Surface (topology)

Summary

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of

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Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

Andrea Bartezzaghi

In this thesis, we consider the numerical approximation of high order geometric Partial Differential Equations (PDEs). We first consider high order PDEs defined on surfaces in the 3D space that are represented by single-patch tensor product NURBS. Then, we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method. With this aim, we consider the construction of periodic NURBS function spaces with high degree of global continuity, even on closed surfaces. As benchmark problems for the proposed discretization, we propose Laplace-Beltrami problems of the fourth and sixth orders, as well as the corresponding eigenvalue problems, and we analyze the impact of the continuity of the basis functions on the accuracy as well as on computational costs. The numerical solution of two high order phase field problems on both open and closed surfaces is also considered: the fourth order Cahn-Hilliard equation and the sixth order crystal equation, both discretized in time with the generalized-alpha method. We then consider the numerical approximation of geometric PDEs, derived, in particular, from the minimization of shape energy functionals by L^2-gradient flows. We analyze the mean curvature and the Willmore gradient flows, leading to second and fourth order PDEs, respectively. These nonlinear geometric PDEs are discretized in time with Backward Differentiation Formulas (BDF), with a semi-implicit formulation based on an extrapolation of the geometry, leading to a linear problem to be solved at each time step. Results about the numerical approximation of the two geometric flows on several geometries are analyzed. Then, we study how the proposed mathematical framework can be employed to numerically approximate the equilibrium shapes of lipid bilayer biomembranes, or vesicles, governed by the Canham-Helfrich curvature model. We propose two numerical schemes for enforcing the conservation of the area and volume of the vesicles, and report results on benchmark problems. Then, the approximation of the equilibrium shapes of biomembranes with different values of reduced volume is presented. Finally, we consider the dynamics of a vesicle, e.g. a red blood cell, immersed in a fluid, e.g. the plasma. In particular, we couple the curvature-driven model for the lipid membrane with the incompressible Navier-Stokes equations governing the fluid. We consider a segregated approach, with a formulation based on the Resistive Immersed Surface method applied to NURBS geometries. After analyzing benchmark fluid simulations with immersed NURBS objects, we report numerical results for the investigation of the dynamics of a vesicle under different flow conditions.

EPFL2017

Situational Awareness Using A Single Omnidirectional Camera

Luigi Bagnato, Eugenio De Vito, Laurent Jacques, Pierre Vandergheynst

To retrieve scene informations using a single omnidirectional camera, we have based our work on a shape from texture method proposed by Lindeberg. To do so, we have adapted the method of Lindeberg, that was developed for planar images, in order to use it on the sphere S2 . The mathematical tools we use are stereographic dilation to implement scale variations for the scale-space representation, and ﬁlter steerability on the sphere to decrease computational order. The texture distortions due to the pro jection from the real world to the image contain the informations that enable shape and orientation to be computed. A multi-scale texture descriptor, the windowed second moment matrix, that contains distorsions informations, is computed and analyzed, with some assumptions about the surface texture to retrieve surface orientation. We have used synthetic signals to evaluate the performances of the adapted method. The obtained results for the distance and shape estimations are good when the textures on the surfaces correspond to the assumptions, generally around the equatorial plane of S2 , but when we move away from the equator, the precision of the estimations decreases signiﬁcantly.

2008

As historical stone masonry structures are vulnerable and prone to damage in earthquakes, investigating their structural integrity is important to reduce injuries and casualties while preserving their historical value. Stone masonry is a composite material that is built with stones and binding mortar. Although experimental campaigns are crucial in understanding the structural behaviour of these walls, the wide spectrum of existing stone masonry typologies and the randomness in geometry and material properties render extensive testing campaigns nearly impossible to account for all the uncertainties and variables. Numerical simulations that explicitly represent the microstructure of the wall (i.e., the geometry and arrangement of the stones) can complement experimental studies. However, methods for generating 3D microstructures were lacking.The focus of this dissertation is to study the geometry of the 3D microstructure of stone masonry walls at multiple scales, which opens unprecedented opportunities for studying the micromechanical behaviour of these walls numerically. The first contribution of this thesis was crafting the first 3D virtual microstructure generator that can cover the main typologies of stone masonry that are frequently found in historical buildings. The herein-created microstructure generator borrowed the conventional packing problem analogy to pack the generated stones inside the boundary of the walls following building and physical placement rules to obtain realistic microstructures. The size spectrum, distribution, interlocking and topology of the generated stones were thoroughly investigated. To quickly and efficiently solve the posed packing problem of each stone, we proposed a general-purpose heuristic algorithm that benefits from Pareto's principle which is commonly known as the

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rule. This heuristic was called the Pareto sequential sampling (PSS) algorithm as it depends on sampling most of the solutions, using any design of experiment (DoE) method, from a domain that is known to be a prominent pool of solutions.The second contribution was related to studying the topology and roughness of stones and surfaces. The traditional spherical harmonics expansion was used to study the morphology of closed surfaces of the nonconvex stones to estimate their fractal dimension and uniformly remesh the triangulated surfaces to be used in numerical simulations. As the roughness of natural stones changes on a single stone, we similarly studied the morphology of locally sampled rough patches. We developed the spherical cap harmonics and disk harmonics spectral approaches to provide us with information on the roughness and fractal dimension of any isotropic self-affine rough surface. To speed up the numerical integration of these spectral expansions, we proposed new algorithms based on the well-known Kaczmarz orthogonal projection in a non-memory-intensive manner and using reasonable computational times. The developed algorithms were hinged on the conjugate symmetry of the harmonic bases and the sparsity of the real signals to lower the dimensionality of the problem and accelerate the rate of convergence. These spectral methods are considered general-purpose and have multiple applications in various fields such as medical imaging and computer graphics. With this dissertation, we put forward geometrical and topological concepts that make detailed micromechanical simulations of stone masonry walls practical.

EPFL2022

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EPFL2022

Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

Andrea Bartezzaghi

EPFL2017