Delaunay Deformable Models: Topology-Adaptive Meshes Based on the Restricted Delaunay Triangulation

International audienceIn this paper, we propose a robust and efficient La- grangian approach, which we call Delaunay Deformable Models, for modeling moving surfaces undergoing large de- formations and topology changes. Our work uses the con- cept of restricted Delaunay triangulation, borrowed from computational geometry. In our approach, the interface is represented by a triangular mesh embedded in the Delau- nay tetrahedralization of interface points. The mesh is it- eratively updated by computing the restricted Delaunay tri- angulation of the deformed objects. Our method has many advantages over popular Eulerian techniques such as the level set method and over hybrid Eulerian-Lagrangian tech- niques such as the particle level set method: localization accuracy, adaptive resolution, ability to track properties as- sociated to the interface, seamless handling of triple junc- tions. Our work brings a rigorous and efficient alternative to existing topology-adaptive mesh techniques such as T- snakes

Nonstandard Finite Element Methods

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Theory and algorithms for swept manifold intersections

Recent developments in such fields as computer aided geometric design, geometric modeling, and computational topology have generated a spate of interest towards geometric objects called swept volumes. Besides their great applicability in various practical areas, the mere geometry and topology of these entities make them a perfect testbed for novel approaches aimed at analyzing and representing geometric objects. The structure of swept volumes reveals that it is also important to focus on a little simpler, although a very similar type of objects - swept manifolds. In particular, effective computability of swept manifold intersections is of major concern. The main goal of this dissertation is to conduct a study of swept manifolds and, based on the findings, develop methods for computing swept surface intersections. The twofold nature of this goal prompted a division of the work into two distinct parts. At first, a theoretical analysis of swept manifolds is performed, providing a better insight into the topological structure of swept manifolds and unveiling several important properties. In the course of the investigation, several subclasses of swept manifolds are introduced; in particular, attention is focused on regular and critical swept manifolds. Because of the high applicability, additional effort is put into analysis of two-dimensional swept manifolds - swept surfaces. Some of the valuable properties exhibited by such surfaces are generalized to higher dimensions by introducing yet another class of swept manifolds - recursive swept manifolds. In the second part of this work, algorithms for finding swept surface intersections are developed. The need for such algorithms is necessitated by a specific structure of swept surfaces that precludes direct employment of existing intersection methods. The new algorithms are designed by utilizing the underlying ideas of existing intersection techniques and making necessary technical modifications. Such modifications are achieved by employing properties of swept surfaces obtained in the course of the theoretical study. The intersection problems is also considered from a little different prospective. A novel, homology based approach to local characterization of intersections of submanifolds and s-subvarieties of a Euclidean space is presented. It provides a method for distinguishing between transverse and tangential intersection points and determining, in some cases, whether the intersection point belongs to a boundary. At the end, several possible applications of the obtained results are described, including virtual sculpting and modeling of heterogeneous materials

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Generating and auto-tuning parallel stencil codes

In this thesis, we present a software framework, Patus, which generates high performance stencil codes for different types of hardware platforms, including current multicore CPU and graphics processing unit architectures. The ultimate goals of the framework are productivity, portability (of both the code and performance), and achieving a high performance on the target platform. A stencil computation updates every grid point in a structured grid based on the values of its neighboring points. This class of computations occurs frequently in scientific and general purpose computing (e.g., in partial differential equation solvers or in image processing), justifying the focus on this kind of computation. The proposed key ingredients to achieve the goals of productivity, portability, and performance are domain specific languages (DSLs) and the auto-tuning methodology. The Patus stencil specification DSL allows the programmer to express a stencil computation in a concise way independently of hardware architecture-specific details. Thus, it increases the programmer productivity by disburdening her or him of low level programming model issues and of manually applying hardware platform-specific code optimization techniques. The use of domain specific languages also implies code reusability: once implemented, the same stencil specification can be reused on different hardware platforms, i.e., the specification code is portable across hardware architectures. Constructing the language to be geared towards a special purpose makes it amenable to more aggressive optimizations and therefore to potentially higher performance. Auto-tuning provides performance and performance portability by automated adaptation of implementation-specific parameters to the characteristics of the hardware on which the code will run. By automating the process of parameter tuning — which essentially amounts to solving an integer programming problem in which the objective function is the number representing the code's performance as a function of the parameter configuration, — the system can also be used more productively than if the programmer had to fine-tune the code manually. We show performance results for a variety of stencils, for which Patus was used to generate the corresponding implementations. The selection includes stencils taken from two real-world applications: a simulation of the temperature within the human body during hyperthermia cancer treatment and a seismic application. These examples demonstrate the framework's flexibility and ability to produce high performance code

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Numerical Geometric Acoustics

Sound propagation in air is accurately described by a small perturbation of the ambient pressure away from a quiescent state. This is the realm of linear acoustics, where the propagation of a time-harmonic wave can be modeled using the Helmholtz equation. When the wavelength is small relative to the size of a scattering obstacle, techniques from geometric optics are applicable. Geometric methods such as raytracing are often used for computational room acoustics simulations in situations where the geometry of the built environment is sufficiently complicated. At the same time, the high-frequency approximation of the Helmholtz equation is described by two partial differential equations: the eikonal equation, whose solution gives the first arrival time of a geometric acoustics/optics wavefront as a field; and a transport equation, the solution of which describes the amplitude of that wavefield. Phenomena related to high-frequency acoustic diffraction are frequently omitted from these models because of their complexity. These phenomena can be modeled using a high-frequency diffraction theory, such as the uniform theory of diffraction. Despite their shortcomings, geometric methods for room acoustics provide a useful trade-off between realism and computational efficiency. Motivated by the limitations of geometric methods, we approach the problem of geometric acoustics using numerical methods for solving partial differential equations. Our focus is offline sound propagation in a high-frequency regime where directly solving the wave or Helmholtz equations is infeasible. To this end, we conduct a broad-based survey of semi-Lagrangian solvers for the eikonal equation, which make the local ray information of the solution explicit. We develop efficient, first-order solvers for the eikonal equation in 3D, called ordered line integral methods (OLIMs). The OLIMs provide intuition about how to design work-efficient semi-Lagrangian eikonal solvers, but their first order accuracy is not sufficient to compute the amplitude consistently. Motivated by the requirements of sound propagation simulations, we develop higher-order semi-Lagrangian eikonal solvers which we term jet marching methods (JMMs). JMMs augment the efficiency of OLIMs by additionally transporting higher-order derivative information of the eikonal in a causal fashion, which allows for high-order solution of the eikonal equation using compact stencils. We use the information made available locally by our JMMs to use paraxial raytracing to simultaneously solve the transport equation yielding the amplitude. We initially develop a JMM which handles a smoothly varying speed of sound on a regular grid in 2D. Motivated by the requirements of room acoustics applications, we develop a second-order JMM for solving the eikonal equation on a tetrahedron mesh for a constant speed of sound as a special case. As before, we use paraxial raytracing to compute the amplitude. Additionally, we compute multiple arrivals by reinitializing the eikonal equation on reflecting walls and diffracting edges. To compute these scattered fields, we devise algorithms which allow us to apply reflection and diffraction boundary conditions for the eikonal and amplitude. For the amplitude, we construct algorithms that allow us to apply the uniform theory of diffraction in a semi-Lagrangian setting efficiently

Concepts for the Representation, Storage, and Retrieval of Spatio-Temporal Objects in 3D/4D Geo-Informations-Systems

The quickly increasing number of spatio-temporal applications in fields like environmental management or geology is a new challenge to the development of database systems. This thesis addresses three areas of the problem of integrating spatio-temporal objects into databases. First, a new representational model for continuously changing, spatial 3D objects is introduced and transferred into a small system of classes within an object-oriented database framework. The model extends simplicial cell complexes to the spatio-temporal setting. The problem of closure under certain operations is investigated. Second, internal data structures are introduced that represent instances of the (user-level) spatio-temporal classes. A new technique provides a compromise between compact storage and efficient retrieval of spatio-temporal objects. These structures correspond to temporal graphs and support updates as well as the maintainance of connected components over time. Third, it is shown how to realise further operations on the new type of objects. Among these operations are range queries, intersection tests, and the Euclidean distance function

Galerkin projection of discrete fields via supermesh construction

Interpolation of discrete FIelds arises frequently in computational physics. This thesis focuses on the novel implementation and analysis of Galerkin projection, an interpolation technique with three principal advantages over its competitors: it is optimally accurate in the L2 norm, it is conservative, and it is well-defined in the case of spaces of discontinuous functions. While these desirable properties have been known for some time, the implementation of Galerkin projection is challenging; this thesis reports the first successful general implementation. A thorough review of the history, development and current frontiers of adaptive remeshing is given. Adaptive remeshing is the primary motivation for the development of Galerkin projection, as its use necessitates the interpolation of discrete fields. The Galerkin projection is discussed and the geometric concept necessary for its implementation, the supermesh, is introduced. The efficient local construction of the supermesh of two meshes by the intersection of the elements of the input meshes is then described. Next, the element-element association problem of identifying which elements from the input meshes intersect is analysed. With efficient algorithms for its construction in hand, applications of supermeshing other than Galerkin projections are discussed, focusing on the computation of diagnostics of simulations which employ adaptive remeshing. Examples demonstrating the effectiveness and efficiency of the presented algorithms are given throughout. The thesis closes with some conclusions and possibilities for future work