Recent advances in topology optimization methods have driven the development of bandgap crystals. These artificial materials with maximized operational bandwidth provide the basis for wave manipulation and investigating the topological phase of matter. However, it is still challenging to efficiently design acoustic bandgap crystals via existing topology optimization methods. Most previous studies considering only a volume fraction constraint on the constituent material may have impractical wide band gaps due to the pseudo-air resonant modes. To solve this issue, this paper establishes a new topology optimization method for creating acoustic bandgap crystals periodically composed of solid and air. We adopt a constraint on air permeability to ensure the connective air channels within the periodic microstructures, which is more applicable in engineering practice. The optimized unit cells from the proposed method are further analyzed to realize topologically protected states, providing opportunities for multi-dimensional wave manipulation in acoustic systems. Numerical examples demonstrate the effectiveness of the proposed method in designing acoustic crystals with broad bandgaps on any given band orders, and gapped/gapless edge states and corner states can be achieved in resulting topological insulators.

We have developed a differentiable programming framework for truncated hierarchical B-splines (THB-splines), which can be used for several applications in geometry modeling, such as surface fitting and deformable image registration, and can be easily integrated with geometric deep learning frameworks. Differentiable programming is a novel paradigm that enables an algorithm to be differentiated via automatic differentiation, i.e., using automatic differentiation to compute the derivatives of its outputs with respect to its inputs or parameters. Differentiable programming has been used extensively in machine learning for obtaining gradients required in optimization algorithms such as stochastic gradient descent (SGD). While incorporating differentiable programming with traditional functions is straightforward, it is challenging when the functions are complex, such as splines. In this work, we extend the differentiable programming paradigm to THB-splines. THB-splines offer an efficient approach for complex surface fitting by utilizing a hierarchical tensor structure of B-splines, enabling local adaptive refinement. However, this approach brings challenges, such as a larger computational overhead and the non-trivial implementation of automatic differentiation and parallel evaluation algorithms. We use custom kernel functions for GPU acceleration in forward and backward evaluation that are necessary for differentiable programming of THB-splines. Our approach not only improves computational efficiency but also significantly enhances the speed of surface evaluation compared to previous methods. Our differentiable THB-splines framework facilitates faster and more accurate surface modeling with local refinement, with several applications in CAD and isogeometric analysis.

In this paper, we present a novel approach to geometry parameterization that we apply to the design of mixing elements for single-screw extruders. The approach uses neural networks of a specific architecture to automatically learn an appropriate parameterization. This stands in contrast to the so far common user-defined parameterizations. Geometry parameterization is crucial in enabling efficient shape optimization as it allows for optimizing complex shapes using only a few design variables. Recent approaches often utilize computer-aided design (CAD) data in conjunction with spline-based methods where the spline’s control points serve as design variables. Consequently, these approaches rely on the design variables specified by the human designer. This approach results in a significant amount of manual tuning to define a suitable parameterization. In addition, despite this effort, many times the optimization space is often limited to shapes in close proximity to the initial shape. In particular, topological changes are usually not feasible. In this work, we propose a method that circumvents this dilemma by providing low-dimensional, yet flexible shape parametrization using a neural network, which is independent of any computational mesh or analysis methods. Using the neural network for the geometry parameterization extends state-of-the-art methods in that the resulting design space is not restricted to user-prescribed modifications of certain basis shapes. Instead, within the same optimization space, we can interpolate between and explore seemingly unrelated designs. To show the performance of this new approach, we integrate the developed shape parameterization into our numerical design framework for dynamic mixing elements in plastics’ extrusion. Finally, we challenge the novel method in a competitive setting against current free-form deformation-based approaches and demonstrate the method’s performance even at this early stage.