Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences

2015 ◽  
pp. 2635-2669 ◽  
Author(s):  
Natasha Flyer ◽  
Grady B. Wright ◽  
Bengt Fornberg
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Finite Differences ◽  
Mesh Free ◽  
Radial Basis ◽  
Mesh Free Method

On the forward modelling of three-dimensional magnetotelluric data using a radial-basis-function-based mesh-free method

2019 ◽  
Vol 219 (1) ◽  
pp. 394-416 ◽  
Author(s):  
Jianbo Long ◽  
Colin G Farquharson
Keyword(s):  
Finite Difference ◽  
Radial Basis Function ◽  
Basis Function ◽  
Weak Form ◽  
Forward Modelling ◽  
Magnetotelluric Data ◽  
Mesh Free ◽  
Radial Basis ◽  
Mesh Free Method ◽  
Mesh Free Methods

SUMMARY The investigation of using a novel radial-basis-function-based mesh-free method for forward modelling magnetotelluric data is presented. The mesh-free method, which can be termed as radial-basis-function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (e.g. grid-based finite difference, finite volume and finite element), the mesh-free method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements is employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjunction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common mesh-free methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard mesh-free methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed mesh-free method is validated with two numerical examples of modelling the magnetotelluric responses over 3-D conductivity models.

Seismic modeling with radial-basis-function-generated finite differences

Geophysics ◽  
2015 ◽  
Vol 80 (4) ◽  
pp. T137-T146 ◽  
Author(s):  
Bradley Martin ◽  
Bengt Fornberg ◽  
Amik St-Cyr
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Finite Differences ◽  
Seismic Modeling ◽  
Radial Basis

scholarly journals A Strong-Form Off-Lattice Boltzmann Method for Irregular Point Clouds

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1802
Author(s):  
Ivan Pribec ◽  
Thomas Becker ◽  
Ehsan Fattahi
Keyword(s):  
Radial Basis Function ◽  
Lattice Boltzmann Method ◽  
Basis Function ◽  
Lattice Boltzmann ◽  
Finite Differences ◽  
Strong Form ◽  
Lattice Boltzmann Methods ◽  
Radial Basis ◽  
Irregular Point ◽  
Boltzmann Method

Radial basis function generated finite differences (RBF-FD) represent the latest discretization approach for solving partial differential equations. Their benefits include high geometric flexibility, simple implementation, and opportunity for large-scale parallel computing. Compared to other meshfree methods, typically based upon moving least squares (MLS), the RBF-FD method is able to recover a high order of algebraic accuracy while remaining better conditioned. These features make RBF-FD a promising candidate for kinetic-based fluid simulations such as lattice Boltzmann methods (LB). Pursuant to this approach, we propose a characteristic-based off-lattice Boltzmann method (OLBM) using the strong form of the discrete Boltzmann equation and radial basis function generated finite differences (RBF-FD) for the approximation of spatial derivatives. Decoupling the discretizations of momentum and space enables the use of irregular point cloud, local refinement, and various symmetric velocity sets with higher order isotropy. The accuracy and computational efficiency of the proposed method are studied using the test cases of Taylor–Green vortex flow, lid-driven cavity, and periodic flow over a square array of cylinders. For scattered grids, we find the polyharmonic spline + poly RBF-FD method provides better accuracy compared to MLS. For Cartesian node layouts, the results are the opposite, with MLS offering better accuracy. Altogether, our results suggest that the RBF-FD paradigm can be applied successfully also for kinetic-based fluid simulation with lattice Boltzmann methods.

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