Plane problems of linear fracture mechanics were solved with the dual boundary element method (DBEM) using the tangential differential operator (TDO) in traction boundary integral equation (BIE). The numerical implementation employed same shape functions for conformal and non-conformal interpolations with nodal parameters fixed at the ends of elements. Different collocation point positions were used in crack surfaces according to continuity requirements related to each BIE type employed. The aim of this paper is presenting a simple implementation for fracture analysis. Results obtained in the literature for stress intensity factors were included to show the accuracy of the formulation.
Airlines flying in Brazil have their regular operations ruled by RBAC, Regulamentos Brasileiros de Aviação Civil, the Brazilian Aviation Civil Regulation, Part 121. The requirement states that any flight must have enough fuel to go from origin to destination (point A to point B). Also, the flight must have fuel to the alternate airport (point B to point C), plus a contingency fuel that equals the fuel quantity required to fly 10% of the flight time from A to B (AAC, 2019). This 10% fuel for contingency is a number defined in the past by the local authority to cover errors during performance calculations, errors in the aircraft navigation, and also due to inadequate or non-existent meteorology forecasting. The sum of these errors requires additional fuel to make in-flight corrections to unpredicted situations (Hao et al., 2016). However, the technical development in aviation brought more accuracy to the air navigation, and more reliability to the computerized flight planning performance calculations and meteorology forecasting. This evolution was possible because nowadays, the systems are integrated with other tools in the airline, increasing the database for predictions and analysis (Altus, 2009). Today, the major commercial aircraft manufacturers equip their airplane models with navigation systems that, in conjunction with the flight plan and existing meteorology forecasting, are capable of precisely predict the atmosphere condition on every flight level and every mile of the flight. These technological enhancements of current aviation are reducing the differences between the planned and actual fuel burn. Companies intend to keep investing in flight planning systems and modern aircraft because, in this way, airlines can save fuel with accurate and optimized flight plans applied to flight operations (Altus, 2009). Problem According to the ANAC, Agência Nacional de Aviação Civil, the Brazilian Aviation Authority, fuel is one of the airlines' highest costs. In Brazil, fuel cost has represented 24.8% to 29.5% of airline costs composition from 2015 to 2017. As shown in Figure 1, we display the cost composition of Brazilian companies, including fuel, rental, maintenance, depreciation, and airport fees, amongst other costs (ANAC, 2018).
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