The Allure of the Boundary Integral Method

Hiroshi Isakari (Associate Professor, Department of System Design Engineering)

The boundary integral method (often called the boundary element method) is a type of numerical analysis method for boundary value problems of partial differential equations. The preface of *Linear Integral Equations* (Rainer Kress, Springer), the bible for researchers in this field, states, "I fell in love with integral equations about twenty years ago when I was working on my thesis, and I am still attracted by their mathematical beauty." Like Kress, I am one of those who continue to be fascinated by this method. Interestingly, and this is not limited to integral equations, numerical analysis based on an elegant formulation often yields good results. I will introduce one such example based on our recent research findings (arXiv:2312.12787). For a certain type of wave scattering problem known as the transmission problem, it was known that a naive formulation of the boundary integral equation becomes ill-conditioned (meaning the condition number of the resulting algebraic equation after discretization becomes large, thus requiring significant computational time to solve). We discovered that by a simple operation of "appropriately rearranging the integral equations and then multiplying some parts by a constant," we could completely eliminate all hypersingular operators from the (square of the) integral operator. Furthermore, we simultaneously revealed that it possesses the beautiful property of having only a single accumulation point of its eigenvalues. Numerical calculations based on the integral equation formulated in this way were several to tens of times more efficient compared to calculations using conventional methods.

Furthermore, in addition to solving boundary value problems, the boundary integral can also be used for purposes such as extracting the geometric features of an object (Figure 1). In fields like CAD and 3D printing, it is common for an object's shape to be given only by its boundary information (for example, as an STL file). Therefore, I believe that we can build an easy-to-use methodology for various fields of engineering, although this is still under research.

I hope to pass on the research of the boundary integral method, which has been continuously developed since the era of Kupradze, to the next generation. At the same time, I look forward to the success of the younger generation (including myself) who will take on the integration of this method with other elemental technologies (such as mathematical optimal design). I offer this as my "Gakumon no susume (An Encouragement of Learning)."

[1] H. Isakari, *Keisan Susu Riko Gaku Ronbunshu*, 2023.

[2] A. Belyaev et al, Computer-Aided Design, 2013.