Abstract
A new numerical tool is presented which models the two-dimensional contaminant transport through saturated porous media using a meshfree method called the radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. An advection-dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using two-dimensional meshfree shape functions constructed using thin plate spline radial basis functions. A computer program is developed for the implementation of the RPIM procedure. Three numerical examples are presented and the results are compared with those obtained from the analytical solution and finite element method. The experimental results are also used to validate the approach. The proposed RPIM has generated results with no oscillations and they are insensitive to Peclet constraints.
Résumé
Un nouvel outil numérique est présenté pour modéliser les transports bidimensionnels de contaminants à travers une matrice poreuse saturée, en utilisant une méthode sans maillage appelée Méthode d'Interpolation de Point Radial (RPIM) avec reproduction polynomiale. Dans la RPIM, une solution approchée est intégralement constituée d'un ensemble de noeuds, et ne nécessite aucune détermination des relations internodales. Afin d'illustrer l'applicabilité de la RPIM, une équation d'advection-dispersion avec sorption a été étudiée. La forme faible de type Galerkin de l'équation directrice est formulée en utilisant des fonctions bidimensionnelles sans maille, construites en utilisant des fonctions spline plaque mince de base radiale. Un programme informatique est développé pour mettre en oeuvre la procédure RPIM. Trois exemples numériques sont présentés, et leurs résultats sont comparés à ceux obtenus par les solutions analytiques et par la méthode des éléments finis. Les résultats expérimentaux sont également utilisés pour valider l'approche. La RPIM proposée génère des résultats sans oscillations, qui sont insensibles aux contraintes de Peclet.
Resumen
Se presenta una nueva herramienta numérica que modela el transporte bidimensional de contaminantes en medios porosos saturados utilizando el método sin malla llamado Interpolación Radial Puntual (MIRP) con reproducción polinomial. En el MIRP se elabora una solución aproximada exclusivamente en función del grupo de nodos, y no se necesita una caracterización de la relación entre nodos. A fin de ilustrar la aplicabilidad del MIRP, se considera una ecuación de advección-dispersión con sorción. La forma débil de Galerkin para la ecuación dominante se formula con funciones de forma bidimensionales sin malla usando funciones básicas radiales del tipo de placa delgada spline. Se desarrolla un programa computacional para la implementación del procedimiento MIRP. Se presentan tres ejemplos numéricos y sus resultados se comparan con aquellos obtenidos con la solución analítica y con el método de los elementos finitos. Los resultados experimentales también se utilizan para validar la aproximación. La propuesta MIRP genera resultados estables y no es sensible a las restricciones impuestas por el número de Peclet.
Resumo
É apresentada uma ferramenta numérica que permite modelar bidimensionalmente o transporte de um contaminante através de um meio poroso saturado utilizando o Método de Interpolação Pontual Radial (MIPR) com reprodução polinomial. No MIPR é determinada uma solução aproximada em todos os nós em que é discretizado o espaço, não sendo necessário conhecer a caracterização das relações entre os nós. Para ilustrar a aplicabilidade deste método é utilizada uma equação de transporte em que é válido um processo advectivo-dispersivo com sorpção. A formulação do tipo Galerkin das equações fundamentais utiliza funções 2D construídas a partir de funções do tipo spline achatadas. Para a implementação do processo MIPR foi desenvolvido um programa computacional. São apresentados três exemplos numéricos e os resultados comparados com os obtidos por um método analítico e por outro numérico. Os resultados experimentais são também utilizados para validar a abordagem. Os resultados gerados pelo MIPR não apresentam oscilações sendo insensíveis às restrições Peclet.
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The authors would like to thank the anonymous reviewers for their careful and insightful review of this paper.
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Praveen Kumar, R., Dodagoudar, G.R. Two-dimensional modelling of contaminant transport through saturated porous media using the radial point interpolation method (RPIM). Hydrogeol J 16, 1497–1505 (2008). https://doi.org/10.1007/s10040-008-0325-y
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DOI: https://doi.org/10.1007/s10040-008-0325-y