## 1. Introduction

[2] One of the focuses of geodynamics is in understanding the long time (e.g. 100 Myr) evolution of the deformation of rocks in the crust, lithosphere, and or mantle. Traditionally, these processes have been studied via field based observations and analogue modeling. In conjunction with these approaches, the use of continuum mechanics to describe the dynamics of such materials, and numerical methods to approximate the underlying partial differential equations has also become a widely accepted technique within the Earth science community to study the deformation of geomaterials.

[3] To study the long time evolution, and large deformation of geomaterials, in the geodynamics community, the underlying continuum is assumed to be a very viscous, incompressible fluid (i.e. Stokes flow). In considering the evolution of brittle, and visco-elastic-plastic material over million year timescales, we invariably require numerical methods which (i) track material history subjected to large deformations, and (ii) continue to follow the material evolution post failure. To accommodate these requirements, a hybrid Eulerian-Lagrangian (HEL) methodology has been exploited by the geodynamics community [*Pracht*, 1971; *Poliakov and Podladchikov*, 1992; *Fullsack*, 1995; *van Keken et al.*, 1997; *Moresi et al.*, 2003; *Gerya and Yuen*, 2003; *Tackley and King*, 2003; *Schreurs et al.*, 2006; *Popov and Sobolev*, 2008; *Thieulot*, 2011]. The principal of the methodology is to separate the discretization used to track the deforming material and the flow variables. The velocity and pressure variables associated with Stokes equations are discretized on an Eulerian grid, while the complete material description is defined via a set of Lagrangian particles. Attributed to each particle with coordinates *x*_{p}, is a material index *χ*, indicating the particular lithology (or composition) which the particle belongs to. We denote this via *χ*(*x*_{p}) = 1, 2, …, *N*_{L}, where *N*_{L} is the maximum number of lithologies in the model. All material history variables are tracked by the Lagrangian particles. Extreme deformation of the material is trivially handled by the particulate representation adopted as there is no connectivity between the particles. Note that in this formulation, material interfaces are not explicitly represented by the particles. An illustration of the HEL methodology is shown in Figure 1.

[4] Due to advances in both computer hardware and software, together with continued research and educational efforts, two-dimensional numerical modeling of geodynamic processes is now common practice in Earth science. Over the last 15 years, much research has been focused on the development of improving the efficiency and scalability of three-dimensional HEL methodologies. Previously, the ability to perform simulations in three-dimensions was restricted to dedicated research groups with access to specialized, high performance computing (HPC) hardware. However, with the advent of affordable, high performance computer clusters, performing 3D computational geodynamic simulations has become widespread and is no longer restricted to the domain of experts, or researchers based at HPC centers.

[5] One of the crucial observables from the output of multiphase geodynamic systems is the evolution of the material configuration. Over Myr timescales, the material within the system will invariably have experienced extremely large deformation, thus large amounts of mixing will have occurred. To illustrate the extend to which stirring and mixing occurs over different length scales in geodynamic models, we refer to simulations results in Figure 2. From inspection of Figure 2, the complexity of the lithological structures (denoted via different colors) which require visualization are inherently more topologically complex than scalar (or vector) fields such as temperature (or velocity). Given the HEL methods currently being adopted in geodynamics, visualization of the lithological structures mandates visualization of particle based data sets. In contrast to the research efforts devoted to improving the 3D modeling capabilities, comparatively less attention in the geodynamics community has been focused on the development of visualization techniques which are appropriate for high-resolution, three-dimensional particle fields. This issue becomes more critical as the usage of three-dimensional geodynamic models continues to increase. In this paper we aim to address this shortcoming by describing a visualization technique which can transform a high-resolution, 3D particle based lithology representation, into a solid volume representation.

[6] The outline of the paper is as follows. In section 2 we briefly overview surface and volume reconstruction techniques which are applicable for analyzing volumetric point based data set which define material lithology. In section 3 we discuss efficient techniques for generating Approximate Voronoi Diagrams (AVDs) to generate a volumetric representation of the material lithology. In section 4we demonstrate the type of images one can generate using the AVD technique by analyzing the simulation results of three-dimensional models of continental collision, salt tectonics and a high resolution synthetic model utilizing one billion particles. Additionally, we also profile the performance characteristics of the AVD implementation used to build the volumetric representations of the material configuration. Here comparisons between the new AVD algorithm and existing Voronoi algorithms from the literature are compared in terms of CPU and memory requirements. Last insection 5 we provide a summary of the AVD methodology.