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Register a new userMeshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach
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We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
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We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $d{X}_t = a({X}_t)dt + {b}({X}_t)d{W}_t$. We consider on a surface domain $Omega$ the statistics $u(mathbf{x}) = mathbb{E}^{mathbf{x}}left[int_0^tau g(X_t)dt right] + mathbb{E}^{mathbf{x}}left[f(X_tau)right]$ with the exit stopping time $tau = inf_t {t > 0 ; |; X_t otin Omega}$. Using Dynkins formula, we compute statistics by developing high-order Generalized Moving Least Squares (GMLS) solvers for the associated surface PDE boundary-value problems. We focus particularly on the mean First Passage Times (FPTs) given by the special case $f = 0,, g = 1$ with $u(mathbf{x}) = mathbb{E}^{mathbf{x}}left[tauright]$. We perform studies for a variety of shapes showing our methods converge with high-order accuracy both in capturing the geometry and the surface PDE solutions. We then perform studies showing how FPTs are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-spaces dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.
We develop and implement a novel lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. Standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write down the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focussing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of a torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.
In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.
Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squares distance $$min {|X_1+ ... + X_N - A_0|: X_j in S(A_j), j = 1, >..., N}.$$ Connections of the results to different pure and applied areas are discussed.
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