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Singular Fields (singular + field)
Selected AbstractsComparison of two wave element methods for the Helmholtz problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2009T. Huttunen Abstract In comparison with low-order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra-weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L-shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill-conditioning of the resulting matrix system. Copyright © 2008 John Wiley & Sons, Ltd. [source] A new variable-order singular boundary element for two-dimensional stress analysisINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002K. M. Lim Abstract A new variable-order singular boundary element for two-dimensional stress analysis is developed. This element is an extension of the basic three-node quadratic boundary element with the shape functions enriched with variable-order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour-integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd. [source] Solution of axisymmetric Maxwell equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003Franck Assous Abstract In this article, we study the static and time-dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25: 49), we investigate the decoupled problems induced in a meridian half-plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H1 component-wise. It is proven that the singular parts are related to singularities of Laplace-like or wave-like operators. We infer from these characterizations: (i) the finite dimension of the space of singular fields; (ii) global space and space,time regularity results for the electromagnetic field. This paper is the continuation of (Modél. Math. Anal. Numér. 1998; 32: 359, Math. Meth. Appl. Sci. 2002; 25: 49). Copyright © 2003 John Wiley & Sons, Ltd. [source] Theoretical tools to solve the axisymmetric Maxwell equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2002F. Assous Abstract In this paper, the mathematical tools, which are required to solve the axisymmetric Maxwell equations, are presented. An in-depth study of the problems posed in the meridian half-plane, numerical algorithms, as well as numerical experiments, based on the implementation of the theory described hereafter, shall be presented in forthcoming papers. In the present paper, the attention is focused on the (orthogonal) splitting of the electromagnetic field in a regular part and a singular part, the former being in the Sobolev space H1 component-wise. It is proven that the singular fields are related to singularities of Laplace-like operators, and, as a consequence, that the space of singular fields is finite dimensional. This paper can be viewed as the continuation of References (J. Comput. Phys. 2000; 161: 218,249, Modél. Math. Anal. Numér, 1998; 32: 359,389) Copyright © 2002 John Wiley & Sons, Ltd. [source] | ||||||||||