Abstract: Publication date: Available online 20 June 2017 Source:Computers & Mathematics with Applications Author(s): Daniele Boffi, Rolf Stenberg In this paper we discuss mixed finite element methods for nearly incompressible elasticity. We show that if a method uses the hydrostatic pressure as unknown, then the finite element spaces have to satisfy the condition of the ellipticity on the kernel, in addition to the well-known Babuška–Brezzi condition. Some known elements are proved to satisfy this condition.

Abstract: Publication date: Available online 19 June 2017 Source:Computers & Mathematics with Applications Author(s): S. Saha Ray In this paper, using the Lie group analysis method, the infinitesimal generators for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained. The new concept of nonlinear self-adjointness of differential equations is used for construction of nonlocal conservation laws. The conservation laws for the (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained by using the new conservation theorem method and the formal Lagrangian approach. Transforming this equation into a system of equations involving with two dependent variables, it has been shown that the resultant system of equations is quasi self-adjoint and finally the new nonlocal conservation laws are constructed by using the Lie symmetry operators.

Abstract: Publication date: Available online 19 June 2017 Source:Computers & Mathematics with Applications Author(s): Sriram Nagaraj, Socratis Petrides, Leszek F. Demkowicz The use of “ideal” optimal test functions in a Petrov–Galerkin scheme guarantees the discrete stability of the variational problem. However, in practice, the computation of the ideal optimal test functions is computationally intractable. In this paper, we study the effect of using approximate, “practical” test functions on the stability of the DPG (discontinuous Petrov–Galerkin) method and the change in stability between the “ideal” and “practical” cases is analyzed by constructing a Fortin operator. We highlight the construction of an “optimal” DPG Fortin operator for H 1 and H ( div ) spaces; the continuity constant of the Fortin operator is a measure of the loss of stability between the ideal and practical DPG methods. We take a two-pronged approach: first, we develop a numerical procedure to estimate an upper bound on the continuity constant of the Fortin operator in terms of the inf–sup constant γ h of an auxiliary problem. Second, we construct a sequence of approximate Fortin operators and exactly compute the continuity constants of the approximate operators, which provide a lower bound on the exact Fortin continuity constant. Our results shed light not only on the change in stability by using practical test functions, but also indicate how stability varies with the approximation order p and the enrichment order Δ p . The latter has important ramifications when one wishes to pursue local h p -adaptivity.

Abstract: Publication date: Available online 19 June 2017 Source:Computers & Mathematics with Applications Author(s): Jingtang Ma, Zhiqiang Zhou, Zhenyu Cui In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump–diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models.

Abstract: Publication date: Available online 19 June 2017 Source:Computers & Mathematics with Applications Author(s): Yang Cao, Shu-Xin Miao, Zhi-Ru Ren We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

Abstract: Publication date: Available online 17 June 2017 Source:Computers & Mathematics with Applications Author(s): P.F. Antonietti, M. Bruggi, S. Scacchi, M. Verani It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the accuracy of the method employed to discretize the underlying differential problem, which may not be able to correctly capture the physics of the problem. In light of the above remarks, in this paper we consider polygonal meshes and employ the virtual element method (VEM) to solve two classes of paradigmatic topology optimization problems, one governed by nearly-incompressible and compressible linear elasticity and the other by Stokes equations. Several numerical results show the virtues of our polygonal VEM based approach with respect to more standard methods.

Abstract: Publication date: Available online 16 June 2017 Source:Computers & Mathematics with Applications Author(s): Chen Li, Ruibin Qin, Ju Ming, Zhongming Wang In this paper, a discontinuous Galerkin method for the stochastic Cahn-Hilliard equation with additive random noise, which preserves the conservation of mass, is investigated. Numerical analysis and error estimates are carried out for the linearized stochastic Cahn-Hilliard equation. The effects of the noises on the accuracy of our scheme are also presented. Numerical examples simulated by Monte Carlo method for both linear and nonlinear stochastic Cahn-Hilliard equations are presented to illustrate the convergence rate and validate our conclusion.

Abstract: Publication date: Available online 9 June 2017 Source:Computers & Mathematics with Applications Author(s): David Mora, Gonzalo Rivera, Rodolfo Rodríguez The paper deals with the a posteriori error analysis of a virtual element method for the Steklov eigenvalue problem. The virtual element method has the advantage of using general polygonal meshes, which allows implementing efficiently mesh refinement strategies. We introduce a residual type a posteriori error estimator and prove its reliability and global efficiency. Local efficiency estimates also hold, although in some elements they involve boundary terms that are not known to be locally negligible. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests, that allow us to assess the performance of this approach.

Abstract: Publication date: Available online 9 June 2017 Source:Computers & Mathematics with Applications Author(s): Gerardo González, Ville Kolehmainen, Aku Seppänen This paper focuses on studying the effects of isotropic and anisotropic total variation (TV) regularization in electrical impedance tomography (EIT). A characteristic difference between these two widely used TV regularization methods is that the isotropic TV is rotationally invariant and the anisotropic TV is not. The rotational variance of the anisotropic TV is known to cause geometric distortions by favoring edge orientations that are aligned with co-ordinate axes. In many applications, such as transmission tomography problems, these distortions often play only a minor role in the overall accuracy of reconstructed images, because the measurement data is sensitive to the shapes of the edges in the imaged domain. In EIT and other diffusive image modalities, however, the data is severely less sensitive to the fine details of edges, and it is an open question how large impact the selection of the TV regularization variant has on the reconstructed images. In this work, this effect is investigated based on a set of experiments. The results demonstrate that the choice between isotropic and anisotropic TV regularization indeed has a significant impact on the properties of EIT reconstructions; especially, the tendency of the anisotropic TV to favor edges aligned with co-ordinate axes is shown to yield large geometric distortions in EIT reconstructions.

Abstract: Publication date: Available online 7 June 2017 Source:Computers & Mathematics with Applications Author(s): Xiu-Bin Wang, Shou-Fu Tian, Hui Yan, Tian Tian Zhang Under investigation in this work is a generalized ( 3 + 1 )-dimensional Kadomtsev–Petviashvili (GKP) equation, which can describe many nonlinear phenomena in fluid dynamics. By virtue of the Bell’s polynomials, an effective and straightforward way is presented to explicitly construct its bilinear form and soliton solution. Furthermore, based on the bilinear formalism and the extended homoclinic test method, the kinky breather wave solutions and rational breather wave solutions of the equation are well constructed. It is hoped that our results can be used to enrich the dynamical behavior of the ( 3 + 1 )-dimensional nonlinear wave fields.

Abstract: Publication date: Available online 7 June 2017 Source:Computers & Mathematics with Applications Author(s): Liangqi Zhang, Shiliang Yang, Zhong Zeng, Jie Chen, Lingquan Wang, Jia Wei Chew A comparative study on four axisymmetric lattice Boltzmann (LB) models, namely, the kinetic theory based model by Guo et al. (2009), the consistent model by Li et al. (2010), the centered scheme model by Zhou (2011), and our model (based on applying the centered scheme to the Guo et al. (2009) model), is conducted both theoretically and numerically. The finite difference interpretation of the LB method by Junk (2001) is applied to evaluate the accuracy of the models under the incompressible limit. Particularly, the finite difference stencils adopted for the spatial gradient terms in the macroscopic axisymmetric Navier–Stokes (N–S) equations are compared. Besides, the numerical performance (i.e., the numerical accuracy, stability and the convergence efficiency) of the models is compared by two benchmark tests, namely, the unsteady-state Womersley flow and the cylindrical cavity flow. The numerical results accord well with the theoretical analysis. Additionally, it is also found that the numerical stability of the axisymmetric LB models is effectively improved by removing the effects from the non-hydrodynamic variables.

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Ola Ragb, L.F. Seddek, M.S. Matbuly A numerical scheme based on differential quadrature methods, is introduced for solving Bratu problem. The problem is firstly reduced to an iterative one. Then, both of differential quadrature method (DQM) and moving least squares differential quadrature method (MLSDQM) are applied to solve iteratively the nonlinear problem. The proposed scheme successfully computes multiple solutions to Bratu’s problem. The obtained results agree with the 1D and 2D closed forms. Further a parametric study is introduced to investigate the computational characteristics of the proposed scheme.

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Jianping Yu, Yongli Sun In this paper, we study the Gaussian solitary waves for some nonlinear evolution equations with logarithmic nonlinearities. These studied logarithmic evolution equations are the generalized logarithmic BBM equations, the logarithmic ( 2 + 1 ) -dimensional KP-like equations, the logarithmic ( 3 + 1 ) -dimensional KP-like equations, the generalized logarithmic ( 2 + 1 ) -dimensional Klein-Gordon equations and the generalized logarithmic ( 3 + 1 ) -dimensional Klein-Gordon equations. We not only prove that they possess Gaussons: solitary wave solutions of Gaussian shape but also derive the relationships among the parameters.

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Hongyu Ye In this paper, we study the existence and the concentration behavior of critical points for the following functional derived from the Schrödinger–Poisson system: E ( u ) = 1 2 ∫ R 3 ∇ u 2 + 1 4 ∫ R 3 ( x − 1 ∗ u 2 ) u 2 − 3 10 ∫ R 3 u 10 3 constrained on the L 2 -spheres S ( c ) = { u ∈ H 1 ( R 3 ) u 2 = c } when c > c ∗ = Q 2 , where Q is up to translations, the unique positive of − Δ Q + 2 3 Q = Q 4 3 Q in R 3 . As such constrained problem is L 2 -critical, E ( u ) is unbounded from below on S ( c ) when c > c ∗ and the existence of critical points constrained on S ( c ) is obtained by a mountain pass argument on S ( c ) . We show that there exists c 1 > ( 9 7 ) 3 4 c ∗ such that E ( u ) has at least one positive critical point restricted to S ( c ) for c ∗ < c ≤ c 1 . As c approaches c PubDate: 2017-06-06T16:30:40Z

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Imtiaz Ahmad, Siraj-ul-Islam, Abdul Q.M. Khaliq In this paper, a local meshless differential quadrature collocation method is utilized to solve multi-dimensional reaction–convection–diffusion PDEs numerically. In some cases, global version of the meshless method is considered as well. The meshless methods approximate solution on scattered and uniform nodes in both local and global sense. In the case of convection-dominated PDEs, the local meshless method is coupled with an upwind technique to avoid spurious oscillations. For this purpose, a physically motivated local domain is utilized in the flow direction. Both regular and irregular geometries are taken into consideration. Numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains.

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Shaolei Ru, Jiecheng Chen In this paper, we first prove the local well-posedness of the fractional Burgers equations in N Dimensions. Combining the local well-posedness and the method of modulus of continuity, we show the global well-posedness of the N-D critical Burgers equation in critical Besov spaces B ̇ p , 1 N p ( R N ) with p ∈ [ N , ∞ ) .

Abstract: Publication date: 15 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 2 Author(s): Andrzej Karafiat A classical mixed boundary-value problem of linear elasticity in two dimensions in the Galerkin boundary integral formulation is considered. We prove a-priori error estimates of the solution to this problem by the isogeometric adaptive method using NURBS. The estimates include approximation of the boundary of the domain.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): D. Fishelov We present a high-order finite difference scheme for Navier–Stokes equations in irregular domains. The scheme is an extension of a fourth-order scheme for Navier–Stokes equations in streamfunction formulation on a rectangular domain (Ben-Artzi et al., 2010). The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x , y and along the diagonals of the element.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Mayken Espinoza-Andaluz, Martin Andersson, Bengt Sundén The purpose of this study is to investigate the computational time required to describe the fluid flow behavior through a porous medium and its relation to the corresponding domain size. The fluid flow behavior is recovered using the lattice Boltzmann method (LBM). The selected methodology has been applied because of its feasibility for mimicking the fluid flow behavior in complex geometries and moving boundaries. In this study, three different porosities are selected to calculate, for several sizes domain, the required computational time to reach the steady state. Two different cases are implemented: (1) increasing the transversal area, but keeping the layer thickness as a constant, and (2) increasing the total volume of the pore domain by increasing all the dimensions of the volume equally. The porous media are digitally generated by placing the solid obstacles randomly, but uniformly distributed in the whole domain. Several relationships relating the computational time, domain size and porosity are proposed. Additionally, an expression that relates the hydraulic tortuosity to the porosity is proposed.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Moritz Braun, Kingsley O. Obodo In this contribution a multi domain finite element density functional code for molecules is presented. The method makes use of higher order elements to enforce the continuity of the orbitals between the spherical domains and the interstitial domain. The salient computational details of the algorithm are described in some detail. Results of calculations for the orbital energies of methane, ethane, water, ammonia and benzene are given and compared with those obtained using GPAW.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Vít Dolejší, Georg May, Filip Roskovec, Pavel Solin We develop a new mesh adaptive technique for the numerical solution of partial differential equations (PDEs) using the h p -version of the finite element method ( h p -FEM). The technique uses a combination of approximation and interpolation error estimates to generate anisotropic triangular elements as well as appropriate polynomial approximation degrees. We present a h p -version of the continuous mesh model as well as the continuous error model which are used for the formulation of a mesh optimization problem. Solving the optimization problem leads to h p -mesh with the smallest number of degrees of freedom, under the constraint that the approximate solution has an error estimate below a given tolerance. Further, we propose an iterative algorithm to find a suitable anisotropic h p -mesh in the sense of the mesh optimization problem. Several numerical examples demonstrating the efficiency and applicability of the new method are presented.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): L. Molina-Espinosa, C.G. Aguilar-Madera, E.C. Herrera-Hernández, C. Verde This work deals with the numerical modeling of single-phase flow in a pipe with one leak. The mathematical model governing mass, momentum and energy transport was established containing three coupled partial differential equations. These governing equations were numerically solved by discretizing time and space with backward finite differences, leading to one implicit scheme at every time step. The numerical solution was validated with available lab data obtaining good agreement. We present relevant results as the time behavior of upstream and downstream flow and pressure, while varying the leak location. Besides, we also show the temperature, pressure and flow profiles along the pipe when the leak is located near the inlet and outlet of the pipe. The predictive capabilities of the numerical model are remarkable to simulate the transient state of volumetric flow rate and pressure when a single leak is provoked at the pipe. A future application of our numerical scheme is for rapid automatic detection of loss of containment in pipes transporting valuable fluids as, for instance: oil, refined hydrocarbons, fuel, liquid catalysts, etc.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Ulrich Wilbrandt, Clemens Bartsch, Naveed Ahmed, Najib Alia, Felix Anker, Laura Blank, Alfonso Caiazzo, Sashikumaar Ganesan, Swetlana Giere, Gunar Matthies, Raviteja Meesala, Abdus Shamim, Jagannath Venkatesan, Volker John ParMooN is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor MooNMD (John and Matthies, 2004): strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of ParMooN. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library PETSc. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Dániel Marcsa, Miklós Kuczmann A major problem in switched reluctance motor is torque ripple, which causes undesirable acoustic noise and vibration. This work focuses on reducing the undesirable torque ripple in 6/4-pole three-phase switched reluctance motor by geometry modification and using control technique. The proposed method combined the specially skewed rotor pole shape with instantaneous torque control with sinusoidal torque sharing function. The results of geometry modification are analysed through the three-dimensional finite element simulation to determine the appropriate skewing angle. The drive performances of conventional and modified motor are compared through the simulations. The effectiveness of the proposed method is also demonstrated and verified by the simulations.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Martin Čermák, Václav Hapla, Jakub Kružík, Alexandros Markopoulos, Alena Vašatová This paper illustrates parallel solution of elastoplastic problems with hardening based on the TFETI domain decomposition method with several preconditioning strategies. We consider von Mises plasticity with isotropic hardening using the return mapping concept for one time step. To treat nonlinearity and nonsmoothness, we use the semismooth Newton method. In each Newton iteration, we solve a linear system of equations using the TFETI domain decomposition method with lumped, Dirichlet, or no preconditioner. Our PERMON software is used for the numerical experiments. The observed times and numbers of iterations are backed up by the regular condition number estimates. Both preconditioners accelerate the solution process in terms of the convergence rate. For the worst conditioned benchmark, the most expensive Dirichlet preconditioner gives the lowest computational times. This lets us suggest the Dirichlet preconditioner as the default choice for engineering problems.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Roland Wittmann, Hans-Joachim Bungartz, Philipp Neumann We describe code optimization and parallelization procedures applied to the sequential overland flow solver FullSWOF2D. Major difficulties when simulating overland flows comprise dealing with high resolution datasets of large scale areas which either cannot be computed on a single node either due to limited amount of memory or due to too many (time step) iterations resulting from the CFL condition. We address these issues in terms of two major contributions. First, we demonstrate a generic step-by-step transformation of the second order finite volume scheme in FullSWOF2D towards MPI parallelization. Second, the computational kernels are optimized by the use of templates and a portable vectorization approach. We discuss the load imbalance of the flux computation due to dry and wet cells and propose a solution using an efficient cell counting approach. Finally, scalability results are shown for different test scenarios along with a flood simulation benchmark using the Shaheen II supercomputer.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): John N. Jomo, Nils Zander, Mohamed Elhaddad, Ali Özcan, Stefan Kollmannsberger, Ralf-Peter Mundani, Ernst Rank The multi-level h p -refinement scheme is a powerful extension of the finite element method that allows local mesh adaptation without the trouble of constraining hanging nodes. This is achieved through hierarchical high-order overlay meshes, a h p -scheme based on spatial refinement by superposition. An efficient parallelization of this method using standard domain decomposition approaches in combination with ghost elements faces the challenge of a large basis function support resulting from the overlay structure and is in many cases not feasible. In this contribution, a parallelization strategy for the multi-level h p -scheme is presented that is adapted to the scheme’s simple hierarchical structure. By distributing the computational domain among processes on the granularity of the active leaf elements and utilizing shared mesh data structures, good parallel performance is achieved, as redundant computations on ghost elements are avoided. We show the scheme’s parallel scalability for problems with a few hundred elements per process. Furthermore, the scheme is used in conjunction with the finite cell method to perform numerical simulations on domains of complex shape.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): E.C. Herrera-Hernández, M. Núñez-López, J.A. González-Calderón In this work, we present a methodological procedure to validate the numerical solution of the diffusive part in a reaction–diffusion model. Uniform explicit finite differences method is used to generate the solution in a confined circular domain with boundary condition of zero flux. For the validation of the numerical solution, we consider three different criteria applied to normal diffusion and sub-diffusive cases: (i) the moments of concentration, (ii) decay of the concentration at the origin and (iii) the mass conservation. The numerical solution fulfills the validation criteria of moments and concentration decay at the origin only in the long-term. The mass conservation criterion is fulfilled when the initial condition is imposed close to the border, whereas when it is set near to the origin a dependence on the diffusion rate appears. Pattern formation is presented after validating the numerical solution for normal diffusive case. Good agreement of stationary spatial pattern against reported results is observed.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Jan Zapletal, Michal Merta, Lukáš Malý In the paper we study the performance of the regularized boundary element quadrature routines implemented in the BEM4I library developed by the authors. Apart from the results obtained on the classical multi-core architecture represented by the Intel Xeon processors we concentrate on the portability of the code to the many-core family Intel Xeon Phi. Contrary to the GP-GPU programming accelerating many scientific codes, the standard x86 architecture of the Xeon Phi processors allows to reuse the already existing multi-core implementation. Although in many cases a simple recompilation would lead to an inefficient utilization of the Xeon Phi, the effort invested in the optimization usually leads to a better performance on the multi-core Xeon processors as well. This makes the Xeon Phi an interesting platform for scientists developing a software library aimed at both modern portable PCs and high performance computing environments. Here we focus at the manually vectorized assembly of the local element contributions and the parallel assembly of the global matrices on shared memory systems. Due to the quadratic complexity of the standard assembly we also present an assembly sparsified by the adaptive cross approximation based on the same acceleration techniques. The numerical results performed on the Xeon multi-core processor and two generations of the Xeon Phi many-core platform validate the proposed implementation and highlight the importance of vectorization necessary to exploit the features of modern hardware.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Jan Březina, Pavel Exner The XFEM and Mortar methods can be used in combination with non-matching or non-conforming grids to deal with problems on complex geometries. However the information about the mesh intersection must be provided. We present algorithms for intersections between 1d and 2d unstructured multi component simplicial meshes and their intersections with a background unstructured 3d mesh. A common algorithm based on the advancing front technique is used for the efficient selection of candidate pairs among simplicial elements. Bounding interval hierarchy (BIH) of axes aligned bounding boxes (AABB) of elements is used to initialize the front tracking algorithm. The family of element intersection algorithms is built upon a line–triangle intersection algorithm based on the Plücker coordinates. These algorithms combined with the advancing front technique can reuse the results of calculations performed on the neighboring elements and reduce the number of arithmetic operations. Barycentric coordinates on each of the intersecting elements are provided for every intersection point. Benchmarks of the element intersection algorithms are presented and three variants of the global intersection algorithm are compared on the meshes raising from hydrogeological applications.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Mohamed Aissa, Tom Verstraete, Cornelis Vuik A computational Fluid Dynamics (CFD) code for steady simulations solves a set of non-linear partial differential equations using an iterative time stepping process, which could follow an explicit or an implicit scheme. On the CPU, the difference between both time stepping methods with respect to stability and performance has been well covered in the literature. However, it has not been extended to consider modern high-performance computing systems such as Graphics Processing Units (GPU). In this work, we first present an implementation of the two time-stepping methods on the GPU, highlighting the different challenges on the programming approach. Then we introduce a classification of basic CFD operations, found on the degree of parallelism they expose, and study the potential of GPU acceleration for every class. The classification provides local speedups of basic operations, which are finally used to compare the performance of both methods on the GPU. The target of this work is to enable an informed-decision on the most efficient combination of hardware and method when facing a new application. Our findings prove, that the choice between explicit and implicit time integration relies mainly on the convergence of explicit solvers and the efficiency of preconditioners on the GPU.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Saúl E. Buitrago Boret, Oswaldo J. Jiménez P. The aim of this research is the design and implementation of an integrated framework to solve numerically the two-dimensional convection diffusion equations on non-rectangular grids formed only by quadrilaterals honoring the internal structures of an oil reservoir. The framework is composed of (1) a preprocessor of the internal boundaries of a 2D oil reservoir structure, (2) a 2D structural quadrilateral grid generator, (3) a solver of the 2D convection diffusion equation (CDE) on the grid, and (4) a visualization module to display the reservoir properties, wells and the simulation results on the grid. An example of typical structures corresponding to an areal view of a hydrocarbon reservoir is presented. Different scenarios were considered varying boundary conditions, source term, and diffusion constant fluid velocity. All the results are consistent with the physical interpretation of each configuration.

Abstract: Publication date: 1 July 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 1 Author(s): Tomáš Krejčí, Jaroslav Kruis, Michal Šejnoha, Tomáš Koudelka Coupled analysis of heat and moisture transport in real world masonry structures deserves a special attention because the spatial discretization by the finite element method leads usually to large number of degrees of freedom. Thin mortar layers and large bricks or stones have very different material properties and the finite element mesh has to be able to describe correct temperature and moisture fields in mortar and in its vicinity in the blocks. This paper describes two possible solutions of such problems. The first solution is based on the domain decomposition method executed on parallel computers, where the Schur complement method is used with respect to non-symmetric systems of algebraic equations. The second alternative method is the application of a multi-scale approach in connection with a processor farming method, where the whole structure is described by a reasonably coarse finite element mesh, called the macro-scale problem, and the material parameters are obtained from the lower-level problems, called the meso-scale problem, by a homogenization procedure. In this procedure, the macro-problem is assigned to the master processor and the meso-scale problems belong to the slave processors in the processor farm.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): A. Gil, J.P.G. Galache, C. Godenschwager, U. Rüde Simulations of the flow field through chaotic porous media are powerful numerical challenges of special interest in science and technology. The simulations are usually done over representative samples which summarise the properties of the material. Several factors affect the accuracy of the results. Firstly the spatial resolution has to be fine enough to be able to capture the smallest geometrical details. Secondly the domain size has to be large enough to contain the large characteristic scale of the porous media. And finally the effects induced by the boundary conditions have to be diluted when more realistic options are not available. This is the case when the geometry is obtained by tomography and the periodic boundary conditions cannot be applied to delimit the sample because its geometry is not periodic. Impermeable boundary conditions are usually chosen to enclose the domain, forcing mass conservation. As a result, the flow field is over-restricted and the total pressure drop can be over-estimated. In this paper a new strategy is presented to optimise the computational resources consumption keeping the restrictions imposed by the accuracy criteria. The effects of the domain size, discretisation thickness and boundary condition disturbances are studied in detail. The study starts with the procedural generation of chaotic porous walls which mimics acicular mullite filters. An advantage of this process is the possibility to create periodic geometries. Periodicity permits the application of advanced techniques such as cyclic cross-correlations between the phase field and the velocity component fields without aliasing. From cross-correlation operations the large characteristic scale is obtained. The result is a lower threshold for the domain size. In second place a mesh independent study is done to find the upper threshold for the lattice spacing. The Minkowski–Bouligand fractal dimension of the fluid–solid interface corroborates the results. It has been demonstrated how the fractal dimension is a good candidate to replace the mesh independent study with lower computational cost for this type of problems. The last step is to compare the results obtained for a periodic geometry applying periodicity and symmetry as boundary conditions. Considering the periodic case as reference the resultant error is analysed. The explanation of the analysis includes how the intensity of the error changes in space and the limitations of symmetric boundary conditions.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Huamin Zhang, Hongcai Yin This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑ j = 1 q A i j X j B i j = F i , i = 1 , 2 , … , p . We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Salman A. Malik, Sara Aziz We consider the inverse problem of determination of the solution and a source term for a time fractional diffusion equation in two dimensional space. The time fractional derivative is the Hilfer derivative. A bi-orthogonal system of functions in L 2 ( Ω ) , obtained from the associated non-self-adjoint spectral problem and its adjoint problem, is used to prove the existence and uniqueness of the solution of the inverse problem. The stability of the solution of the inverse problem on the given data is proved.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Lyubomir Boyadjiev, Yuri Luchko In this paper, a multi-dimensional α -fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α / 2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α -fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α -fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Zujin Zhang, Zheng-an Yao This paper concerns with the regularity criteria for the 3 D axisymmetric MHD system. It is proved that the control of swirl component of vorticity can ensure the smoothness of the solution.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Hong-Ru Xu, Shui-Lian Xie In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported.

Abstract: Publication date: Available online 12 May 2017 Source:Computers & Mathematics with Applications Author(s): José L. Galán-García, Gabriel Aguilera-Venegas

Abstract: Publication date: Available online 6 May 2017 Source:Computers & Mathematics with Applications Author(s): Hailong Yuan, Jianhua Wu, Yanling Li In this paper, a two-species cooperative model with diffusion and under homogeneous Dirichlet boundary conditions is investigated. It is shown the existence, stability, uniqueness and multiplicity of positive solutions. In particular, we study the global asymptotical stability of the unique positive solution when a ∈ ( λ 1 ( − b θ d 1 + β θ d ) , ∞ ) and α is large. Our method of analysis is based on perturbation technique, the Lyapunov–Schmidt procedure and the bifurcation theory.

Abstract: Publication date: Available online 5 May 2017 Source:Computers & Mathematics with Applications Author(s): Sashikumaar Ganesan, Shangerganesh Lingeshwaran A finite element scheme for the solution of a cancer invasion model is proposed. The cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density, extra cellular matrix and the matrix degrading enzymes. The model incorporates proliferation and haptotaxis effect of cancer cells, their interaction with extracellular matrix, the production of matrix degrading enzymes and consequent degradation of the extracellular matrix. The coupled partial differential equations are discretized in space with the standard Galerkin finite elements and in time with the Crank–Nicolson method. Moreover, the nonlinear terms in the coupled equations are treated semi-implicitly in the finite element scheme. The numerical scheme is validated with numerical results taken from the literature. In addition to the mesh convergence study, the effects of haptotactic rate, proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated.

Abstract: Publication date: Available online 2 May 2017 Source:Computers & Mathematics with Applications Author(s): Marc Bakry, Sébastien Pernet, Francis Collino In this work we construct a new reliable, efficient and local a posteriori error estimate for the single layer and hyper-singular boundary integral equations associated to the Helmholtz equation in two dimensions. It uses a localization technique based on a generic operator Λ which is used to transport the residual into L 2 . Under appropriate conditions on the construction of Λ , we show that it is asymptotically exact with respect to the energy norm of the error. The single layer equation and the hyper-singular equation are treated separately. While the current analysis requires the boundary to be smooth, numerical experiments show that the new error estimators also perform well for non-smooth boundaries.