Abstract-The Morse-Smale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalar-valued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closure-finite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divide-and-conquer strategy allows for memory-efficient computation of the MS complex and simplification on-the-fly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementation-specific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on off-the-shelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory.Index Terms-Topology-based analysis, Morse-Smale complex, large scale data.

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Abstract-Analysis of the results obtained from material simulations is important in the physical sciences. Our research was motivated by the need to investigate the properties of a simulated porous solid as it is hit by a projectile. This paper describes two techniques for the generation of distance fields containing a minimal number of topological features, and we use them to identify features of the material. We focus on distance fields defined on a volumetric domain considering the distance to a given surface embedded within the domain. Topological features of the field are characterized by its critical points. Our first method begins with a distance field that is computed using a standard approach, and simplifies this field using ideas from Morse theory. We present a procedure for identifying and extracting a feature set through analysis of the MS complex, and apply it to find the invariants in the clean distance field. Our second method proceeds by advancing a front, beginning at the surface, and locally controlling the creation of new critical points. We demonstrate the value of topologically clean distance fields for the analysis of filament structures in porous solids. Our methods produce a curved skeleton representation of the filaments that helps material scientists to perform a detailed qualitative and quantitative analysis of pores, and hence infer important material properties. Furthermore, we provide a set of criteria for finding the "difference" between two skeletal structures, and use this to examine how the structure of the porous solid changes over several timesteps in the simulation of the particle impact.

Abstract-The Morse-Smale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the Morse-Smale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.