Analysis and Processing of Irregularly Distributed Point Clouds

We address critical issues arising in the practical implementation of processing real point cloud data that exhibits irregularities. We develop an adaptive algorithm based on Learning Theory for processing point clouds from a stationary sensor that standard algorithms have difficulty approximating. Moreover, we build the theory of distribution-dependent subdivision schemes targeted at representing curves and surfaces with gaps in the data. The algorithms analyze aggregate quantities of the point cloud over subdomains and predict these quantities at the finer level from the ones at the coarser level

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

A geometric framework for immersogeometric analysis

The purpose of this dissertation is to develop a geometric framework for immersogeometric analysis that directly uses the boundary representations (B-reps) of a complex computer-aided design (CAD) model and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. Using the non-boundary-fitted mesh which does not need to conform to the shape of the object can alleviate the challenge of mesh generation for complex geometries. This also reduces the labor-intensive and time-consuming work of geometry cleanup for the purpose of obtaining watertight CAD models in order to perform boundary-fitted mesh generation. The Dirichlet boundary conditions in the fluid domain are enforced weakly over the immersed object surface in the intersected elements. The surface quadrature points for the immersed object are generated on the parametric and analytic surfaces of the B-rep models. In the case of trimmed surfaces, adaptive quadrature rule is considered to improve the accuracy of the surface integral. For the non-boundary-fitted mesh, a sub-cell-based adaptive quadrature rule based on the recursive splitting of quadrature elements is used to faithfully capture the geometry in intersected elements. The point membership classification for identifying quadrature points in the fluid domain is based on a voxel-based approach implemented on GPUs. A variety of computational fluid dynamics (CFD) simulations are performed using the proposed method to assess its accuracy and efficiency. Finally, a fluid--structure interaction (FSI) simulation of a deforming left ventricle coupled with the heart valves shows the potential advantages of the developed geometric framework for the immersogeomtric analysis with complex moving domains

Flexible Object Manipulation

Flexible objects are a challenge to manipulate. Their motions are hard to predict, and the high number of degrees of freedom makes sensing, control, and planning difficult. Additionally, they have more complex friction and contact issues than rigid bodies, and they may stretch and compress. In this thesis, I explore two major types of flexible materials: cloth and string. For rigid bodies, one of the most basic problems in manipulation is the development of immobilizing grasps. The same problem exists for flexible objects. I have shown that a simple polygonal piece of cloth can be fully immobilized by grasping all convex vertices and no more than one third of the concave vertices. I also explored simple manipulation methods that make use of gravity to reduce the number of fingers necessary for grasping. I have built a system for folding a T-shirt using a 4 DOF arm and a fixed-length iron bar which simulates two fingers. The main goal with string manipulation has been to tie knots without the use of any sensing. I have developed single-piece fixtures capable of tying knots in fishing line, solder, and wire, along with a more complex track-based system for autonomously tying a knot in steel wire. I have also developed a series of different fixtures that use compressed air to tie knots in string. Additionally, I have designed four-piece fixtures, which demonstrate a way to fully enclose a knot during the insertion process, while guaranteeing that extraction will always succeed

An isogeometric coupled boundary element method and finite element method for structural-acoustic analysis through loop subdivision surfaces

This present thesis proposes a novel approach for coupling finite element and boundary element formulations using a Loop subdivision surface discretisation to allow efficient acoustic scattering analysis over shell structures. The analysis of underwater structures has always been a challenge for engineers because it couples shell structural dynamics and acoustic scattering. In the present work, a finite element implementation of the Kirchhoff-Love formulation is used for shell structural dynamic analysis and the boundary element method is adopted to solve the Helmholtz equation for acoustic scattering analysis. The boundary element formulation is chosen as it can handle infinite domains without volumetric meshes. In the conventional engineering workflow, generating meshes of complex geometries to represent the underwater structures, e.g. submarines or torpedoes, is very time consuming and costly even if it is only a data conversion process. Isogeometric analysis (IGA) is a recently developed concept which aims to integrate computer aided design (CAD) and numerical analysis by using the same geometry model. Non-uniform rational B-splines~(NURBS), the most commonly used CAD technique, were considered in early IGA developments. However, NURBS have limitations when used in analysis because of their tensor-product nature. Subdivision surfaces discretisation is an alternative to overcome NURBS limitation. The new method adopts a triangular Loop subdivision surface discretisation for both geometry and analysis. The high order subdivision basis functions have $C^1$ continuity, which satisfies the requirements of the Kirchhoff-Love formulation and are highly efficient for the acoustic field computations. The control meshes for the shell analysis and the acoustic analysis have the same resolution, which provides a fully integrated isogeometric approach for coupled structural-acoustic analysis of shells. The method is verified by the example of an acoustic plane wave which scatters over an elastic spherical shell. The ability of the presented method to handle complex geometries is also demonstrated

Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar−joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body−hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas

Body-fitting Meshes for the Discontinuous Galerkin Method

Abstract In this dissertation, a scheme capable of generating highly accurate, body fitting meshes and its application with the Discontinuous Galerkin Method is introduced. Unlike most other mesh generators, this scheme generates meshes consisting of quadrilateral or hexahedral elements exclusively. The high approximation quality is achieved by means of high order curved elements. The resulting meshes are highly suited for the application with high order methods, specifically the Discontinuous Galerkin Method, as large portions of the mesh are fully structured while matching the high order of the numerical method with a high order geometry representation at object and domain boundaries. The mesh scheme works in two steps. It chooses an interior volume that can be represented using a fully structured Cartesian mesh and connects it to the embedded objects and domain boundaries with a so called buffer-layer in a second step. Inside the layer, high order curved elements are applied for yielding high geometric representation accuracy. The resulting meshes are ideal for the application of high order methods. In the interior part the Cartesian structure can be exploited for obtaining high efficiency of the numerical method while the accuracy potential can be realized also in the presence of curved objects and boundaries. After introducing the mesh scheme, the Discontinuous Galerkin Method is described and applied for solving Maxwell’s equations. As a high order method it achieves exponential convergence under p-refinement. It is shown that using meshes produced by the novel scheme this property is achieved for curved domains as well. As an example, optimal convergence rates are demonstrated in a cylindrical cavity problem. In another example, the abilities of the method to produce correct spectral properties of closed resonator problems are investigated. To this end, a time-domain signal is recorded during the transient analysis. After applying the Fourier transform accurate frequency spectra are observed, which are free of spurious modes

Frame Fields for Hexahedral Mesh Generation

As a discretized representation of the volumetric domain, hexahedral meshes have been a popular choice in computational engineering science and serve as one of the main mesh types in leading industrial software of relevance. The generation of high quality hexahedral meshes is extremely challenging because it is essentially an optimization problem involving multiple (conflicting) objectives, such as fidelity, element quality, and structural regularity. Various hexahedral meshing methods have been proposed in past decades, attempting to solve the problem from different perspectives. Unfortunately, algorithmic hexahedral meshing with guarantees of robustness and quality remains unsolved. The frame field based hexahedral meshing method is the most promising approach that is capable of automatically generating hexahedral meshes of high quality, but unfortunately, it suffers from several robustness issues. Field based hexahedral meshing follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a feature-aligned frame field and (2) generation of an integer-grid map that best aligns with the frame field. The main robustness issue stems from the fact that smooth frame fields frequently exhibit singularity graphs that are inappropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The thesis aims at analyzing the gap between the topologies of frame fields and hexahedral meshes and developing algorithms to realize a more robust field based hexahedral mesh generation. The first contribution of this work is an enumeration of all local configurations that exist in hexahedral meshes with bounded edge valence and a generalization of the Hopf-Poincaré formula to octahedral (orthonormal frame) fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field. In the collaboration work with colleagues [BRK+22], the dataset HexMe consisting of practically relevant models with feature tags is set up, allowing a fair evaluation for practical hexahedral mesh generation algorithms. The extendable and mutable dataset remains valuable as hexahedral meshing algorithms develop. The results of the standard field based hexahedral meshing algorithms on the HexMesh dataset expose the fragility of the automatic pipeline. The major contribution of this thesis improves the robustness of the automatic field based hexahedral meshing by guaranteeing local meshability of general feature aligned smooth frame fields. We derive conditions on the meshability of frame fields when feature constraints are considered, and describe an algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite the fact that local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 57%, when compared on the challenging HexMe dataset