Glyphs for space-time Jacobians of time-dependent vector fields

Glyphs have proven to be a powerful visualization technique for general tensor fields modeling physical phenomena such as diffusion or the derivative of flow fields. Most glyph constructions, however, do not provide a way of considering the temporal derivative, which is generally nonzero in non-stationary vector fields. This derivative offers a deeper understanding of features in time-dependent vector fields. We introduce an extension to 2D and 3D tensor glyph design that additionally encodes the temporal information of velocities, and thus makes it possible to represent time-dependent Jacobians. At the same time, a certain set of requirements for general tensor glyphs is fulfilled, such that the new method provides a visualization of the steadiness or unsteadiness of a vector field at a given instance of time

Visual Analysis of Second and Third Order Tensor Fields in Structural Mechanics

This work presents four new methods for the analysis and visualization of tensor fields. The focus is on tensor fields which arise in the context of structural mechanics simulations. The first method deals with the design of components made of short fiber reinforced polymers using injection molding. The stability of such components depends on the fiber orientations, which are affected by the production process. For this reason, the stresses under load as well as the fiber orientations are analyzed. The stresses and fiber orientations are each given as tensor fields. For the analysis four features are defined. The features indicate if the component will resist the load or not, and if the respective behavior depends on the fiber orientation or not. For an in depth analysis a glyph was developed, which shows the admissible fiber orientations as well as the given fiber orientation. With these visualizations the engineer can rate a given fiber orientation and gets hints for improving the fiber orientation. The second method depicts gradients of stress tensors using glyphs. A thorough understanding of the stress gradient is desirable, since there is some evidence that not only the stress but also its gradient influences the stability of a material. Gradients of stress tensors are third order tensors, the visualization is therefore a great challenge and there is very little research on this subject so far. The objective of the third method is to analyse the complete invariant part of the tensor field. Scalar invariants play an important role in many applications, but proper selection of such invariants is often difficult. For the analysis of the complete invariant part the notion of 'extremal point' is introduced. An extremal point is characterized by the fact that there is a scalar invariant which has a critical point at this position. Moreover it will be shown that the extrema of several common invariants are contained in the set of critical points. The fourth method presented in this work uses the Heat Kernel Signature (HKS) for the visualization of tensor fields. The HKS is computed from the heat kernel and was originally developed for surfaces. It characterizes the metric of the surface under weak assumptions. i.e. the shape of the surfaces is determined up to isometric deformations. The fact that every positive definite tensor field can be considered as the metric of a Riemannian manifold allows to apply the HKS on tensor fields

Numerical simulation of fracture pattern development and implications for fuid flow

Simulations are instrumental to understanding flow through discrete fracture geometric representations that capture the large-scale permeability structure of fractured porous media. The contribution of this thesis is threefold: an efficient finite-element finite-volume discretisation of the advection/diffusion flow equations, a geomechanical fracture propagation algorithm to create fractured rock analogues, and a study of the effect of growth on hydraulic conductivity. We describe an iterative geomechanics-based finite-element model to simulate quasi-static crack propagation in a linear elastic matrix from an initial set of random flaws. The cornerstones are a failure and propagation criterion as well as a geometric kernel for dynamic shape housekeeping and automatic remeshing. Two-dimensional patterns exhibit connectivity, spacing, and density distributions reproducing en echelon crack linkage, tip hooking, and polygonal shrinkage forms. Differential stresses at the boundaries yield fracture curving. A stress field study shows that curvature can be suppressed by layer interaction effects. Our method is appropriate to model layered media where interaction with neighbouring layers does not dominate deformation. Geomechanically generated fracture patterns are the input to single-phase flow simulations through fractures and matrix. Thus, results are applicable to fractured porous media in addition to crystalline rocks. Stress state and deformation history control emergent local fracture apertures. Results depend on the number of initial flaws, their initial random distribution, and the permeability of the matrix. Straightpath fracture pattern simplifications yield a lower effective permeability in comparison to their curved counterparts. Fixed apertures overestimate the conductivity of the rock by up to six orders of magnitude. Local sample percolation effects are representative of the entire model flow behaviour for geomechanical apertures. Effective permeability in fracture dataset subregions are higher than the overall conductivity of the system. The presented methodology captures emerging patterns due to evolving geometric and flow properties essential to the realistic simulation of subsurface processes

Multiscale and multiphysics computational models of processes in shock wave lithotripsy

This thesis presents two computational models applied to processes in shock wave lithotripsy. The first is a multiphysics model of the focusing of an acoustic pulse and the subsequent shock wave formation that occurs in a refracting electromagnetic lithotripter. This model solves both the linear elasticity equations and the Euler equations with a Tait equation of state in arbitrary subsets of the full computational domain. It is implemented within BEARCLAW and uses a finite-volume Riemann solver approach. The model is validated using a standard lens design and is shown to accurately predict the effects of a lens modification. This model is also extended to include a kidney stone simulant in the domain in which a simple isotropic damage law is included. The second computational model is a 3D multiscale fracture model which predicts crack formation and propagation within a kidney stone simulant by utilizing a continuum-mesoscopic interaction. The simulant included in the model is realistic in that the data representing the stone is drawn from MicroCT image data. At the continuum scale the linear elasticity equations are solved while incorporating an anisotropic damage variable, again using a finite-volume Riemann solver within BEARCLAW. At the mesoscale, damage accumulates based on experimentally informed probability distributions and on predefined surfaces representing a granular structure. In addition to the computational models, some experimental results are discussed. These include probability distributions of fracture properties found from MicroCT images of kidney stone simulants and corresponding image processing procedures.Doctor of Philosoph

Glyphs for space-time Jacobians of time-dependent vector fields

Glyphs have proven to be a powerful visualization technique for general tensor fields modeling physical phenomena such as diffusion or the derivative of flow fields. Most glyph constructions, however, do not provide a way of considering the temporal derivative, which is generally nonzero in non-stationary vector fields. This derivative offers a deeper understanding of features in time-dependent vector fields. We introduce an extension to 2D and 3D tensor glyph design that additionally encodes the temporal information of velocities, and thus makes it possible to represent time-dependent Jacobians. At the same time, a certain set of requirements for general tensor glyphs is fulfilled, such that the new method provides a visualization of the steadiness or unsteadiness of a vector field at a given instance of time

Improving Filtering for Computer Graphics

When drawing images onto a computer screen, the information in the scene is typically more detailed than can be displayed. Most objects, however, will not be close to the camera, so details have to be filtered out, or anti-aliased, when the objects are drawn on the screen. I describe new methods for filtering images and shapes with high fidelity while using computational resources as efficiently as possible. Vector graphics are everywhere, from drawing 3D polygons to 2D text and maps for navigation software. Because of its numerous applications, having a fast, high-quality rasterizer is important. I developed a method for analytically rasterizing shapes using wavelets. This approach allows me to produce accurate 2D rasterizations of images and 3D voxelizations of objects, which is the first step in 3D printing. I later improved my method to handle more filters. The resulting algorithm creates higher-quality images than commercial software such as Adobe Acrobat and is several times faster than the most highly optimized commercial products. The quality of texture filtering also has a dramatic impact on the quality of a rendered image. Textures are images that are applied to 3D surfaces, which typically cannot be mapped to the 2D space of an image without introducing distortions. For situations in which it is impossible to change the rendering pipeline, I developed a method for precomputing image filters over 3D surfaces. If I can also change the pipeline, I show that it is possible to improve the quality of texture sampling significantly in real-time rendering while using the same memory bandwidth as used in traditional methods