# Your search: "author:"Sethian, James A""

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## Scholarly Works (16 results)

The simulation of the dynamics of thin elastic surfaces is an essential component in applications ranging from cardiovascular medicine to flapping wing aerospace engineering. In this thesis we introduce a fully Eulerian method for accurately simulating elastic surfaces. Our approach can be applied to membranes and shells with nonlinear elastic properties, either moving under their own dynamics or immersed in a Fluid. By representing the surface with a level set function and a set of advected reference coordinates, we are able to evaluate the full elastic surface forces. Our approach is compatible with compatible with implicit interface representations, including level set techniques, and can make use of high-order finite difference stencils or more sophisticated techniques. We introduce our method and its implementation, including new solution techniques for several problems associated with immersed interface representations. We demonstrate second order accurate numerical results for a range of model problems in two and three dimensions.

The volume fraction data structure describes the location of a given material within space. It partitions a domain into a set of discrete cells, and for each cell stores the ratio of material contained within it to the total volume of the cell. The task of transforming a grid of volume fractions into a surface representation of the boundary is called the Material Interface Reconstruction Problem. We investigate a variational formulation to this problem, implemented via a level set method, that produces a surface representation of the boundary. We start with an initial guess and iteratively refine the surface by computing a surface-area minimizing curvature flow, followed by an approximate L2 projection of that flow onto a volume-preserving space. We find that the method yields satisfactory results for both two-phase and multiphase data. In the cases where there are C0 but not C1 boundaries between only two materials, the exact solution produces oscillations with period equal to the volume fraction grid size, and amplitude exponentially decreasing away from the location of the loss of smoothness. To reduce oscillations, an alternative method is proposed where in place of the volume fraction constraint, we minimize a weighted sum of the total surface area, plus the sum of squared error in reconstruction. We discuss the results and compare with the projection algorithm.

In this thesis, the Voronoi Implicit Interface Method (VIIM) is presented together with several applications in multiphase curvature flow, multiphase incompressible fluid flow, mesh generation for interconnected surfaces, and multiscale modelling of foam dynamics. The VIIM tracks the evolution of multiple interacting regions ("phases") whose motion may be determined by geometry, complex physics, intricate jump conditions, internal constraints, and boundary conditions. From a mathematical point of view, the method provides a theoretical framework to evolve interconnected interfaces with junctions. Discretising this theoretical framework leads to an efficient Eulerian-based numerical method that uses a single unsigned distance function, together with a region indicator function, to represent a multiphase system. The VIIM works in any number of spatial dimensions, accurately represents complex geometries involving triple and higher-order junctions, and automatically handles topological changes in the evolving interface, including creation and destruction of phases. Here, the central ideas behind the method are presented, implementation is discussed, and convergence tests are performed to illustrate the accuracy of the method. Several applications of the VIIM are shown, including in constant speed normal driven flow; multiphase curvature flow with constraints; and multiphase incompressible fluid flow in which density, viscosity, and surface tension can be defined on a per-phase basis and membranes can be permeable.

An efficient and robust mesh generation algorithm for interconnected surfaces is also presented. The algorithm capitalises on a geometric construction used in the VIIM, known as the "Voronoi interface", to generate high-quality triangulated meshes that are topologically consistent, such that mesh elements meet precisely at junctions without gaps, overlaps, or hanging nodes. The generated meshes can be used in finite element methods for solving partial differential equations on a network of evolving interconnected curved surfaces.

Finally, a scale-separated, multiscale model for the dynamics of a soap bubble foam is presented. The model leads to a computational framework for studying the interlinked effects of drainage, rupture, and rearrangement in a foam of bubbles, coupling microscale fluid flow in a network of thin-film membranes ("lamellae") and junctions ("Plateau borders") to macroscale gas dynamics driven by surface tension. Here, thin-film equations for fluid flow inside curved lamellae and Plateau borders are derived, flux boundary conditions which conserve liquid mass are developed, and local conservation laws for transport of film thickness during rearrangement are designed. From a numerical perspective, several new numerical methods are developed, including Lagrangian-based schemes for conserving liquid in the membranes during rearrangement, finite element methods to solve fourth-order nonlinear partial different equations on curved surfaces, methods to accurately solve coupled flux boundary conditions at Plateau borders and quadruple points, and projection methods to couple gas dynamics to the VIIM. Convergence tests are performed to demonstrate the accuracy of the numerical methods, and results of the multiscale model are shown for a variety of problems, including collapsing foam clusters displaying thin-film interference effects.

X-ray nanocrystallography is an emerging technique for imaging nanoscale objects that alleviates the large crystallization requirement of conventional crystallography by collecting diffraction patterns from a large ensemble of smaller and easier to build nanocrystals, which are typically delivered to the x-ray beam via a liquid jet. In order to determine the structure of an imaged object, several parameters must first be determined, including the crystal sizes, incident photon flux densities, and crystal orientations. Autoindexing techniques, which have been used extensively to orient conventional crystals, only determine the orientation of the nanocrystals up to symmetry of the crystal lattice, which is often greater than the symmetry of the diffraction information, resulting in what is known as the twinning problem. In addition, the image data is corrupted by large degrees of shot noise due to low collected signal, background signal due to the liquid jet and detector electronics, as well as other sources of noise. Furthermore, diffraction only measures the magnitudes of the Fourier transform of the object and, thus, one must recover phase information in order to invert the data and recover a three-dimensional reconstruction of the constituent molecular structure. Previous approaches for handling the twinning problem have mainly relied on having a known similar structure available, which may not be present for fundamentally new structures. We present a series of techniques to determine the crystal sizes, incident photon flux densities, and crystal orientations in the presence of large amounts of noise common in experiments. Additionally, by using a new sampling strategy, we demonstrate that phase information can be computed from nanocrystallographic diffraction images using only Fourier magnitude information, via a compressive phase retrieval algorithm. We demonstrate the feasibility of this new approach by testing it on simulated data with parameters and noise levels common in current experiments.

We present a general framework for accurately evaluating finite difference operators in the presence of known discontinuities across an interface. Using these techniques, we develop simple-to-implement, second-order accurate methods for elliptic problems with interfacial discontinuities and for the incompressible Navier-Stokes equations with singular forces. To do so, we first establish an expression relating the derivatives being evaluated, the finite difference stencil, and a compact extrapolation of the jump conditions. By representing the interface with a level set function, we show that this extrapolation can be constructed using dimension- and coordinate-independent normal Taylor expansions with arbitrary order of accuracy. Our method is robust to non-smooth geometry, permits the use of symmetric positive-definite solvers for elliptic equations, and also works in 3D with only a change in finite difference stencil. We rigorously establish the convergence properties of the method and present extensive numerical results. In particular, we show that our method is second-order accurate for the incompressible Navier-Stokes equations with surface tension.

In this thesis, we discuss various techniques for improving exploration for deep reinforcement learning. We begin with a brief review of reinforcement learning (RL) and the fundamental v.s. exploitation trade-off. Then we review how deep RL has improved upon classical and summarize six categories of the latest exploration methods for deep RL, in the order increasing usage of prior information. We then explore representative works in three categories discuss their strengths and weaknesses. The first category, represented by Soft Q-learning, uses regularization to encourage exploration. The second category, represented by count-based via hashing, maps states to hash codes for counting and assigns higher exploration to less-encountered states. The third category utilizes hierarchy and is represented by modular architecture for RL agents to play StarCraft II. Finally, we conclude that exploration by prior knowledge is a promising research direction and suggest topics of potentially impact.