Naísses Z. Lima

The Nonconforming Point Interpolation Method Applied to Electromagnetic Problems

Nonconforming point interpolation method (NPIM) is a meshless method that has been applied to problems in mechanics in the last years. In this paper, we investigate NPIM in electromagnetism. We present its formulation and shape functions, which are generated by the radial point interpolation method with polynomial terms. The numerical results are compared to the ones obtained by the finite flement method (FEM) and the element-free Galerkin method (EFG).

Handling Material Discontinuities in the Generalized Finite Element Method to Solve Wave Propagation Problems

To solve wave propagation problems involving change of medium, many authors employ the generalized finite element method with plane wave enrichment and Lagrange multipliers to ensure interface constraints. However this approach produces ill conditioned and nonpositive definite systems, making it hard to solve. This paper presents an approach based on the mortar element method that substitutes the Lagrange multipliers with the advantage of generating sparse and positive definite systems. Various numerical aspects affecting the generalized finite element method efficiency are analyzed by solving a 2-D scattering problem.

Application of Local Point Interpolation Method to Electromagnetic Problems With Material Discontinuities Using a New Visibility Criterion

In this paper, the local point interpolation method (LPIM) is used with a modified visibility criterion to handle material discontinuities. In general, visibility criterion is applied only to shape function generation over support nodes selection. We present a modified version where it is also applied to the integration process. The method is simpler and more robust than other techniques often employed on multimaterials problems, with a straight-forward implementation.

A Modified Meshless Local Petrov–Galerkin Applied to Electromagnetic Axisymmetric Problems

A modified meshless local Petrov-Galerkin for an electromagnetic axisymmetric problem is presented in this paper. The method uses the shape functions generated by the radial point interpolation method with a modified T-scheme to select the support nodes, and also a new and malleable strategy to determine the test domains. The convergence of the method is evaluated using a coaxial cavity problem and it is compared with the finite-element method for two different meshes: one with a good quality mesh and another partially composed to bad quality elements. The total execution time using both methods is also compared.

A framework for meshless methods using generic programming

This article describes a framework for meshless methods using the generic programming paradigm. The framework is developed in C++ language with support of template mechanism. The idea is to build a set of extensible tools so that the framework is able to instantiate the main meshless methods such as Element Free Galerkin Method (EFG), Meshless Local Petrov Galerkin Method (MLPG), Point Interpolation Methods (PIM) and others.

Face-Based Gradient Smoothing Point Interpolation Method Applied to 3-D Electromagnetics

This paper presents the face-based gradient smoothing point interpolation method (FS-PIM), a numerical method derived from the PIM that solves 3-D boundary value problems. FS-PIM is supported by the theory of G-space, weakened-weak formulations and the gradient smoothing operation. The method is applied in the analysis of electrostatic problems. The obtained results show that both convergence rate and accuracy of the approximation generated by FS-PIM are better than the ones presented by the finite element method, indicating that the technique is suitable for electromagnetic applications.

Point interpolation methods based on weakened-weak formulations

This paper presents a study of meshless Point Interpolation Methods based on weakened-weak forms. The mathematical formulations of the methods are presented as well as the procedures for the support nodes selection called T-schemes. The numerical results are shown for four different types of electromagnetic static problems in order to ponctuate the characteristics of the approximation generated by these new methods.

Meshless Vector Radial Basis Functions With Weak Forms

Meshless methods construct their shape functions based on scattered nodes in the domain. One drawback of this approach is the presence of nonphysical modes in the numerical solution when dealing with vector problems due to the lack of the divergence free condition, in a similar way that occurs with the node-based finite-element method. On the other hand, vector radial basis functions were developed to produce numerical approximations that satisfy the divergence free condition. This paper presents the usage of those functions in conjunction with weak forms to solve vector electromagnetic problems. Numerical tests involving the Maxwell eigenvalue problem and the wave propagation in a waveguide are solved to demonstrate that the numerical solution is not corrupted with spurious modes.

Edge Meshless Method Applied to Vector Electromagnetic Problems

A challenge in meshless methods dealing with vector electromagnetic problems is to produce numerical solutions that are free of spurious modes given that the generated vector field does not satisfy the condition of zero divergence. The edge meshless method (EMM) constructs its approximations using special shape functions based on edges to produce vector fields that are divergence free and to guarantee the continuity of the tangential field components. This paper presents the application of the EMM to solve vector electromagnetic problems. The 2-D Maxwell eigenvalue problem with anisotropic medium is tested to demonstrate that the technique produces correct numerical solution without spurious modes.

Meshless vector radial basis functions with weak forms

Meshless methods construct their shape functions based on scattered nodes in the domain. One drawback of this approach is the presence of nonphysical modes in the numerical solution when dealing with vector problems due to the lack of the divergence free condition, in a similar way that occurs with the node based finite element method. On the other hand, vector radial basis functions were developed to produce numerical approximations that satisfies the divergence free condition. This paper presents the usage of those functions in conjunction with weak forms to solve vector electromagnetic problems. An eigenvalue problem is solved to demonstrate that the numerical solution is not corrupted with spurious modes.