SMI 2014Combined shape and topology optimization of 3D structures
Graphical abstract
Introduction
Topology optimization is the discipline of finding the optimal shape and topology of a structure [1], [2]. It can be used to solve a wide variety of design problems arising when producing diverse products such as cars, houses, computer chips and antennas. The manufacturers are often concerned with finding the stiffest structure, the lightest structure which does not break, the structure with the highest cooling effect, or the structure with the best flow or highest efficiency.
With the advances in 3D printing technology, topology optimization is not just of interest to manufacturers, but to anyone who has access to a 3D printer. Most consumers lack formal training in structural mechanics, which can hinder the process with many iterations and costly failed attempts. Consumers can under-engineer a design unsuitable for the intended load, or over-engineer a design that wastes expensive construction material. Topology optimization offers consumers a tool for designing shapes that meet their structural needs while using minimal construction resources.
In this paper, we present a fully automated design tool for designing structurally sound structures which can be manufactured, constructed or printed. The modeler only has to specify boundary conditions, the optimization objective, constraints and an initial structure. In other words, the designer specifies a set of requirements (the functionality of the structure and not the structure itself) and the method automatically designs a structure which fits those requirements. Note that this design process is significantly different from today where a designer manually models a structure and requirements are taken into account during this design process.
The proposed method for topology optimization is based on the Deformable Simplicial Complex (DSC) method [3]. The DSC method represents a solid structure with a conforming tetrahedral mesh (a simplicial complex) whose tetrahedral elements either lie entirely inside or outside the structure. The interface between solid and void (the surface) is represented explicitly by the triangular faces shared by an interior and exterior tetrahedral element. Furthermore, the DSC method ensures well-formed tetrahedral elements by constantly performing mesh improvement routines while the surface is being deformed. Finally, it provides adaptive resolution, allowing fine details where and when needed.
The method uses two optimization strategies:
- Discrete optimization
Relabels elements from solid to void to improve the objective or constraints which are not satisfied. The relabeling is based on topological derivatives [4], [5], [6], [7], [8], i.e. the change in the objective or constraints by introducing an infinitesimal hole.
- Continuous optimization
Performs a non-parametric shape optimization [9], [10], [11], [12], [13]. First, an improved shape, which is within a small perturbation of the current shape, is found by solving a constrained optimization problem using the Method of Moving Asymptotes (MMA) [14]. The surface is then deformed to this improved shape using the DSC method [3]. While the surface is deformed, the mesh is adapted such that its tetrahedral elements are well-formed at all times.
We will show that this tool is of interest to both engineers and designers. For example, we show that it can be used to improve stiffness and balance of a 3D model, to save material and to generate functional as well as, in our opinion, visually pleasing designs.
Recent trends in the computer graphics society are to add mechanical properties to 3D models. Prévost et al. have been concerned with the balance of printed models [15], Skouras et al. about printing deformable characters using a stiff and soft material [16] and several research teams have focused on self-supporting masonry structures [17], [18], [19].
A major concern has been to improve the stiffness of 3D models. Umetani et al. perform a cross-sectional structural analysis and visualize the result [20]. A user can then manually edit the model to improve the stiffness while getting almost instant feedback. The instant feedback is only possible because the analysis is limited to cross-sections. Stava et al. present a more automated method for improving stiffness [21]. They perform a complete worst-case structural analysis on a tetrahedral mesh to determine the structurally weak regions. Based on this analysis, it is decided whether to improve the model by thickening, hollowing or adding a strut. Finally, Zhou et al. [22] also perform a worst-case structural analysis with more precise determination of the worst-case loads than in [21]. Furthermore, they conclude that solving a shape optimization problem to minimize stress is impractical due to the non-linearity and non-convexity of the problem. Therefore, they make do with visualizing the structurally weak regions.
Topology optimization problems are indeed non-convex. However, the topology optimization community has been solving these problems to at least local optimality for decades and the resulting designs usually perform better than designs optimized by humans [2]. Feasible solutions to these problems are often found by standard numerical gradient-based optimization algorithms. However, note that the smooth compliance functional is often chosen as the objective function to ease the optimization instead of the non-smooth, but often more interesting, maximal stress as Zhou et al. propose.
A key ingredient in a topology optimization method is the shape representation which is required to be able to handle topology changes. Hence, topological optimization has focused primarily on implicit representations over uniform voxel grids. Such representations can handle topology changes but lead to fixed-resolution results with cuberille artifacts. The most popular implicit topology optimization approaches are the density and level set approaches. The density approach [23], [2] represents the structure by assigning a density value between 0 (void) and 1 (material) to each cell in a fixed grid or mesh. The structure is now deformed by changing these density values. The level set approach uses the level set method [24] evaluated on a fixed grid or mesh [25], [26]. Here, the structure is represented by the zero level set and deformed by changes to the level set function. Both methods iteratively change the shape to approach the optimum.
We propose to represent the surface explicitly. An explicit representation, for example a triangle mesh, has previously been used for shape optimization [9], [10]. However, shape optimization does not allow for topology changes and often only small shape deformations. Furthermore, it has been used in combination with the level set method [27], [28], [29], [30], [31] where it is necessary to constantly switch between the implicit and explicit representations. An explicit representation has also been used in combination with a computationally expensive remeshing of the entire design domain at each iteration [4], [32]. Finally, it has previously been shown that using the DSC method for topology optimization works in 2D and therefore has potential [33]. However, here, we show that this concept is able to solve real-world topology optimization problems in 3D.
Note that this list of structural optimization methods is far from exhaustive.
The main contributions of this paper are as follows.
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As opposed to previous methods introduced in computer graphics, our method automatically optimizes the shape and topology of structure given boundary conditions, an objective function, constraints and an initial shape. This completely eliminates the manual editing which has been characteristic for the current approaches.
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Compared to current methods from the topology optimization community, the method uses a single explicit representation to represent the structure and, at the same time, is able to handle topology changes. This gives rise to several advantages including a single mesh for shape representation and finite element calculations, possibility of both continuous and discrete optimization strategies and both the initial and optimized structure are in the form of surface triangle meshes. Finally, the adaptive mesh makes it possible to achieve a much more detailed result within reasonable time on an ordinary laptop than otherwise possible using the standard fixed grid methods.
- •
To be able to solve real-world topology optimization problems in 3D, it was necessary to make significant changes compared to the 2D proof-of-concept by Christiansen et al. [33]. Consequently, the discrete step relabels elements based on an optimization procedure which takes constraints into account instead of based on a simple threshold of the objective. Furthermore, the presented method handles self-weight, it is initialized by any surface triangle mesh, areas can be fixed to either solid or void and several global constraints have been implemented and utilized. Finally, the requirements for computational efficiency is much higher in 3D than 2D. Therefore, the mesh adaptivity of the DSC method is utilized and the computations are distributed on multiple cores.
Section snippets
Method
The proposed method uses a simplicial complex to represent the shape of a structure. A simplicial complex discretizes a domain into tetrahedral elements. In 3D it consists of the simplices; nodes (points), edges (line pieces), faces (triangles) and tetrahedra (triangular pyramids). Furthermore, the tetrahedra do not overlap and any point in the discretized domain is either inside a tetrahedron or on the boundary between tetrahedra. In addition, all tetrahedra are labeled as being either void
Results
The proposed method can be used in the fabrication design process in areas such as construction, manufacturing and design. In this section, we will illustrate this statement by solving problems within each of these fields. The results are generated on a laptop with a 2.4 GHz quad-core Intel Core i7 processor and 8 GB of 1333 MHz DDR3 RAM. Parameters and performance measures are depicted in Table 1. Furthermore, the objective of all examples is to minimize compliance subject to constraints as
Conclusion
The presented method is the first to optimize both the 3D shape and topology of a surface triangle mesh without the use of an implicit representation. This is achieved by embedding the triangle mesh in a simplicial complex and using the Deformable Simplicial Complex method. Consequently, the method accepts a surface triangle mesh as input and outputs another surface triangle mesh which is only different from the input mesh where it has been optimized. Furthermore, as opposed to standard fixed
Acknowledgments
The authors appreciate the support from the Villum Foundation through the grant: “NextTop”.
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