International Journal of Computational Methods

In the current paper, we introduce a different version of the variational iteration method (VIM) that provide an easy and effective procedure for solving nonlinear two-point boundary-value problems appearing in applied and engineering applications. For problems where convergence rate of the original VIM is slow, the method can be readily improved to increase convergence rate. Convergence analysis and error bounds are discussed. Finally, applicability of the proposed method and accuracy are examined by solving three problems.

An improved $r$-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.

This study presents an extension of the cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular elements for the static and free vibration analyses of carbon nanotube reinforced composite (CNTRC) plates. The single-walled carbon nanotubes (SWCNTs) are assumed to be uniformly distributed (UD) and functionally graded (FG) distributed along the thickness direction. The material properties of carbon nanotube-reinforced composite plates are estimated according to the rule of mixture. The governing equations are developed based on the first-order shear deformation plate theory (FSDT). In the CS-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized discrete shear gap method is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on the whole triangular element is used to smooth the strains on these three sub-triangles. Effects of several parameters, such as the different distribution of carbon nanotubes (CNTs), nanotube volume fraction, boundary condition and width-to-thickness ratio of plates are investigated. In addition, the effect of various orientation angles of CNTs is also examined in detail. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available results in the literature.

Modal analysis is widely used to investigate the dynamic characteristics of large and complex structures. For finite element models, iterative solvers are needed to precisely calculate eigenpairs or frequency and vibration mode. However, in cases such as large-scale analysis or reanalysis studies, or optimization design of a huge structure, computational cost may become too time consuming. This paper focuses on the quick structural modal analysis based on the reanalysis technique for large complex structures. Based on the stiffness and mass matrix of the analytical structures, a high precision and efficiency eigensolution is generated by the proposed modal analysis method (the Pseudo Random Independent and Coupling Inverse Iteration (PRICII) method), which combines the pseudo random number initialization, ICII (Independent and Coupling Inverse Iteration) strategy with the double Rayleigh–Ritz analysis. By comparing with the Subspace iteration method, Lanczos method, etc. the large-scale numerical examples show that the actual computational savings of the proposed method are usually higher than 75% with sufficient precision. Also, its applications in topology optimization are greatly effective.

A fully coupled soil–water-structure interaction algorithm was presented in the framework of smoothed particle hydrodynamics (SPH). In this algorithm, soil–water interaction was simulated based on the two-phase mixture theory. Each phase of the mixture occupies part of the macroscopic mixture and satisfies its own conservation equations of mass and momentum. The Drucker–Prager model with nonassociated plastic flow rule was used to describe the constitutive behavior of soil. The water was treated as Newtonian fluid. Interaction between soil and water was modeled by the pore water pressure and the viscous drag force. The structure was considered as rigid and the interaction with soil/water was modeled by the frictional sliding contact algorithm. With this algorithm, it is possible to investigate pore water pressure, the effective stress and deformation of the soil undergoing large deformation. Moreover, the effect of the temporal and spatial evolution of soil porosity was taken into consideration. This study first examined the proposed algorithm for a U-tube seepage problem and a two-dimensional consolidation problem. Afterwards, the continuous deep penetrating process of the spudcan, which involved large soil deformation and complex soil–water-structure interaction, was simulated under axisymmetric conditions. The comparison with previous research indicates the robustness and applicability of the proposed algorithm. Furthermore, the proposed approach could be a potentially efficient tool helping to reveal the mechanism of soil failure in geotechnical, costal and ocean engineering.

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

A sequential model of multiple-shot impacts has been established to investigate the shot peening process. Shot groups are proposed and designed with different patterns to obtain full surface coverage in the impacted region and a satisfactory computational efficiency. The sequential model was applied for the prediction of residual stress on a GH4169 alloy specimen. The results showed that uniform and saturated states of residual stress along the surface and depth profile were obtained in the impacted region when the numerical order of shot patterns reached 4. Furthermore, the numerical results of compressive residual stress in the subsurface were compared with the experimental results obtained using the X-ray diffraction (XRD) analysis and the incremental hole drilling method. The maximum relative error between the numerical results and XRD measurement was 11.6%. Furthermore, the stress profile measured using the incremental hole drilling method was consistent with the numerical results. The established finite element model demonstrated its robustness and effectiveness for the evaluation of residual stress in the shot-peened GH4169 alloy, and it may be applied to other metallic materials with simple modifications.

This paper addresses the dynamic output feedback control problem for a class of nonlinear multiple time-delays system. First, the system is decomposed into two subsystems based on the input matrix and output matrix. Second, a dynamic compensator is employed for the first subsystem, and then the output feedback controller is designed based on the second subsystem and compensator. With a new Lyapunov–Krasovskii functional, we show that the closed-loop time-delays system is stable and the designed controller make the solutions of the system convergent to an adjustable bounded region. Compared with the previous works, the proposed dynamic output feedback technique is more flexible and the required conditions on the considered systems are less conservative. Finally, the simulations for a discrete-time chemical stirred tank reactor case are performed to verify the effectiveness of the method.

In this paper, simultaneous effects of viscous dissipation and Joule heating on unsteady two-phase blood flow through a stenosed artery in the presence of variable applied magnetic field have been investigated. The present two-layered model of blood flow consists of a central core of suspended erythrocytes and a cell-depleted plasma layer surrounding the core. It is assumed that the viscosity of the cell-free plasma layer is constant while the viscosity of the core is a function of the hematocrit level. A consistent system of nonlinear partial differential equations is solved numerically using shooting methods to estimate the flow rate, flow resistance and wall shear stress. The quantitative profile analysis of velocity, temperature and concentration as well as the Nusselt number and Sherwood number is carried out over the entire arterial segment. To validate the model, a comparative study has been done between the present results and the experimental results for the cell velocity distribution of 40% RBC containing blood which exhibits that the present results are in fairly good agreement with the experimental results. The velocity contours have been plotted to understand the flow pattern in the diseased narrowed artery, which alters significantly in the downstream of the stenosis under the influence of magnetic field.

This paper is devoted to a continuous-time stochastic differential system which is derived by incorporating white noise to a deterministic $SIRI$ epidemic model with mass action incidence, cure and relapse. We focus on the impact of a relapse on the asymptotic properties of the stochastic system. We show that the relapse encourages the persistence of the disease in the population and we determine the threshold of the relapse rate, above the threshold the disease prevails in the population. Furthermore, we show that there exists a unique density function of solutions which converges in $L^1$, under certain conditions of the parameters to an invariant density.

This work presents a quadratic boundary integral equation formulation for isotropic damage analysis of components subjected to mechanical, thermal and centrifugal loads. To evaluate domain-related integrals due to the damage effects, the radial integration method (RIM) based on the use of the approximating the normalized displacements in the domain integrals by a series of prescribed radial basis functions (RBF) is adopted. The density of micro-defects is assumed to be small in the material. The scalar damage parameter expressed in an exponential evolution equation is utilized. Numerical examples including a rectangular plate, thick-walled cylinder and rotating disk problems under the thermal loads are given.

Recently, FE-SEA was proposed especially for the analysis of vibro-acoustic problems in mid-frequency regime. In practical applications, the use of conventional FEM in this hybrid FE-SEA method entails some inherent drawbacks, which is closely associated with well-known “overly-stiff” feature of FEM and the balance of mass matrix and stiffness matrix. In this work, a modified integration rule (MIR) is applied in the acoustic FEM model to readjust the mass matrix. Then, the MIR-FEM is embedded to achieve a hybrid MIR-FE-SEA framework, aiming to further improve the accuracy of mid-frequency response prediction. In the proposed MIR-FE-SEA model, the plate structure which often has a higher modal density is considered as a statistical subsystem and modeled statistically using statistical energy analysis (SEA). The acoustical cavity with a relative lower modal density is modeled deterministically using MIR-FEM. The coupling between these two different types of subsystems is achieved through the diffuse field reciprocity relation. The proposed MIR-FE-SEA is verified by various numerical examples.

The direct and optimal control solution of laminar fully developed, steady Magnetohydrodynamics (MHD) flow of an incompressible, electrically conducting power-law non-Newtonian fluid in a square duct is considered with the heat transfer. The fluid is subjected to an external uniform magnetic field as well as a constant pressure gradient. The apparent fluid viscosity is both a function of the unknown velocity and the flow index which makes the momentum equation nonlinear. Viscous and Joule dissipation terms are also included. The direct problem is solved by using Galerkin finite element method (FEM) with mixed finite elements and the control problem approach is the discretize-then-optimize procedure. The control formulations with the flow index parameter and the Hartmann number are given to regain the desired velocity profile and temperature isolines of the MHD flow.

Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson’s ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.