SUMMARYThe 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 ( ux-BIE) for Laplace's equation. Selfregular formulations lead to highly e ective 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 ux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, ux 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 ux 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 ux results converge monotonically to the exact answer. In the ux-BIE implementation, where all integrals are regularized, ux results accuracy improves systematically, even with some oscillations, when reÿning the mesh or increasing the order of the interpolating function. The ux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and reÿnement. Accurate results for the potential and the ux 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 ( and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems.
This chapter describes the importance of one of the major areas within Structural HealthMonitoring and Health and Usage Monitoring Systems: Sensor Placement Optimization (SPO). In this sense, the most important SPO metrics are presented and discussed with the reader. The methodological procedure of the optimization problem statement is also considered and discussed. he significance of rotary-wing aircraft, damages that can occur in the blade, works related to damage identification, modern methods and algorithms that have been used in this context, and a comprehensive SPO study for a helicopter main rotor blade will be presented. Finally, a new technique that considers the number of sensors as an objective associated with some of the main metrics used in SPO. The technique uses the Multiobjective Lichtenberg Algorithm and Feature Selection, an important area used in Data Mining.
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