Adaptive methodology for meshless finite point method

In this work, a posteriori error estimator and an adaptive refinement process for the meshless finite point method (FPM), which is based on point collocation, are presented. The error indicator is formulated by the least-squares functional evaluation, used in the shape function development. New degrees of freedom or additional points can be incorporated without difficulty, in zones where the error estimator presents a high value, by means of h–p refinement processes. The validity of the proposed error estimator can be demonstrated by developments of numerical problems in mechanics of solids, using an adaptive refinement process of the solution

Adaptive meshless centres and RBF stencils for Poisson equation

We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method

Direct solution of Navier-Stokes equations by using an upwind local RBF-DQ method

The differential quadrature (DQ) method is able to obtain quite accurate numerical solutions of differential equations with few grid points and less computational effort. However, the traditional DQ method is convenient only for regular regions and lacks upwind mechanism to characterize the convection of the fluid flow. In this paper, an upwind local radial basis function-based DQ (RBF-DQ) method is applied to solve the Navier-Stokes equations, instead of using an iterative algorithm for the primitive variables. The non-linear collocated equations are solved using the Levenberg-Marquardt method. The irregular regions of 2D channel flow with different obstructions situations are considered. Finally, the approach is validated by comparing the results with those obtained using the well-validated Fluent commercial package

Comparative analysis between themaxent and the weighted least square shape functions in acollocation meshless method

En el presente artículo se analiza el comportamiento de la función de forma basada en el principio de máxima entropía (maxent), en el contexto de un método sin malla con un esquema de colocación, comparando su resultado con la función de forma tradicional basada en mínimos cuadrados ponderados fijos (FWLS). La función de forma maxent considerada en el presente trabajo posee ciertas propiedades deseables para formulaciones sin malla basadas en un esquema de colocación, como lo son su positividad, suavidad y aspecto uniforme, para distintos tipos de discretizaciones. Además, en los contornos, la aproximación no depende de las funciones de forma de los nodos interiores del dominio, propiedad que se conoce como reducción de la función de forma sobre el contorno. Para comparar este tipo de funciones se han desarrollado ejemplos que incluyen la resolución de ecuaciones elípticas de segundo orden, en 1D y 2D. Los resultados numéricos muestran un mejor comportamiento de la función de forma maxent en comparación con la de FWLS, en particular en cuanto a la convergencia y estabilidad del método sin malla de colocación resultante.In this article the behavior of a shape function based on the maximum entropy principle (maxent) is analyzed in a meshless collocation method, compared with a traditional fixed weighted least square shape function (FWLS). The maxent shape function used in this work has certain properties that are desired in a meshless collocation method, for example the positivity, the smooth and uniform aspect for different discretizations. Further, in the boundary, the approximation not depends of the shape function of the interior nodes, this property is know as a reduction of the shape function on the boundary. To compare this type of function, it was developed examples that include the solution of eliptical second order equations in 1D and 2D. The numerical results shown a better behavior of the maxent shape function compared with the FWLS, particularly in terms of the convergence and stability of the meshless collocations method that result.Peer Reviewe

A-posteriori error estimation for the finite point method with applications to compressible flow

An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented.Peer ReviewedPostprint (author's final draft

A-posteriori error estimation for the finite point method with applications to compressible flow (preprint)

An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented

A finite point method for adaptive three-dimensional compressible flow calculation

The Finite Point Method (FPM) is a meshless technique which is based on both, a Weighted Least-Squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a major investigation into the capabilities of the FPM to deal with threedimensional applications concerning real compressible fluid flow problems. In the first part of this work, the upwind biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting meshless capabilities, an h-adaptive methodology for two and three-dimensional compressible flow calculations is developed. This adaptive technique applies a solution-based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are highly encouraging. Several numerical examples are provided throughout the article in order to illustrate their performance

A finite point method for adaptive three-dimensional compressible flow calculation

The Finite Point Method (FPM) is a meshless technique which is based on both, a Weighted Least-Squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a major investigation into the capabilities of the FPM to deal with threedimensional applications concerning real compressible fluid flow problems. In the first part of this work, the upwind biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting meshless capabilities, an h-adaptive methodology for two and three-dimensional compressible flow calculations is developed. This adaptive technique applies a solution-based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are highly encouraging. Several numerical examples are provided throughout the article in order to illustrate their performance.Preprin

An adaptive finite point method for the shallow water equations

An adaptive Finite Point Method (FPM) for solving shallow water problems is presented. The numerical methodology we propose, which is based on weighted‐least squares approximations on clouds of points, adopts an upwind‐biased discretization for dealing with the convective terms in the governing equations. The viscous and source terms are discretized in a pointwise manner and the semi‐discrete equations are integrated explicitly in time by means of a multi‐stage scheme. Moreover, with the aim of exploiting meshless capabilities, an adaptive h‐refinement technique is coupled to the described flow solver. The success of this approach in solving typical shallow water flows is illustrated by means of several numerical examples and special emphasis is placed on the adaptive technique performance. This has been assessed by carrying out a numerical simulation of the 26th December 2004 Indian Ocean tsunami with highly encouraging results. Overall, the adaptive FPM is presented as an accurate enough, cost‐effective tool for solving practical shallow water problems. Copyright © 2011 John Wiley & Sons, Ltd

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented.&nbsp