International Journal of Computational Methods

In order to solve partial differential equations (PDEs) numerically, one first needs to approximate the field functions (such as the displacement functions), and then obtain the derivatives of the field functions (such as the strains), by directly differentiating the field functions. Using such direct-derivatives in formulating a numerical method is common and is used in the standard finite element method (FEM), but such models are often found to be “stiff”. In the weakened weak (W2) formulations, it is found that the use of properly re-constructed derivatives can be beneficial in ways because the model can become “softer”. This paper presents a novel “pick-out” theory and technique for re-constructing the derivatives (such as the strains) of functions defined in a local domain, using smoothing operations. The local domain can be a smoothing domain used in the smoothed finite element methods (S-FEMs), smoothed point interpolation methods (S-PIMs), and smoothed particle hydrodynamics (SPH). It is discovered that through the use of a set of linearly independent smoothing functions that are continuous in the local domain, one can simply pick out various orders of smoothed derivatives (at the center of a domain) from any given function that may discontinuous (strictly) inside the local domain. As long as the smoothing function is continuous in the smoothing domain, the picked out “smoothed derivatives” are equivalent (in a local integral sense) to the compatible direct-derivatives, which ensures the convergence of the smoothed model (such as the S-FEM) when the smoothing domains shrinking to zero. The pick-out technique can be used in strong, weak, local weak, weak-strong, or weakened weak formulations to create stable and convergent numerical models. It may offer a new window of opportunity to develop new effective numerical models using smoothed derivatives (strains) that are “softer” and can produce accurate solutions also in the derivatives (strains and stresses) of the field functions (displacements).

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance ${𝜀}^{\prime }$ when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

The purpose of this paper is to present and illustrate a smoothed particle hydrodynamics (SPH) method to study the process of a drop impacting on a dry solid surface. SPH is a Lagrangian mesh-free particle method that offers advantages in modeling the evolution of the liquid surface during drop impact. A new surface tension model is used. The artificial viscosity is also used, which is demonstrated to be, approximately, a linear function of the dynamic viscosity of the liquid. The SPH method is used to simulate different liquid drops impacting on dry surfaces. The numerical results agree with experimental data obtained from the literature. The influence of various parameters on the drop impact, including impact velocity, diameter, viscosity, surface tension, and density of the drop, is also studied. The results show that the dimensionless spreading diameter of the drop increases if the impact velocity, diameter, or density increases, while the increase in viscosity and surface tension decreases the spreading diameter. The results indicate that the drop impact depends more strongly on the viscosity and impact velocity than on the diameter, surface tension, and density of the drop. In addition to the impact of a spherical drop, the impact of an ellipsoidal drop on a dry surface is also studied. The results show that the aspect ratio of the drop has a significant influence on the outcome of drop impact.

Setting initial values of parameters of mixture distributions estimated by using the EM recursive algorithm is very important to the overall quality of estimation. None of the existing methods are suitable for heteroscedastic mixtures with a large number of components. We present relevant novel methodology of estimating the initial values of parameters of univariate, heteroscedastic Gaussian mixtures, on the basis of dynamic programming partitioning of the range of observations into bins. We evaluate variants of the dynamic programming method corresponding to different scoring functions for partitioning. We demonstrate the superior efficiency of the proposed method compared to existing techniques for both simulated and real datasets.

In this paper, we investigate numerical solution of Allen–Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen–Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.

The smoothed boundary method has emerged as an attractive method for solving partial differential equations in domains of complex morphology. In the current work, we present two schemes for incorporating Dirichlet boundary conditions on complex domains into partial differential equations through an energy penalty formulation of the smoothed boundary method, which may be straightforwardly combined with, e.g., variational phase-field methods. Convergence to the exact solution is guaranteed in a certain limit, and it is shown how parameters of the smoothed boundary method should be chosen to minimize error, and avoid additional error introduced by poor spatial resolution during numerical solution. An example demonstrating the simplicity of the implementation of the methods is presented, and the scheme believed to have most general merit is determined.

An algorithm has been developed to identify the position of voids in structures by coupling level set method and finite element method (FEM). The identification problem is transformed into a minimum problem whose objective function is defined as a least square form of displacement error. A perimeter constraint term is also added in the objective function to make the solution well posed. The level set is applied in the present algorithm to represent the position and geometry of the voids. The velocity field of level set function is obtained by analyzing the shape derivative of objective function. FEM based on Euler description is employed for solving the forward problem. The same fixed meshes adopted by the solution of forward problem are used for finite difference computation of the level set function. The procedure of this algorithm has been applied to the voids identification of two-dimensional (2D) and three-dimensional (3D) structures, the examples of single void and multiple voids are considered. The results indicate that the voids in structure can be identified effectively by the present algorithm and the algorithm is also stable to noise.

In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

This paper presents a Quorum Sensing Driven Bacterial Swarm Optimization (QBSO) to solve Dynamic Power Economic Ecological Emission Dispatch (DPEEED) problem with valve point loading effects. The exploration capability of the bacterial swarm is improved in this paper using quorum sensing and the anti-predatory activity. The performance of the proposed QBSO method is verified for optimality, practicability, convergence and robustness on a 5, 10-unit system. Furthermore, a practical south Indian 20-thermal generating units with recorded load demand profiles of three different weather situations. The numerical dispatch result indicates that the presented technique finds the minimized generating cost and consistently progresses the computational time. The presented technique outperforms the Bacterial Swarm Optimization (BSO) and the previous presented methods.

The evaluation of collective modes is fundamental in the analysis of molecular dynamics simulations. Several methods are available to extract that information, i.e., normal mode analysis, principal component and spectral analysis of trajectories, basically differing by the quantity considered as the nodal one (frequency, amplitude, or pattern of displacement) and leading to the definition of different kinds of collective excitations and physical spectral observables. The different views converge in the harmonic regime and/or for homo-atomic systems. However, for anharmonic and out of equilibrium dynamics, different quantities bring different information, and only their comparison can give a complete view of the system behavior. To allow such a comparative analysis, we review and compare the different approaches, applying them in diverse combinations to two examples of physical relevance: graphene and fullerene C${}_{60}$.