The Natural Boundary Integral Equation Of The Two-dimensional Orthotropic Potential Problem

Many practical engineering problems, such as heat conduction, fluid flow, torsion problems etc., can be all concluded as potential problems. In this paper, the two-dimensional orthotropic potential problem is solved by using the boundary element method (BEM).A new potential derivative boundary integral equation termed natural boundary integral equation (NBIE) is introduced. The natural boundary integral equation only contains the strongly singular integral and it can be solved perfectly, by contrast, the conventional potential derivative boundary integral equation (CDBIE) has the hypersingular integral. Thus the natural boundary integral equation reduces the singularity by one order.The governing equation of the two-dimensional orthotropic potential problem is transformed into standard Laplace equation by the coordinate transformation method. Then by using the natural boundary integral equation of isotropic potential problem, the potential derivatives of boundary nodes and interior points in the new coordinate system are calculated. Finally, by the back substitution, the potential derivatives of the two-dimensional orthotropic potential problem can be obtained successfully.Compared with the conventional potential derivative boundary integral equation, the natural boundary integral equation can obtain more accurate potential derivatives for boundary notes. When calculating the potential derivatives of interior points very close to the boundary, another form of natural boundary integral equation is discussed and the analytical integral algorithm of nearly singular integrals is applied. The results by two forms of natural boundary integral equation are more precise. Besides, the natural boundary integral equation can take effect for inner points closer to the boundary. Furthermore, the results obtained by second form of natural boundary integral equation are more accurate than those by the original natural boundary integral equation.Numerical examples of heat conduction with orthotropic media verify the efficiency and accuracy of the natural boundary integral equation.