On the solution arising in two-cylinders electrostatics.

The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski[2] and Lekner[4] for identity.

Indirect boundary element method combining extra fundamental solutions for solving exterior acoustic problems with fictitious frequencies.

The authors propose an alternative approach to solve the problem of fictitious frequencies. It is different from the mixed potential approach in the indirect method as well as the Burton and Miller approach in the direct boundary element method (BEM). The authors add some fundamental solutions with unknown strength in the solution representation to complete the solution space. From the viewpoint of the adding source, the present idea is similar to the combined Helmholtz interior integral equation formulation (CHIEF) method. The difference between the added source point and null-field point of CHIEF is their role. The former supplies the deficient basis due to the fictitious frequency while the latter provides the extra constraint equation. It can be alternatively found by adding the right unitary vectors of zero singular value. The bordered matrix is invertible for the fictitious frequency if the extra source points do not locate at the failure position. This is the reason why the property is analogous to the CHIEF method in the direct BEM. Therefore, it can fill in the blank area of why there is no CHIEF method in the indirect method. The authors also analytically derive the locations of possible failure source points by using the degenerate kernel.