Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications

Nonlocal gradient operators are prototypical nonlocal differential operators thatare very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on nonlocal gradient operators with a non-spherical interaction neighborhood. We show that the truncation of the spherical interaction neighborhood to a half sphere helps making nonlocal gradient operators well-defined and the associated nonlocal Dirichlet energies coercive. These become possible, unlike the case with full spherical neighborhoods, without any extra assumption on the strengths of the kernels near the origin. We then present some applications of the nonlocal gradient operators with non-spherical interaction neighborhoods. These include nonlocal linear models in mechanics such as nonlocal isotropic linear elasticity and nonlocal Stokes equations, and a nonlocal extension of the Helmholtz decomposition.Comment: Corrected typo

Surface-Only Liquids

© ACM, 2016. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Da, F., Hahn, D., Batty, C., Wojtan, C., & Grinspun, E. (2016). Surface-Only Liquids. Acm Transactions on Graphics, 35(4), 78. https://doi.org/10.1145/2897824.2925899We propose a novel surface-only technique for simulating incompressible, inviscid and uniform-density liquids with surface tension in three dimensions. The liquid surface is captured by a triangle mesh on which a Lagrangian velocity field is stored. Because advection of the velocity field may violate the incompressibility condition, we devise an orthogonal projection technique to remove the divergence while requiring the evaluation of only two boundary integrals. The forces of surface tension, gravity, and solid contact are all treated by a boundary element solve, allowing us to perform detailed simulations of a wide range of liquid phenomena, including waterbells, droplet and jet collisions, fluid chains, and crown splashes.European Research Council, National Science Foundation, Natural Sciences and Engineering Research Council of Canad

Surface-Only Simulation of Fluids

Surface-only simulation methods for fluid dynamics are those that perform computation only on a surface representation, without relying on any volumetric discretization. Such methods have superior asymptotic complexity in time and memory than the traditional volumetric discretization approaches, and thus are more tractable for simulation of complex fluid phenomena. Although for most computer graphics applications and many engineering applications, the interior flow inside the fluid phases is typically not of interest, the vast majority of existing numerical techniques still rely on discretization of the volumetric domain. My research first tackles the mesh-based surface tracking problem in the multimaterial setting, and then proposes surface-only simulation solutions for two scenarios: the soap-films and bubbles, and the general 3D liquids. Throughout these simulation approaches, all computation takes place on the surface, and volumetric discretization is entirely eliminated

Numerical Relativity: A review

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.Comment: 2 References added; 1 corrected. 67 pages. To appear in Classical and Quantum Gravity. (uses iopart.cls

Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves

The purpose of this dissertation is to develop numerical methods for fluid–structure interaction (FSI) analysis that are suitable for modeling and simulating bioprosthetic heart valves (BHVs). BHVs are prosthetic replacements for the valves that regulate blood flow through the heart. BHVs reproduce natural hemodynamic conditions by mimicking the structure of native heart valves: they consist of thin flexible leaflets, passively driven by interaction with surrounding fluid. Current designs frequently require replacement 10–15 years after implantation. Computer simulation may help identify causes of and solutions to durability issues. Despite much previous research into computer simulation of heart valve FSI, inconvenience or inaccuracy of readily available numerical methods have prevented widespread incorporation of FSI into models of heart valve mechanics. Challenges associated with heart valve FSI simulation include large deformations of the region occupied by fluid, with changes of topology as the valve opens and closes, and low mass of the structure relative to the fluid, which necessitates careful treatment of fluid–structure coupling. The presence of large pressure gradients also requires special attention to the treatment of fluid mass conservation. Further, a useful numerical method for studying and improving designs of BHVs should be able to capture variations of valve geometry without requiring major effort to construct geometry-specific discretizations. To meet these challenges, I develop a new numerical approach, combining the immersed boundary concept of capturing fluid–structure interfaces on unfitted discretizations with recent developments in isogeometric analysis (IGA), which directly uses geometrical designs of engineered systems as discrete analysis meshes. In this work, I immerse an isogeometric structure discretization into an unfitted analysis mesh of the fluid subproblem. I refer to the immersion of design geometries into unfitted analysis meshes as immersogeometric analysis. To reliably couple unfitted discretizations of the fluid and structure subproblems, I introduce a new semi-implicit time integration procedure and analyze its stability and convergence in the context of linear model problems. I verify that this analysis extrapolates to the nonlinear setting through numerical experiments and explore the validity of my modeling assumptions by comparing computer simulations with observations from an in vitro experiment.Computational Science, Engineering, and Mathematic