Interactive Isocontouring of High-Order Surfaces

Scientists and engineers are making increasingly use of hp-adaptive discretization methods to compute simulations. While techniques for isocontouring the high-order data generated by these methods have started to appear, they typically do not facilitate interactive data exploration. This work presents a novel interactive approach for approximate isocontouring of high-order data. The method is based on a two-phase hybrid rendering algorithm. In the first phase, coarsely seeded particles are guided by the gradient of the field for obtaining an initial sampling of the isosurface in object space. The second phase performs ray casting in the image space neighborhood of the initial samples. Since the neighborhood is small, the initial guesses tend to be close to the isosurface, leading to accelerated root finding and thus efficient rendering. The object space phase affects the density of the coarse samples on the isosurface, which can lead to holes in the final rendering and overdraw. Thus, we also propose a heuristic, based on dynamical systems theory, that adapts the neighborhood of the seeds in order to obtain a better coverage of the surface. Results for datasets from computational fluid dynamics are shown and performance measurements for our GPU implementation are given

ElVis: A system for the accurate and interactive visualization of high-order finite element solutions

pre-printThis paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through run time plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of uncertainty for engineers using ElVis

GPU-based volume visualization from high-order finite element fields

pre-printThis paper describes a new volume rendering system for spectral/hp finite-element methods that has as its goal to be both accurate and interactive. Even though high-order finite element methods are commonly used by scientists and engineers, there are few visualization methods designed to display this data directly. Consequently, visualizations of high-order data are generally created by first sampling the high-order field onto a regular grid and then generating the visualization via traditional methods based on linear interpolation. This approach, however, introduces error into the visualization pipeline and requires the user to balance image quality, interactivity, and resource consumption. We first show that evaluation of the volume rendering integral, when applied to the composition of piecewise-smooth transfer functions with the high-order scalar field, typically exhibits second-order convergence for a wide range of high-order quadrature schemes, and has worst case first-order convergence. This result provides bounds on the ability to achieve high-order convergence to the volume rendering integral. We then develop an algorithm for optimized evaluation of the volume rendering integral, based on the categorization of each ray according to the local behavior of the field and transfer function. We demonstrate the effectiveness of our system by running performance benchmarks on several high-order fluid-flow simulations

ElVis: A System of the Accurate and Interactive Visualization of High-Order Finite Element Solutions

This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with highorder finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA’s OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations–high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular–will demonstrate the superiority of ElVis’ pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers’ debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of uncertainty for engineers using ElVi

Doctor of Philosophy

dissertationHigh-order finite element methods, using either the continuous or discontinuous Galerkin formulation, are becoming more popular in fields such as fluid mechanics, solid mechanics and computational electromagnetics. While the use of these methods is becoming increasingly common, there has not been a corresponding increase in the availability and use of visualization methods and software that are capable of displaying visualizations of these volumes both accurately and interactively. A fundamental problem with the majority of existing visualization techniques is that they do not understand nor respect the structure of a high-order field, leading to visualization error. Visualizations of high-order fields are generally created by first approximating the field with low-order primitives and then generating the visualization using traditional methods based on linear interpolation. The approximation step introduces error into the visualization pipeline, which requires the user to balance the competing goals of image quality, interactivity and resource consumption. In practice, visualizations performed this way are often either undersampled, leading to visualization error, or oversampled, leading to unnecessary computational effort and resource consumption. Without an understanding of the sources of error, the simulation scientist is unable to determine if artifacts in the image are due to visualization error, insufficient mesh resolution, or a failure in the underlying simulation. This uncertainty makes it difficult for the scientists to make judgments based on the visualization, as judgments made on the assumption that artifacts are a result of visualization error when they are actually a more fundamental problem can lead to poor decision-making. This dissertation presents new visualization algorithms that use the high-order data in its native state, using the knowledge of the structure and mathematical properties of these fields to create accurate images interactively, while avoiding the error introduced by representing the fields with low-order approximations. First, a new algorithm for cut-surfaces is presented, specifically the accurate depiction of colormaps and contour lines on arbitrarily complex cut-surfaces. Second, a mathematical analysis of the evaluation of the volume rendering integral through a high-order field is presented, as well as an algorithm that uses this analysis to create accurate volume renderings. Finally, a new software system, the Element Visualizer (ElVis), is presented, which combines the ideas and algorithms created in this dissertation in a single software package that can be used by simulation scientists to create accurate visualizations. This system was developed and tested with the assistance of the ProjectX simulation team. The utility of our algorithms and visualization system are then demonstrated with examples from several high-order fluid flow simulations

A parametric modeling concept for predicting biomechanical compatibility in total hip arthroplasty

This work attempts to predict the long-term outcome of total hip arthroplasty based on available patient-specific information and possible installation positions of the prosthesis. For this purpose, a holistic modeling approach for the numerical simulation of osseointegration and long-term stability of endoprostheses, including possible prosthesis positions, is developed. In addition, new, efficient, and reliable methods for the numerical description of adaptive bone remodeling and osseointegration are proposed: The adaptive bone remodeling is described as a geometric-linear, material-nonlinear finite element model, following thermodynamically consistent material modeling guidelines. The resulting constitutive equations are expanded to describe osseointegration and transferred into a contact interface between bone and prosthesis. Finally, the results are projected to an imaging format that is easier to interpret for medical professionals, using a newly developed simulation for X-ray images. The inclusion of possible prosthesis positions spans an infinite-dimensional event space. Therefore, the model is reduced to a finite-dimensional surrogate model sampled with an adaptive sparse-grid collocation method. Without clinical validation, reliable statements cannot be made, and therefore the numerical examples given in this thesis can be regarded as proof of correct implementation and feasibility studies. This dissertation thus provides an answer to how much computational effort is required to provide a real digital decision aid in orthopedic surgery

Visualization of two-phase flow dynamics : techniques for droplet interactions, interfaces, and material transport

Computational visualization allows scientists and engineers to better understand simulation data and gain insights into the studied natural processes. Particularly in the field of computational fluid dynamics, interactive visual presentation is essential in the investigation of physical phenomena related to gases and liquids. To ensure effective analysis, flow visualization techniques must adapt to the advancements in the field of fluid dynamics that benefits substantially from the growing computational power of both commodity desktops and supercomputers on the one hand, and steadily expanding knowledge about fluid physics on the other. A prominent example of these advances can be found in the research of two-phase flow with liquid droplets and jets, where high performance computation and sophisticated algorithms for phase tracking enable well resolved and physically accurate simulations of liquid dynamics. Yet, the field of two-phase flow has remained largely unexplored in visualization research so far, leaving the scientists and engineers with a number of challenges when analyzing the data. These include the difficulty in tracking and investigating topological events in large droplet groups, high complexity of droplet dynamics due to the involved interfaces, and a limited choice of high quality interactive methods for the analysis of related transport phenomena. It is therefore the aim of this thesis to address these challenges by providing a multi-scale approach for the visual investigation of two-phase flow, with the focus on the analysis of droplet interaction, fluid interfaces, and material transport. To address the problem of analyzing highly complex two-phase flow simulations with droplet groups and jets, a linked-view approach with three-dimensional and abstract space-time graph representation of droplet dynamics is proposed. The interactive brushing and linking allows for general exploration of topological events as well as detailed inspection of dynamics in terms of oscillations and rotations of droplets. Another approach further examines the separation of liquid phases by segmenting liquid volumes according to their topological changes in future time. For visualization, boundary surfaces of these volume segments are extracted that reveal intricate details of droplet topology dynamics. Additionally, within this framework, visualization of advected particles corresponding to arbitrarily selected segment provides useful insights into the spatio-temporal evolution of the segment. The analysis of interfaces is necessary to understand the interplay of interface dynamics and the dynamics of droplet interactions. A commonly used technique for interface tracking in the volume of fluid-based simulations is the piecewise linear approximation which, although accurate, can affect the quality of the simulation results. To study the influence of the interface reconstruction on the phase tracking procedure, a visualization method is presented that extracts the interfaces by means of the first-order Taylor approximation, and provides several derived quantities that help assess the simulation results in relation to the interface reconstruction quality. The liquid interface is further investigated from the physical standpoint with an approach based on quantities derived from velocity and surface tension gradients. The developed method supports examination of surface tension forces and their impact on the interface instability, as well as detailed analysis of interface deformation characteristics. A line of research important for engineering applications is the analysis of electric fields on droplet interfaces. It is, however, complicated by higher-order elements used in the simulations to preserve field discontinuities. A visualization method has been developed that correctly visualizes these discontinuities at material boundaries. Additionally, the employed space-time representation of the droplet-insulator contact line reveals characteristics of electric field dynamics. The dynamics of droplets are often examined assuming single-phase flow, for instance when the internal material transport is of interest. From the visualization perspective, this allows for adaption of traditional vector field visualization techniques to the investigation of the studied phenomena. As one such concept, dye based visualization is proposed that extends the transport analysis to advection-diffusion problems, therefore revealing true transport behavior. The employed high quality advection preserves fine details of the dye, while the implementation on graphics processing units ensures interactive visualization. Several streamline-based concepts are applied in space-time representation of 2D unsteady flow. By interpreting time as the third spatial dimension, many 3D streamline-based visualization techniques can be applied to investigate 2D unsteady flow. The introduced vortex core ribbons support the examination of vortical flow behavior by revealing rotation near the core lines. For the study of topological structures, a method has been developed that extracts separatrices implicitly as boundaries of regions with different flow behavior, and therefore avoids potentially complicated explicit extraction of various topological structures. All proposed techniques constitute a novel multi-scale approach for visual analysis of two-phase flow. The analysis of droplet interactions is addressed with visualization of the phenomena leading to breakups and with detailed visual inspection of these breakups. On the interface level, techniques for the interface analysis give insights into the simulation quality, mechanisms behind topology changes, as well as the behavior of electrically charged droplets. Further down the scale, the dye-based visualization, streamline-based concepts for space-time analysis, and the implicit extraction of flow topology allow for the investigation of droplet internal transport as well as general single-phase flow scenarios. The applicability of the proposed methods extends, in a varying degree, beyond the use in two-phase flow. Their usability is demonstrated on data from simulations based on Navier-Stokes equations that exemplify practical problems in the research of fluid dynamics.Die numerische Visualisierung ermöglicht Wissenschaftlern und Ingenieuren, Simulationsergebnisse besser zu verstehen und Einblicke in Naturprozesse zu gewinnen. Insbesondere ist die visuelle Darstellung von Ergebnissen numerischer Strömungsmechanik für die Untersuchung physikalischer Phänomene bei Gasen und Flüssigkeiten äußerst wichtig. Die numerische Strömungsmechanik profitiert einerseits von wachsender Rechenleistung handelsüblicher Desktops und Supercomputer, andererseits von den neuen Entwicklungen in der Strömungsforschung. Um eine effektive Analyse von Strömungen zu gewährleisten, müssen sich die Visualisierungstechniken kontinuierlich den Fortschritten in der Strömungsmechanik anpassen. Ein bemerkenswertes Beispiel hierfür ist die Forschung in der Zweiphasenströmung, in der Hochleistungsrechner und effiziente Algorithmen zur Phasenverfolgung hochaufgelöste und physikalisch genaue Simulationen der Flüssigkeitsdynamik ermöglichen. Dennoch ist die Zweiphasenströmung seitens der Visualisierung weitgehend unerforscht geblieben. Insbesondere sehen sich Wissenschaftler und Ingenieure mit verschiedenen Problemen konfrontiert, die mit angepassten Visualisierungstechniken vermieden werden können. Zu den Problemen zählen beispielweise die Verfolgung und Untersuchung der topologischen Ereignisse in Tropfengruppen, hohe Komplexität der Tropfendynamik und die begrenzte Auswahl an interaktiven Methoden zur Untersuchung der Transportphänomene. Demzufolge ist das Ziel dieser Dissertation, die Entwicklung eines Ansatzes zur visuellen Analyse von Zweiphasenströmung auf mehreren Skalen mit dem Fokus auf Interaktionen zwischen den Tropfen, Dynamik der Oberfläche und Materialtransport. Um die Analyse hochkomplexer Simulationsdaten der Zweiphasenströmung zu behandeln, wird eine auf Linked-View-Verfahren basierte Visualisierungstechnik präsentiert, in der die Tropfen sowohl in einer 3D Darstellung als auch in einer abstrakten Graph-Repräsentation visualisiert werden. Der interaktive Brushing-and-Linking-Ansatz ermöglicht eine globale Exploration der topologischen Ereignisse sowie eine detaillierte Inspektion der Dynamik im Hinblick auf die Oszillation und Rotation der Tropfen. Eine andere Technik zeigt die Aufteilung des Tropfenvolumens im zeitlichen Verlauf. Somit ermöglicht diese Methode eine ausführliche Untersuchung der Topologiedynamik mit Hilfe einer statischen Visualisierung. Dafür werden Grenzflächen erzeugt, die das ursprüngliche Volumen des Tropfens hinsichtlich der sich entwickelnden Zerfallskomponenten aufzeigen. Zusätzlich werden die zur Verfolgung der Tropfen benutzten Partikel visualisiert, um Einblicke in die Dynamik der Separation zu gewähren. Die Analyse der Oberfläche ist notwendig, um die Wechselwirkung zwischen der Oberflächendynamik und der Dynamik der Tropfeninteraktion besser zu verstehen. Eine häufig angewendete Technik zur Verfolgung der Phasengrenzen im Volume-of-Fluid-Verfahren ist die zellenweise planare Approximation. Obwohl diese einen guten Kompromiss zwischen Genauigkeit und Performanz bietet, kann die Approximation die Qualität der Simulationsergebnisse erheblich beeinflussen. Es wird deshalb eine Visualisierungsmethode präsentiert, die die Oberfläche mit Hilfe der Taylor-Approximation erster Ordnung extrahiert und unter anderem darauf basierte Größen bereitstellt, die die Relation zwischen der Simulationsapproximation und Qualität der Ergebnisse zeigt. Die Tropfenoberfläche wird weiterhin mit einer Visualisierungsmethode analysiert, die von den Geschwindigkeits- und Oberflächenspannungsgradienten abgeleitete Größen verwendet. Die entwickelte Methode unterstützt die Untersuchung der Deformation der Oberfläche sowie die Untersuchung der Oberflächenspannung und deren Auswirkung auf die Oberflächenstabilität. Eine wichtige Forschungsrichtung in der Zweiphasenströmung ist die Analyse elektrischer Felder auf der Tropfenoberfläche. Die in der Simulation angewendeten Elemente höherer Ordnung ermöglichen physikalische Diskontinuitäten, die für die visuelle Analyse eine gesonderte Behandlung benötigen. Im Zuge dessen wird eine Methode präsentiert, welche die Diskontinuitäten visuell korrekt darstellt und zusätzlich eine Raum-Zeit-Darstellung anwendet, um Einblicke in die Phänomene an der Kontaktlinie zwischen den Tropfen und dem untersuchten Isolator zu gewähren. Die Tropfendynamik wird oft mit der Annahme einer Einphasenströmung analysiert, beispielsweise für die Untersuchung der internen Strömung des Tropfens. Dies ermöglicht eine Anpassung und Verwendung traditioneller Visualisierungsmethoden für Vektorfelder. Eine solche Technik ist die ,,Dye-Advection'', die in dieser Dissertation nicht nur zur Analyse der Advektion, sondern auch zur Untersuchung der Diffusion verwendet wird. Die eingesetzte hochqualitative Rekonstruktion des virtuellen Pigments bewahrt feine Details, während die Implementierung auf der Grafikkarte eine interaktive Visualisierung ermöglicht. Überdies werden einige auf Stromlinien basierende Konzepte in Raum-Zeit-Darstellung angewendet, in der die Zeit als die dritte Raumachse interpretiert wird. Demzufolge können diese Methoden zur Analyse der zeitabhängigen zweidimensionalen Strömung verwendet werden. Die eingeführten ,,Vortex Core Ribbons" unterstützen die Analyse der rotierenden Strömung um die Wirbelkernlinien. Für die Analyse der topologischen Strukturen wurde eine Methode entwickelt, die die Separatrizen implizit als Ränder einer Segmentierung des Vektorfeldes extrahiert. Damit wird eine möglicherweise komplexe direkte Extraktion der Separatrizen vermieden. Die präsentierten Visualisierungsmethoden bilden ein neuartiges Multiskalen-Verfahren zur visuellen Analyse von Zweiphasenströmungen. Die Tropfeninteraktionen werden mit Hilfe einer Visualisierung dargestellt, die sich auf die Ursache des Tropfenzerfalls und deren Ablauf konzentriert. Für die Untersuchung der Oberfläche zeigen die vorgeschlagenen Techniken die Qualität der Ergebnisse hinsichtlich der Oberflächenrekonstruktion, die Mechanismen hinter den topologischen Ereignissen, als auch die Dynamik der elektrisch geladenen Tropfen auf. Andererseits werden unter Annahme der Einphasenströmung neue Techniken basierend auf Dye-Advection, Stromlinien-basierte Konzepte, sowie Verfahren zur Extraktion der Topologie untersucht, um einen besseren Einblick in den Materialtransport zu gewinnen. Die Anwendung dieser Methoden wird in dieser Dissertation auf Daten demonstriert, die durch Simulation, basierend auf Navier-Stokes-Gleichungen, erzeugt wurden

Ray Casting Curved-Quadratic Elements

We present a method for ray casting curved-quadratic elements in 3D. The advantages of this approach is that a curved element can be directly visualized. Conventionally, higher-order elements are tessellated with several linear elements so that standard visualization techniques can be applied to the linear elements. Our method primarily focuses on how to find an approximation to the intersection between a ray and a curved-quadratic element. Once this approximation is found, conventional accumulation and color mapping techniques can be applied to the approximation to produce a volumetric visualization of the element. A cutting plane implementation is also shown that leverages the ray casting technique

Ray Casting Curved-Quadratic Elements

We present a method for ray casting curved-quadratic elements in 3D. The advantages of this approach is that a curved element can be directly visualized. Conventionally, higher-order elements are tessellated with several linear elements so that standard visualization techniques can be applied to the linear elements. Our method primarily focuses on how to find an approximation to the intersection between a ray and a curved-quadratic element. Once this approximation is found, conventional accumulation and color mapping techniques can be applied to the approximation to produce a volumetric visualization of the element. A cutting plane implementation is also shown that leverages the ray casting technique