Shape optimisation with multiresolution subdivision surfaces and immersed finite elements (1510.02719v1)

Abstract: We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors. Based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness. During shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information. By virtue of the multiresolution editing semantics, updating the coarse control mesh vertex coordinates leads to large-scale geometry changes and, conversely, updating the fine control mesh coordinates leads to small-scale geometry changes. In our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum. This approach effectively prevents the appearance of non-physical boundary geometry oscillations and control mesh pathologies, like inverted elements. Independent of the fineness of the control mesh used for optimisation, on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution. With the immersed finite element method there is no need to maintain an analysis suitable domain mesh. In some of the presented two- and three-dimensional elasticity examples the topology derivative is used for creating new holes inside the domain.

• The paper introduces an optimization framework that integrates multiresolution subdivision surfaces with immersed finite elements for robust shape optimization.
• The methodology differentiates coarse and fine controls to induce large-scale and localized changes while avoiding non-physical boundary fluctuations.
• Computational examples in 2D and 3D validate improved design quality and scalability, with potential extensions to topology optimization.

Shape Optimization with Multiresolution Subdivision Surfaces and Immersed Finite Elements

This paper by Bandara, Rüberg, and Cirak presents an optimization methodology that marries multiresolution subdivision surfaces with immersed finite elements to effectively conduct shape optimization in two- and three-dimensional scenarios. The core premise is to leverage the advantages of subdivision surfaces for geometric representation and immersed finite elements for efficient numerical solving of boundary and adjoint problems involved in shape optimization.

Methodology

The authors have developed an algorithm where domain boundaries are represented using a combination of coarse control meshes and detail vectors, akin to multiresolution wavelet decompositions. This multiresolution approach allows them to efficiently reconstruct control meshes of varying granularity, which is crucial for managing large-scale geometric changes during optimization. A key aspect of the approach is the distinction in influence when updating vertex coordinates on control meshes: coarse mesh edits induce large-scale changes, while fine mesh edits result in localized adjustments.

To compute shape gradients, the authors adopt an immersed finite element method that retains a finer representation of the boundary irrespective of the resolution used for optimization. The immersed approach avoids the need for mesh regeneration or smoothing typically required in conventional finite element methods. The combination of multiresolution subdivision editing techniques with immersed finite element analysis underpins the robustness and efficiency of the optimization process.

Results

The approach has been validated through several computational examples, involving two-dimensional and three-dimensional elasticity problems with applications to compliance minimization. Notably, the proposed technique prevents non-physical boundary oscillations and mesh pathologies such as inverted elements, ensuring stable and realistic optimized designs. Additionally, the authors explored topology optimization with the use of topology derivatives to introduce new holes within the domain, which is a valuable extension allowing granularity in optimizing structural designs.

Implications and Future Developments

The use of multiresolution representations in optimization opens several exciting avenues for future research. In practical applications, this approach facilitates a seamless transition from design to analysis with CAD systems, potentially reducing design iteration times. Furthermore, the adaptability of control mesh resolutions has implications for aesthetic and manufacturability considerations in engineering design.

Theoretically, the method offers a pathway to integrating more complex topologies and boundary conditions within optimization loops without encountering traditional numerical instabilities. The authors suggest that further exploration into non-uniform rational B-splines (NURBS) and adaptive geometry refinements may extend the reach of these methods into more sophisticated engineering problems. Additionally, automating the generation and adaptation of subdivision control meshes based on topology derivatives could enhance scalability and efficiency.

In conclusion, the paper presents a comprehensive framework that effectively utilizes multiresolution subdivision surfaces and immersed finite elements in shape optimization. The methodology not only addresses computational efficiency but also aligns closely with modern geometrical design practices, highlighting its potential significance in engineering and design optimization domains.