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.@en

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.@en

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.@en

Density-based topology optimization using global and local volume constraints is a key technique to automatically design lightweight structures. It is known that stiffness optimal structures comprise spatially varying geometric patterns that span multiple length scales. However, both variants of topology optimization have challenges to efficiently converge to such a structural layout. In this paper, we investigate material layouts that are generated from stress trajectories, i.e., to compile a globally consistent structure by tracing the stress trajectories from finite element simulation of the solid design domain under external loads. This is particularly appealing from a computational perspective, since it avoids iterative optimization that involves finite element analysis on fine meshes. By regularizing the thickness of each trajectory using derived strain energy measures along them, stiff structural layouts can be generated in a highly efficient way. We then shed light on the use of the resulting structures as initial density fields in density-based topology optimization, i.e., to generate an initial density field that is then further optimized via topology optimization. We demonstrate that by using a stress trajectory guided density initialization in lieu of a uniform density field, convergence issues in density-based topology optimization can be significantly relaxed at comparable stiffness of the resulting structural layouts.

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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.

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We present a visualization technique for 2D stress tensor fields based on the construction of a globally conforming lattice. Conformity ensures that the lattice edges follow the principal stress directions and the aspect ratio of lattice elements represents the stress anisotropy. Since such a lattice structure cannot be space-filling in general, it is constructed from multiple intersecting lattice beams. Conformity at beam intersections is ensured via a constrained optimization problem, by computing the aspect ratio of elements at intersections so that their edges meet when continued along the principal stress lines. In combination with a coloring scheme that encodes relative stress magnitudes, a global visualization is achieved. By introducing additional constraints on the positional variation of the beam intersections, coherent visualizations are achieved when external loads or material parameters are changed. In a number of experiments using non-trivial scenarios, we demonstrate the capability of the proposed visualization technique to show the global and local structure of a given stress field.

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