Speaker
Description
A recent work by Areias et al. [ASA23] shows how it's possible to detect contact by solving the scalar Screened Poisson equation $$ \Delta \phi (\mathbf{x}) - \operatorname{k}^2 \phi(\mathbf{x}) = 0 \qquad \text{for } \operatorname{k} \in \mathbb{R}$$ with constant boundary conditions. Solving this equation in a domain $\Omega$ of arbitrary shape allows us to obtain an Approximate Distance Function (ADF) to uniquely establish whether the boundary of the domain contacts itself at certain points. In this work we explore a BEM-based formulation of this approach, which avoids the discretization of the whole bulk of the domain. For this purpose, we successfully implement the resolution of the Screened Poisson equation in $\pi$-BEM, a parallel boundary element method solver developed by Giuliani et al. [GMH18].
We notice that in the near contact configuration two cells that are topologically distant from each other become geometrically close. This highlights the fact that an inadequate numerical treatment of quasi-singular integrals may increase substantially the global computational effort. To address this issue, we develop a specific quasi-singular quadrature strategy based on a variable transformation technique, to handle all cases close to contact. That is, we change the coordinates of integration from cartesian to spherical. This way, the Jacobian from the change of coordinates compensates the singular kernel, regularizing the integrand.
References
[ASA23] P. Areias, N. Sukumar, and J. Ambrosio. Continuous gap contact formulation based on the screened poisson equation. Computational Mechanics, 72:707–723, 2023.
[GMH18] N. Giuliani, A. Mola, and L. Heltai. π-BEM: A flexible parallel implementation for adaptive, geometry aware, and high order boundary element methods. Advances in Engineering Software, 121:39–58, 2018.