Chia-Cheng Tsai¹

CMES-Computer Modeling in Engineering & Sciences, Vol.45, No.3, pp. 249-272, 2009, DOI:10.3970/cmes.2009.045.249

Abstract Analytical particular solutions of Chebyshev polynomials are obtained for problems of Reissner plates under arbitrary loadings, which are governed by three coupled second-ordered partial differential equation (PDEs). Our solutions can be written explicitly in terms of monomials. By using these formulas, we can obtain the approximate particular solution when the arbitrary loadings have been represented by a truncated series of Chebyshev polynomials. In the derivations of particular solutions, the three coupled second-ordered PDE are first transformed into a single six-ordered PDE through the Hörmander operator decomposition technique. Then the particular solutions of this six-ordered PDE More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.3, pp. 253-276, 2009, DOI:10.3970/cmes.2009.043.253

Abstract Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a₀ + a₁x + ... + a_nxⁿ expanded by the monomials is the simplest one, which is easy to handle mathematically. For higher accuracy, one always attempts to use a higher-order polynomial as an interpolant. But, Runge gave a counterexample, demonstrating that the polynomial interpolation problem may be ill-posed. Very high-order polynomial interpolation is very hard to realize by numerical computations. In this paper we propose a… More >

M. Azreg-Aïnou¹

CMES-Computer Modeling in Engineering & Sciences, Vol.42, No.1, pp. 1-18, 2009, DOI:10.3970/cmes.2009.042.001

Abstract Adomian polynomials (AP's) are expressed in terms of new objects called reduced polynomials (RP's). These new objects, which carry two subscripts, are independent of the form of the nonlinear operator. Apart from the well-known two properties of AP's, curiously enough no further properties are discussed in the literature. We derive and discuss in full detail the properties of the RP's and AP's. We focus on the case where the nonlinear operator depends on one variable and construct the most general analytical expressions of the RP's for small values of the difference of their subscripts. It More >

G.J.A. Loeven^1,2, H. Bijl³

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.3, pp. 193-212, 2008, DOI:10.3970/cmes.2008.036.193

Abstract In this paper a Two-Step approach is presented for uncertainty quantification for expensive problems with multiple uncertain parameters. Both steps are performed using the Probabilistic Collocation method. The first step consists of a sensitivity analysis to identify the most important parameters of the problem. The sensitivity derivatives are obtained using a first or second order Probabilistic Collocation approximation. For the most important parameters the probability distribution functions are propagated using the Probabilistic Collocation method using higher order approximations. The Two-Step approach is demonstrated for flow around a NACA0012 airfoil with eight uncertain parameters in the More >

Chia-Cheng Tsai¹

CMES-Computer Modeling in Engineering & Sciences, Vol.27, No.3, pp. 151-162, 2008, DOI:10.3970/cmes.2008.027.151

Abstract In this paper we develop analytical particular solutions for the polyharmonic and the products of Helmholtz-type partial differential operators with Chebyshev polynomials at right-hand side. Our solutions can be written explicitly in terms of either monomial or Chebyshev bases. By using these formulas, we can obtain the approximate particular solution when the right-hand side has been represented by a truncated series of Chebyshev polynomials. These formulas are further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). Numerical experiments, which include More >

L. Parussini¹, V. Pediroda²

CMES-Computer Modeling in Engineering & Sciences, Vol.23, No.1, pp. 29-52, 2008, DOI:10.3970/cmes.2008.023.029

Abstract In this paper different Polynomial Chaos methods coupled to Fictitious Domain approach have been applied to one- and two- dimensional elliptic problems with multi uncertain variables in order to compare the accuracy and convergence of the methodologies. Both intrusive and non-intrusive methods have been considered, with particular attention to their employment for quantification of geometric uncertainties. A Fictitious Domain approach with Least-Squares Spectral Element approximation has been employed for the analysis of differential problems with uncertain boundary domains. Its main advantage lies in the fact that only a Cartesian mesh, that represents the enclosure, needs More >

L. Parussini¹, V. Pediroda²

CMES-Computer Modeling in Engineering & Sciences, Vol.22, No.1, pp. 41-64, 2007, DOI:10.3970/cmes.2007.022.041

Abstract In this paper the Non-Intrusive Polynomial Chaos Method coupled to a Fictitious Domain approach has been applied to one- and two-dimensional elliptic problems with geometric uncertainties, in order to demonstrate the accuracy and convergence of the methodology. The main advantage of non-intrusive formulation is that existing deterministic solvers can be used. A new Least-Squares Spectral Element method has been employed for the analysis of deterministic differential problems obtained by Non-Intrusive Polynomial Chaos. This algorithm employs a Fictitious Domain approach and for this reason its main advantage lies in the fact that only a Cartesian mesh More >

V. Sladek¹, J. Sladek¹, M. Tanaka²

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.1, pp. 69-84, 2005, DOI:10.3970/cmes.2005.007.069

Abstract An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material coefficients. The method is based on the local integral equations (utilizing a fundamental solution) and meshfree approximation of field variable. A lot of numerical experiments are carried out in order to study the numerical stability, accuracy, convergence and efficiency of several approaches utilizing various interpolations. More >