficiently execute these strategies and validate the stiffness of their results. This facilitates the systematic analysis and comparison of design strategies based on the mechanical properties they achieve. SGLDBench enables the evaluation of diverse load conditions and, through the tight integration of the solver, supports high-resolution designs and stiffness analysis. Additionally, SGLDBench emphasizes visual analysis to explore the relationship between the geometric structure of a design and the distribution of stresses, offering insights into the specific properties and behaviors of different design strategies. SGLDBench's specific features are highlighted through several experiments, comparing the results of reference strategies with respect to geometric and mechanical properties.

Concrete casting, a cornerstone of construction, relies on formwork to shape structures and has been used to create topology-optimized lightweight designs. The interplay between cast structures and formwork necessitates balancing casting constraints, particularly filling efficiency in topological channels, to avoid defects that compromise performance. However, filling efficiency is often empirically addressed, limiting optimization potential. Traditional methods require extensive post-processing to improve filling efficiency, increasing costs and design time. This work introduces a process simulation-informed reverse topology optimization framework, integrating casting constraints into the design process. The framework combines topology optimization, Discrete Element Method (DEM) simulation, and filling ratio identification. Its effectiveness is demonstrated through 2D numerical examples, experimental validation, and a preliminary 3D extension. Results show that the optimized structures improve filling efficiency and allow customizable casting positions. This approach offers a novel strategy for formwork optimization, enhancing efficiency and reducing costs in the building industry.

Additive manufacturing (AM) facilitates the production of complex structures. It is often combined with structural design by topology optimization to create lightweight structures with minimal material and maximum stiffness. In this paper, we consider the design of lightweight structures to be created by casting in formwork that is produced by additive manufacturing. This problem, arising from the building industry, relates to but differs from prior work on topology optimization for structures that are directly produced by additive manufacturing. Specifically, formwork is not permitted to contain extra supports in casting space since otherwise it results in casting blockages. Moreover, topological structures with cavities cannot be produced through casting. This work presents a topology optimization method for designing structures to be cast in AM-produced formwork. This approach addresses these two key challenges: (i) ensuring the formwork is self-supporting during printing to eliminate the need for additional supports, and (ii) designing the structure to be free of internal enclosed cavities, which would otherwise lead to disconnected or floating parts in the formwork. The effectiveness of the proposed method was demonstrated through several numerical examples and experimental evaluations. Results show that the formwork can be printed without extra supports, and internal enclosed cavities in optimized structures can be fully eliminated. The findings provide a new strategy to produce the lightweight structure and corresponding structural formwork.

Maintaining the maximum stiffness of components with as little material as possible is an overarching objective in computational design and engineering. It is well-established that in stiffness-optimal designs, material is aligned with orthogonal principal stress directions. In the limit of material volume, this alignment forms micro-structures resembling quads or hexahedra. Achieving a globally consistent layout of such orthogonal micro-structures presents a significant challenge, particularly in three-dimensional settings. In this paper, we propose a novel geometric algorithm for compiling stress-aligned hexahedral lattice structures. Our method involves deforming an input mesh under load to align the resulting stress field along an orthogonal basis. The deformed object is filled with a hexahedral grid, and the deformation is reverted to recover the original shape. The resulting stress-aligned mesh is used as basis for a final hollowing procedure, generating a volume-reduced stiff infill composed of hexahedral micro-structures. We perform quantitative comparisons with structural optimization and hexahedral meshing approaches and demonstrate the superior mechanical performance of our designs with finite element simulation experiments.

In additive manufacturing, the fabrication sequence has a large influence on the quality of manufactured components. While planning of the fabrication sequence is typically performed after the component has been designed, recent developments have demonstrated the possibility and benefits of simultaneous optimization of both the structural layout and the corresponding fabrication sequence. This is particularly relevant in multi-axis additive manufacturing, where rotational motion offers enhanced flexibility compared to planar fabrication. The simultaneous optimization approach, called space–time topology optimization, introduces a pseudo-time field to encode the manufacturing process order, alongside a pseudo-density field representing the structural layout. To comply with manufacturing principles, the pseudo-time field needs to be monotonic, i.e., free of local minima. However, explicitly formulated constraints proposed in prior work are not always effective, particularly for complex structural layouts that commonly result from topology optimization. In this paper, we introduce a novel method to regularize the pseudo-time field in space–time topology optimization. We conceptualize the monotonic additive manufacturing process as a virtual heat conduction process starting from the surface upon which a component is constructed layer by layer. The virtual temperature field, which shall not be confused with the actual temperature field during manufacturing, serves as an analogy for encoding the fabrication sequence. In this new formulation, we use local virtual heat conductivity coefficients as optimization variables to steer the temperature field and, consequently, the fabrication sequence. The virtual temperature field is inherently free of local minima due to the physics it resembles. We numerically validate the effectiveness of this regularization in space–time topology optimization under process-dependent loads, including gravity and thermomechanical loads.

Mechanical metamaterials signify a groundbreaking leap in material science and engineering. The intricate and experience-dependent design process poses a challenge in uncovering architectural material sequences with exceptional mechanical properties. This study introduces inverse-designed 3D sequential metamaterials with outstanding mechanical attributes, achieved through a novel computational framework. The explored sequences based on Schoen's I-graph–wrapped package (IWP) and Schwarz Primitive (Schwarz P) surpass the Hashin-Shtrikman upper bound of Young's modulus at relative densities of 0.24 and 0.43, outperforming previous records. Optimized Body-Centered-Cubic (BCC) truss-based sets outperform traditional ones by 72.7%. This innovative approach can be extended for metamaterial customization, involving the optimization of multi-directional Young's modulus, total stiffness, and the addition of isotropy constraints. The paper explores the characteristics and implications of this innovation, emphasizing the impact of geometric and topological variations on mechanical performance. These metamaterial sequences offer unparalleled adaptability, and hold significant potential in structural engineering and adaptive mechanical systems, opening avenues for technological advancements.

Recent years have seen a growing interest in topology optimization of functionally graded microstructures, characterized by an array of microstructures with varying volume fractions. However, microstructures optimized at slightly different volume fractions do not necessarily connect well when placed adjacently. Furthermore, optimization is commonly performed on a finite set of volume fractions, limiting the number of microstructure configurations. In this paper, we introduce the concept of differentiable microstructures, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. To construct such microstructures, we propose a novel formulation for topology optimization. In this approach, a series of 2-dimensional microstructures is represented using a height field, and the objective is to maximize the bulk modulus of the entire series. Through this optimization process, an initial microstructure with a small volume fraction undergoes non-uniform transformations, generating a series of microstructures with progressively increasing volume fractions. Notably, when compared to traditional uniform morphing methods, our proposed optimization approach yields a series of microstructures with bulk moduli that closely approach the theoretical limit.

Aligning lattice infills with the principal stress directions in loaded objects is crucial for improving stiffness. However, this principle only works for a single loading condition, where the stress field in 2D is described by two orthogonal principal stress directions. In this paper, we introduce a novel approach for designing and optimizing triangular lattice structures to accommodate multiple loading conditions, i.e., multiple stress fields need to be considered. Our method comprises two main steps: homogenization-based topology optimization and geometry-based de-homogenization. To ensure geometric regularity of the triangular lattices, we propose a simplified version of the general rank-3 laminate and parameterize the design domain using equilateral triangles with unique edge thickness. During optimization, edge thicknesses and orientations are adjusted based on the homogenized properties of the lattice. Our numerical findings demonstrate that this simplification introduces only a slight decrease in stiffness of less than 5% compared to using the general rank-3 laminate, and results in lattice structures with compelling geometric regularity. For geometry-based de-homogenization, we adopt a field-aligned triangulation approach to generate a globally consistent triangle mesh in which each triangle is oriented according to the optimized orientation field. Our approach for handling multiple loading conditions, akin to de-homogenization techniques for single loading conditions, yields highly detailed, optimized and spatially varying lattice structures. The method is computationally efficient, as simulations and optimizations are conducted at a low-resolution discretization of the design domain. Furthermore, since our approach is geometry-based, obtained structures are encoded into a compact geometric format that facilitates downstream operations such as editing and fabrication.

Wire and Arc Additive Manufacturing (WAAM) has great potential for efficiently producing large metallic components. However, like other additive manufacturing techniques, materials processed by WAAM exhibit anisotropic properties. Assuming isotropic material properties in design optimization thus leads to less efficient material utilization. Instead of viewing WAAM-induced material anisotropy as a limitation, we consider it an opportunity to improve structural performance. This requires the integration of process planning into structural design. In this paper, we propose a novel method to utilize material anisotropy to enhance the performance of structures both during fabrication and in their use. Our approach is based on space–time topology optimization, which simultaneously optimizes the structural layout and the fabrication sequence. To model material anisotropy in space–time topology optimization, we derive the material deposition direction from the gradient of the pseudo-time field, which encodes the fabrication sequence. Numerical results demonstrate that leveraging material anisotropy effectively improves the performance of intermediate structures during fabrication as well as the overall structure.

Effective treatment of large acetabular defects remains among the most challenging aspects of revision total hip arthroplasty (THA), due to the deficiency of healthy bone stock and degradation of the support columns. Generic uncemented components, which are favored in primary THA, are often unsuitable in revision cases, where the bone-implant contact may be insufficient for fixation, without significant reaming of the limited residual bone. This study presents a computational design strategy for automatically generating patient-specific implants that simultaneously maximize the bone-implant contact area, and minimize bone reaming while ensuring insertability. These components can be manufactured using the same additive manufacturing methods as porous components and may reduce cost and operating-time, compared to existing patient-specific systems. This study compares the performance of implants generated via the proposed method to optimally fitted hemispherical implants, in terms of the achievable bone-implant contact surface, and the volume of reamed bone. Computer-simulated results based on the reconstruction of a set of 15 severe pelvic defects (Paprosky 2A-3B) suggest that the patient-specific components increase bone-implant contact by 63% (median: 63%; SD: 44%; 95% CI: 52.3%–74.0%; RMSD: 42%), and reduce the volume of reamed bone stock by 97% (median: 98%; SD: 4%; 95% CI: 95.9%–97.4%; RMSD: 3.7%).

We present a novel dehomogenization approach for the efficient design of high-resolution load-bearing structures. The proposed approach builds upon a streamline-based parametrization of the design domain, using a set of space-filling and evenly spaced streamlines in the two mutually orthogonal direction fields that are obtained from homogenization-based topology optimization. Streamlines in these fields are converted into a graph, which is then used to construct a quad-dominant mesh whose edges follow the direction fields. In addition, the edge width is adjusted according to the density and anisotropy of the optimized orthotropic cells. In a number of numerical examples, we demonstrate the mechanical performance and regular appearance of the resulting structural designs and compare them with those from classic and contemporary approaches.

Additive manufacturing of metal parts involves phase transformations and high temperature gradients which lead to uneven thermal expansion and contraction, and, consequently, distortion of the fabricated components. The distortion has a great influence on the structural performance and dimensional accuracy, e.g., for assembly. It is therefore of critical importance to model, predict and, ultimately, reduce distortion. In this paper, we present a computational framework for fabrication sequence optimization to minimize distortion in multi-axis additive manufacturing (e.g., robotic wire arc additive manufacturing), in which the fabrication sequence is not limited to planar layers only. We encode the fabrication sequence by a continuous pseudo-time field, and optimize it using gradient-based numerical optimization. To demonstrate this framework, we adopt a computationally tractable yet reasonably accurate model to mimic the material shrinkage in metal additive manufacturing and thus to predict the distortion of the fabricated components. Numerical studies show that optimized curved layers can reduce distortion by orders of magnitude as compared to their planar counterparts.

Patient-specific implants offer a host of benefits over their generic counterparts. Nonetheless, the design and optimization of these components present several technical challenges, among them being the need to ensure their insertability into the host bone tissue. This presents a significant challenge due to the tight-fitting nature of the bone-implant interface. This paper presents a novel insertability metric designed to efficiently assess whether a rigid body can be inserted into a tight-fitting cavity, without interference. In contrast to existing solutions, the metric is fully differentiable and can be incorporated as a design constraint into shape optimization routines. By exploiting the tight-fitting condition, the problem of planning an interference-free insertion path is reformulated as the search for a single interference-free movement, starting from the inserted configuration. We prove that if there exists any outward movement for which no interference is indicated, then the body can be fully extracted from or, equivalently, inserted into the cavity. This formulation is extremely efficient and highly robust with respect to the complexity of the geometry. We demonstrate the effectiveness and efficiency of our method by applying it to the optimization of two-dimensional (2D) and three-dimensional (3D) designs for insertability, subject to various design requirements. We then incorporate the proposed metric into the optimization of an acetabular cup used in total hip replacement (THR) surgery where geometric and structural requirements are considered.

The summer school is realized as a joint teaching project between several partner universities under the auspices of the IDEA League alliance. The school trains students and young researchers on exploring and applying the potential of computational-based strategies in design for additive manufacturing, while engaging in dedicated team working activities. The aim of this work is to report on and discuss the success of the summer school in terms of the learning goals achievement. The work is based on the results of a one-day workshop with the summer school alumni showing how the acquired knowledge, skills and competences impacted their studies, research, and professional careers in a long-term after the course is finished, and outlining possible course content improvements to plan the summer school in 2023 that will take place with physical presence.

In this paper, we present novel algorithms for visualizing the three mutually orthogonal principal stress directions in 3D solids under load and we discuss the efficient integration of these algorithms into the 3D Trajectory-based Stress Visualizer (3D-TSV), a visual analysis tool for the exploration of the principal stress directions of 3D stress field. In the design of 3D-TSV, several perceptual problems have been solved. We present a novel algorithm for generating a space-filling and evenly spaced set of stress lines. The algorithm obtains a more
regular appearance by considering the locations of lines, and enables the extraction of a level-of-detail representation with adjustable sparseness of the trajectories along a certain stress direction. A new combined visualization of two principal directions via oriented ribbons enables to convey ambiguities in the orientation of the principal stress directions. Additional depth cues have been added to improve the perception of the spatial relationships between trajectories. 3D-TSV provides a modular and generic implementation of key algorithms required for a trajectory-based visual analysis of principal stress directions, including the automatic seeding of space-filling stress lines, their extraction using numerical schemes, their mapping to an effective renderable
representation, and rendering options to convey structures with special mechanical properties. 3D-TSV is accessible to end users via a C++- and OpenGL-based rendering frontend that is seamlessly connected to a
MatLab-based extraction backend. The code (BSD license) of 3D-TSV as well as scripts to make ANSYS and ABAQUS simulation results accessible to the 3D-TSV backend are publicly available.

Personalized designs bring added value to the products and the users. Meanwhile, they also pose challenges to the product design process as each product differs. In this paper, with the focus on personalized fit, we present an overview as well as details of the personalized design process based on design practice. The general workflow of personalized product design is introduced first. Then different steps in the workflow such as human data/parameters acquisition, computational design, design for digital fabrication, and product evaluation are presented. Tools and methods that are often used in different steps in the process are also outlined where in human data acquisition, 3D scanning, and digital human models are addressed. For computational design, the use of computational thinking tools such as abstraction, decomposition, pattern recognition and algorithms are discussed. In design for digital fabrication, additive manufacturing methods (e.g. FDM), and their requirements on the design are highlighted. For product evaluation, both functional evaluation and usability evaluation are considered and the evaluation results can be the starting point of the next design iteration. Finally, several case studies are presented for a better understanding of the workflow, the importance of different steps in the workflow and the deviations in the approach regarding different contexts. In conclusion, we intend to provide designers a holistic view of the design process in designing personalized products as well as help practitioners trigger innovations regarding each step of the process.

The optimization of porous infill structures via local volume constraints has become a popular approach in topology optimization. In some design settings, however, the iterative optimization process converges only slowly, or not at all even after several hundreds or thousands of iterations. This leads to regions in which a distinct binary design is difficult to achieve. Interpreting intermediate density values by applying a threshold results in large solid or void regions, leading to sub-optimal structures. We find that this convergence issue relates to the topology of the stress tensor field that is simulated when applying the same external forces on the solid design domain. In particular, low convergence is observed in regions around so-called trisector degenerate points. Based on this observation, we propose an automatic initialization process that prescribes the topological skeleton of the stress field into the density field as solid simulation elements. These elements guide the material deposition around the degenerate points, but can also be remodelled or removed during the optimization. We demonstrate significantly improved convergence rates in a number of use cases with complex stress topologies. The improved convergence is demonstrated for infill optimization under homogeneous as well as spatially varying local volume constraints.