A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow

Radhakrishnan, Anand; Xu, Michael; Shahane, Shantanu; Vanka, Surya Pratap

doi:10.48550/arxiv.2104.13758

Cited by 5 publications

(6 citation statements)

References 60 publications

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“…We have used here a simple single-grid SOR scheme for its simplicity and economy. However, it can be made to converge much faster by using a multilevel strategy, as in our recent work [32].…”

Section: Discussionmentioning

confidence: 99%

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Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

Shahane¹,

Vanka²

2022

Preprint

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Computations of incompressible flows with all velocity boundary conditions require solution of a Poisson equation for pressure or pressure-corrections with all Neumann boundary conditions. Discretization of such a Poisson equation results in a rank-deficient ill-conditioned matrix of coefficients. When a non-conservative discretization method such as finite difference, finite element, or spectral scheme is used, the ill-conditioned matrix also generates an inconsistency which makes the residuals in the iterative solution to saturate at a threshold level that depends on the spatial resolution and the order of the discretization scheme. In this paper, we examine this inconsistency for a high-order meshless discretization scheme suitable for solving the equations on a complex domain. The high order meshless method uses polyharmonic spline radial basis functions (PHS-RBF) with appended polynomials to interpolate scattered data and constructs the discrete equations by collocation.

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Section: Discussionmentioning

confidence: 99%

“…The SOR solver can be easily coupled with a multilevel method to accelerate convergence. We are in the process of developing such an algorithm [32]…”

Section: Poisson Equation With Manufactured Solution Using Meshless D...mentioning

confidence: 99%

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

Shahane¹,

Vanka²

2022

Preprint

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“…Flatter functions give higher accuracy but result in coefficient matrices of high condition numbers [39][40][41][42][43][44][45][46][47][48][49]. Appending polynomials to the RBF has been shown to give high accuracy, depending on the degree of the highest monomial appended [19][20][21][22][50][51][52][53][54][55][56]. Ideally, high order of accuracy can be achieved by appending suitably high order monomials.…”

Section: Multiquadrics (Mq)mentioning

confidence: 99%

Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

Naman¹,

Shahane²,

Roy³

et al. 2021

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In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.

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“…earlier [45,49,50] that when the RBFs are appended with polynomials and the derivatives of a smooth function are evaluated, the discretization errors converge exponentially per the degree of the appended polynomial. RBFs have been originally used as global interpolants of scattered data [51] and to solve partial differential equations [52][53][54], but the size of the problem has been limited because of the condition number of the solution matrix.…”

Section: Introductionmentioning

confidence: 99%

A Semi-Implicit Meshless Method for Incompressible Flows in Complex Geometries

Shahane¹,

Vanka²

2021

Preprint

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We present an exponentially convergent semi-implicit meshless algorithm for the solution of Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives at scattered points using radial basis functions as interpolants. Higher-order polynomials are appended to polyharmonic splines (PHS-RBF) and a collocation method is used to derive the interpolation coefficients. The interpolating kernels are then differentiated and the partial-differential equations are satisfied by collocation at the scattered points. The PHS-RBF interpolation is shown to be exponentially convergent with discretization errors decreasing as a high power of a representative distance between points. We present here a semi-implicit algorithm for timedependent and steady state fluid flows in complex domains. At each time step, several iterations are performed to converge the momentum and continuity equations. A Poisson equation for pressure corrections is formulated by imposing divergence free condition on the iterated velocity field. At each time step, the momentum and pressure correction equations are repeatedly

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A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow

Cited by 5 publications

References 60 publications

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

A Semi-Implicit Meshless Method for Incompressible Flows in Complex Geometries

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