Analysis of time dependent problems using exponential basis functions

In this research we present a method based on using Exponential Basis Functions (EBFs) to solve a class of time dependent engineering problems. The solution is first approximated by a summation of EBFs satisfying the differential equation and then completed by satisfying the time dependent boundary conditions as well as the initial conditions through a collocation method. This can be performed by considering two approaches. In the first one the solution is split into three parts, i.e. a homogeneous solution obtained by homogeneous boundary conditions, a homogeneous solution obtained by non-homogeneous solution and finally a particular solution induced by source terms. In the second approach the solution is split into two parts, i.e. a homogeneous solution and a particular solution induced by source terms. The two approaches are then employed to construct a time marching algorithm for the solution of problems over a long period of time. We shall present the details of the application of the two approaches introduced to some mathematical and engineering problems. The details of the time marching algorithm proposed are explained. Several problems are solved to show the capabilities of the approaches used. Some benchmark problems are also devised and solved for further studies. It is shown that the one of the introduced approaches is capable of solving a class of problems with moving boundaries

On the use of exponential basis functions in the analysis of shear deformable laminated plates

In this report, we introduce a meshfree approach for static analysis of isotropic/orthotropic crossply laminated plates with symmetric/non-symmetric layers. Classical, first and third order shear deformation plate theories are employed to perform the analyses. In this method, the solution is first split into homogenous and particular parts and then the homogenous part is approximated by the summation of an appropriately selected set of exponential basis functions (EBFs) with unknown coefficients. In the homogenous solution the EBFs are restricted to satisfy the governing differential equation. The particular solution is derived using a similar approach and another series of EBFs. The imposition of the boundary conditions and determination of the unknown coefficients are performed by a collocation method through a discrete transformation technique. The solution method allows us to obtain semi-analytical solution of plate problems with various shapes and boundary conditions. The solutions of several benchmark plate problems with various geometries are presented to validate the results

The generalized finite point method

In this paper we propose a new mesh-less method based on a sub-domain collocation approach. By reducing the size of the sub-domains the method becomes similar to the well-known finite point method (FPM) and thus it can be regarded as the generalized form of finite point method (GFPM). However, unlike the FPM, the equilibrium equations are weakly satisfied on the sub-domains. It is shown that the accuracy of the results is dependent on the sizes of the sub-domains. To find an optimal size for a sub-domain we propose a patch test procedure in which a set of polynomials of higher order than those chosen for the approximations/interpolations are used as the exact solution and a suitable error norm is minimized through a size tuning procedure. In this paper we have employed the GFPM in elasto-static problems. We give the results of the size optimization in a series of tables for further use. Also the results of the integrations on a generic sub-domain are given as a series of library functions for those who want to use GFPM as a cheap and fast integral-based mesh-less method. The performance of GFPM has been demonstrated by solving several sample problems

Simple modifications for stabilization of the finite point method

A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill‐conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non‐monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd

Exponential basis functions in solution of incompressible fluid problems with moving free surfaces

In this report, a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries. A Lagrangian formulation established on pressure, as a potential equation, is employed. In this method, the approximate solution is expressed by a linear combination of exponential basis functions (EBFs), with complex-valued exponents, satisfying the governing equation. Constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique. The numerical solution is performed in a time marching approach using an implicit algorithm. In each time step, the governing equation is solved at the beginning and the end of the step, with the aid of an intermediate geometry. The use of EBFs helps to find boundary velocities with high accuracy leading to a precise geometry updating. The developed Lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems. Non-linear sloshing fluids in rigid rectangular two-dimensional basins are particularly addressed

Simulation of Nonlinear Free Surface Waves using a Fixed Grid Method

The simulation of nonlinear surface waves is of significant importance in safety studies of fluid containers and reservoirs. In this paper, nonlinear free surface flows are simulated using a fixed grid method which employs local exponential basis functions (EBFs). Assuming the flow to be inviscid and irrotational, the velocity potential Laplace’s equation is spatially discretized and solved by considering the nonlinear Bernoulli’s equation for irrotational flow as the boundary condition on the free surface. The nonlinear boundary conditions are imposed through a semi-implicit iterative time marching. The fixed grid feature of the method, based on a Lagrangian description of fluid flow, allows for retaining the portion of the discretization performed in the first time step for the bulk of the fluid. Thus, the portion which pertains to the regions near the moving boundaries is reprocessed during the time marching.  The accuracy and efficiency of the existing solution is shown by simulating various problems such as liquid sloshing induced by external excitation of the reservoir or initial deformed shape of liquid, seiche phenomena and solitary wave propagation in a basin with constant depth or with a step, and comparing the results with those which are analytically available or those from available codes such as Abaqus.  The proposed method shows far better stability of the results when compared with those of Abaqus which sometimes exhibit divergence after a relatively large number of time steps. For instance, in the propagation of the considered solitary wave in an infinite-like domain problem, the wave height is calculated by the maximum error of 1.6% and 9% using the present method and Abaqus, respectively

Simulating Fluid and Structure Interaction using Exponential Basis Functions

In this paper a meshless method using exponential basis functions is developed for fluid-structure interaction in liquid tanks undergoing non-linear sloshing. The formulation in the fluid part is based on the use of Navier-Stokes equations, presented in Lagrangian description as Laplacian of the pressure, for inviscid incompressible fluids. The use of exponential basis functions satisfying the Laplace equation leads to a strong form of volume preservation which has a direct effect on the accuracy of the pressure field. In a boundary node style, the bases are used to incrementally solve the fluid part in space and time. The elastic structure is discretized by the finite elements and analyzed by the Newmark method. The direct use of the pressure, as the potential of the acceleration, helps to find the loads acting on the structure in a straight-forward manner. The interaction equations are derived and used in the analysis of a tank with elastic walls. The overall formulation may be implemented simply. To demonstrate the efficiency of the solution, the obtained results are compared with those obtained from a finite elements solution using Lagrangian description. The results show that while the wave height and the oscillations of elastic walls of the two analyses are in good agreement with each other; the use of the proposed meshless analysis not only leads to accurate hydrodynamic pressure but also reduces the computational time to one-eighth of the time needed for the finite elements analysis

Molekularna i serološka istraživanja invazije vrstom Neospora caninum u golubova u jugozapadnom Iranu.

Neospora caninum is a protozoan parasite with worldwide distribution, mainly implicated as responsible for bovine abortion. There are indications that the presence of birds on cattle farms could be associated with the increase in seroprevalence and abortions related to N. caninum. The present study reports the serological (Neospora agglutination test) and molecular (PCR) presence of N. caninum in pigeons. From the 102 samples analyzed, 31 samples (30.39%) were seropositive for N. caninum and the overall molecular prevalence of N. caninum in the brains of the same pigeons was 9.8% (10/102). This is the first report of detection of N. caninum in Iranian pigeons. The results indicate soil contamination due to N. caninum oocysts, because pigeons feed from the ground, and suggest that the meat from the pigeons may be an important source for infection of dogs.Neospora caninum praživotinja je proširena diljem svijeta koja pretežito uzrokuje pobačaj u goveda. Upozorava se da prisutnost ptica na goveđim farmama može biti povezana s povećanjem serološke prevalencije i broja pobačaja uzrokovanih tom vrstom. U ovom je istraživanju N. caninum dokazana u golubova serološki testom aglutinacije i molekularno lančanom reakcijom polimerazom. Od 102 pretražena uzorka 31 (30,39%) je bio serološki pozitivan na prisutnost N. caninum, a molekularna prevalencija u mozgu istih golubova iznosila je 9,8% (10/102). To je prvo izvješće o dokazu N. caninum u iranskih golubova. Rezultati pokazuju da je onečišćenje tla oocistama N. caninum izvor uzročnika za golubove jer se oni hrane na tlu. Također se upozorava na činjenicu da meso golubova može biti važan izvor zaraze za pse

Molekularna i serološka istraživanja invazije vrstom Neospora caninum u golubova u jugozapadnom Iranu.

Neospora caninum is a protozoan parasite with worldwide distribution, mainly implicated as responsible for bovine abortion. There are indications that the presence of birds on cattle farms could be associated with the increase in seroprevalence and abortions related to N. caninum. The present study reports the serological (Neospora agglutination test) and molecular (PCR) presence of N. caninum in pigeons. From the 102 samples analyzed, 31 samples (30.39%) were seropositive for N. caninum and the overall molecular prevalence of N. caninum in the brains of the same pigeons was 9.8% (10/102). This is the first report of detection of N. caninum in Iranian pigeons. The results indicate soil contamination due to N. caninum oocysts, because pigeons feed from the ground, and suggest that the meat from the pigeons may be an important source for infection of dogs.Neospora caninum praživotinja je proširena diljem svijeta koja pretežito uzrokuje pobačaj u goveda. Upozorava se da prisutnost ptica na goveđim farmama može biti povezana s povećanjem serološke prevalencije i broja pobačaja uzrokovanih tom vrstom. U ovom je istraživanju N. caninum dokazana u golubova serološki testom aglutinacije i molekularno lančanom reakcijom polimerazom. Od 102 pretražena uzorka 31 (30,39%) je bio serološki pozitivan na prisutnost N. caninum, a molekularna prevalencija u mozgu istih golubova iznosila je 9,8% (10/102). To je prvo izvješće o dokazu N. caninum u iranskih golubova. Rezultati pokazuju da je onečišćenje tla oocistama N. caninum izvor uzročnika za golubove jer se oni hrane na tlu. Također se upozorava na činjenicu da meso golubova može biti važan izvor zaraze za pse

Wave Propagation in Unbounded Domains under a Dirac Delta Function with FPM

Wave propagation in unbounded domains is one of the important engineering problems. There have been many attempts by researchers to solve this problem. This paper intends to shed a light on the finite point method, which is considered as one of the best methods to be used for solving problems of wave propagation in unbounded domains. To ensure the reliability of finite point method, wave propagation in unbounded domain is compared with the sinusoidal unit point stimulation. Results indicate that, in the case of applying stimulation along one direction of a Cartesian coordinate, the results of finite point method parallel to the stimulation have less error in comparison with the results of finite element method along the same direction with the same stimulation