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Tangential Derivatives (tangential + derivative)
Selected AbstractsThe boundary element method for solving the Laplace equation in two-dimensions with oblique derivative boundary conditionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2007D. Lesnic Abstract In this communication, we extend the Neumann boundary conditions by adding a component containing the tangential derivative, hence producing oblique derivative boundary conditions. A variant of Green's formula is employed to translate the tangential derivative to the fundamental solution in the boundary element method (BEM). The two-dimensional steady-state heat conduction with the imposed oblique boundary condition has been tested in smooth, piecewise smooth and multiply connected domains in which the Laplace equation is the governing equation, producing results at the boundary in excellent agreement with the available analytical solutions. Convergence of the normal and tangential derivatives at the boundary is also achieved. The numerical boundary data are then used to successfully calculate the values of the solution at interior points again. The outlined test cases have been repeated with various boundary element meshes, indicating that the accuracy of the numerical results increases with increasing boundary discretization. Copyright © 2006 John Wiley & Sons, Ltd. [source] Self-regular boundary integral equation formulations for Laplace's equation in 2-DINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001A. B. Jorge Abstract The purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form (flux-BIE) for Laplace's equation. Self-regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self-regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd. [source] The boundary element method for solving the Laplace equation in two-dimensions with oblique derivative boundary conditionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2007D. Lesnic Abstract In this communication, we extend the Neumann boundary conditions by adding a component containing the tangential derivative, hence producing oblique derivative boundary conditions. A variant of Green's formula is employed to translate the tangential derivative to the fundamental solution in the boundary element method (BEM). The two-dimensional steady-state heat conduction with the imposed oblique boundary condition has been tested in smooth, piecewise smooth and multiply connected domains in which the Laplace equation is the governing equation, producing results at the boundary in excellent agreement with the available analytical solutions. Convergence of the normal and tangential derivatives at the boundary is also achieved. The numerical boundary data are then used to successfully calculate the values of the solution at interior points again. The outlined test cases have been repeated with various boundary element meshes, indicating that the accuracy of the numerical results increases with increasing boundary discretization. Copyright © 2006 John Wiley & Sons, Ltd. [source] Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicatorsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002K. H. Muci-Küchler Abstract In this work, a new global reanalysis technique for the efficient computation of stresses and error indicators in two-dimensional elastostatic problems is presented. In the context of the boundary element method, the global reanalysis technique can be viewed as a post-processing activity that is carried out once an analysis using Lagrangian elements has been performed. To do the reanalysis, the functional representation for the displacements is changed from Lagrangian to Hermite, introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom. Next, assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using Lagrangian elements, the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced. Under this scheme, a second system of equations is generated and, once it is solved, the nodal values of the tangential derivatives of the displacements are obtained. This approach gives more accurate results for the stresses at the nodes since it avoids the need to differentiate the shape functions in order to obtain the normal strain in the tangential direction. When compared with the use of Hermite elements, the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements. Another important feature of the proposed approach is that an efficient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of Lagrangian elements and the global reanalysis technique. Copyright © 2001 John Wiley & Sons, Ltd. [source] On the diffraction of Poincaré wavesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2001P. A. Martin Abstract The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two-dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary-value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary-value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non-empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd. [source] | ||||||||||