International Journal of Computational Methods

The general series solution (GSS) approach is presented, in order to determine the stress and displacement fields in disks under arbitrarily distributed normal and tangential loads. An Airy stress function in series form is selected. Stresses are expressed by infinite coefficients. Thus displacements are expressed by the infinite stress coefficients. And self-equilibrated loads acting on the side edge are extended to Fourier series. Stress coefficients are related to loading coefficients by stress boundary conditions. Then five examples show the validity of this approach. The GSS approach might lead to industrial applications in rock mechanics, petroleum and mining engineering, etc.

On the basis of overlapping domain decomposition, a new parallel mixed finite element algorithm is constructed to solve the parabolic problem. The algorithm is combined with splitting positive definite mixed element procedure using the method of subspace correction and is fully parallel. The a priori error estimate and some numerical experiments are presented.

This paper is concerned with the application of the discontinuous Galerkin (DG) method to the solution of unsteady linear hyperbolic conservation laws on Cartesian grids. We present several superconvergence results and we construct a robust recovery-type a posteriori error estimator for the directional derivative approximation based on an enhanced recovery technique. We first identify a special numerical flux and a suitable initial discretization for which the $L^2$-norm of the solution is of order $p+1$, when tensor product polynomials of degree at most $p$ are used. Then, we prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be $p+1/2$. Moreover, we establish an ${𝒪}(h^{2p+1})$ global superconvergence for the solution flux at the outflow boundary of the domain. We also provide a simple derivative recovery formula which is ${𝒪}(h^{p+1})$ superconvergent approximation to the directional derivative. We use the superconvergence results to construct asymptotically exact a posteriori error estimate for the directional derivative approximation by solving a local steady problem on each element. Finally, we prove that the a posteriori DG error estimate at a fixed time converges to the true error in the $L^2$-norm at ${𝒪}(h^{p+1})$ rate. Our results are valid without the flow condition restrictions. Numerical examples validating these theoretical results are presented.

This paper presents the development of an accurate and robust numerical modeling of instability of an interface separating two-phase system, such as liquid–gas and/or solid–gas systems. The instability of the interface can be refereed to the buoyancy and capillary effects in liquid–gas system. The governing unsteady Navier–Stokes along with the stress balance and kinematic conditions at the interface are solved separately in each fluid using the finite-volume approach for the liquid–gas system and the Hamilton–Jacobi equation for the solid–gas phase. The developed numerical model represents the surface and the body forces as boundary value conditions on the interface. The adapted approaches enable accurate modeling of fluid flows driven by either body or surface forces. The moving interface is tracked and captured using the level set function that initially defined for both fluids in the computational domain. To asses the developed numerical model and its versatility, a selection of different unsteady test cases including oscillation of a capillary wave, sloshing in a rectangular tank, the broken-dam problem involving different density fluids, simulation of air/water flow, and finally the moving interface between the solid and gas phases of solid rocket propellant combustion were examined. The latter case model allowed for the complete coupling between the gas-phase physics, the condensed-phase physics, and the unsteady nonuniform regression of either liquid or the propellant solid surfaces. The propagation of the unsteady nonplanar regression surface is described, using the Essentially-Non-Oscillatory (ENO) scheme with the aid of the level set strategy. The computational results demonstrate a remarkable capability of the developed numerical model to predict the dynamical characteristics of the liquid–gas and solid–gas flows, which is of great importance in many civilian and military industrial and engineering applications.

The efficiency of any numerical scheme measures on the accuracy of the scheme and its computational time. In this work, an efficient augmented local radial basis function finite difference (RBF-FD) scheme has been developed for steady convection–diffusion problems. The accommodative full approximation scheme–full multigrid (FAS-FMG) analogy with local refinement is adopted to achieve this efficiency. The efficiency of the method is illustrated through linear convection–diffusion equation, coupled nonlinear equation and for the incompressible Navier–Stokes (NS) equations. Numerical studies are made by using multiquadric (MQ) RBF. The upwind type nodes are considered to handle convective terms effectively for NS equations. The developed scheme saves 70–80% of the CPU time for the first two model problems and at least 50% of the CPU time for NS equations than the usual RBF-FD method found in the literature. It is found that there are optimum sets of nodes which give accurate and stable solution. It is also found that the value of the shape parameter will be lower when the number of nodes is higher. But higher the value of Re, higher will be the shape parameter.

In order to consider corner configurations with straight corner fronts in three-dimensional (3D) solids, a super polygonal prismatic element containing a straight corner front is established by using the numerical eigensolutions of singular stress fields and the Hellinger–Reissner variational principle. Singular stresses near the corner front subject to far-field boundary conditions can be obtained by incorporating the super singular element with conventional 3D brick elements. The numerical studies are conducted to demonstrate the simplicity of the proposed technique in handling fracture problems of 3D corner configurations and cracks. The usage of the super singular element can avoid mesh refinement near the corner front domain that is necessary for conventional and enriched finite element methods, and lead to high accuracy and fast convergence. Compared with the conventional finite element methods and existing analytical methods, the present method is more suitable for dealing with complicated problems of stress singularity in elasticity including multiple defects.

The discretized virtual internal bond (DVIB) is a newly developed lattice bond method for fracture simulation. The discrete structure of the DVIB method is composed of unit bond cells, which can take any geometry with any number of bonds. Hence, the DVIB is able to represent the meso-structural characters of a material composed of micro-grains. The regular hexagon is fully symmetrical in geometry, isotropic with respect to fracture propagation and has more failure directions than the usual triangular cell. Due to its geometric advantages, the regular hexagon is adopted as the unit cell of DVIB to simulate the dynamic fracture propagation. It is shown that the fracture paths simulated by this method are independent of the mesh scheme and the simulation results are reliable. The interaction between particles is characterized by a hyperelastic bond potential, and most characters of the dynamic fracture behaviors are captured. It is suggested that the DVIB with regular hexagon cells is a reliable and efficient method for dynamic fracture simulation.

A stable localized meshless method (SLMM) is proposed for convection dominated PDEs exhibiting boundary layer. Some cases of continuous and discontinuous boundary data as well as continuous and discontinuous source function with constant and variable convection coefficients are considered. In this approach, the localized meshless method is implemented on specialized sub-domains embedded with flow direction of underlying fluid. The proposed method is based on flow featured overlapping sub-domains, called stencils. The concept of flow direction is used to construct good quality stencils having the ability to capture flow features, such as boundary layer, accurately. Numerical experiments are presented to compare the proposed method with the finite-difference method on special grid (FDSG), the standard finite-element method, hybridized SUPG method, hybridized upwind method, residual-free bubbles (RFB) method and other meshless methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one-, two-, and three-dimensional convection-dominated PDEs. In some cases, performance of the SLMM is comparable and sometimes better than the mesh-based finite-element and finite-difference methods.

Immersed methods are considered as a class of nonboundary-fitted meshing technique for simulating fluid–structure interactions. However, the conventional approach of coupling the fluid and solid domains, as in the immersed boundary method and the immersed finite element method, often cannot handle high Reynolds number flows interacting with moving and deformable solids. As the solid dynamics is imposed by the fluid dynamics, it often leads to unrealistically large deformation of the solid in cases of high Reynolds number flows. The first attempt in resolving this issue was proposed in the modified immersed finite element method (mIFEM), however, some terms were determined heuristically. In this paper, we provide a full and rigorous derivation for the mIFEM with corrections to the previously proposed terms, which further extends the accuracy of the algorithm. In the “swapped” coupling logic, we solve for the dynamics of the solid, and then numerically impose it to the background fluid, which allows the solid to control its own dynamics and governing laws instead of following that of the fluid. A few examples including a biomedical engineering application are shown to demonstrate the capability in handling large Reynolds number flows using the derived mIFEM.

The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M. [2013] “Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type,” Appl. Math. Mod. 37, 7539–7548; Turkyilmazoglu, M. [2014] “An effective approach for numerical solutions of high-order Fredholm integro-differential equations,” Appl. Math. Comput. 227, 384–398; Turkyilmazoglu, M. [2015] “Parabolic partial differential equations with nonlocal initial and boundary values,” Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.

In the present study, Gravitational Search Algorithm (GSA) is combined with Finite Element Method (FEM) for optimizing laminated composites vibration behavior. The fiber orientation angle of layers is considered as design variable. The 8-layerd and 12-layerd plates with both of square and rectangular shapes are investigated. Twenty distinct boundary conditions and $5^{\circ }$ of fiber angle increment are considered. The results of the proposed method are in good agreement with reference methods, and in some cases the GSA-FEM is more efficient. Moreover, the simple structure of GSA and its exploration and exploitation features avoids trapping in a local optimum.

In this work, a cut cell-based sharp interface method is developed to deal with the interaction between linearly elastic solids in Eulerian framework. The material interface is represented by cut faces, and evolved by level set equation. Finite volume scheme is applied and strong coupling is achieved by using the Riemann solver for linearly elastic solid–solid interaction on the material interface. Original Ghost Solid Method (OGSM) and Modified Ghost Solid Method (MGSM) are realized in the Eulerian framework for comparison. Simulation results show that OGSM can lead to severe nonphysical oscillations when the density ratio is high, which is same as the results in KK. The location of wave front calculated by Ghost Solid Methods (GSMs) deviates from the exact location, because the evaluation of the ghost points deviates from the real interface location. The developed cut cell-based method gives accurate wave front location, and is stable for high density and acoustic impedance ratio and high-order reconstruction.