Adaptive meshless methods in electromagnetic modeling: A gradient-based refinement strategy

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.

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.

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.

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.

A numerical method is developed for consolidation analysis of lumpy clay fillings by using the double porosity model and the meshless method. Lumpy clay fillings consist of inter-lump voids and a clay matrix, which further consists of soil frame and pores. When a load is applied to these fillings submerged in the water, pore water pressures are generated and dissipate in the inter-lump voids and matrix. The model accounts for the coupling between the deformation of the soil skeleton and the excess pore water pressures in both the matrix and inter-lump voids, and also the fluid exchange between the inter-lump voids and matrix. The meshless method based on the radial point interpolation method (radial PIM) is used for spatial discretizations of displacement and pore water pressures. The order of interpolation function for the displacement is one order higher than that for pore water pressure to improve the numerical problem. Time domain is discretized through the backward Euler algorithm. The developed method is verified through a benchmark problem and two centrifuge tests. Finally, the effects of variations of various key parameters on the consolidation process are numerically studied for one-dimensional and plane strain (two-dimensional) problems. It is found that the developed numerical approach can successfully simulate the consolidation behavior of lumpy clay fillings despite their complex behaviour.

Mesh-free method for numerical solution of the multi-group discrete ordinate neutron transport equation

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.

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

Compact RBF meshless methods for photonic crystal modelling

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.

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.

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.

Computational simulation of cellular proliferation using a meshless method

When numerical methods such as the finite element method (FEM) are used to solve the Helmholtz equation, the solutions suffer from the so‐called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences.In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D‐case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem‐dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright © 2008 John Wiley &amp; Sons, Ltd.