A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

Optimal Direction-Dependent Path Planning for Autonomous Vehicles

The focus of this thesis is optimal path planning. The path planning problem is posed as an optimal control problem, for which the viscosity solution to the static Hamilton-Jacobi-Bellman (HJB) equation is used to determine the optimal path. The Ordered Upwind Method (OUM) has been previously used to numerically approximate the viscosity solution of the static HJB equation for direction-dependent weights. The contributions of this thesis include an analytical bound on the convergence rate of the OUM for the boundary value problem to the viscosity solution of the HJB equation. The convergence result provided in this thesis is to our knowledge the tightest existing bound on the convergence order of OUM solutions to the viscosity solution of the static HJB equation. Only convergence without any guarantee of rate has been previously shown. Navigation functions are often used to provide controls to robots. These functions can suffer from local minima that are not also global minima, which correspond to the inability to find a path at those minima. Provided the weight function is positive, the viscosity solution to the static HJB equation cannot have local minima. Though this has been discussed in literature, a proof has not yet appeared. The solution of the HJB equation is shown in this work to have no local minima that is not also global. A path can be found using this method. Though finding the shortest path is often considered in optimal path planning, safe and energy efficient paths are required for rover path planning. Reducing instability risk based on tip-over axes and maximizing solar exposure are important to consider in achieving these goals. In addition to obstacle avoidance, soil risk and path length on terrain are considered. In particular, the tip-over instability risk is a direction-dependent criteria, for which accurate approximate solutions to the static HJB equation cannot be found using the simpler Fast Marching Method. An extension of the OUM to include a bi-directional search for the source-point path planning problem is also presented. The solution is found on a smaller region of the environment, containing the optimal path. Savings in computational time are observed. A comparison is made in the path planning problem in both timing and performance between a genetic algorithm rover path planner and OUM. A comparison in timing and number of updates required is made between OUM and several other algorithms that approximate the same static HJB equation. Finally, the OUM algorithm solving the boundary value problem is shown to converge numerically with the rate of the proven theoretical bound

Higher-Order DGFEM Transport Calculations on Polytope Meshes for Massively-Parallel Architectures

In this dissertation, we develop improvements to the discrete ordinates (S_N) neutron transport equation using a Discontinuous Galerkin Finite Element Method (DGFEM) spatial discretization on arbitrary polytope (polygonal and polyhedral) grids compatible for massively-parallel computer architectures. Polytope meshes are attractive for multiple reasons, including their use in other physics communities and their ease in handling local mesh refinement strategies. In this work, we focus on two topical areas of research. First, we discuss higher-order basis functions compatible to solve the DGFEM S_N transport equation on arbitrary polygonal meshes. Second, we assess Diffusion Synthetic Acceleration (DSA) schemes compatible with polytope grids for massively-parallel transport problems. We first utilize basis functions compatible with arbitrary polygonal grids for the DGFEM transport equation. We analyze four different basis functions that have linear completeness on polygons: the Wachspress rational functions, the PWL functions, the mean value coordinates, and the maximum entropy coordinates. We then describe the procedure to extend these polygonal linear basis functions into the quadratic serendipity space of functions. These quadratic basis functions can exactly interpolate monomial functions up to order 2. Both the linear and quadratic sets of basis functions preserve transport solutions in the thick diffusion limit. Maximum convergence rates of 2 and 3 are observed for regular transport solutions for the linear and quadratic basis functions, respectively. For problems that are limited by the regularity of the transport solution, convergence rates of 3/2 (when the solution is continuous) and 1/2 (when the solution is discontinuous) are observed. Spatial Adaptive Mesh Refinement (AMR) achieved superior convergence rates than uniform refinement, even for problems bounded by the solution regularity. We demonstrated accuracy in the AMR solutions by allowing them to reach a level where the ray effects of the angular discretization are realized. Next, we analyzed DSA schemes to accelerate both the within-group iterations as well as the thermal upscattering iterations for multigroup transport problems. Accelerating the thermal upscattering iterations is important for materials (e.g., graphite) with significant thermal energy scattering and minimal absorption. All of the acceleration schemes analyzed use a DGFEM discretization of the diffusion equation that is compatible with arbitrary polytope meshes: the Modified Interior Penalty Method (MIP). MIP uses the same DGFEM discretization as the transport equation. The MIP form is Symmetric Positive De_nite (SPD) and e_ciently solved with Preconditioned Conjugate Gradient (PCG) with Algebraic MultiGrid (AMG) preconditioning. The analysis from previous work was extended to show MIP's stability and robustness for accelerating 3D transport problems. MIP DSA preconditioning was implemented in the Parallel Deterministic Transport (PDT) code at Texas A&M University and linked with the HYPRE suite of linear solvers. Good scalability was numerically verified out to around 131K processors. The fraction of time spent performing DSA operations was small for problems with sufficient work performed in the transport sweep (O(10^3) angular directions). Finally, we have developed a novel methodology to accelerate transport problems dominated by thermal neutron upscattering. Compared to historical upscatter acceleration methods, our method is parallelizable and amenable to massively parallel transport calculations. Speedup factors of about 3-4 were observed with our new method

Cortical Surface Registration and Shape Analysis

A population analysis of human cortical morphometry is critical for insights into brain development or degeneration. Such an analysis allows for investigating sulcal and gyral folding patterns. In general, such a population analysis requires both a well-established cortical correspondence and a well-defined quantification of the cortical morphometry. The highly folded and convoluted structures render a reliable and consistent population analysis challenging. Three key challenges have been identified for such an analysis: 1) consistent sulcal landmark extraction from the cortical surface to guide better cortical correspondence, 2) a correspondence establishment for a reliable and stable population analysis, and 3) quantification of the cortical folding in a more reliable and biologically meaningful fashion. The main focus of this dissertation is to develop a fully automatic pipeline that supports a population analysis of local cortical folding changes. My proposed pipeline consists of three novel components I developed to overcome the challenges in the population analysis: 1) automatic sulcal curve extraction for stable/reliable anatomical landmark selection, 2) group-wise registration for establishing cortical shape correspondence across a population with no template selection bias, and 3) quantification of local cortical folding using a novel cortical-shape-adaptive kernel. To evaluate my methodological contributions, I applied all of them in an application to early postnatal brain development. I studied the human cortical morphological development using the proposed quantification of local cortical folding from neonate age to 1 year and 2 years of age, with quantitative developmental assessments. This study revealed a novel pattern of associations between the cortical gyrification and cognitive development.Doctor of Philosoph

Simulation and Analysis of Unconventional Reservoirs Using Fast Marching Method and Transient Drainage Volume

Unconventional tight/shale reservoirs have become an important component of the world’s energy map in the recent decade and have been attracting a lot of interests in both academia and industry. However, the industry today still faces significant challenges in understanding the fundamental mechanisms. Unconventional tight/shale reservoirs are characterized by low or ultra-low permeability, such that the transient pressure behavior might last throughout the production lifetime. Recent research has proposed a novel approach for unconventional reservoir analysis based on the high-frequency asymptotic approximation of diffusivity equation. By solving the Eikonal equation with the Fast Marching Method (FMM), one can rapidly obtain the diffusive time of flight (DToF) which depicts the pressure transient propagation process. A fast DToF-based forward simulation is further proposed to solve the fluid flow equation in a 1D equivalent coordinate system, with the DToF as the spatial coordinate. In this study, we first adopt the DToF-based simulation as a rapid forward simulator to formulate an efficient hydraulic fracture design and optimization workflow. The DToF-based simulation can be orders of magnitude faster than the conventional finite difference/volume based simulation, and is ideal for optimization process where hundreds or thousands of simulations are necessary. Our workflow focuses on optimizing the number of hydraulic fracture stages, their spacing, and the allocation of proppant. The workflow also accounts for the geologic uncertainty, which given by different natural fracture distributions. Next, we extend this DToF-based simulation from Cartesian and corner point grid system to unstructured grids to better characterize the complex fracture geometry induced by hydraulic fracturing job. Two different constructions of the local Eikonal equation solver, based on Fermat’s principle and Eulerian discretization, are investigated and compared. Numerical examples are presented to illustrate the power and validity of this extended DToF-based simulation workflow. Finally, we propose a model-free production data analysis method to analyze the performance of unconventional reservoirs when a full simulation model is not available. The transient drainage volume is derived directly based on bottom-hole pressure and production rate. We further define the drainage volume derivative and instantaneous recovery ratio, which can measure how effectively the hydraulic fractures have stimulated the reservoir. This technique is then applied to select candidate wells for refracturing

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal