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The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically.

For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time.

By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated.

Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

The purpose of this paper is to perform two-dimensional numerical simulation involving fluid-structure interaction of flexible filament. The filament is tethered to the bottom of a rectangular channel with oscillating fluid flow inlet conditions at low Reynolds number. The simulations are performed using a temporal second-order finite volume-based immersed boundary method (IBM). Further, to understand the relation between different aspect ratios i.e. ratio of filament length to channel height (Len/H) and fixed channel geometry ratio, i.e. ratio of channel height to channel length (H/Lc) on mixing and pumping capabilities.

The discretization of governing continuity and Navier–Stokes equation is done by finite-volume method on a staggered Cartesian grid. SIMPLE algorithm is used to solve fluid velocity and pressure terms. Two cases of oscillatory flow conditions are used with the flexible filament tethered at the center of bottom channel wall. The first case is sinusoidal oscillatory flow with phase shift (SOFPS) and second case is sinusoidal oscillatory flow without phase shift (SOF). The simulation results are validated with filament dynamics studies of previous researchers. Further, parametric analysis is carried to study the effect of filament length (aspect ratio), filament bending rigidity and Reynolds number on the complex deformation and behavior of flexible filament interacting with nearby oscillating fluid motion.

It is found that selection of right filament length and bending rigidity is crucial for fluid mixing scenarios. The phase shift in fluid motion is also found to critically effect filament displacement dynamics, especially for rigid filaments. Aspect ratio, suitable for mixing applications is dependent on channel geometry ratio. Symmetric deformation is observed for filaments subjected to SOFPS condition irrespective of bending rigidity, whereas medium and low rigidity filaments placed in SOF condition show severe asymmetric behavior. Two key findings of this study are: symmetric filament conformity without appreciable bending produces sweeping motion in fluid flow, which is highly suited for mixing application; and asymmetric behavior shown by the filament depicts antiplectic metachronism commonly found in beating cilia. As a result, it is possible to pin point the type of fluid motion governing fluid mixing and fluid pumping. The developed computational model can, thus, successfully demonstrate filament-fluid interaction for a wide variety of similar problems.

The present study uses a temporal second-order finite volume-based IBM to examine flexible filament dynamics for various applications such as fluid mixing. Also, it highlights the relationship between channel geometry ratio and filament aspect ratio and its effect on filament sweep patterns. The study further reports the effect of filament displacement dynamics with or without phase shift for inlet oscillating fluid flow condition.

Understanding the factors that contribute to the growth of sediment delta lobes in river systems has significant benefit towards protecting civil and social infrastructure from severe weather events. To develop this understanding, this paper aims to construct a three‐dimensional numerical model of a sediment delta depositing on to a two‐dimensional bedrock basement entering an ocean at a constant sea‐level.

The approach used adapts and applies techniques and schemes previously used in building numerical heat transfer models of melting systems. Particular emphasis is placed on modifying fixed grid enthalpy like schemes.

The resulting model provides important insight on the features that control the partition of sediment delta deposition between the land and ocean domains. The model also illustrates how tectonic subsidence may control the rate of delta growth.

This is the first numerical heat transfer inspired model of a three‐dimensional sediment delta deposit over both land and ocean domains. The problem has scientific merit in that it represents a melting‐like moving boundary problem with two distinct moving boundaries and a space/time dependent latent heat. Further, this work is a necessary first step towards building a comprehensive understanding of how to restore delta systems to protect civil and social infrastructure.

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.

The purpose of this paper is to present the composite finite element method (CFEM), with Cⁿ(n≥0) continuity so it improves the accuracy of the finite element method (FEM) for solving second‐order partial differential equations (PDEs) and also, can be used for solving higher order PDEs.

In this method, the nodal values in the conventional FEM have been replaced by the appropriate nodal functions. Based on this idea, a procedure has been proposed for obtaining the CFEM−Cⁿ shape functions based on the CFEM−Cⁿ⁻¹ shape functions as follows: the nodal values in the CFEM−Cⁿ⁻¹ have been replaced by deliberately selected nodal functions so that the smoothness of the CFEM−Cⁿ⁻¹ shape functions increase.

The proposed method has the following properties: first, its shape functions have simple explicit forms with respect to the natural coordinates of elements and consequently, the required integrals for calculation of stiffness matrix can be evaluated numerically by low‐order Gauss quadratures; second, numerical investigations show that the CFEM with Cⁿ(n>1) continuity leads to more accurate results in comparison with the FEM; third, in multi‐dimensional problems, the curved boundaries are modeled more accurately by the proposed method in comparison with the FEM; fourth, this method can treat with the weak discontinuities such as the interface between different materials, as simple as the FEM does; and fifth, this method can successfully model Kirchhoff plate problems.

This method is an improvement of the moving particle FEM and reproducing kernel element method. Despite these two methods, CFEM shape functions have simple explicit forms with respect to the natural coordinates of elements.

The purpose of this paper is to describe the implementation of discrete singular convolution (DSC) method to steady seepage flow while presenting one of the possible uses of DSC method in geotechnical engineering. It also aims to present the implementation of DSC to the problems with mixed boundary conditions.

Second order spatial derivatives of potential and stream functions in Laplace's equation are discretized using the DSC method in which the regularized Shannon's delta kernel is used as an approximation to delta distribution. After implementation of boundary conditions, the system of equations is solved for the unknown terms.

The results are compared with those obtained from the finite element method and the finite difference method.

The method is applied to the flow problem through porous medium for the first time.

The purpose of this paper is to extend the natural neighbour radial point interpolation method (NNRPIM) to the dynamic analysis (free vibrations and forced vibrations) of two‐dimensional, three‐dimensional and bending plate problems.

The NNRPIM shape‐function construction is briefly presented, as are the dynamic equations and the mode superposition method is used in the forced vibration analysis. Several benchmark examples of two‐dimensional and plate bending problems are solved and compared with the three‐dimensional NNRPIM formulation. The obtained results are compared with the available exact solutions and the finite element method (FEM) solutions.

The developed NNRPIM approach is a good alternative to the FEM for the solution of dynamic problems, once the obtained results with the EFGM shows a high similarity with the obtained FEM results and for the majority of the studied examples the NNRPIM results are more close to the exact solution results.

Comparing the FEM and the NNRPIM, the computational cost of the NNRPIM is higher.

The paper demonstrates extension of the NNRPIM to the dynamic analysis of two‐dimensional, three‐dimensional and bending plate problems. The elimination of the shear‐locking phenomenon in the NNRPIM plate bending formulation. The various solved examples prove a high convergence rate and accuracy of the NNRPIM.

The purpose of this paper is to introduce the double-periodic lattice, composed of bending-resistant fibers. The essence of the model is that the filaments are of infinite length and withstand tension and bending. The constitutive equations of the lattice in discrete and differential formulations are derived. Two complementary systems of loads, which cause different deformation two orthogonal families of fibers, occur in the lattice. The fracture behavior of the material containing a semi-infinite crack is investigated. The crack problem reduces to the exactly solvable Riemann-Hilbert problem. The solution demonstrates that the behavior of material cardinally depends upon the tension in the orthogonal family of fibers. If tension in fibers exists, opening of the crack under action of loads in two-dimensional lattice is similar to those in elastic solid. In the absence of tension, contrarily, there is a finite angle between edges at the crack tip.

The description of stress state in the crack vicinity is reduced to the solution of mixed boundary value problem for simultaneous difference equations. In terms of Fourier images for unknown functions the problem is equivalent to a certain Riemann-Hilbert problem.

The analytical solution of the problem shows that fracture behavior of the material depends upon the presence of stabilizing tension in fibers, parallel to crack direction. In the presence of tension in parallel fibers fracture character of two-dimensional lattice is similar to behavior of elastic solid. In this case the condition of crack grows can be formulated in terms of critical stress intensity factor. Otherwise, in the absence of stabilizing tension, the crack surfaces form a finite angle at the tip.

Linear behavior of fibers until rupture. Small deflections. Perfect two-dimensional lattice.

The model provides exact analytical estimation of stresses on the crack tip as the function of fibers’ stiffness.

The model is the extension of known lattice models, taking into account the semi-infinite crack in the lattice. This is the first known closed form solution for an infinite lattice model with the crack.

We describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multigrid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. To demonstrate the wavelet technique, we consider Poisson's equation in two dimensions. By comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund and others. The convergence of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method.

The finite strip method has been shown to apply to many problems in continuum mechanics. Within the constraints of the method, it has been shown to be superior to the finite element method in terms of data preparation, program complexity and execution time. The finite strip method has been recently extended to groundwater flow problems. The orthogonality of appropriately selected shape functions gives the finite strip method its computational efficiency. The uncoupling achieved from this orthogonality also produces a numerical method which is intrinsically parallel. Consequently, additional efficiencies can be obtained in a parallel environment. Numerical studies of the finite strip method to model a two‐dimensional groundwater flow problem demonstrate the accuracy of the solution and the superior performance of the numerical procedure in a parallel environment.