International Journal of Computational Methods

In this paper, a Laguerre method is presented to solve singularly perturbated two-point boundary value problems. By means of the matrix relations of the Laguerre polynomials and their derivatives, original problem is transformed into a matrix equation. Later, we use collocation points in the matrix equation and thus the considered problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of the desired approximate solution. Also, an error estimation based on the residual function is introduced for the method. The Laguerre polynomial solution is improved by using this error estimation. Finally, error estimation and residual improvement are illustrated by examples and comparisons are given with other methods.

Realizing the importance of widely used technique of plating for flexural retrofitting of reinforced concrete (RC) beams and its drawbacks due to premature failure(s), present work concentrates in developing a finite element tool model capable of successfully capturing multiple premature failure modes and their corresponding behaviors. The model is simple but focused; the capability and accuracy of the results have been validated through test literature, particularly focusing on the load capacities of beams at progressive stages of failure modes; which is from crack initiation through to complete failure, such as the load of crack initiation, first crack and complete failure. Acceptable accuracy is shown in terms of crack type(s), crack patterns, sequence, location and direction of propagation through the innovative use of cohesive zone model (CZM). The model clearly explains that debonding and peeling, although originating from a same location for most cases, are extensions of different types of cracks.

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

A three-dimensional (3D) Computational Fluid Dynamics (CFD) solver based on the gradient smoothing method (GSM) is developed for compressible flows based on previous research. The piecewise constant smoothing function with one-point integration scheme is implemented for gradient approximation of field variables and convective fluxes. The matrix-based method for gradient approximations is also developed to improve the numerical efficiency. Numerical examples of gradient approximations of several given functions have shown that the proposed GSM is more accurate and robust to mesh distortion. A transonic ONERA M6 wing is used to demonstrate the effectiveness of the proposed GSM-CFD solver.

The aim of the present study is to calculate the process of detonation combustion of gas mixtures in engines. Development and verification of 3D transient mathematical model of chemically reacting gas mixture flows incorporating hydrogen was performed. Development of a computational model based on the mathematical one for parallel computing on supercomputers incorporating CPU and GPU units was carried out. Investigation of the influence of computational grid size on simulation precision and computational speed was performed. Investigation of calculation runtime acceleration was carried out subject to variable number of parallel threads on different architectures and implying different strategies of parallel computation.

In the standard smoothed particle hydrodynamics (SPH) method, the interaction between two particles might be not pairwise when the support domain varies, which can result in a reduction of accuracy. To deal with this problem, a modified SPH approach is presented in this paper. First of all, a Lagrangian kernel is introduced to eliminate spurious distortions of the domain of material stability, and the gradient is corrected by a linear transformation so that linear completeness is satisfied. Then, concepts of support and dual-support are defined to deal with the unbalanced interactions between the particles with different support domains. Several benchmark problems in one, two and three dimensions are tested to verify the accuracy of the modified SPH model and highlight its advantages over the standard SPH method through comparisons.

This paper presents an exact pointwise error estimator applied to the transport equation. The error estimate, which is represented by a combination of residual-free bubble and Green’s functions, is computed by postprocessing the finite element solution. The fine or unresolved scales are analyzed applying the variational multiscale method. The technology can be applied to linear and high-order elements. Numerical examples are presented in which the satisfactory results corroborate the theoretical foundations.

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

This paper presents a methodology to use the software ANSYS in modeling and active vibration control of a functionally graded (FGM) plate with upper and lower surface-bonded piezoelectric layers. First a FGM plate with piezoelectric layers is designed using APDL ANSYS. Then a modal analysis has been carried out to get the first five rank frequencies and mode shapes. A proportional–integral–derivative (PID) and a linear-quadratic-based output feedback controller are introduced to realize the vibration control through a closed loop. Results for various volume fraction indexes are presented.

Numerical solutions to inverse Sturm–Liouville problems by three spectra, on a finite interval $[0,a]$ are considered. Dirichlet boundary conditions are assumed to hold at the end-points of the interval $[0,a]$, and either a Dirichlet or a Robin-type interior point condition is imposed at an arbitrary interior point $a_0$ of $[0,a]$. We demonstrate that the numerical solution of the inverse three spectra problem corresponding to an interior point $a_0$ depends continuously on $a_0$. This statement is particularly useful when measurements of the input data cannot be obtained at the interior point $a_0$, but they can be obtained at a nearby interior point. Consequently, we could solve approximately the original inverse problem by departing slightly from $a_0$. We validate this statement through numerical experiments for sequences of interior points $a_n$ converging to $a_0$.

In this work, the natural neighbor radial point interpolation method (NNRPIM) is extended to the numeric analysis of crack propagation problems. Here, the advanced discretization meshless technique is combined with a linear elastic crack growth algorithm. The algorithm simulates the crack propagation by displacing iteratively the crack tip, which consequently requires a local remeshing. In each iteration, it is estimated the stress state in the crack tip and afterwards the direction of the crack propagation is obtained considering the maximum circumferential stress criterion.

The required local remeshing does not represent a numeric difficulty for the NNRPIM. The main advantage of the NNRPIM is its capability to fully discretize the problem domain using only an unstructured nodal distribution. Being a truly meshless method, the NNRPIM is able to define autonomously the nodal connectivity and the background integration mesh.

The classic NNRPIM formulation permits to enforce the nodal connectivity by means of two kind of influence-cells: first degree influence-cells or second degree influence-cells. This work investigates the influence of the nodal connectivity on the simulated crack propagation path. Thus, demanding benchmark crack propagation examples are studied and the obtained results are compared with reference solutions available in the literature.

Aluminized explosives have been applied in military industry since decades ago. Compared with ideal explosives such as TNT, HMX, RDX, aluminized explosives feature both fast detonation and slow metal combustion chemistry, generating a complex multi-phase reactive flow. Though aluminized explosives have been employed for a long time, the mechanism underneath the chemical process is still not thoroughly understood. In this paper, a smooth particle hydrodynamics (SPH) method incorporated ignition and growth model, and afterburning model has been proposed for the simulation of aluminized explosive. Ignition and growth model is currently the most popular model for the simulation of high explosives, which is capable of accurately reproducing arrival time of detonation front and pressure history of high explosives. It has been integrated in commercial software such as ANSYS-LS DYNA. In addition, an afterburning model has been integrated in the SPH code to simulate the combustion of aluminum particles. Simulation is compared with experiment and good agreement is observed. The proposed mathematical model can be used to study the detonation of aluminized explosives.