A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e.,~a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.

We present a GPU algorithm for deformable simulation. Our method offers good computational efficiency and penetration-free guarantee at the same time, which are not common with existing techniques. The main idea is an algorithmic integration of projective dynamics (PD) and incremental potential contact (IPC). PD is a position-based simulation framework, favored for its robust convergence and convenient implementation. We show that PD can be employed to handle the variational optimization with the interior point method e.g., IPC. While conceptually straightforward, this requires a dedicated rework over the collision resolution and the iteration modality to avoid incorrect collision projection with improved numerical convergence. IPC exploits a barrier-based formulation, which yields an infinitely large penalty when the constraint is on the verge of being violated. This mechanism guarantees intersection-free trajectories of deformable bodies during the simulation, as long as they are apart at the rest configuration. On the downside, IPC brings a large amount of nonlinearity to the system, making PD slower to converge. To mitigate this issue, we propose a novel GPU algorithm named A-Jacobi for faster linear solve at the global step of PD. A-Jacobi is based on Jacobi iteration, but it better harvests the computation capacity on modern GPUs by lumping several Jacobi steps into a single iteration. In addition, we also re-design the CCD root finding procedure by using a new minimum-gradient Newton algorithm. Those saved time budgets allow more iterations to accommodate stiff IPC barriers so that the result is both realistic and collision-free. Putting together, our algorithm simulates complicated models of both solids and shells on the GPU at an interactive rate or even in real time.