Conformal geometry processing

This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. Conformal transformations are valuable in applications because they naturally preserve the integrity of geometric data. To date, however, there has been no clearly stated and consistent theory of conformal transformations that can be used to develop general-purpose geometry processing algorithms: previous methods for computing conformal maps have been restricted to the flat two-dimensional plane, or other spaces of constant curvature. In contrast, our formulation can be used to produce---for the first time---general surface deformations that are perfectly conformal in the limit of refinement. It is for this reason that we commandeer the title Conformal Geometry Processing.

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in our theory. Given an immersed surface, we wish to construct new immersions that (i) induce a conformally equivalent metric and (ii) exhibit a prescribed change in extrinsic curvature. Curvature determines the potential in the Dirac equation; the solution of this equation determines the geometry of the new surface. We derive the precise conditions under which curvature is allowed to evolve, and develop efficient numerical algorithms for solving the Dirac equation on triangulated surfaces.

From a practical perspective, this theory has a variety of benefits: conformal maps are desirable in geometry processing because they do not exhibit shear, and therefore preserve textures as well as the quality of the mesh itself. Our discretization yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications. We also present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces.

Geochemistry and resonance ionization of platinum-group elements

Experimental studies were conducted with the goals of 1) determining the origin of Pt- group element (PGE) alloys and associated mineral assemblages in refractory inclusions from meteorites and 2) developing a new ultrasensitive method for the in situ chemical and isotopic analysis of PGE. A general review of the geochemistry and cosmochemistry of the PGE is given, and specific research contributions are presented within the context of this broad framework.

An important step toward understanding the cosmochemistry of the PGE is the determination of the origin of POE-rich metallic phases (most commonly εRu-Fe) that are found in Ca, AJ-rich refractory inclusions (CAI) in C3V meteorites. These metals occur along with γNi-Fe metals, Ni-Fe sulfides and Fe oxides in multiphase opaque assemblages. Laboratory experiments were used to show that the mineral assemblages and textures observed in opaque assemblages could be produced by sulfidation and oxidation of once homogeneous Ni-Fe-PGE metals. Phase equilibria, partitioning and diffusion kinetics were studied in the Ni-Fe-Ru system in order to quantify the conditions of opaque assemblage formation. Phase boundaries and tie lines in the Ni-Fe-Ru system were determined at 1273, 1073 and 873K using an experimental technique that allowed the investigation of a large portion of the Ni-Fe-Ru system with a single experiment at each temperature by establishing a concentration gradient within which local equilibrium between coexisting phases was maintained. A wide miscibility gap was found to be present at each temperature, separating a hexagonal close-packed εRu-Fe phase from a face-centered cubic γNi-Fe phase. Phase equilibria determined here for the Ni-Fe-Ru system, and phase equilibria from the literature for the Ni-Fe-S and Ni-Fe-O systems, were compared with analyses of minerals from opaque assemblages to estimate the temperature and chemical conditions of opaque assemblage formation. It was determined that opaque assemblages equilibrated at a temperature of ~770K, a sulfur fugacity 10 times higher than an equilibrium solar gas, and an oxygen fugacity 10⁶ times higher than an equilibrium solar gas.

Diffusion rates between -γNi-Fe and εRu-Fe metal play a critical role in determining the time (with respect to CAI petrogenesis) and duration of the opaque assemblage equilibration process. The diffusion coefficient for Ru in Ni (D^Ru_Ni) was determined as an analog for the Ni-Fe-Ru system by the thin-film diffusion method in the temperature range of 1073 to 1673K and is given by the expression:

D^Ru_Ni (cm² sec^-1) = 5.0(±0.7) x 10^-3 exp(-2.3(±0.1) x 10¹² erg mole^-1/RT) where R is the gas constant and T is the temperature in K. Based on the rates of dissolution and exsolution of metallic phases in the Ni-Fe-Ru system it is suggested that opaque assemblages equilibrated after the melting and crystallization of host CAI during a metamorphic event of ≥ 10³ years duration. It is inferred that opaque assemblages originated as immiscible metallic liquid droplets in the CAI silicate liquid. The bulk compositions of PGE in these precursor alloys reflects an early stage of condensation from the solar nebula and the partitioning of V between the precursor alloys and CAI silicate liquid reflects the reducing nebular conditions under which CAI were melted. The individual mineral phases now observed in opaque assemblages do not preserve an independent history prior to CAI melting and crystallization, but instead provide important information on the post-accretionary history of C3V meteorites and allow the quantification of the temperature, sulfur fugacity and oxygen fugacity of cooling planetary environments. This contrasts with previous models that called upon the formation of opaque assemblages by aggregation of phases that formed independently under highly variable conditions in the solar nebula prior to the crystallization of CAI.

Analytical studies were carried out on PGE-rich phases from meteorites and the products of synthetic experiments using traditional electron microprobe x-ray analytical techniques. The concentrations of PGE in common minerals from meteorites and terrestrial rocks are far below the ~100 ppm detection limit of the electron microprobe. This has limited the scope of analytical studies to the very few cases where PGE are unusually enriched. To study the distribution of PGE in common minerals will require an in situ analytical technique with much lower detection limits than any methods currently in use. To overcome this limitation, resonance ionization of sputtered atoms was investigated for use as an ultrasensitive in situ analytical technique for the analysis of PGE. The mass spectrometric analysis of Os and Re was investigated using a pulsed primary Ar⁺ ion beam to provide sputtered atoms for resonance ionization mass spectrometry. An ionization scheme for Os that utilizes three resonant energy levels (including an autoionizing energy level) was investigated and found to have superior sensitivity and selectivity compared to nonresonant and one and two energy level resonant ionization schemes. An elemental selectivity for Os over Re of ≥ 10³ was demonstrated. It was found that detuning the ionizing laser from the autoionizing energy level to an arbitrary region in the ionization continuum resulted in a five-fold decrease in signal intensity and a ten-fold decrease in elemental selectivity. Osmium concentrations in synthetic metals and iron meteorites were measured to demonstrate the analytical capabilities of the technique. A linear correlation between Os⁺ signal intensity and the known Os concentration was observed over a range of nearly 10⁴ in Os concentration with an accuracy of ~ ±10%, a millimum detection limit of 7 parts per billion atomic, and a useful yield of 1%. Resonance ionization of sputtered atoms samples the dominant neutral-fraction of sputtered atoms and utilizes multiphoton resonance ionization to achieve high sensitivity and to eliminate atomic and molecular interferences. Matrix effects should be small compared to secondary ion mass spectrometry because ionization occurs in the gas phase and is largely independent of the physical properties of the matrix material. Resonance ionization of sputtered atoms can be applied to in situ chemical analysis of most high ionization potential elements (including all of the PGE) in a wide range of natural and synthetic materials. The high useful yield and elemental selectivity of this method should eventually allow the in situ measurement of Os isotope ratios in some natural samples and in sample extracts enriched in PGE by fire assay fusion.

Phase equilibria and diffusion experiments have provided the basis for a reinterpretation of the origin of opaque assemblages in CAI and have yielded quantitative information on conditions in the primitive solar nebula and cooling planetary environments. Development of the method of resonance ionization of sputtered atoms for the analysis of Os has shown that this technique has wide applications in geochemistry and will for the first time allow in situ studies of the distribution of PGE at the low concentration levels at which they occur in common minerals.

Structural and clumped-isotope constraints on the mechanisms of displacement along low-angle detachments

Despite years of research on low-angle detachments, much about them remains enigmatic. This thesis addresses some of the uncertainty regarding two particular detachments, the Mormon Peak detachment in Nevada and the Heart Mountain detachment in Wyoming and Montana.

Constraints on the geometry and kinematics of emplacement of the Mormon Peak detachment are provided by detailed geologic mapping of the Meadow Valley Mountains, along with an analysis of structural data within the allochthon in the Mormon Mountains. Identifiable structures well suited to constrain the kinematics of the detachment include a newly mapped, Sevier-age monoclinal flexure in the hanging wall of the detachment. This flexure, including the syncline at its base and the anticline at its top, can be readily matched to the base and top of the frontal Sevier thrust ramp, which is exposed in the footwall of the detachment to the east in the Mormon Mountains and Tule Springs Hills. The ~12 km of offset of these structural markers precludes the radial sliding hypothesis for emplacement of the allochthon.

The role of fluids in the slip along faults is a widely investigated topic, but the use of carbonate clumped-isotope thermometry to investigate these fluids is new. Faults rocks from within ~1 m of the Mormon Peak detachment, including veins, breccias, gouges, and host rocks, were analyzed for carbon, oxygen, and clumped-isotope measurements. The data indicate that much of the carbonate breccia and gouge material along the detachment is comminuted host rock, as expected. Measurements in vein material indicate that the fluid system is dominated by meteoric water, whose temperature indicates circulation to substantial depths (c. 4 km) in the upper crust near the fault zone.

Slip along the subhorizontal Heart Mountain detachment is particularly enigmatic, and many different mechanisms for failure have been proposed, predominantly involving catastrophic failure. Textural evidence of multiple slip events is abundant, and include multiple brecciation events and cross-cutting clastic dikes. Footwall deformation is observed in numerous exposures of the detachment. Stylolitic surfaces and alteration textures within and around “banded grains” previously interpreted to be an indicator of high-temperature fluidization along the fault suggest their formation instead via low-temperature dissolution and alteration processes. There is abundant textural evidence of the significant role of fluids along the detachment via pressure solution. The process of pressure solution creep may be responsible for enabling multiple slip events on the low-angle detachment, via a local rotation of the stress field.

Clumped-isotope thermometry of fault rocks associated with the Heart Mountain detachment indicates that despite its location on the flanks of a volcano that was active during slip, the majority of carbonate along the Heart Mountain detachment does not record significant heating above ambient temperatures (c. 40-70°C). Instead, cold meteoric fluids infiltrated the detachment breccia, and carbonate precipitated under ambient temperatures controlled by structural depth. Locally, fault gouge does preserve hot temperatures (>200°C), as is observed in both the Mormon Peak detachment and Heart Mountain detachment areas. Samples with very hot temperatures attributable to frictional shear heating are present but rare. They appear to be best preserved in hanging wall structures related to the detachment, rather than along the main detachment.

Evidence is presented for the prevalence of relatively cold, meteoric fluids along both shallow crustal detachments studied, and for protracted histories of slip along both detachments. Frictional heating is evident from both areas, but is a minor component of the preserved fault rock record. Pressure solution is evident, and might play a role in initiating slip on the Heart Mountain fault, and possibly other low-angle detachments.