Zhilin Li^1,∗, Baiying Dong², Fenghua Tong³, Weilong Wang³

CMES-Computer Modeling in Engineering & Sciences, Vol.119, No.1, pp. 63-72, 2019, DOI:10.32604/cmes.2019.04635

Abstract The immersed boundary method is well-known, popular, and has had vast areas of applications due to its simplicity and robustness even though it is only first order accurate near the interface. In this paper, an immersed boundary-augmented method has been developed for linear elliptic boundary value problems on arbitrary domains (exterior or interior) with a Dirichlet boundary condition. The new method inherits the simplicity, robustness, and first order convergence of the IB method but also provides asymptotic first order convergence of partial derivatives. Numerical examples are provided to confirm the analysis. More >

Chia-Cheng Tsai¹, Po-Ho Lin²

CMC-Computers, Materials & Continua, Vol.28, No.3, pp. 231-260, 2012, DOI:10.3970/cmc.2012.028.231

Abstract Recently, Liu (CMES 21(2007), 53) developed the modified collocation Trefftz method (MCTM) by setting a characteristic length slightly larger than the maximum radius of the computational domain. In this study, we find that the range of admissible characteristic length can be significantly enlarged if the LU decomposition is applied for solving the resulted dense unsymmetric matrix. Furthermore, we discover a range formula for admissible characteristic length, in which the number of the T-complete functions, the shape of the computation domain, and the exponent bits of the involved floating-point arithmetic have been taken into consideration. In order to validate the prescribed… More >

Mustafa Turkyilmazoglu^1,*

CMES-Computer Modeling in Engineering & Sciences, Vol.120, No.1, pp. 63-81, 2019, DOI:10.32604/cmes.2019.06858

Abstract A ratio approach based on the simple ratio test associated with the terms of homotopy series was proposed by the author in the previous publications. It was shown in the latter through various comparative physical models that the ratio approach of identifying the range of the convergence control parameter and also an optimal value for it in the homotopy analysis method is a promising alternative to the classically used h-level curves or to the minimizing the residual (squared) error. A mathematical analysis is targeted here to prove the equivalence of both the ratio approach and the traditional residual approach, especially… More >

Jiaquan Xie^1,3,*, Fuqiang Zhao^1,3, Zhibin Yao^1,3, Jun Zhang^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.115, No.1, pp. 67-84, 2018, DOI:10.3970/cmes.2018.115.067

Abstract In this paper, the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of three-dimensional multi-term fractional-order PDEs with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by three-variable shifted Jacobi polynomials are compared with the exact solutions. Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm. Lastly, several numerical examples are presented… More >

M.T.C. Araújo Fernandes¹, C.O. Cardoso², W.J. Mansur³

CMES-Computer Modeling in Engineering & Sciences, Vol.113, No.3, pp. 325-342, 2017, DOI:10.3970/cmes.2017.113.341

Abstract A new adaptive (automatic) time stepping algorithm, called RCA (Rate of Convergence Algorithm) is presented. The new algorithm was applied in nonlinear finite element analysis of path-dependent problems. The step size is adjusted by monitoring the estimated convergence rate of the nonlinear iterative process. The RCA algorithm is relatively simple to implement, robust and its performance is comparable to, and in some cases better than, the automatic load incrementaion algorithm existent in commercial codes. Discussions about the convergence rate of nonlinear iterative processes, an estimation of the rate and a study of the parameters of the RCA algorithm are presented.… More >

Jinsheng Wang¹, Liqing Liu², Yiming Chen², Lechun Liu², Dayan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.1, pp. 69-85, 2015, DOI:10.3970/cmes.2015.104.069

Abstract The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials. The fractional derivative is described in the Caputo sense. Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable. By solving the algebraic equations, the numerical solutions are acquired. The method in general is easy to implement and yields good results. Numerical examples are provided to demonstrate the validity and applicability of the method. More >

Jason F. Hammond¹, Elizabeth J. Stewart², John G. Younger³, Michael J.Solomon², David M. Bortz^4,5

CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.3, pp. 295-340, 2014, DOI:10.32604/cmes.2014.098.295

Abstract The overall goal of this work is to develop a numerical simulation which correctly describes a bacterial biofilm fluid-structure interaction and separation process. In this, the first of a two-part effort, we fully develop a convergent scheme and provide numerical evidence for the method order as well as a full 3D separation simulation. We use an immersed boundary-based method (IBM) to model and simulate a biofilm with density and viscosity values different from than that of the surrounding fluid. The simulation also includes breakable springs connecting the bacteria in the biofilm which allows the inclusion of erosion and detachment into… More >

Xiaolin Li¹

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.6, pp. 483-507, 2014, DOI:10.3970/cmes.2014.097.483

Abstract Combining moving least square (MLS) approximations and boundary integral equations, a symmetric and boundary-only meshless method, the Galerkin boundary node method (GBNM), is developed in this paper for two- and threedimensional elasticity problems with mixed boundary conditions. Unlike other MLS-based meshless methods, boundary conditions in this meshless method can be applied directly and easily. In the GBNM, the stiffness matrices so obtained are symmetric. The property of symmetry is an added advantage in coupling the GBNM with the finite element method (FEM). Thus, a symmetric coupling of the GBNM and the FEM is also discussed for elasticity problems. Error analysis… More >

Jin Li^1,2, Hongxing Ru¹, Dehao Yu^3,4

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.6, pp. 463-482, 2014, DOI:10.3970/cmes.2014.097.463

Abstract The computation with Simpson’s rule for the supersingular integrals on circle is discussed. When the singular point coincides with some priori known point, the convergence rate of the Simpson rule is higher than the globally one which is considered as the superconvergence phenomenon. Then the error functional of density function is derived and the superconvergence phenomenon of composite Simpson rule occurs at certain local coordinate of each subinterval. Based on the error functional, a modify quadrature is presented. At last, numerical examples are provided to validate the theoretical analysis and show the efficiency of the algorithms. More >

Ji-Chuan Liu¹, Jun-Gang Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.5, pp. 437-462, 2014, DOI:10.3970/cmes.2014.097.437

Abstract We consider the determination of heat flux within a body from the Cauchy data. The aim of this paper is to seek an approach to solve the onedimensional heat equation in a bounded domain without initial value. This problem is severely ill-posed and there are few theoretic results. A quasi-reversibility regularization method is used to obtain a regularized solution and convergence estimates are given. For numerical implementation, we apply a method of lines to solve the regularized problem. From numerical results, we can see that the proposed method is reasonable and feasible. More >