Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

A low complexity algorithm for non-monotonically evolving fronts

A new algorithm is proposed to describe the propagation of fronts advected in the normal direction with prescribed speed function F. The assumptions on F are that it does not depend on the front itself, but can depend on space and time. Moreover, it can vanish and change sign. To solve this problem the Level-Set Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching Method [Carlini et al.; 2008] has recently been introduced. The novelty of our method is that its overall computational complexity is predicted to be comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky; 2006] in most instances. This latter algorithm is O(N^n log N^n) if the computational domain comprises N^n points. Our strategy is to use it in regions where the speed is bounded away from zero -- and switch to a different formalism when F is approximately 0. To this end, a collection of so-called sideways partial differential equations is introduced. Their solutions locally describe the evolving front and depend on both space and time. The well-posedness of those equations, as well as their geometric properties are addressed. We then propose a convergent and stable discretization of those PDEs. Those alternative representations are used to augment the standard Fast Marching Method. The resulting algorithm is presented together with a thorough discussion of its features. The accuracy of the scheme is tested when F depends on both space and time. Each example yields an O(1/N) global truncation error. We conclude with a discussion of the advantages and limitations of our method.Comment: 30 pages, 12 figures, 1 tabl

Finite Element Methods in Smart Materials and Polymers

Functional polymers show unique physical and chemical properties, which can manifest as dynamic responses to external stimuli such as radiation, temperature, chemical reaction, external force, and magnetic and electric fields. Recent advances in the fabrication techniques have enabled different types of polymer systems to be utilized in a wide range of potential applications in smart structures and systems, including structural health monitoring, anti‐vibration, and actuators. The progress in these integrated smart structures requires the implementation of finite element modelling using a multiphysics approach in various computational platforms. This book presents finite element methods applied in modeling of the smart structures and materials with particular emphasis on hydrogels, metamaterials, 3D-printed and anti-vibration constructs, and fibers

Further development of Level Set method (modified level set equation and its numerical assessment)

Pas de résuméThe level set method was introduced by Osher & Sethian (1988) as a general technique to capture moving interfaces. It has been used to study crystal growth, to simulate water and fire for computer graphics applications, to study two-phase flows and in many other fields. The wellknown problem of the level set method is the following: if the flow velocity is not constant, the level set scalar may become strongly distorted. Thus, the numerical integration may suffer from loss of accuracy. In level set methods, this problem is remedied by the reinitialization procedure, i.e. by reconstruction of the level set function in a way to satisfy the eikonal equation. We propose an alternative approach. We modify directly the level set equation by embedding a source term. The exact expression of this term is such that the eikonal equation is automatically satisfied. Furthermore on the interface, this term is equal to zero. In the meantime, the advantage of our approach is this: the exact expression of the source term allows for the possibility of derivation of its local approximate forms, of first-and-higher order accuracy. Compared to the extension velocity method, this may open the simplifications in realization of level set methods. Compared to the standard approach with the reinitialization procedure, this may give the economies in the number of level set re-initializations, and also, due to reduced number of reinitializations, one may expect an improvement in resolution of zero-set level. Hence, the objective of the present dissertation is to describe and to assess this approach in different test cases.LYON-Ecole Centrale (690812301) / SudocSudocFranceF

Implicit boundary integral methods

Boundary integral methods (BIMs) solve constant coefficient, linear partial differential equations (PDEs) which have been formulated as integral equations. Implicit BIMs (IBIMs) transform these boundary integrals in a level set framework, where the boundaries are described implicitly as the zero level set of a Lipschitz function. The advantage of IBIMs is that they can work on a fixed Cartesian grid without having to parametrize the boundaries. This dissertation extends the IBIM model and develops algorithms for problems in two application areas. The first part of this dissertation considers nonlinear interface dynamics driven by bulk diffusion, which involves solving Dirichlet Laplace Problems for multiply connected regions and propagating the interface according to the solutions of the PDE at each time instant. We develop an algorithm that inherits the advantages of both level set methods (LSMs) and BIMs to simulate the nonlocal front propagation problem with possible topological changes. Simulation results in both 2D and 3D are provided to demonstrate the effectiveness of the algorithm. The second part considers wave scattering problems in unbounded domains. To obtain solutions at eigenfrequencies, boundary integral formulations use a combination of double and single layer potentials to cover the null space of the single layer integral operator. However, the double layer potential leads to a hypersingular integral in Neumann problems. Traditional schemes involve an interpretation of the integral as its Hadamard's Finite Part or a complicated process of element kernel regularization. In this thesis, we introduce an extrapolatory implicit boundary integral method (EIBIM) that evaluates the natural definition of the BIM. It is able to solve the Helmholtz problems at eigenfrequencies and requires no extra complication in different dimensions. We illustrate numerical results in both 2D and 3D for various boundary shapes, which are implicitly described by level set functions.Mathematic