A three-dimensional implementation of the boundary element and level set based structural optimisation

Abstract

a b s t r a c tThis paper presents a three-dimensional structural optimisation approach based on the boundary element and level set methods. The structural geometry is implicitly represented with the level set method, which evolves an initial structural model towards an optimal configuration using an evolutionary structural optimisation approach. The boundary movements in the three-dimensional level set based optimisation method allow automatic hole nucleation through the intersection of two surfaces moving towards each other. This suggests that perturbing only the boundary can give rise to changes not only in shape, but also in topology. At each optimisation iteration, the Marching Cubes algorithm is used to extract the modified geometry (i.e. the zero level set contours) in the form of a triangular mesh. As the boundary element method is based on a boundary discretisation approach, the extracted geometry (in the form of a triangular mesh) can be directly analysed within it. However, some mesh smoothing is required; HC-Laplacian smoothing is a useful algorithm that overcomes the volumetric loss associated with simpler algorithms. This eliminates the need for an additional discretisation tool and provides a natural link between the implicitly represented geometry and its structural model throughout the optimisation process. A complete algorithm is proposed and tested for the boundary element and level set methods based topology optimisation in three-dimensions. Optimal geometries compare well against those in the literature for a range of benchmark examples.

A boundary element and level set based topology optimisation using sensitivity analysis

“…The stress based hole insertion criterion is further investigated for a possible correlation with a topological derivative based hole insertion mechanism in [23]. The BEM and LSM based evolutionary optimisation method is further extended for the solution of three-dimensional problems in [24].…”

A boundary element and level set based bi-directional evolutionary structural optimisation with a volume constraint

“…The three-dimensional BEM and LSM based BESO approach for a specific strain energy objective function without a constant volume implementation has been studied in the authors previous work as reported in [12]. Where it has been shown that the proposed approach can be easily extended to three-dimensions.…”

“…• the boundary element method (BEM) based boundary only discretisation The reduction of problem dimensionality with the use of BEM based body conforming mapping is very attractive as compared to the FEM based domain discretisation. In the literature of structural optimisation, researchers combined the BEM with the LSM for the solution of optimisation problems in both two and three-dimensions, e.g, compliance minimisation [10,11,12,13], sound scattering [14,15], heat conduction [16], etc.…”

“…However, due to the explicit geometry representation adopted in [26], special care is always required when hole merges with each other and with the boundary. The BE based BESO approach has been further integrated with the LSM for the solution of both two and three-dimensional optimisation problems in [12,28], which allows for the complex topological changes to take place automatically. As reported in the literature, the BE based BESO approaches largely eliminate the common problems occur in the FE based approaches, e.g.…”

Order-Randomized Laplacian Mesh Smoothing

“…Since Laplacian smoothing is simple, general and well-understood, it is often the technique of choice when a smoothing or noise suppression algorithm needs to be incorporated into a general mesh processing framework [10], or into a more complex algorithm, as for example the machine learning algorithm for surface reconstruction in [1], or the Boundary Element Method based evolutionary structural optimization in [17].…”