Authors:S. Kollmannsberger; A. Özcan; M. Carraturo; N. Zander; E. Rank Pages: 1483 - 1497 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): S. Kollmannsberger, A. Özcan, M. Carraturo, N. Zander, E. Rank Computational heat transfer analysis often involves moving fluxes which induce traveling fronts of phase change coupled to one or more field variables. Examples are the transient simulation of melting, welding or of additive manufacturing processes, where material changes its state and the controlling fields are temperature and structural deformation. One of the challenges for a numerical computation of these processes is their multi-scale nature with a highly localized zone of phase transition which may travel over a large domain of a body. Here, a transient local adaptation of the approximation, with not only a refinement at the phase front, but also a de-refinement in regions, where the front has passed is of advantage because the de-refinement can assure a bounded number of degrees of freedom which is independent from the traveling length of the front. We present a computational model of this process which involves three novelties: (a) a very low number of degrees of freedom which yet yields a comparatively high accuracy. The number of degrees of freedom is, additionally, kept practically constant throughout the duration of the simulation. This is achieved by means of the multi-level h p -finite element method. Its exponential convergence is verified for the first time against a semi-analytic, three-dimensional transient linear thermal benchmark with a traveling source term which models a laser beam. ( b) A hierarchical treatment of the state variables. To this end, the state of the material is managed on a separate, octree-like grid. This material grid may refine or coarsen independently of the discretization used for the temperature field. This methodology is verified against an analytic benchmark of a melting bar computed in three dimensions in which phase changes of the material occur on a rapidly advancing front. (c) The combination of these technologies to demonstrate its potential for the computational modeling of selective laser melting processes. To this end, the computational methodology is extended by the finite cell method which allows for accurate simulations in an embedded domain setting. This opens the new modeling possibility that neither a scan vector nor a layer of material needs to conform to the discretization of the finite element mesh but can form only a fraction within the discretization of the field- and state variables.

Authors:Marcella Bonazzoli; Victorita Dolean; Frédéric Hecht; Francesca Rapetti Pages: 1498 - 1514 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Marcella Bonazzoli, Victorita Dolean, Frédéric Hecht, Francesca Rapetti In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell’s equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell’s equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.

Authors:Rumeng Zheng; Xiaoyun Jiang; Hui Zhang Pages: 1515 - 1530 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Rumeng Zheng, Xiaoyun Jiang, Hui Zhang In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and ( 2 − α ) order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.

Authors:Alexander V. Chichurin; Elena M. Ovsiyuk; Viktor M. Red’kov Pages: 1550 - 1565 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Alexander V. Chichurin, Elena M. Ovsiyuk, Viktor M. Red’kov Generalized Schrödinger equation for Cox spin zero particle is studied in presence of magnetic field on the background of Lobachevsky space. Separation of the variables is performed. An equation describing motion along the axis z turns out to be much more complicated than when describing the Cox particle in Minkowski space. The form of the effective potential curve indicates that we have a quantum-mechanical problem of the tunneling type. The derived equation has 6 regular singular points. To physical domains z = ± ∞ there correspond the singular points 0 and 1 of the derived equation. Frobenius solutions of the equation are constructed, convergence of the relevant series is examined by Poincaré–Perron method. These series are convergent in the whole physical domain z ∈ ( − ∞ , + ∞ ) . Visualization of constructed solutions and numerical study of the tunneling effect are performed.

Authors:H. Bagheri; Y. Kiani; M.R. Eslami Pages: 1566 - 1581 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): H. Bagheri, Y. Kiani, M.R. Eslami In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.

Authors:Keunsoo Park; Marc Gerritsma; Maria Fernandino Pages: 1582 - 1594 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Keunsoo Park, Marc Gerritsma, Maria Fernandino To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a C 1 continuous h -adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples.

Authors:Fuqi Yin; Xueli Li Pages: 1595 - 1615 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Fuqi Yin, Xueli Li This paper considers random attractor and its fractal dimension for Benjamin–Bona–Mahony equation driven by additive white noise on unbounded domains . Firstly, we investigate the existence of random attractor for the random dynamical system defined on an unbounded domain. Secondly, we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space. Finally, we apply expectations of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic Benjamin–Bona–Mahony equation driven by additive white noise.

Authors:Song-Ping Zhu; Xin-Jiang He Pages: 1635 - 1647 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Song-Ping Zhu, Xin-Jiang He In this paper, a modified Black–Scholes (B–S) model is proposed, based on a revised assumption that the range of the underlying price varies within a finite zone, rather than being allowed to vary in a semi-infinite zone as presented in the classical B–S theory. This is motivated by the fact that the underlying price of any option can never reach infinity in reality; a trader may use our new formula to adjust the option price that he/she is willing to long or short. To develop this modified option pricing formula, we assume that a trader has a view on the realistic price range of a particular asset and the log-returns follow a truncated normal distribution within this price range. After a closed-form pricing formula for European call options has been successfully derived, some numerical experiments are conducted. To further demonstrate the meaning of the proposed model, empirical studies are carried out to compare the pricing performance of our model and that of the B–S model with real market data.

Authors:Paweł Dłotko; Thomas Wanner Pages: 1648 - 1666 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Paweł Dłotko, Thomas Wanner The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.

Authors:Rehab M. El-Shiekh Pages: 1676 - 1684 Abstract: Publication date: Available online 11 January 2018 Source:Computers & Mathematics with Applications Author(s): Rehab M. El-Shiekh In this paper, generalized models for both ( 2 + 1 )-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and ( 3 + 1 )-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both ( 2 + 1 )-dimensional cmKdV and ( 3 + 1 )-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given

Authors:Wei Zhao; Yiu-chung Hon; Martin Stoll Pages: 1685 - 1704 Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Wei Zhao, Yiu-chung Hon, Martin Stoll Spectral/pseudo-spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations. In this paper, we will present the use of a localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for solving 2D nonlocal problems with radial nonlocal kernels. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. Numerical implementation indicates that the proposed LRBF-PSM is simple to use, efficient and robust to solve various nonlocal problems.

Authors:Eugeniusz Zieniuk; Krzysztof Szerszeń Pages: 1067 - 1094 Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Eugeniusz Zieniuk, Krzysztof Szerszeń In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.

Authors:Xu Li; Hai-Feng Huo; Ai-Li Yang Pages: 1095 - 1106 Abstract: Publication date: 15 February 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 4 Author(s): Xu Li, Hai-Feng Huo, Ai-Li Yang Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.

Authors:Chi Young Ahn; Sangwoon Yun Pages: 1143 - 1158 Abstract: Publication date: 15 February 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 4 Author(s): Chi Young Ahn, Sangwoon Yun We study on the reconstruction of 3D left ventricle(LV) using only 2D echocardiography data and information on apical long-axis views. Especially, this paper focuses on determining the 3D position of LV contours extracted from 2D echocardiography images. First we mathematically model the relationship between LV contours on the apical views and their corresponding 3D positions. The relationship is expressed as a linear equation in which the right-hand side is the measured data consisting of all the LV contour points on each view and the coefficient matrix is an unknown matrix that transforms the unknown 3D positions into contour points on their related apical view, with distance and orthogonality conditions on the coefficient matrix and the 3D positions. Next we consider a non-convex constrained minimization problem to determine the coefficient matrix and the 3D positions. To solve this minimization problem, we adopt two block coordinate descent method with a solver in OPTI for quadratically constrained quadratic program. For validating the proposed method, some numerical experiments are performed with synthetic data. The experimental results show that the proposed model is promising and available for real echocardiographydata.

Authors:A. Elmoussaoui; P. Argoul; M. El Rhabi; A. Hakim Pages: 1159 - 1180 Abstract: Publication date: 15 February 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 4 Author(s): A. Elmoussaoui, P. Argoul, M. El Rhabi, A. Hakim This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on kinetic and stochastic game theories. The proposed model is a mesoscopic probabilistic approach that retains features obtained from both micro- and macro-scale representations; pedestrian interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical model’s solution is given for large times. A numerical resolution scheme based on the splitting method is implemented and then applied to crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to the time obtained by a deterministic approach by means of randomly varying some of its parameters.

Authors:Thomas Abstract: Publication date: 1 March 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 5 Author(s): Thomas Führer In this work we study a DPG method for an ultra-weak variational formulation of a reaction–diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one. Furthermore, we introduce a simple elementwise postprocessing of the solution and prove superconvergence. Numerical experiments indicate that the obtained results are valid beyond the underlying model problem.

Authors:Michal Dzikowski; Lukasz Jasinski; Marcin Dabrowski Abstract: Publication date: Available online 24 February 2018 Source:Computers & Mathematics with Applications Author(s): Michal Dzikowski, Lukasz Jasinski, Marcin Dabrowski We use Lattice Boltzmann Method (LBM) MRT and Cumulant schemes to study the performance and accuracy of single-phase flow modeling for propped fractures. The simulations are run using both the two- and three-dimensional Stokes equations, and a 2.5D Stokes–Brinkman approximate model. The LBM results are validated against Finite Element Method (FEM) simulations and an analytical solution to the Stokes–Brinkman flow around an isolated circular obstacle. Both LBM and FEM 2.5D Stokes–Brinkman models are able to reproduce the analytical solution for an isolated circular obstacle. In the case of 2D Stokes and 2.5D Stokes–Brinkman models, the differences between the extrapolated fracture permeabilities obtained with LBM and FEM simulations for fractures with multiple obstacles are below 1%. The differences between the fracture permeabilities computed using 3D Stokes LBM and FEM simulations are below 8% . The differences between the 3D Stokes and 2.5 Stokes–Brinkman results are less than 7% for FEM study, and 8% for the LBM case. The velocity perturbations that are introduced around the obstacles are not fully captured by the parabolic velocity profile inherent to the 2.5D Stokes–Brinkman model.

Authors:F. Dassi; L. Mascotto Abstract: Publication date: Available online 24 February 2018 Source:Computers & Mathematics with Applications Author(s): F. Dassi, L. Mascotto We present numerical tests of the virtual element method (VEM) tailored for the discretization of a three dimensional Poisson problem with high-order “polynomial” degree (up to p = 10 ). Besides, we discuss possible reasons for which the method could return suboptimal/wrong error convergence curves. Among these motivations, we highlight ill-conditioning of the stiffness matrix and not particularly “clever” choices of the stabilizations. We propose variants of the definition of face/bulk degrees of freedom, as well as of stabilizations, which lead to methods that are much more robust in terms of numerical performances.

Authors:Weizheng Xu; Weiguo Wu Abstract: Publication date: Available online 22 February 2018 Source:Computers & Mathematics with Applications Author(s): Weizheng Xu, Weiguo Wu In this paper, we present an improved third-order WENO-Z scheme to improve the order of convergence at critical points. This scheme is constructed through slightly modifying the local smoothness indicators of the conventional WENO-Z scheme with the way of Taylor expansion for third-order convergence. The present scheme is proved to be close to third-order by several standard accuracy tests. The performance enhancement of the WENO scheme through this modification is verified on a variety of one-dimensional and two-dimensional standard numerical experiments. Numerical results indicate that the present scheme provides better results in comparison with the earlier third-order WENO schemes like WENO-JS3, WENO-Z3.

Authors:Chao Qian; Jiguang Rao; Dumitru Mihalache; Jingsong He Abstract: Publication date: Available online 22 February 2018 Source:Computers & Mathematics with Applications Author(s): Chao Qian, Jiguang Rao, Dumitru Mihalache, Jingsong He The nonlocal Davey–Stewartson (DS) I equation with a parity-time-symmetric potential with respect to the y -direction, which is called the y -nonlocal DS I equation, is a two-dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation. The multi-breather solutions for the y -nonlocal DS I equation are derived by using the Hirota bilinear method. Lump-type solutions and hybrid solutions consisting of lumps sitting on periodic line waves are generated by long wave limits of the obtained soliton solutions. Also, various types of analytical solutions for the nonlocal NLS equation with negative nonlinearity, including both the Akhmediev breathers and the Peregrine rogue waves sitting on periodic line waves, can be generated with appropriate constraints on the parameters of the obtained exact solutions of the y -nonlocal DS I equation. Particularly, we show that a family of hybrid solitons describing the Peregrine rogue wave that coexists with the Akhmediev breather, both of them sitting on a spatially-periodic background can be thus obtained.

Authors:Xuehua Yang; Haixiang Zhang; Da Xu Abstract: Publication date: Available online 21 February 2018 Source:Computers & Mathematics with Applications Author(s): Xuehua Yang, Haixiang Zhang, Da Xu The fourth-order diffusion systems depict the wave and photon propagation in intense laser beams and play an important role in the phase separation in binary mixture. In this paper, by using orthogonal spline collocation (OSC) method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of fourth-order fractional reaction–diffusion equations. For the original unknown u and auxiliary variable v = Δ u , the full-discrete unconditional stabilities based on a priori analysis are derived by virtue of properties of OSC. Moreover, the convergence rates in L 2 -norm for unknown u are strictly investigated. At the same time, the optimal error estimates in H 1 -norm for unknown u and in L 2 -norm for variable v , are also derived, respectively. For further verifying the theoretical analysis, some numerical examples are provided.

Authors:Leitao Chen; Laura Schaefer Abstract: Publication date: Available online 21 February 2018 Source:Computers & Mathematics with Applications Author(s): Leitao Chen, Laura Schaefer A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.

Authors:Sitong Chen; Xianhua Tang Abstract: Publication date: Available online 19 February 2018 Source:Computers & Mathematics with Applications Author(s): Sitong Chen, Xianhua Tang This paper is concerned with the following Klein–Gordon–Maxwell system: − △ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , x ∈ R 3 , △ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 , where ω > 0 is a constant, V ∈ C ( R 3 , R ) , f ∈ C ( R 3 × R , R ) , and f is superlinear at infinity. Using some weaker superlinear conditions instead of the common super-cubic conditions on f , we prove that the above system has (1) infinitely many solutions when V ( x ) is coercive and sign-changing; (2) a least energy solution when V ( x ) is positive periodic. These results improve the related ones in the literature.

Authors:Zhaqilao Abstract: Publication date: Available online 19 February 2018 Source:Computers & Mathematics with Applications Author(s): Zhaqilao A symbolic computation approach to constructing higher order rogue waves with a controllable center of the nonlinear systems is presented, making use of their Hirota bilinear forms. As some examples, it turns out that some higher order rogue wave solutions of the Kadomtsev–Petviashvili (KP) type equations in ( 3 + 1 ) and ( 2 + 1 ) -dimensions are obtained. Some features of controllable center of rogue waves are graphically discussed.

Authors:Zhao-Zheng Liang; Guo-Feng Zhang Abstract: Publication date: Available online 19 February 2018 Source:Computers & Mathematics with Applications Author(s): Zhao-Zheng Liang, Guo-Feng Zhang Two new preconditioners, which can be viewed as variants of the deteriorated positive definite and skew-Hermitian splitting preconditioner, are proposed for solving saddle point problems. The corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks. The choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated Cayley transform, which result in faster convergence rate and more clustered spectrum. Compared with some preconditioners of similar structures, the new preconditioners have better convergence properties and spectrum distributions. In addition, more practical preconditioning variants of the new preconditioners are considered. Numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate GMRES.

Authors:Jun Ji; Yimin Wei Abstract: Publication date: Available online 19 February 2018 Source:Computers & Mathematics with Applications Author(s): Jun Ji, Yimin Wei The notion of the Moore–Penrose inverses of matrices was recently extended from matrix space to even-order tensor space with Einstein product in the literature. In this paper, we further study the properties of even-order tensors with Einstein product. We define the index and characterize the invertibility of an even-order square tensor. We also extend the notion of the Drazin inverse of a square matrix to an even-order square tensor. An expression for the Drazin inverse through the core-nilpotent decomposition for a tensor of even-order is obtained. As an application, the Drazin inverse solution of the singular linear tensor equation A ∗ X = B will also be included.

Authors:Nikolaos Papageorgiou Abstract: Publication date: Available online 17 February 2018 Source:Computers & Mathematics with Applications Author(s): Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.

Authors:Daewook Kim; Jong Yeoul Park; Yong Han Kang Abstract: Publication date: Available online 16 February 2018 Source:Computers & Mathematics with Applications Author(s): Daewook Kim, Jong Yeoul Park, Yong Han Kang In this paper, we show the energy decay rate for a von Karman system with a boundary nonlinear delay term. This work is devoted to investigate the influence of kernel function g and the effect of the boundary nonlinear term μ 1 u t ( t ) m − 1 u t ( t ) , a boundary nonlinear time delay term μ 2 u t ( t − τ ) m − 1 u t ( t − τ ) and prove energy decay rates of solutions when g do not necessarily decay exponentially and the boundary condition has a time delay.

Authors:Joana P. Neto; Rui Moura Coelho; Duarte Valério; Susana Vinga; Dominik Sierociuk; Wiktor Malesza; Michal Macias; Andrzej Dzieliński Abstract: Publication date: Available online 16 February 2018 Source:Computers & Mathematics with Applications Author(s): Joana P. Neto, Rui Moura Coelho, Duarte Valério, Susana Vinga, Dominik Sierociuk, Wiktor Malesza, Michal Macias, Andrzej Dzieliński Bone is a living tissue that is constantly being renewed, where different cell types can induce a remodeling action to its structure. These mechanisms are typically represented through differential equations, accounting for the biochemical coupling between osteoclastic and osteoblastic cells. Remodeling models have also been extended to include the effects of tumorous disruptive pathologies in the bone dynamics. This article provides a novel approach to existing biochemical models, acting on two different stages. First, the models are said to physiologically better explain an osteolytic metastatic disease to the bone than the multiple myeloma previously considered. Second, and most importantly, variable order derivatives were introduced, for the first time in biochemical bone remodeling models. This resulted in a set of equations with less parameters that describe tumorous remodeling, and provide similar results to those of the original formulation. A more compact model, that promptly highlights tumorous bone interactions, is then achieved. Comparison of simulations and parameters is provided. Such results are a one-step-closer insight to, in a near future, easily provide clinical decision systems ensuring tailored personalized therapy schemes, for more efficient and targeted therapies.

Authors:Tao Zhang; Xiaolin Li Abstract: Publication date: Available online 16 February 2018 Source:Computers & Mathematics with Applications Author(s): Tao Zhang, Xiaolin Li A generalized element-free Galerkin (GEFG) method is developed in this paper for solving Stokes problem in primitive variable form. To obtain stable numerical results for both velocity and pressure, extended terms are only introduced into the approximate space of velocity in a special way as that in the generalized finite element method. Theoretical analysis shows that the GEFG method implies a stabilized formulation similar to that in the variational multiscale element-free Galerkin (VMEFG) method. Numerical results show the efficiency of the present method and reveal that both computational errors and CPU times of the present method are less than those of the VMEFG and the finite element methods.

Authors:Fuzhen Pang; Haichao Li; Xueren Wang; Xuhong Miao; Shuo Li Abstract: Publication date: Available online 16 February 2018 Source:Computers & Mathematics with Applications Author(s): Fuzhen Pang, Haichao Li, Xueren Wang, Xuhong Miao, Shuo Li In this paper, a semi analytical method is used to investigate the free vibration of doubly-curved shells of revolution with arbitrary boundary conditions. The doubly-curved shells of revolution are divided into their segments in the meridional direction, and the theoretical model for vibration analysis is formulated by applying Flügge’s thin shell theory. Regardless of the boundary conditions, the displacement functions of shell segments are composed by the Jacobi polynomials along the revolution axis direction and the standard Fourier series along the circumferential direction. The boundary conditions at the ends of the doubly-curved shells of revolution and the continuous conditions at two adjacent segments were enforced by the penalty method. Then, the natural frequencies of the doubly-curved shells are obtained by using the Rayleigh–Ritz method. For arbitrary boundary conditions, this method does not require any changes to the mathematical model or the displacement functions, and it is very effective in the analysis of free vibration for doubly-curved shells of revolution. The credibility and exactness of proposed method are compared with the results of finite element method (FEM), and some numerical results are reported for free vibration of the doubly-curved shells of revolution under classical and elastic boundary conditions. Results of this paper can provide reference data for future studies in related field.

Authors:Sergio Abstract: Publication date: 15 February 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 4 Author(s): Sergio González-Andrade, Sofía López-Ordóñez In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the p -Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems.

Authors:Tang Abstract: Publication date: 15 February 2018 Source:Computers & Mathematics with Applications, Volume 75, Issue 4 Author(s): De Tang, Li Ma It is well known that the studies of the evolution of biased movement along a resource gradient could create very interesting phenomena. This paper deals with a general two-species Lotka–Volterra competition model for the same resources in an advective nonhomogeneous environment, where the individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. It is assumed that the two species have the same population dynamics but different diffusion and advection rates. It is shown that at least five scenarios can occur (i) If one with a very strong biased movement relative to diffusion and the other with a more balanced approach, the species with much larger advection dispersal rate is driven to extinction; (ii) If one with a very strong biased movement and the other is smaller compare to its diffusion, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; (iii) If both of the species random dispersal rates are sufficiently large (respectively small), two competing species coexist; (iv) If one with a sufficiently large random dispersal rate and the other with a sufficiently small one, two competing species still coexist; (v) If one with a sufficiently small random dispersal rate and the other with a suitable diffusion, which causes the extinction of the species with smaller random movement. Where (iii), (iv) and (v) show the global dynamics of (5) when both of the species dispersal rates are sufficiently large or sufficiently small. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients (respectively, random dispersal rate), and an intermediate biased movement rate (respectively, random dispersal rate) may evolve. Finally, we also apply a perturbation argument to illustrate the evolution of these rates.

Authors:Carlos Abstract: Publication date: Available online 30 December 2017 Source:Computers & Mathematics with Applications Author(s): José G. López-Salas, Carlos Vázquez SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan and Lesniewsk (2008), Mercurio and Morini (2009) and Rebonato and White (2008). Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.

Authors:Alireza Rahimi; Abbas Kasaeipoor Ali Amiri Mohammad Hossein Doranehgard Emad Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): Alireza Rahimi, Abbas Kasaeipoor, Ali Amiri, Mohammad Hossein Doranehgard, Emad Hasani Malekshah, Lioua Kolsi In the present study, the three-dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice Boltzmann method. The enclosure is filled with CuO–water nanofluid. To predict thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. In lattice Boltzmann simulation, two different MRT models are used to solve the problem. The D3Q7-MRT model is used to solve the temperature filed, and the D3Q19 is employed to solve the fluid flow of natural convection within the enclosure. The influences of different Rayleigh numbers 1 0 3 < R a < 1 0 6 and solid volume fractions 0 < φ < 0 . 04 and four different arrangements of discrete active walls on the fluid flow, heat transfer, total entropy generation, local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively.

Authors:Alireza Rahimi; Mohammad Sepehr Milad Janghorban Lariche Abbas Kasaeipoor Emad Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): Alireza Rahimi, Mohammad Sepehr, Milad Janghorban Lariche, Abbas Kasaeipoor, Emad Hasani Malekshah, Lioua Kolsi Two-dimensional natural convection and entropy generation in a square cavity filled with CuO–water nanofluid is performed. The lattice Boltzmann method is employed to solve the problem numerically. The influences of different Rayleigh numbers 1 0 3 < R a < 1 0 6 and solid volume fractions 0 < φ < 0 . 05 on the fluid flow, heat transfer and total/local entropy generation are presented comprehensively. Also, the heatline visualization is employed to identify the heat energy flow. To predict the thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. It is concluded that the configurations of active fins have pronounced effect on the fluid flow, heat transfer and entropy generation. Furthermore, the Nusselt number has direct relationship with Rayleigh number and solid volume fraction, and the entropy generation has direct and reverse relationships with Rayleigh number and solid volume fraction, respectively.

Authors:Dirk Praetorius; Michele Ruggeri Bernhard Stiftner Abstract: Publication date: Available online 28 December 2017 Source:Computers & Mathematics with Applications Author(s): Dirk Praetorius, Michele Ruggeri, Bernhard Stiftner Based on lowest-order finite elements in space, we consider the numerical integration of the Landau–Lifschitz–Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams–Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.

Authors:A.A. Dosiyev Abstract: Publication date: Available online 26 December 2017 Source:Computers & Mathematics with Applications Author(s): A.A. Dosiyev The solution of the Dirichlet problem for Laplace’s equation on a special polygon is harmonically extended to a sector with the center at the singular vertex. This is followed by an integral representation of the extended function in this sector, which is approximated by the mid-point rule. By using the extension properties for the approximate values at the quadrature nodes, a well-conditioned and exponentially convergent, with respect to the number of nodes algebraic system of equations are obtained. These values determine the coefficients of the series representation of the solution around the singular vertex of the polygonal domain, which are called the generalized stress intensity factors (GSIFs). The comparison of the results with those existing in the literature, in the case of Motz’s problem, show that the obtained GSIFs are more accurate. Moreover, the extremely accurate series segment solution is obtained by taking an appropriate number of calculated GSIFs.

Authors:Sihua Liang; Binlin Zhang Abstract: Publication date: Available online 26 December 2017 Source:Computers & Mathematics with Applications Author(s): Sihua Liang, Dušan Repovš, Binlin Zhang In this paper, we consider the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity ε 2 s M ( [ u ] s , A ε 2 ) ( − Δ ) A ε s u + V ( x ) u = u 2 s ∗ − 2 u + h ( x , u 2 ) u , x ∈ R N , u ( x ) → 0 , as x → ∞ , where ( − Δ ) A ε s is the fractional magnetic operator with 0 < s < 1 , 2 s ∗ = 2 N ∕ ( N − 2 s ) , M : R 0 + → R + is a continuous nondecreasing function, V : R N → R 0 + and A : R N → R N are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε < E ; and (ii) for any m ∗ ∈ N , has m ∗ pairs of solutions if ε < E m ∗ , where E and E m ∗ are sufficiently small positive numbers. Moreover, these solutions u ... PubDate: 2017-12-27T11:44:35Z

Authors:Yao-Ming Zhang; Fang-Ling Sun Wen-Zhen Yan Der-Liang Young Abstract: Publication date: Available online 21 December 2017 Source:Computers & Mathematics with Applications Author(s): Yao-Ming Zhang, Fang-Ling Sun, Wen-Zhen Qu, Yan Gu, Der-Liang Young The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.

Authors:Chuanjun Chen; Yanping Chen Xin Zhao Abstract: Publication date: Available online 21 December 2017 Source:Computers & Mathematics with Applications Author(s): Chuanjun Chen, Yanping Chen, Xin Zhao In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in H 1 -norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.

Authors:Pan Zheng; Chunlai Robert Willie Xuegang Abstract: Publication date: Available online 16 December 2017 Source:Computers & Mathematics with Applications Author(s): Pan Zheng, Chunlai Mu, Robert Willie, Xuegang Hu This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity u t = Δ u − χ ∇ ⋅ u ∇ ln v + r u − μ u 2 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t = Δ v − v + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 , where the parameters χ , μ > 0 and r ∈ R . Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.