Summary In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.

The water penetration in soil is investigated numerically using the finite element method (FEM) in a novel way. In the suggested method, new spherical Hankel shape functions are used and the finite element method is reformulated based on them. These new functions are obtained from the first and second kind of Bessel functions. The properties of Hankel shape functions lead to having more accuracy and robustness for the proposed method with low number of elements. To validate the suggested approach, at first, a boundary value problem is solved and the results are compared with the available analytical solution. Then, in order to prove the efficiency and applicability of the present model in the seepage problems, five examples including saturated and unsaturated flow in porous media are studied and the hydraulic head is calculated. Afterward, the results obtained from the classical and new method are compared together. The comparisons indicate that the suggested method with the low number of elements is more precise than the classic FEM with the same mesh.

In this paper, the Incompressible Smoothed Particle Hydrodynamics (ISPH) method is presented to simulate flood waves in uneven beds. The SPH method is a mesh free particle modeling approach that is capable of tracking the large deformation of free surfaces in an easy and accurate manner. Wave breaking is one of the phenomena that its free surface is complicated. Therefore, ISPH method is robust tool for the modeling of this kind of free surface. The basic equations are the incompressible mass conservation and Navier-Stokes equations that are solved using a two-step fractional method. In the first step, these equations are solved to compute velocity components by omitting the pressure term and in the absence of incompressible condition. In the second step, the continuity constraint is satisfied and the Poisson equation is solved to calculate pressure terms. In the present model, a new technique is applied to allocate density of the particles for the calculations. By employing this technique, ISPH method is stabled. The validation by comparison with laboratory data is conducted for bumpy channel with various boundary conditions. The numerical results showed good agreement with available experimental data. Also relative error is calculated for two numerical cases.

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A new finite element approach is developed here for the modeling of boundary value problems. In the present model, the finite element method (FEM) is reformulated by new shape functions called spherical Hankel shape functions. The mentioned functions are derived from the first and second kind of Bessel functions that have the properties of both of them. These features provide an improvement in the solution accuracy with number of elements which are equal or lower than the ones used by the classic FEM. The efficiency and accuracy of the suggested model in the potential problems are examined by several numerical examples. Then, the obtained results are compared with the analytical and numerical solutions. The comparisons indicate the high accuracy of the present method.