Processing Elastic Surfaces and Related Gradient Flows

Surface processing tools and techniques have a long history in the fields of computer graphics, computer aided geometric design and engineering. In this thesis we consider variational methods and geometric evolution problems for various surface processing applications including surface fairing, surface restoration and surface matching. Geometric evolution problems are often based on the gradient flow of geometric energies. The Willmore functional, defined as the integral of the squared mean curvature over the surface, is a geometric energy that measures the deviation of a surface from a sphere. Therefore, it is a suitable functional for surface restoration, where a destroyed surface patch is replaced by a smooth patch defined as the minimizer of the Willmore functional with boundary conditions for the position and the normal at the patch boundary. However, using the Willmore functional does not lead to satisfying results if an edge or a corner of the surface is destroyed. The anisotropic Willmore energy is a natural generalization of the Willmore energy which has crystal-shaped surfaces like cubes or octahedra as minimizers. The corresponding L2-gradient flow, the anisotropic Willmore flow, leads to a fourth-order partial differential equation that can be written as a system of two coupled second second order equations. Using linear Finite Elements, we develop a semi-implicit scheme for the anisotropic Willmore flow with boundary conditions. This approach suffer from significant restrictions on the time step size. Effectively, one usually has to enforce time steps smaller than the squared spatial grid size. Based on a natural approach for the time discretization of gradient flows we present a new scheme for the time and space discretization of the isotropic and anisotropic Willmore flow. The approach is variational and takes into account an approximation of the L2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the anisotropic Willmore functional at the new time step. To evaluate the anisotropic Willmore energy on the unknown surface of the next time step, we first ask for the solution of an inner, secondary variational problem describing a time step of anisotropic mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the anisotropic mean curvature vector and enters the approximation of the actual anisotropic Willmore functional. The resulting two step time discretization of the Willmore flow is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size. The Willmore functional of a surface is referred to as the elastic surface energy. Another interesting application of modeling elastic surfaces as minimizers of elastic energies is surface matching, where a correspondence between two surfaces is subject of investigation. There, we seek a mapping between two surfaces respecting certain properties of the surfaces. The approach is variational and based on well-established matching methods from image processing in the parameter domains of the surfaces instead of finding a correspondence between the two surfaces directly in 3D. Besides the appropriate modeling we analyze the derived model theoretically. The resulting deformations are globally smooth, one-to-one mappings. A physically proper morphing of characters in computer graphic is capable with the resulting computational approach

Ocean acidification alleviates low-temperature effects on growth and photosynthesis of the red alga Neosiphonia harveyi (Rhodophyta)

This study aimed to examine interactive effects between ocean acidification and temperature on the photosynthetic and growth performance of Neosiphonia harveyi. N. harveyi was cultivated at 10 and 17.5 °C at present (~380 µatm), expected future (~800 µatm), and high (~1500 µatm) pCO2. Chlorophyll a fluorescence, net photosynthesis, and growth were measured. The state of the carbon-concentrating mechanism (CCM) was examined by pH-drift experiments (with algae cultivated at 10 °C only) using ethoxyzolamide, an inhibitor of external and internal carbonic anhydrases (exCA and intCA, respectively). Furthermore, the inhibitory effect of acetazolamide (an inhibitor of exCA) and Tris (an inhibitor of the acidification of the diffusive boundary layer) on net photosynthesis was measured at both temperatures. Temperature affected photosynthesis (in terms of photosynthetic efficiency, light saturation point, and net photosynthesis) and growth at present pCO2, but these effects decreased with increasing pCO2. The relevance of the CCM decreased at 10 °C. A pCO2 effect on the CCM could only be shown if intCA and exCA were inhibited. The experiments demonstrate for the first time interactions between ocean acidification and temperature on the performance of a non-calcifying macroalga and show that the effects of low temperature on photosynthesis can be alleviated by increasing pCO2. The findings indicate that the carbon acquisition mediated by exCA and acidification of the diffusive boundary layer decrease at low temperatures but are not affected by the cultivation level of pCO2, whereas the activity of intCA is affected by pCO2. Ecologically, the findings suggest that ocean acidification might affect the biogeographical distribution of N. harveyi

On a new conformal functional for simplicial surfaces

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $\beta$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of $W_2$ and $W_{2,w}$ are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and Surfaces, 8th International Conference", June 201

Biochemical composition of temperate and Arctic populations of Saccharina latissima after exposure to increased pCO2 and temperature reveals ecotypic variation

Previous research suggested that the polar and temperate populations of the kelp Saccharina latissima represent different ecotypes. The ecotypic differentiation might also be reflected in their biochemical composition (BC) under changing temperatures and pCO2. Accordingly, it was tested if the BC of Arctic (Spitsbergen) and temperate S. latissima (Helgoland) is different and if they are differently affected by changes in temperature and pCO2. Thalli from Helgoland grown at 17 °C and 10 °C and from Spitsbergen at 10 °C and 4 °C were all tested at either 380, 800, or 1,500 latm pCO2, and total C-, total N-, protein,soluble carbohydrate, and lipid content, as well as C/Nratio were measured. At 10 °C, the Arctic population had a higher content of total C, soluble carbohydrates, and lipids, whereas the N- and protein content was lower. At the lower tested temperature, the Arctic ecotype had particularly higher contents of lipids, while content of soluble carbohydrates increased in the Helgoland population only. In Helgoland-thalli, elevated pCO2 caused a higher content of soluble carbohydrates at 17 °C but lowered the content of N and lipids and increased the C/N-ratio at 10 °C. Elevated pCO2 alone did not affect the BC of the Spitsbergen population. Conclusively, the Arctic ecotype was more resilient to increased pCO2 than the temperate one, and both ecotypes differed in their response pattern to temperature. This differential pattern is discussed in the context of the adaptation of the Arctic ecotype to low temperature and the polar night

Divergent Responses in Growth and Nutritional Quality of Coastal Macroalgae to the Combination of Increased pCO\u3csub\u3e2\u3c/sub\u3e and Nutrients

Coastal ecosystems are subjected to global and local environmental stressors, including increased atmospheric carbon dioxide (CO2) (and subsequent ocean acidification) and nutrient loading. Here, we tested how two common macroalgal species in the Northwest Atlantic (Ulva spp. and Fucus vesiculosus Linneaus) respond to the combination of increased CO2 and nutrient loading. We utilized two levels of pCO2 with two levels of nutrients in a full factorial design, testing the growth rates and tissue quality of Ulva and Fucus grown for 21 days in monoculture and biculture. We found that the opportunistic, fast-growing Ulva exhibited increased growth rates under high pCO2 and high nutrients, with growth rates increasing three-fold above Ulva grown in ambient pCO2 and ambient nutrients. By contrast, Fucus growth rates were not impacted by either environmental factor. Both species exhibited a decline in carbon to nitrogen ratios (C:N) with elevated nutrients, but pCO2 concentration did not alter tissue quality in either species. Species grown in biculture exhibited similar growth rates to those in monoculture conditions, but Fucus C:N increased significantly when grown with Ulva, indicating an effect of the presence of Ulva on Fucus. Our results suggest that the combination of ocean acidification and nutrients will enhance abundance of opportunistic algal species in coastal systems and will likely drive macroalgal community shifts, based on species-specific responses to future conditions

Acidification increases abundances of Vibrionales and Planctomycetia associated to a seaweed-grazer system: potential consequences for disease and prey digestion efficiency

Ocean acidification significantly affects marine organisms in several ways, with complex interactions. Seaweeds might benefit from rising CO2 through increased photosynthesis and carbon acquisition, with subsequent higher growth rates. However, changes in seaweed chemistry due to increased CO2 may change the nutritional quality of tissue for grazers. In addition, organisms live in close association with a diverse microbiota, which can also be influenced by environmental changes, with feedback effects. As gut microbiomes are often linked to diet, changes in seaweed characteristics and associated microbiome can affect the gut microbiome of the grazer, with possible fitness consequences. In this study, we experimentally investigated the effects of acidification on the microbiome of the invasive brown seaweed Sargassum muticum and a native isopod consumer Synisoma nadejda. Both were exposed to ambient CO2 conditions (380 ppm, pH 8.16) and an acidification treatment (1,000 ppm, pH 7.86) for three weeks. Microbiome diversity and composition were determined using high-throughput sequencing of the variable regions V5-7 of 16S rRNA. We anticipated that as a result of acidification, the seaweed-associated bacterial community would change, leading to further changes in the gut microbiome of grazers. However, no significant effects of elevated CO2 on the overall bacterial community structure and composition were revealed in the seaweed. In contrast, significant changes were observed in the bacterial community of the grazer gut. Although the bacterial community of S. muticum as whole did not change, Oceanospirillales and Vibrionales (mainly Pseudoalteromonas) significantly increased their abundance in acidified conditions. The former, which uses organic matter compounds as its main source, may have opportunistically taken advantage of the possible increase of the C/N ratio in the seaweed under acidified conditions. Pseudoalteromonas, commonly associated to diseased seaweeds, suggesting that acidification may facilitate opportunistic/pathogenic bacteria. In the gut of S. nadejda, the bacterial genus Planctomycetia increased abundance under elevated CO2. This shift might be associated to changes in food (S. muticum) quality under acidification. Planctomycetia are slow-acting decomposers of algal polymers that could be providing the isopod with an elevated algal digestion and availability of inorganic compounds to compensate the shifted C/N ratio under acidification in their food. In conclusion, our results indicate that even after only three weeks of acidified conditions, bacterial communities associated to ungrazed seaweed and to an isopod grazer show specific, differential shifts in associated bacterial community. These have potential consequences for seaweed health (as shown in corals) and isopod food digestion. The observed changes in the gut microbiome of the grazer seem to reflect changes in the seaweed chemistry rather than its microbial composition.Erasmus Mundus Doctoral Programme MARES on Marine Ecosystem Health Conservation [MARES_13_08]; FCT (Foundation for Science and Technology, Portugal) [SFRH/BPD/63703/2009, SFRH/BPD/107878/2015, SFRH/BPD/116774/2016]; EU SEAS-ERA project INVASIVES [SEAS-ERA/0001/2012]; [CCMAR/Multi/04326/2013

On a linear programming approach to the discrete Willmore boundary value problem and generalizations

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature Our purpose is to address the delicate task of approximating global minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that: - One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well; - Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution. Due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem