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  • Open Access


    The Equal-Norm Multiple-Scale Trefftz Method for Solving the Nonlinear Sloshing Problem with Baffles

    Chao-Feng Shih1, Yung-Wei Chen1,3,*, Jiang-Ren Chang2, Shih-Ping Soon1

    CMES-Computer Modeling in Engineering & Sciences, Vol.127, No.3, pp. 993-1012, 2021, DOI:10.32604/cmes.2021.012702


    In this paper, the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles. When considering solving sloshing problems with baffles by using boundary integral methods, degenerate geometry and problems of numerical instability are inevitable. To avoid numerical instability, the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance. Again, the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme. A weighting factor of the group-preserving scheme is introduced into a linear system… More >

  • Open Access


    A Meshless Regularized Integral Equation Method (MRIEM) for Laplace Equation in Arbitrary Interior or Exterior Plane Domains

    Chein-Shan Liu1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 57-68, 2007, DOI:10.3970/icces.2007.003.057

    Abstract A new method is developed to solve the interior and exterior Dirichlet problems for the two-dimensional Laplace equation, namely the meshless regularized integral equation method (MRIEM), which consists of three parts: Fourier series expansion, the second kind Fredholm integral equation and an analytically regularized solution of the unknown boundary condition on an artificial circle. We find that the new method is powerful even for the problem with very complex boundary shape and with boundary noise. More >

  • Open Access


    A Multiple-Precision Study on the Modified Collocation Trefftz Method

    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 >

  • Open Access


    A Note on Solving the Generalized Dirichlet to Neumann Map on Irregular Polygons using Generic Factored Approximate Sparse Inverses

    E-N.G. Grylonakis1, C.K. Filelis-Papadopoulos1, G.A. Gravvanis1

    CMES-Computer Modeling in Engineering & Sciences, Vol.109-110, No.6, pp. 505-517, 2015, DOI:10.3970/cmes.2015.109.505

    Abstract A new transform method for solving boundary value problems in two dimensions was proposed by A.S. Fokas, namely the unified transform. This approach seeks a solution to the unknown boundary values by solving a global relation, using the known boundary data. This relation can be used to characterize the Dirichlet to Neumann map. For the numerical solution of the global relation, a collocation-type method was recently introduced. Hence, the considered method is used for solving the 2D Laplace equation in several irregular convex polygons. The linear system, resulting from the collocation-type method, was solved by the Explicit Preconditioned Generalized Minimum… More >

  • Open Access


    On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations

    Christina Babenko1, Roman Chapko2, B. Tomas Johansson3

    CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.5, pp. 299-317, 2014, DOI:10.3970/cmes.2014.101.299

    Abstract We consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary integral equations. The second one is the Cauchy problem having Dirichlet and Neumann data given on a part of the boundary of the solution domain. This problem is similarly transformed into a system of ill-posed boundary integral equations. For both systems, to numerically solve them, a mesh grading transformation is employed together with trigonometric quadrature… More >

  • Open Access


    Boundary Layer Effect in Regularized Meshless Method for Laplace Equation

    Weiwei Li1, Wen Chen1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 347-362, 2014, DOI:10.3970/cmes.2014.100.347

    Abstract This paper presents an efficient strategy for the accurate evaluation of near-boundary solutions in the regularized meshless method (RMM), also known as the boundary layer effect associated with the boundary element method. The RMM uses the double layer potentials as its interpolation basis function. When the field point is close to the boundary, the basis function will present nearly strongand hyper-singularities, respectively, for potentials and its derivative. This paper represents the first attempt to apply a nonlinear transformation, based on sinh function, to the accurate evaluation of nearly singular kernels associated with the RMM. The accuracy and efficiency of the… More >

  • Open Access


    The Cell Method: Quadratic Interpolation with Tetrahedra for 3D Scalar Fields

    Martino Pani1, Fulvia Taddei1

    CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.4, pp. 279-300, 2013, DOI:10.3970/cmes.2013.094.279

    Abstract The Cell Method (CM) is a numerical method to solve field equations starting from its direct algebraic formulation. For two-dimensional problems it has been demonstrated that using simplicial elements with an affine interpolation, the CM obtains the same fundamental equation of the Finite Element Method (FEM); using the quadratic interpolation functions, the fundamental equation differs depending on how the dual cell is defined. In spite of that, the CM can still provide the same convergence rate obtainable with the FEM. Particularly, adopting a uniform triangulation and basing the dual cells on the Gauss points of the primal edges, the CM… More >

  • Open Access


    Cauchy Problem for the Laplace Equation in 2D and 3D Doubly Connected Domains

    Ji-Chuan Liu1, Quan-Guo Zhang2

    CMES-Computer Modeling in Engineering & Sciences, Vol.93, No.3, pp. 203-220, 2013, DOI:10.3970/cmes.2013.093.203

    Abstract In this paper, we propose an algorithm to solve a Cauchy problem of the Laplace equation in doubly connected domains for 2D and 3D cases in which the Cauchy data are given on the outer boundary. We want to seek a solution in the form of the single-layer potential and discrete it by parametrization to yield an ill-conditioned system of algebraic equations. Then we apply the Tikhonov regularization method to solve this ill-posed problem and obtain a stable numerical solution. Based on the regularization parameter chosen suitably by GCV criterion, the proposed method can get the approximate temperature and heat… More >

  • Open Access


    A Fully Coupled Model of Non-linearWave in a Harbor

    Daguo Wang1

    CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.4, pp. 289-312, 2013, DOI:10.3970/cmes.2013.091.289

    Abstract A 2-D time-domain numerical coupled model for non-linear wave forces acting on a fixed ship is developed in the present study. The whole domain is divided into the inner domain and the outer domain. The inner domain is the area around the ship section and the flow is described by the Laplace equation. The remaining area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions. The matching conditions on the interfaces between the inner domain and the outer domain are the continuation of volume flux… More >

  • Open Access


    Stable MFS Solution to Singular Direct and Inverse Problems Associated with the Laplace Equation Subjected to Noisy Data

    LiviuMarin 1

    CMES-Computer Modeling in Engineering & Sciences, Vol.37, No.3, pp. 203-242, 2008, DOI:10.3970/cmes.2008.037.203

    Abstract In this paper, a meshless method for the stable solution of direct and inverse problems associated with the two-dimensional Laplace equation in the presence of boundary singularities and noisy boundary data is proposed. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS), whilst the existence of the boundary singularity is taken into account by subtracting from the original MFS solution the corresponding singular solutions, as given by the asymptotic expansion of the solution near the singular point. However, even in the case when the boundary singularity is accounted for, the numerical solutions obtained by… More >

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