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

Shankar, Varun; Narayan, Akil; Kirby, Robert M.

doi:10.1016/j.jcp.2018.07.015

Cited by 34 publications

(32 citation statements)

References 40 publications

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“…where is selected to be the minimum to prove unisolvency for the RBF interpolant with φ(r) = r m . This approach was also recently used for PHS-based methods for advection on the sphere [31], and advection and reaction-diffusion equations on more general manifolds [32]. However, modern PHS RBF-FD methods for Euclidean domains decouple m and [4][5][6]24].…”

Section: A New Scaling Law For M Andmentioning

confidence: 99%

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

Shankar

Fogelson

2018

Journal of Computational Physics

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We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advectiondiffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit-explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

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Section: A New Scaling Law For M Andmentioning

confidence: 99%

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

Shankar

Fogelson

2018

Journal of Computational Physics

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“…RBF kernel function has good application range and fitness, and low complexity. 34,35 It is not only suitable for large samples, but also for small samples. Therefore, we choose the RBF kernel function here.…”

Section: Svmmentioning

confidence: 99%

Classification of corn kernels grades using image analysis and support vector machine

Zhu

Yang

et al. 2018

Advances in Mechanical Engineering

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In order to classify the quality of corn kernels in an affordable, convenient, and accurate manner, a method based on image analysis and support vector machine is proposed. A total of 129 corn kernels with Grade A, Grade B, and Grade C are used for the experiments. Six typical characteristic parameters of samples are extracted as the characteristic groups. Four different classifiers are applied and compared: support vector machine-genetic algorithm, support vector machine-particle swarm optimization, support vector machine-grid search optimization, and back-propagation neural networks. Experimental results show that the support vector machine and back-propagation neural networks without parameter optimization have the same classification accuracy rates of 92.31%. The classification accuracies are improved using the support vector machine optimization algorithms. The average correct classification rates of support vector machine-genetic algorithm and support vector machine-particle swarm optimization are all 97.44%, while the correct classification rate of support vector machine-grid search achieves 94.87%. It is concluded that the support vector machine algorithm based on parameter optimization is superior to back-propagation neural networks algorithm, and the parameter optimization effects of genetic algorithm and particle swarm optimization are better than grid search method. With a relatively small number of samples, the support vector machine-genetic algorithm and support vector machineparticle swarm optimization algorithms can improve the grading accuracy of corn kernels.

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“…In the broader meshfree discretization literature, a number of works have developed discretizations of general surface PDE [45,111,62,96,104,105,81,110,95,73,43,88,68]. Such schemes treat the manifold extrinsically, working in ambient space and projecting to manifold, or intrinsically, using compact reconstructions of the manifold to obtain local coordinates.…”

Section: Introductionmentioning

confidence: 99%

A General-Purpose, Inelastic, Rotation-Free Kirchhoff-Love Shell Formulation for Peridynamics

Behzadinasab,

Alaydin,

Trask

et al. 2021

Preprint

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We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.

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

Cited by 34 publications

References 40 publications

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

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

Classification of corn kernels grades using image analysis and support vector machine

A General-Purpose, Inelastic, Rotation-Free Kirchhoff-Love Shell Formulation for Peridynamics

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