Meshless Local Collocation Method Research Articles

Pharmacokinetic parameter identification using a meshless method approach for transdermal drug delivery

The Localized Collocation Meshless Method (LCMM) is applied to the solution of a set of partial differential equations governing the diffusive transport of a dermally injected compound as described by the Compartmental Pharmacokinetic (CPK) model. This efficient and accurate numerical technique then provides the framework for the estimation of pharmacokinetic (PK) parameters by the comparison of two models, both containing information derived from experimental results, using the iterative Golden Section search algorithm. This comparison is quantified using a norm between the two solution spaces. While this method does provide an estimate of the effective-diffusion coefficient, it ultimately demonstrates that a simple, free-diffusion model is inadequate for quantifying the spatial and temporal distribution of a dermally-injected compound if elimination effects are not accounted for properly.

Three-dimensional application of the meshless generalized finite difference method for solving the extended Fisher–Kolmogorov equation

In this article, a novel meshless local collocation method is proposed for the numerical solution of the three-dimensional (3D) extended Fisher–Kolmogorov (EFK) equation. The second-order Crank–Nicolson scheme and the meshless generalized finite difference method (GFDM) are respectively adopted to discretize the time and spatial derivatives of the EFK equation. A different setting of collocation nodes is introduced to the meshless GFDM for solving the nonlinear fourth order system resulting after the time discretization process. Two numerical experiments are carried out to verify the accuracy and the convergence of the developed numerical meshless algorithm.

Meshless modeling of coupled transdermal pharmacokinetics with analytical validation

An efficient localized collocation meshless computational approach is applied to the study of transdermal drug delivery (TDD). To validate the numerical technique, a one-dimensional analytical solution is first derived for the set of coupled differential equations that govern the pharmacokinetic process in the two-compartment compound diffusion model. These equations consist of one partial differential equation (PDE) and one ordinary differential equation (ODE). These equations are solved in one-dimensional space via superposition and separation of variables to arrive at the field solution for the compound concentration in the tissue. The convergence of the numerical result to the analytical result indicates the viability of the localized collocation meshless method (LCMM) as a valid approach to the solution of TDD problems. The analytical solution is used to benchmark the numerical technique for several cases with differing pharmacokinetic coefficients, demonstrating the robustness and accuracy of the meshless numerical technique. Furthermore, the highly efficient LCCM implementation is extended to address multi-dimensional TDD cases achieving stable and physically consistent results.

Meshless Modeling of Flow Dispersion and Progressive Piping in Poroelastic Levees

Performance data on earth dams and levees continue to indicate that piping is one of the major causes of failure. Current criteria for prevention of piping in earth dams and levees have remained largely empirical. This paper aims at developing a mechanistic understanding of the conditions necessary to prevent piping and to enhance the likelihood of self-healing of cracks in levees subjected to hydrodynamic loading from astronomical and meteorological (including hurricane storm surge-induced) forces. Systematic experimental investigations are performed to evaluate erosion in finite-length cracks as a result of transient hydrodynamic loading. Here, a novel application of the localized collocation meshless method (LCMM) to the hydrodynamic and poroelastic problem is introduced to arrive at high-fidelity field solutions. Results from the LCMM numerical simulations are designed to be used as an input, along with the soil and erosion parameters obtained experimentally, to characterize progressive piping.

A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations

In this study, the direct meshless local Petrov–Galerkin (DMLPG) method has been employed to solve the stochastic Cahn–Hilliard–Cook and Swift–Hohenberg equations. First of all, we discretize the temporal direction by a finite difference scheme. In order to obtain a fully discrete scheme the direct meshless local collocation method is used to discretize the spatial variable and the Euler–Maruyama method is used for time discretization. The used method is a truly meshless technique. In order to illustrate the efficiency and accuracy of the explained numerical technique, we study two stochastic models with their applications in biology and engineering, i.e., the stochastic Cahn–Hilliard–Cook equation and a stochastic Swift–Hohenberg model.

The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations

The collocation technique based on the radial basis functions (RBFs) method is simple and efficient for solving a wide area of problems. But the mentioned technique is poor for solving problems that have shock (advection problems) or the discontinuous initial condition. The local RBFs collocation technique is a meshless method based on the strong form. The use of local collocation RBFs method overcomes the mentioned important issue. In the current paper, based on the proposed idea in Wang (2015) [54], we consider a linear combination of shape functions of local radial basis functions collocation method and moving Kriging interpolation technique. For showing the efficiency of new technique, some multi-dimensional problems such as Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations have been chosen. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

A Meshless Local Collocation Method for Natural Frequencies and Mode Shapes of Functionally Graded Plates

The novelty of this paper is the use of a meshless local collocation method based on the multiquadric radial basis functions for free vibration analysis of functionally graded plates. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. The natural frequencies computed by the present method are in good agreement with the element-free solutions of Zhao et al.

Meshless local collocation method for natural frequencies and mode shapes of laminated composite shells

Meshless local collocation method produces much better conditioned matrices than meshless global collocation methods. In this paper, the meshless local collocation method based on thin plate spline radial basis function and first-order shear deformation theory are used to calculate the natural frequencies and mode shapes of laminated composite shells. Through numerical experiments, the accuracy and efficiency of present method are demonstrated.

Analysis of Laminated Composite Plates by Local Inverse Multiquadrics Collocation Method

In present paper, deflection and stress of laminated composite plates are analyzed by a meshless local collocation method based on inverse multiquadrics radial basis function. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. Transverse displacement, normal stresses, and shear stresses of the simply supported laminated composite plates under sinusoidal load are computed by the present method. The convergence characteristics are studied by several numerical examples. The present results are compared with available published results which demonstrate the accuracy and efficiency of present method.

A model-integrated localized collocation meshless method for large scale three-dimensional heat transfer problems

We present a Model Integrated Meshless Solver (MIMS) tailored to solve practical large-scale industrial problems. This is accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without the need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input such as small features, poorly formed faces, and other such pathologies often generated from solid models in the course of design and in the end to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

Local thin plate spline collocation for free vibration analysis of laminated composite plates

Abstract This paper focuses for the first time on free vibration analysis of laminated composite plates by a meshless local collocation method based on thin plate spline radial basis function. This method approximates the governing equations using the nodes in the support domain of any data center. Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties, and lamination schemes are computed by present method. The choice of shape parameter, effect of dimensionless sizes of the support domain on accuracy, convergence characteristics are studied by several numerical examples. The results are compared with available published results which demonstrate the accuracy and efficiency of present method.

A localized collocation meshless method (LCMM) for incompressible flows CFD modeling with applications to transient hemodynamics

The current paper reports on the development and validation of a localized collocation meshless method (LCMM) to model laminar incompressible flows. A high order upwinding scheme was devised to dampen the numerical oscillations arising in convection-dominated flows. Subsequently, the LCMM was analytically validated and demonstrated to yield third-order accurate solutions when compared to a benchmark analytical decaying vortex solution. Numerical validations are provided by comparison with the finite volume commercial (FVM) solver Fluent 6.2. The flow geometry for the numerical validation arises from a biomedical application that consists of modeling blood flow in the inter-connection between a bypass graft and an artery. Very good agreement was found between the LCMM and the FVM.