400 research outputs found
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations
Meshfree solution schemes for the incompressible Navier--Stokes equations are
usually based on algorithms commonly used in finite volume methods, such as
projection methods, SIMPLE and PISO algorithms. However, drawbacks of these
algorithms that are specific to meshfree methods have often been overlooked. In
this paper, we study the drawbacks of conventionally used meshfree Generalized
Finite Difference Method~(GFDM) schemes for Lagrangian incompressible
Navier-Stokes equations, both operator splitting schemes and monolithic
schemes. The major drawback of most of these schemes is inaccurate local
approximations to the mass conservation condition. Further, we propose a new
modification of a commonly used monolithic scheme that overcomes these problems
and shows a better approximation for the velocity divergence condition. We then
perform a numerical comparison which shows the new monolithic scheme to be more
accurate than existing schemes
Higher-Order GFDM for Linear Elliptic Operators
We present a novel approach of discretizing diffusion operators of the form
in the context of meshfree generalized finite
difference methods. Our ansatz uses properties of derived operators and
combines the discrete Laplace operator with reconstruction functions
approximating the diffusion coefficient . Provided that the
reconstructions are of a sufficiently high order, we prove that the order of
accuracy of the discrete Laplace operator transfers to the derived diffusion
operator. We show that the new discrete diffusion operator inherits the
diagonal dominance property of the discrete Laplace operator and fulfills
enrichment properties. Our numerical results for elliptic and parabolic partial
differential equations show that even low-order reconstructions preserve the
order of the underlying discrete Laplace operator for sufficiently smooth
diffusion coefficients. In experiments, we demonstrate the applicability of the
new discrete diffusion operator to interface problems with point clouds not
aligning to the interface and numerically prove first-order convergence
A Quasi-Conforming Embedded Reproducing Kernel Particle Method for Heterogeneous Materials
We present a quasi-conforming embedded reproducing kernel particle method
(QCE-RKPM) for modeling heterogeneous materials that makes use of techniques
not available to mesh-based methods such as the finite element method (FEM) and
avoids many of the drawbacks in current embedded and immersed formulations
which are based on meshed methods. The different material domains are
discretized independently thus avoiding time-consuming, conformal meshing. In
this approach, the superposition of foreground (inclusion) and background
(matrix) domain integration smoothing cells are corrected by a quasi-conforming
quadtree subdivision on the background integration smoothing cells. Due to the
non-conforming nature of the background integration smoothing cells near the
material interfaces, a variationally consistent (VC) correction for domain
integration is introduced to restore integration constraints and thus optimal
convergence rates at a minor computational cost. Additional interface
integration smoothing cells with area (volume) correction, while
non-conforming, can be easily introduced to further enhance the accuracy and
stability of the Galerkin solution using VC integration on non-conforming
cells. To properly approximate the weak discontinuity across the material
interface by a penalty-free Nitsche's method with enhanced coercivity, the
interface nodes on the surface of the foreground discretization are also shared
with the background discretization. As such, there are no tunable parameters,
such as those involved in the penalty type method, to enforce interface
compatibility in this approach. The advantage of this meshfree formulation is
that it avoids many of the instabilities in mesh-based immersed and embedded
methods. The effectiveness of QCE-RKPM is illustrated with several examples
Meshfree Methods for PDEs on Surfaces
This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software.
In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost.
The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative methods for solving linear systems that arise in numerical PDEs. The key components for multilevel methods: \grid coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution. We finish with several application problems.
We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
In this work, we apply stochastic collocation methods with radial kernel
basis functions for an uncertainty quantification of the random incompressible
two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the
existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase
Navier-Stokes equation for each given realization. We are able to empirically
show that the resulting kernel-based stochastic collocation is highly
competitive in this setting and even outperforms some other standard methods
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