Varun Shankar

A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction---Diffusion Equations on Surfaces

In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) on closed surfaces embedded in

A radial basis function (RBF) finite difference method for the simulation of reaction–diffusion equations on stationary platelets within the augmented forcing method

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The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

Rd. Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

Radial basis function (RBF)-based parametric models for closed and open curves within the method of regularized stokeslets

The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid-structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmitting information (forces and velocities) between these two representations. Most IB simulations represent their structures with piecewise linear approximations and utilize Hookean spring models to approximate structural forces. Our specific motivation is the modeling of platelets in hemodynamic flows. In this paper, we study two alternative representations - radial basis functions (RBFs) and Fourier-based (trigonometric polynomials and spherical harmonics) representations - for the modeling of platelets in two and three dimensions within the IB framework, and compare our results with the traditional piecewise linear approximation methodology. For different representative shapes, we examine the geometric modeling errors (position and normal vectors), force computation errors, and computational cost and provide an engineering trade-off strategy for when and why one might select to employ these different representations.

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

SUMMARY We present a computational method for solving the coupled problem of chemical transport in a fluid (blood) with binding/unbinding of the chemical to/from cellular (platelet) surfaces in contact with the fluid, and with transport of the chemical on the cellular surfaces. The overall framework is the augmented forcing point method (AFM) (L. Yao and A.L. Fogelson, Simulations of chemical transport and reaction in a suspension of cells I: An augmented forcing point method for the stationary case, IJNMF (2012) 69, 1736–52.) for solving fluid-phase transport in a region outside of a collection of cells suspended in the fluid. We introduce a novel radial basis function–finite difference (RBF-FD) method to solve reaction–diffusion equations on the surface of each of a collection of 2D stationary platelets suspended in blood. Parametric RBFs are used to represent the geometry of the platelets and give accurate geometric information needed for the RBF-FD method. Symmetric Hermite-RBF interpolants are used for enforcing the boundary conditions on the fluid-phase chemical concentration, and their use removes a significant limitation of the original AFM. The efficacy of the new methods is shown through a series of numerical experiments; in particular, second-order convergence for the coupled problem is demonstrated. Copyright

Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

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RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computa...