Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

Mai‐Duy, N.; Tran-Cong, T.

doi:10.1016/s0955-7997(01)00092-3

Cited by 49 publications

(36 citation statements)

References 13 publications

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“…Furthermore, the networks were also developed successfully to solve PDEs in procedures which are regarded as truly mesh-free methods (e.g. Kansa [31]; Zerroukat et al [32]; Mai-Duy and Tran-Cong [33,1,24]). However, it should be noted that it is still very hard to achieve such an universal approximation RBFN in practice due to the difficulties associated with choosing the network parameters such as the number of radial basis functions, their positions and widths.…”

Section: Irbfn Interpolationmentioning

confidence: 99%

“…However, it should be noted that it is still very hard to achieve such an universal approximation RBFN in practice due to the difficulties associated with choosing the network parameters such as the number of radial basis functions, their positions and widths. In previous works, Mai-Duy and Tran-Cong [24,25] proposed indirect RBFNs (IRBFNs) which are based on the integration process, and their results showed that the IRBFNs perform better than the usual direct RBFNs (DRBFNs) in terms of accuracy and convergence rate for both function and its derivatives. In this paper, the IRBFN is introduced into the BEM scheme to approximate the boundary solution for the analysis of 2D steady natural convection flow problems.…”

Section: Irbfn Interpolationmentioning

confidence: 99%

“…In this paper, the IRBFN is introduced into the BEM scheme to approximate the boundary solution for the analysis of 2D steady natural convection flow problems. In contrast to previous works (Mai-Duy and Tran-Cong [33,1,24]) where the neural networks were used to approximate globally (meshless) the strong form of the governing equations (PDE's), the present work deals with the use of neural networks in the boundary element part of the mesh which discretises the inverse statement of the governing equations. In view of the fact that the BEM allows the reduction of the problem dimensionality by one, only the IRBFN for function and its derivatives (e.g.…”

Section: Irbfn Interpolationmentioning

confidence: 99%

“…The detailed implementation and accuracy of the IRBFN method were reported previously (Mai-Duy and Tran-Cong [24,25]). In the following section, the IRBFN is coupled with boundary integral equations for analysis of natural convection flows.…”

Section: Irbfn Interpolationmentioning

confidence: 99%

“…Recently, Mai-Duy and Tran-Cong [24,25] have shown that the indirect radial basis function networks (IRBFNs) perform better than element-based methods for function interpolation. The IRBFNs were then successfully introduced into the BEM scheme to represent boundary values for the analysis of viscous flow in a lid-driven cavity (Mai-Duy and Tran-Cong [26]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An effective RBFN‐boundary integral approach for the analysis of natural convection flow

Mai‐Duy

Tran-Cong

2004

Numerical Methods in Fluids

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Section: Irbfn Interpolationmentioning

confidence: 99%

Section: Irbfn Interpolationmentioning

confidence: 99%

Section: Irbfn Interpolationmentioning

confidence: 99%

Section: Irbfn Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An effective RBFN‐boundary integral approach for the analysis of natural convection flow

Mai‐Duy

Tran-Cong

2004

Numerical Methods in Fluids

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A localized meshless approach for modeling spatial–temporal calcium dynamics in ventricular myocytes

Yao

2011

Numer Methods Biomed Eng

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SUMMARY Spatial-temporal calcium dynamics due to calcium release, buffering and re-uptaking plays a central role in studying excitation-contraction (E-C) coupling in both normal and diseased cardiac myocytes. In this paper, we employ a meshless method, namely, the local radial basis function collocation method (LRBFCM) to model such calcium behaviors by solving a nonlinear system of reaction-diffusion partial differential equations. In particular, a simplified structural unit containing a single transverse-tubule (or t-tubule) and its surrounding half sarcomeres is investigated using the meshless method. Numerical results are compared to those generated by finite element methods, showing the capability and efficiency of the LRBFCM in modeling calcium dynamics in ventricular myocytes. The single t-tubule model is also extended to the whole-cell scale with t-tubules excluded to demonstrate the scalability of the proposed meshless method in handling very large domains. The experiments have shown that the LRBFCM is suitable to multi-scale modeling of calcium dynamics in ventricular myocytes with high accuracy and efficiency.

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Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques

Salazar

Arcila

Villa

2020

Numer Methods Biomed Eng

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Electroporation is a widely used method consisting of application of high‐voltage, short‐duration electric pulses to increase cell membrane permeability, allowing cellular internalization of medications. In this work, the influence of two primary parameters, voltage level (V) and pulse spacing (N), on electroporation efficiency, uniformity and aggressiveness, as quantified by the total mass transport to viable cells, intracellular concentration gradients and an aggressiveness factor introduced here, is studied by means of numerical simulations of drug transport in electroporated tissues. The global method of approximate particular solutions (Global MAPS) is used to solve the governing equations, together with domain scaling, singular value decomposition and smoothing algorithms, to address the ill‐conditioning of the final system and suppress small scale oscillations. The accuracy of Global MAPS is evaluated by comparing the initial extracellular concentration, Ce, and final intracellular concentration, Ci, with previous finite volume method results, obtaining similar behavior of Ce and Ci along the tissue domain, with some differences for Ci in high‐gradient zones. According to the Global MAPS results, the influence of V and N on Ci is only significant over a certain range, within which the largest drug transport to viable cells occurs. In general, both electroporation efficiency and aggressiveness change in nonuniform manner with V and decrease with N, whereas the electroporation uniformity decreases as V increases and N decreases. The contour plots obtained here can be considered useful tools to compare electroporation‐based treatments in terms of their efficiency, aggressiveness and uniformity, assisting in the selection of a suitable treatment plan for cancer.

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Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

Cited by 49 publications

References 13 publications

An effective RBFN‐boundary integral approach for the analysis of natural convection flow

An effective RBFN‐boundary integral approach for the analysis of natural convection flow

A localized meshless approach for modeling spatial–temporal calcium dynamics in ventricular myocytes

Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques

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