Application of moving least squares algorithm for solving systems of Volterra integral equations

The numerical method developed in the current paper is based on the moving least squares (MLS) method. To this end, the MLS approximation method has been used, and a program has been made which can solve the system of Volterra integral equations (VIEs) with any number of equations and unknown functions. And then the proposed method is implemented on the system of linear VIEs with variable coefficients. The numerical examples are given that show the acceptable accuracy and efficiency of the proposed scheme.

Bending response of FGPM plates under voltage load

In this paper, we investigate the transient response of thin FGPM plates under voltage load within classical theory of the plates. The material properties of plate (such us elastic, piezoelectric, dielectric coefficients and mass density) are functionally graded in the in-plane direction. The strong form meshless formulations for solution of considered initial-boundary value problems are developed in combination with Moving Least Squares (MLS) approximation technique. Several numerical experiments are presented for the investigation of the influence of in-plane gradations of material coefficients on the transient response of the FGPM plates.

Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

Transient analysis of FGM plates bending under thermal loading: comparative study within classical and generalized thermoelasticity

In the classical thermoelasticty, there is coupling between thermal and elastic fields, in general, but the elastic and thermal excitations spread according to physically different laws. In classical thermodynamics, the temperature change propagates according to diffusion law. However, the heat pulses at low temperature propagate evidently with a finite velocity in view of waves similar to spreading of elastic excitations. A great effort has been spent in this matter and there are two main non-classical theories: Lord-Shulman theory and Green-Lindsay theory. In this paper, we compare the response of thin FGM plates under thermal load within the classical and non-classical theories of thermoelasticity. The variable material properties of plate (such as the Young’s modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The strong form meshless formulations for solution of considered initial-boundary value problems is developed in combination with Moving Least Squares (MLS) approximation scheme. The response of FGM plates on thermal loading is studied via numerical simulations with focusing on comparison of results obtained within the classical and generalized thermoelasticity. The numerical results concern also the parametric study of influence of gradation of material coefficients on bending of FGM plates

A Meshfree Technique for Numerical Simulation of Reaction-Diffusion Systems in Developmental Biology

In this work, element free Galerkin (EFG) method is posed for solving nonlinear, reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. A predicator-corrector scheme is applied, to avoid directly solving of coupled nonlinear systems. The EFG method employs the moving least squares (MLS) approximation to construct shape functions. This method uses only a set of nodal points and a geometrical description of the body to discretize the governing equation. No mesh in the classical sense is needed. However a background mesh is used for integration purpose. Numerical solutions for two cases of interest, the Schnakenberg model and the Gierer-Meinhardt model, in various regions is presented to demonstrate the effects of various domain geometries on the resulting biological patterns.

Development of two-dimensional groundwater flow simulation model using meshless method based on MLS approximation function in unconfined aquifer in transient state

In recent decades, due to reduction in precipitation, groundwater resource management has become one of the most important issues considered to prevent loss of water. Many solutions are concerned with the investigation of groundwater flow behavior. In this regard, development of meshless methods for solving the groundwater flow system equations in both complex and simple aquifers' geometry make them useful tools for such investigations. The independency of these methods to meshing and remeshing, as well as its capability in both reducing the computation requirement and presenting accurate results, make them receive more attention than other numerical methods. In this study, meshless local Petrov–Galerkin (MLPG) is used to simulate groundwater flow in Birjand unconfined aquifer located in Iran in a transient state for 1 year with a monthly time step. Moving least squares and cubic spline are employed as approximation and weight functions respectively and the simulated head from MLPG is compared to the observation results and finite difference solutions. The results clearly reveal the capability and accuracy of MLPG in groundwater simulation as the acquired root mean square error is 0.757. Also, with using this method there is no need to change the geometry of aquifer in order to construct shape function.