The assessment of Tsunami propagation in the sea and ocean is the primary aim of this research. Tsunami propagation is generally simulated by solitary waves. In this research, for the simulation of solitary wave propagation in fluid with free surface, an Exponential basis functions meshless method is introduced. The formulation of this method is based on Lagrangian form of Navier-stokes equations for non-viscous fluids based in pressure. In this regard, the Laplace equation of pressure is solved at each time step. Then, the geometry is updated based on the Lagrangian formulation of the motion through an implicit algorithm. Considering the changes of geometry in simulation time, the introduced meshless method is efficient and very quick in calculations. The results for water surface profile before breaking are in good agreement with experimental data.

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 䐀 䐀 ential 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 e☮ leKmeeynwtso radnsa

Existence of singular points inside the solution domain or on its boundary deteriorates the accuracy and convergence rate of numerical methods. This phenomenon usually happens due to discontinuities in the boundary conditions or abrupt changes in the domain shape. This study has focused on the solution of singular plate problems using the Exponential basis functions method. In this method, unknown functions are considered as a linear combination of Exponential basis functions and the coefficients are calculated by approximate satisfaction of the boundary conditions. To increase the accuracy and convergence rate in problems with singular points, a series of singular, quasi-Exponential functions are added to the method’s Exponential basis functions. These functions have proper discontinuity in location of the singular points and satisfy the homogenous differential equation. The results obtained from the solution of three cracked plate problems show considerable increase in the accuracy and convergence rate of the proposed method compared with the Exponential basis functions methodwithout any noticeable increase in the computational cost.

In this paper, we present a novel method based on using Exponential basis functions (EBFs) to solve the heat conduction problem in axially layered materials. In the first step, we have considered each layer of material as a separate element. Then the solution in each element was approximated by a summation of EBFs satisfying the differential equation of the transient heat conduction problem. The unknown coefficients of the series solution were related to initial condition and Dirichlet side conditions of each layer employing a discrete transformation technique. Finally, the general solution of material was completed by satisfying the continuity conditions between adjacent layers in a manner similar to the conventional finite element method. In this hybrid method, a collocation scheme was used for satisfying the time dependent boundary conditions as well as the initial conditions. The capability of the presented technique was investigated in the solution of some benchmark problems.

Despite the success and versatility of mesh based methods --finite element method in particular- there has been a growing demand in last decades towards the development and adoption of methods which can eliminate using the mesh, i.e. the so called meshless or mesh-free methods. Difficulties in the generation of high quality meshes, in terms of computational cost, technical problems such as serial nature of the meshgeneration process and the urge of parallel processing for today’s huge problems have been the main motivation for the implementation of new researches. Apart from these, the human required expertise can never be completely omitted from the analysis process. However, the problem is much more pronounced in 3D problems. To this end, many meshless methods have been developed in recent years among which SPH, EFG, MLPG, RKPM, FPM and RBF-based methods could be named. The Exponential basis functions method (EBF) is one of these methods which has been successfully employed in various engineering problems, ranging from heat transfer and various plate theories to classical and non-local elasticity and fluid dynamics. The method uses a linear combination of Exponential basis functions to approximate the field variables. It is shown that these functions have very good approximation capabilities and their application guarantees a high convergence rate. These Exponential bases are chosen such that they satisfy the homogenous form of the differential equation. This leads to an algebraic characteristic equation in terms of exponents of basic functions. From this point of view, this method may be categorized as an extension to the well-known Trefftz family of methods. These methods rely on a set of the so called T-complete bases for their approximation of the field variables. These bases should satisfy the homogenous form of the governing equation. They have been used with various degrees of success in a wide range of problems. The main drawback of these methods –however- lies in the determination of the basis, which should be found for every problem. This problem has been reduced to the solution of the algebraic characteristic equation in the Exponential basis functions method. The method is readily applicable to linear, constant coefficient operators, and has been recently extended to more general cases of linear and also non-linear problems with variable coefficients. The relative performance of usual programming languages such as C++in comparison with mathematical software packages -like Mathematica and/or Matlab- is one of the major questions when using such packages to develop new numerical methods. This can affect the interpretation of the performance of newly developed methods compared to established ones. In this paper, the implementation of the Exponential basis functions method on various software platforms has been discussed. C++and Mathematica programming have been examined as a representative of different software platforms. The Exponential basis function method is implemented in each platform, using various options available. Results show that with a proper implementation, the numerical error of the method can be decreased considerably. Regarding the results of this research, optimal implementations of C++and Mathematica platforms, error ratio is between 2.5 and 6, respectively.

Sloshing phenomenon is one of the complex problems in free surface flow phenomena. Numerical meshless methods as a new method can be used to solve this problem. In these methods, the lack of a mesh and complex elements for the domain of problems due to the change in geometry of the solution over time provides a lot of flexibility in solving numerical problems. In the previous researches, the sloshing problem in reservoirs was solved, using the Laplace equation with respect to the velocity potential, but the solution to this problem with pressure equations has not much considered; therefore, using the pressure equations and a suitable lagrangian time algorithm, generalized Exponential basis function method has been developed for dynamic stimulation reservoirs. The approximation is solved, using a meshless method of generalized Exponential basis functions and the entire domain of problem will discrete to a number of nodes and then with appropriate boundary conditions, the unknowns are approximated. In this study, linear and nonlinear examples have been solved under harmonic stimulation, in two-dimensional form of rectangular cube tanks, and the results of them have been compared with the analysis solving methods, other numerical methods, and experimental data. The results show that the present method in two-dimensional mode is very noticeable compared with other available lagrangian methods because of accuracy in solving problem and spending time.

In this paper, we study the concept of Exponential convex functions with respect to $s$ and prove Hermite-Hadamard type inequalities for the newly introduced this class of functions. In addition, we get some refinements of    the Hermite-Hadamard (H-H) inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is Exponential convex with respect to $s$. Our results coincide with the results obtained previously in special cases.

In this paper, the authors study and introduce some new integral forms of Hermite-Hadamard inequalities in the form of harmonically convex functions known as Exponential type harmonically $ (\alpha, s)_{h}$-convex function. Additionally, several special characteristics of this class of functions are examined. More precisely, the authors provide some properties and characteristics related to the Hermite-Hdamard inequality for harmonically $ (\alpha, s)_{h}$-convex function, applications of this work with certain examples are made to establish results obtained.

Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.

In this article, a meshless method based on Exponential basis functions (EBFs) is presented to simulate the harmonic waves with moving free-surfaces generated by the piston-type wave maker. Accordingly, velocity potential is adopted in a Mixed Eulerian-Lagrangian (MEL) approach. Boundary conditions are met through a point-wise collocation approach. In order to update the geometry in the simulation time, the free surface points are only moved vertically. To reduce the reflection in the wave flume, a damping zone is added at the far end opposite to the wave maker, where the velocity is modified by adding an artificial damping term. The results indicated the ability of this numerical method in simulating free surface flow problems like non-linear waves with a good accuracy, as well as suitable performances and the least run time calculation.