International Journal of Computational Methods

In this paper, we investigate the solitary wave solutions for the two-dimensional modified Korteweg–de Vries–Burgers (mKdV-B) equation in shallow water model. Despite that Painlevé test fails to prove the integrability of the mKdV-B equation by using the WTC-Kruskal algorithm, the Bäcklund transformation is obtained via the truncation expansion. The exact solutions of the mKdV-B equation are found using factorization techniques, Exp-function and energy integral approach of the corresponding ordinary differential equation. We found that the dispersion relation of the linearized mKdV-B equation lies on the complex plane yielding a damping character. By keeping the water height relatively small, we illustrate the resulting solutions in several figures showing the shock and solitary wave nature in the flow. The stability for the mKdV-B equation is analyzed by using the phase plane method.

This paper consists to study numerically the heat transfer and fluid flow in a vertical vented divided channel, which is heated asymmetrically. A specifically developed numerical model, based on the finite-volume method, is used for the solutions of the governing differential-equations. The SIMPLER algorithm was adopted for the treatment of velocity-pressure coupling. Concerning the radiation exchange, we consider that the working fluid is transparent, so only the solid surfaces contribute to the radiation exchange and assumed to be diffuse-gray. A special attention is given to study the effects of Rayleigh numbers, aspect ratio of the channel, thermal conductivity ratio, rib geometry (length and width) on streamlines, temperature fields and average Nusselt number. A parametric study is made for different Rayleigh numbers, aspect ratio of the channel, thermal conductivity ratio, rib geometry (length and width). Results are presented in terms of isotherms, streamlines, and average Nusselt number.

In many extended versions of the finite element method (FEM) the mesh does not conform to the physical domain. Therefore, discontinuity of variables is expected when some elements are cut by the boundary. Thus, the integrands are not continuous over the whole integration domain. Apparently, none of the well developed integration schemes such as Gauss quadrature can be used readily. This paper investigates several modifications of the Gauss quadrature to capture the discontinuity within an element and to perform a more precise integration. The extended method used here is the finite cell method (FCM), an extension of a high-order approximation space with the aim of simple meshing. Several examples are included to evaluate different modifications.

This study discussed the effects of shape parameters on the radial point interpolation method (RPIM) accuracy in 2D geometrically nonlinear problems. Four finite deformation problems with compressible Neo-Hookean material are numerically solved with the RPIM algorithm using the multi-quadric (MQ) radial basis function. Both regular and irregular node distributions are used. Their displacements and Cauchy stresses are compared for different values of shape parameters and monomial basis. It is found that the shape parameters proposed for linearly elastic problems (q = 1.03, α_c = 4) can still be applicable to 2D geometrically nonlinear problems but careful selections should be made for the calculation of stress. For example, when q is used as 1.75 with irregular node distributions, stresses can be calculated more precisely.

Based on the use of radial basis functions (RBFs), we present in this paper a meshless collocation method to compute both European and American option prices by solving the variance gamma (VG) model. The valuation of the financial derivatives is performed by solving a corresponding partial integro-differential equation (PIDE). In the case of European option, numerical comparison with the analytical solution shows that the proposed scheme achieves a higher accurate approximation than most existing numerical methods. When analytical solution is not available in the case of American option, we use a dividend process to obtain an alternative characterization of the American option so that solution to the PIDE can be achieved in the entire computational region. Since the RBFs used in this paper are infinitely differentiable, the approximation of the derivatives of option prices can be obtained at no extra interpolation cost. In addition, the leave-one-out cross validation (LOOCV) algorithm is generalized for obtaining a local optimal choice of the shape parameter contained in the RBFs for superior convergence. Several numerical examples are given to verify the efficiency and stability of the proposed method.

In this research, a general solution of volume constancy differential equation is presented based on the equation of deformation field for a general process. As the Bezier method is suitable for construction of complex geometries, the solution is used in conjunction with the Bezier method to analyze the equal channel angular extrusion (ECAE) process of rectangular cross section. Thus, a generalized kinematically admissible velocity field is derived from the equation of deformation zone such that the compatibility of the surface representing the deformation zone is fulfilled. The effects of die angle, friction between the billet and die wall, and the angle of outer curved corner, on extrusion pressure are all considered in the analysis. It is found that extrusion pressure decreases with increasing both the die angle and the outer curved corner angle and with decreasing the friction coefficient. Also, the effect of die curvature on inhemogenity of strain is assessed. It is exhibited that increasing the angle of outer curved corner decreases the extrusion pressure and increases the inhomogeniety of strain field of deformation zone. A good agreement is found between the predicted and experimental results pertaining to two dies of different outer curved corner.

The free energy-based lattice Boltzmann (FEB-LB) model is a very powerful method and is fully consistent with Maxwell's equal-area construction, especially for low temperature conditions. Drawbacks of this model are its massive amounts of calculations and large computation time (i.e., iteration number), therefore it is often executed on supercomputers or parallel computing systems. In this paper, the three-dimensional FEB-LB model has been optimized using the Runge-Kutta fourth-order method. It is shown that using the fourth-order Runge–Kutta method, the computation time is decreased about four times. In addition, in the present work, using the optimized FEB-LB model, the effects of surface tension and interface width on spurious velocities in the liquid–gas interface have been investigated.

A simplified meshless methods for brittle fracture and nonlinear material is presented. In this method, the crack is modeled by a set of discrete crack segments crossing the entire domain of influence of the meshless shape functions. The key advantage of this method is its simplicity since no representation of the crack topology is needed. A nonlocal stress tensor around the crack tip is used as fracture criterion. A neo-Hooke material in the bulk material is used and a cohesive zone model is employed once discrete cracks occur. We also present consistent linearization of the cohesive zone model. The method is applied to fracture modeling in concrete that is accompanied by excessive cracking and therefore methods that represent the crack path have major drawbacks. We demonstrate the accuracy of the proposed method for complex problems involving mode-I and mixed mode failure.

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data.

Finite element (FE) model updating belongs to the class of inverse problems in mechanics and is a constrained optimization problem. In FE model updating, the difference between the modal parameters (the frequencies, damping ratios and the mode shapes) obtained from the FE model of the structure and those from the vibration measurements are minimized within an optimization algorithm. The design variables of the optimization problem are the stiffness reduction factors, which represent the damage. In this study, the Genetic Algorithms (GA), the Parallel GA, the local search algorithms, the Trust Region Gauss Newton, the Sequential Quadratic Programming, the Levenberg–Marquardt Techniques and the hybrid versions of these methods are applied within the FE Model Updating Technique for updating the Young's modulus of different FEs of a reinforced concrete beam. Different damage scenarios and different noise levels are taken into account. The results of the study show that the local search algorithms cannot detect, locate and quantify damage in reinforced concrete beam type structures while the GA together with the hybrid and the parallel versions detect, localize and identify the damage very accurately. It is apparent that the hybrid GA & Trust Region Gauss Newton Technique is best in terms of the computation speed as well as accuracy.

Further development of the novel meshfree Galerkin Kriging method for bending and buckling analysis of laminated composite plates is the main objective of the present work. The present formulation follows the classical plate theory while the discrete equations are derived from the standard Galerkin weak form. The moving Kriging interpolation technique is used for constructing shape functions, which possess the delta property and thus no special techniques are required for imposing the essential boundary conditions. Various numerical examples with different geometries of plates for both bending and buckling analysis are solved to demonstrate the applicability and the effectiveness of the present approach. The calculated results are compared with reference solutions available in the literature and very good agreements are obtained. It obviously illustrates that the proposed method can be considered as an alternative numerical approach for buckling and bending of laminated plates.

An axisymmetric smoothed particle hydrodynamics (ASPH) with contact algorithm is presented to simulate the normal perforation of aluminum plates. Traditional ASPH considering contact implicitly by conservation equations has the problem of virtual tensile stresses and shear stresses for perforation simulation. To overcome the problem, a particle-to-particle contact algorithm is employed to treat contact interfaces explicitly in the present method. An artificial stress method is extended for the method to remove tensile instability. The present method is firstly validated by three test cases. Then, it is used to simulate the normal perforation of aluminum plates with ogive-nose steel projectiles. Numerical simulation results show that the method predicted residual velocities of projectiles in good agreement with the experimental data.

Driven by applications in the design of protective structure systems, the need to model high velocity impact is becoming of great importance. This paper presents a Smoothed Particle Hydrodynamics (SPH) procedure for 3D simulation of high velocity impacts where high rate hydrodynamics and material strength are of great concern. The formulations and implementations of the Johnson–Cook strength and damage model considering temperature effect, and Mie–Gruneison and Tilloton equations of state are discussed. The performance of the procedure is demonstrated through two example analyses, one modeling a cubic tungsten projectile penetrating a multi-layered target panel and the other involving a sphere perforating a thin plate. The results obtained, with comparisons made to both experimental results and other numerical solutions previously reported, show that our SPH-3D implementation is accurate and reliable for modeling the overall behavior of the high rate hydrodynamics with material strength.