Combining the radial basis function Eulerian and Lagrangian schemes with geostatistics for modeling of radionuclide migration through the geosphere

To assess the long-term safety of a radioactive waste disposal system, mathematical models are used to describe groundwater flow, chemistry, and potential radionuclide migration through geological formations. A number of processes need to be considered, when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity as an input parameter varies from place to place. In such cases, geostatistical science offers a variety of spatial estimation procedures. Methods for solving the solute transport equation can also be classified as Eulerian, Lagrangian and mixed. The numerical solution of partial differential equations (PDE) has usually been obtained by finite-difference methods (FDM), finite-element methods (FEM), or finite-volume methods (FVM). Kansa introduced the concept of solving partial differential equations using radial basis functions (RBF) for hyperbolic, parabolic, and elliptic PDEs. The aim of this study was to present a relatively new approach to the modeling of radionuclide migration through the geosphere using radial basis function methods in Eulerian and Lagrangian coordinates. In this study, we determine the average and standard deviation of radionuclide concentration with regard to variable hydraulic conductivity, which was modelled by a geostatistical approach. Radionuclide concentrations will also be calculated in heterogeneous and partly heterogeneous 2D porous media. (C) 2004 Elsevier Ltd. All rights reserved

The use of the mesh free methods (radial basis functions) in the modeling of radionuclide migration and moving boundary value problems

Recently, the mesh free methods (radial basis functions-RBFs) have emerged as a novel computing method in the scientific and engineering computing community. The numerical solution of partial differential equations (PDEs) has been usually obtained by finite difference methods (FDM), finite element methods (FEM) and boundary elements methods (BEM). These conventional numerical methods still have some drawbacks. For example, the construction of the mesh in two or more dimensions is a nontrivial problem. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the mesh free method, which uses radial basis functions, with the traditional finite difference scheme and analytical solutions. We will present some examples of using RBFs in geostatistical analysis of radionuclide migration modeling. The advection-dispersion equation will be used in the Eulerian and Lagrangian forms. Stefan's or moving boundary value problems will also be presented. The position of the moving boundary will be simulated by the moving data centers method and level set method

Modelling of radionuclide migration through the geosphere with radial basis function method and geostatistics

The modelling of radionuclide transport through the geosphere is necessary in the safety assessment of repositories for radioactive waste. A number of key geosphere processes need to be considered when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity, as input parameter, varies from place to place. In such cases geostatistical science offers a variety of spatial estimation procedures. To assess the a long term safety of a radioactive waste disposal system, mathematical models are used to describe the complicated groundwater flow, chemistry and potential radionuclide migration through geological formations. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). Kansa introduced the concept of solving PDEs using radial basis functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. The aim of this study was to present a relatively new approach to the modelling of radionuclide migration through the geosphere using radial basis functions methods and to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by a geostatistical approach. We will also explore residual errors and their influence on optimal shape parameters

Solving moving-boundary problems with the wavelet adaptive radial basis functions method

Moving boundaries are associated with the time-dependent problems where the momentary position of boundaries needs to be determined as a function of time. The level set method has become an effective tool for tracking, modelling and simulating the motion of free boundaries in fluid mechanics, computer animation and image processing. This work extends our earlier work on solving moving boundary problems with adaptive meshless methods. In particular, the objective of this paper is to investigate numerical performance the radial basis functions (RBFs) methods, with compactly supported basis and with global basis, coupled with a wavelet node refinement technique and a greedy trial space selection technique. Numerical simulations are provided to verify the effectiveness and robustness of RBFs methods with different adaptive techniques

Uporaba brezmrežnih metod (radialnih baznih funkcij) za modeliranje migracije radionuklidov in problemov s spremljajočo se mejo

Recently, the mesh free methods (radial basis functions-RBFs) have emerged as a novel computing method in the scientific and engineering computing community. The numerical solution of partial differential equations (PDEs) has been usually obtained by finite difference methods (FDM), finite element methods (FEM) and boundary elements methods (BEM). These conventional numerical methods still have some drawbacks. For example, the construction of the mesh in two or more dimensions is a nontrivial problem. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the mesh free method, which uses radial basis functions, with the traditional finite difference scheme and analytical solutions. We will present some examples of using RBFs in geostatistical analysis of radionuclide migration modeling. The advection-dispersion equation will be used in the Eulerian and Lagrangian forms. Stefan\u27s or moving boundary value problems will also be presented. The position of the moving boundary will be simulated by the moving data centers method and level set method.Zadnje čase se, kot novejše računske metode, v znanstveni in inženirski skupnosti pojavljajo brezmrežne metode (radialne bazne funkcije). Numerično reševanje parcialnih diferencialnih enačb običajno poteka s pomočjo metode končnih diferenc, metode končnih elementov in metode robnih elementov. Te metode imajo še vedno nekaj pomanjkljivosti, in sicer npr. konstrukcija mreže v dveh ali treh prostorskih dimenzijah ni enostaven problem. Reševanje parcialnih diferencialnih enačb z uporabo radialne bazne kolokacije je primerna alternativa tradicionalnim numeričnim metodam, ker ne zahteva obsežnega generiranja mreže. Rezultati se bodo primerjali z rezultati dobljenimi s pomočjo metode končnih diferenc in analitičnih rešitev. Predstavili bomo nekaj primerov: najprej uporabo radialnih baznih funkcij v geostatistični analizi modeliranja migracije radionuklidov. Migracija radionuklidov se bo simulirala s pomočjo advekcijske-disperzijske enačbe, in sicer v Eulerjevi in Lagrangeovi obliki. V nadaljevanju bodo predstavljeni tudi Stefanovi problemi oz. problem primikajočih se meja (površin). Položaje primikajoče meje bomo simulirali s pomočjo metode primikajočih se centrov in nivojne metode

A comparison of the effectiveness of using the meshless method and the finite difference method in geostatistical analysis of transport modeling

Disposal of radioactive waste in geological formations is a great concern with regards to nuclear safety. The general reliability and accuracy of transport modeling depends predominantly on input data such as hydraulic conductivity, water velocity, radioactive inventory, and hydrodynamic dispersion. The most important input data are obtained from field measurernents, but they are not always available. One way to study the spatial variability of hydraulic conductivit,y is geostatistics. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). These methods require a mesh to support the localized approximations. The multiquadric (MQ) radial basis function meihod is a recent meshless collocation method with global basis functions. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the meshless method, which uses radial basis functions, with the traditional finite difference scheme. ln our case we determine the average and standard deviation of radionuclide concentration with regard to spatial variability of hydraulic conductivity that was modeled by a geostatistical approach

International Conference Nuclear Energy for New Europe 2006 Solving One-Dimensional Phase Change Problems with Moving Grid Method and Mesh Free Radial Basis Functions

ABSTRACT Many heat-transfer problems involve a change of phase of material due to solidification or melting. Applications include: the safety studies of nuclear reactors (molten core concrete interaction), the drilling of high ice-content soil, the storage of thermal energy, etc. These problems are often called Stefan's or moving boundary value problems. Mathematically, the interface motion is expressed implicitly in an equation for the conservation of thermal energy at the interface (Stefan's conditions). This introduces a non-linear character to the system which treats each problem somewhat uniquely. The exact solution of phase change problems is limited exclusively to the cases in which e.g. the heat transfer regions are infinite or semiinfinite one dimensional-space. Therefore, solution is obtained either by approximate analytical solution or by numerical methods. Finite-difference methods and finite-element techniques have been used extensively for numerical solution of moving boundary problems. Recently, the numerical methods have focused on the idea of using a mesh-free methodology for the numerical solution of partial differential equations based on radial basis functions. In our case we will study solid-solid transformation. The numerical solutions will be compared with analytical solutions. Actually, in our work we will examine usefulness of radial basis functions (especially multiquadric-MQ) for one-dimensional Stefan's problems. The position of the moving boundary will be simulated by moving grid method. The resultant system of RBF-PDE will be solved by affine space decomposition