Computational simulation of cellular proliferation using a meshless method

This paper presents a numerical model of two-dimensional (2D) contaminant transport through unsaturated porous media using a mesh-free method, called the radial point interpolation method (RPIM), with polynomial reproduction. A 2D form of the advection– dispersion equation, involving a reactive contaminant with linear first-order degradation, is considered in the analysis. In the RPIM, the Galerkin weak form is used to establish the system equation using 2D mesh-free shape functions constructed using thin plate spline radial-basis functions. A Matlab code is developed to obtain the numerical solution. Experimental results are used to validate the proposed RPIM. The applicability and performance of the RPIM are studied using three numerical examples. The results of the RPIM are compared with those obtained from the finite-element method. The RPIM has generated results with no oscillations, and which are insensitive to Péclet constraints.

This article is devoted to showing the efficiency and accuracy of two numerical procedures for the numerical solution of the capillary formation in tumor angiogenesis. A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason for choosing MLRPI approach is that it does not require any background integration cells; instead, integrations are implemented over local quadrature domains, which are further simplified for reducing the complication of computation using regular and simple shape. The spectral meshless radial point interpolation (SMRPI) method does not need any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI, whereas the operational matrices convert easily the governing equations (even high order) into a linear system of equations in SMRPI. The finite difference technique is employed to approximate the time derivatives. A stability analysis technique based on the eigenvalue of the matrices is employed to check the stability condition for the presented meshless methods. Furthermore, the methods are successfully applied to solve the problem with high values of the cell diffusion constant which many of the available methods did not investigate for solving these cases. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for surveying the convergence of the fully discrete scheme. From numerical results, it is considerable that the present methods provide the similar results. Moreover, less CPU time and less computational complexity are two advantages of the SMRPI method with respect to the MLRPI method.

The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and successfully applied to many engineering problems. Recently, a new meshfree method was proposed based on the universal moving Kriging interpolation. This paper studies the difference between the meshfree shape functions created based on the point interpolation and the Kriging interpolation. It is found that both the two methods yield the same shape function as long as the same radial basis function or semivariogram is adopted for interpolation. Although the two methods lead to the same shape function, the theorem in Kriging formulation may provide an alternative theoretical support for the RPIM. Some common semivariograms used in Kriging may also be incorporated in the RPIM. In addition, in order to satisfy the conformability requirements, a penalty technique is introduced in this paper to form a conforming Kriging, which can pass the standard patch test exactly.

The finite difference time domain (FDTD) method is considered a prominent approach in computational electromagnetics for the wave propagation analysis of closed geometries in time domain. However, since this uses a structured mesh, the staircase approximation at the curved boundaries is a computational challenge in FDTD. Recently, meshless methods based on radial point interpolation method (RPIM) on unstructured mesh have gained importance in electromagnetic simulations due to their capability of modeling arbitrary shaped geometries. In the RPIM, the field variable at a node is interpolated by the values at the surrounding nodes which fall in the support domain. After assembly, this results in a banded matrix of size N χ N, where N is the total number of mesh cells. The computational cost of solving this is high for large structures. On the other hand, FDTD is a simple method in which the field variables are directly calculated using update equations. So, the FDTD and Meshless RPIM methods can be combined to model structures with partial curved geometries, with meshless RPIM method at the curved regions and the Cartesian FDTD for the rest of the domain. To validate the method numerically, a WR-229 90 degree waveguide bend is modelled using meshless method at the curved region and FDTD at the straight waveguide regions. Gaussian radial basis functions are used for the interpolation and to generate the shape function matrices in the meshless region. Interface matching conditions for the electric and magnetic fields are matched across the interface boundaries. The external domains are truncated with 2nd order MUR absorbing boundary conditions. The results of the hybrid approach are compared with those from the regular FDTD, with the Meshless approaches and the improvement in the computational cost is reported.

In this paper, we introduce a two-dimensional (2-D) Node-based Meshless Radial Point Interpolation Method (RPIM) for modeling nonmagnetized plasma. The problem domain is firstly discretized by a set of electric field nodes (E-nodes) and a set of magnetic field nodes (H-nodes). E-nodes and H-nodes are defined in a manner similar to the point-matched time-domain finite-element method. With the RPIM scheme, the field variable at the node is interpolated by a scalar expansion function which combines radial basis function with monomial basis function. After some mathematical manipulations, the shape function of the node in the support domain is obtained. The field variables and corresponding spatial derivatives in field equations can be obtained analytically and the central finite-difference is used to approximate the time derivatives. The leapfrog RPIM meshless method formulations are then obtained. TM wave propagation in isotropic nonmagnetized plasma is simulated to validate the accuracy and efficiency of the proposed method.

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.

Numerical electromagnetic computations must often accommodate random geometric representations while handling biological tissues, and engineered components with manufacturing tolerances. Meshless time-domain radial point interpolation method (RPIM) offers advantages to quantitatively analyze such geometric uncertainties using polynomial chaos expansion (PCE). Formulations for geometric uncertainties may require variations in mesh or node distribution for each analyzed sample, leading to high computational requirement for re-meshing. The proposed geometric stochastic RPIM (G-SRPIM) overcomes this with a single domain model by expressing the shape function matrix of RPIM in a stochastic framework. The uncertainty is quantified in G-SRPIM through a novel way by which its random support domain moment matrices are organized in a block structure, and inverted using Schur's complement and Neumann approximation, exploiting the underlying symmetry. The proposed method is validated by analyzing a parallel plate waveguide with a slit exhibiting random variations, a realistic 3D bio-electromagnetic problem involving a section of human head, and an iris filter with random variations in its iris dimensions. Standard deviation upto 45% of the average inter-node distance is tested without jeopardizing the stability. The accuracy of our approach is compared with Monte-Carlo (MC) simulations on a deterministic RPIM using the Kolmogorov-Smirnov (KS) test. Additionally, results are compared with MC simulation on CST<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\text{\text{®}}}$</tex-math></inline-formula> Studio Suit 2018<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\text{™}}$</tex-math></inline-formula> and stochastic collocation (SC). The proposed method exhibits superior execution time compared to SC and MC-based non-intrusive implementations, underscoring its efficiency and reliability in handling geometric uncertainties in microwave components.

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method

Throughout the years, the health of the mouth and the smile have gain an increasing importance, due to the fact that is one of the most important communication skills of a person. Yet, dental implant treatments offers a solution that may generate more satisfactory outcomes for patients who are not able to adapt to conventional dentures or who have already compromised local host bone. In this work, advanced discretization numerical techniques were used, more specifically, the Natural Neighbor Radial Point Interpolation Method (NNRPIM) and the Radial Point Interpolation Method (RPIM), and also the Finite Element Method (FEM). The main goal of this work is to understand the mechanical behavior of zirconia implants and the bone response in the presence of such implants and also, compare the three numerical methods used in this work, FEM, RPIM and NNRPIM. The results obtained show that meshless methods are capable to produce more accurate and smooth displacement and stress fields when compared to finite element technique.

Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates

For acoustic computations in the mid-frequency range the finite element method (FEM) is a well-known standard tool. Unfortunately, for increasing frequencies, i.e. higher wavenumbers, the FEM suffers from the so-called pollution effect which is mainly a consequence of the dispersion, meaning that the numerical wavenumber and the exact wavenumber disagree. Using meshfree methods as, e.g., the radial point interpolation method (RPIM) or the element-free Galerkin method (EFGM) can reduce this effect significantly. Moreover, meshfree methods allow the usage of shape functions that can be adapted to the differential equation to be solved. Consequently, an iterative method can be derived, which uses a standard meshfree method to compute a first approximation for the given problem. In a second step this approximation is taken to construct new shape functions that are able to better reproduce the wave-like character of the solution. If a few requirements are considered, this method leads to better results in a more efficient way. In this paper two meshfree methods, namely an iterative RPIM and an iterative EFGM, are examined. The methods are compared to the FEM and restrictions for an efficient applicability are shown.

This paper proposes the usage of an adaptive meshless method – radial point interpolation method (RPIM) for the analysis of armchair graphene nano ribbon (GNR) to be used for interconnect applications. The validity and efficiency of the proposed method are demonstrated using the following three examples: one-dimensional problem, two-dimensional problem and the cantilever beams under various forces and boundary conditions. Electrostatic analysis of the GNR is performed using RPIM. The delay of the GNR is calculated using the quasi-static capacitance. It is found that the delay is reduced compared to the RKPM analysis, making it as a suitable candidate for RF interconnects.

This paper presents the solution of the kinematic wave equation using a meshless radial point interpolation method (RPIM). The partial differential equation is discretized using a Galerkin weighted residual method employing RPIM shape functions. A forward difference scheme is used for temporal discretization, while the direct substitution method is employed to solve the nonlinear system at each time step. The formulation is validated against solutions from conventional numerical techniques and physical observation. In all cases, excellent agreements are achieved and hence the validation of the proposed formulation. Optimum values of the multi-quadrics shape parameters were then determined before the assessment of the performance of the method. Based on the convergence rate, it has been shown that the proposed method performs better than the finite difference method and equivalent to the finite element method. This highlights the potential of RPIM as an alternative method for hydrologic modeling.

A meshfree method with dynamic node reconfiguration for analysis of thermo-elastic problems with moving concentrated heat sources