Volume 7, Issue 4
Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 413-434.

Published online: 2014-07

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• Abstract

The study of vesicles, capsules and red blood cells (RBCs) under flow is a field of active research, belonging to the general  problematic of fluid/structure interactions. Here, we are interested in  modeling vesicles, capsules and RBCs using a boundary integral formulation, and focus on exact singularity subtractions of the kernel of the integral equations in 3D. In order to increase the precision of singular and near-singular integration, we propose here a refinement procedure in the vicinity of the pole of the Green-Oseen kernel. The refinement is performed homogeneously everywhere on the source surface in order to reuse the additional quadrature nodes when calculating boundary integrals in multiple target points. We also introduce a multi-level look-up algorithm in order to select the additional quadrature nodes in vicinity of the pole of the Green-Oseen kernel. The expected convergence rate of the proposed algorithm is of order $\mathcal{O}(1/N^2)$ while the computational complexity is of order $\mathcal{O}$($N^2$ln$N$), where $N$ is the number of degrees of freedom used for surface discretization. Several numerical tests are presented to demonstrate the convergence and the efficiency of the method.

• Keywords

Stokes flow, fluid structure interaction, boundary integral method, red blood cells, singularity subtraction.

64N38, 65N80, 74F10, 76Z05

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@Article{NMTMA-7-413, author = {}, title = {Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {4}, pages = {413--434}, abstract = {

The study of vesicles, capsules and red blood cells (RBCs) under flow is a field of active research, belonging to the general  problematic of fluid/structure interactions. Here, we are interested in  modeling vesicles, capsules and RBCs using a boundary integral formulation, and focus on exact singularity subtractions of the kernel of the integral equations in 3D. In order to increase the precision of singular and near-singular integration, we propose here a refinement procedure in the vicinity of the pole of the Green-Oseen kernel. The refinement is performed homogeneously everywhere on the source surface in order to reuse the additional quadrature nodes when calculating boundary integrals in multiple target points. We also introduce a multi-level look-up algorithm in order to select the additional quadrature nodes in vicinity of the pole of the Green-Oseen kernel. The expected convergence rate of the proposed algorithm is of order $\mathcal{O}(1/N^2)$ while the computational complexity is of order $\mathcal{O}$($N^2$ln$N$), where $N$ is the number of degrees of freedom used for surface discretization. Several numerical tests are presented to demonstrate the convergence and the efficiency of the method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1308si}, url = {http://global-sci.org/intro/article_detail/nmtma/5882.html} }
TY - JOUR T1 - Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 413 EP - 434 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1308si UR - https://global-sci.org/intro/article_detail/nmtma/5882.html KW - Stokes flow, fluid structure interaction, boundary integral method, red blood cells, singularity subtraction. AB -

The study of vesicles, capsules and red blood cells (RBCs) under flow is a field of active research, belonging to the general  problematic of fluid/structure interactions. Here, we are interested in  modeling vesicles, capsules and RBCs using a boundary integral formulation, and focus on exact singularity subtractions of the kernel of the integral equations in 3D. In order to increase the precision of singular and near-singular integration, we propose here a refinement procedure in the vicinity of the pole of the Green-Oseen kernel. The refinement is performed homogeneously everywhere on the source surface in order to reuse the additional quadrature nodes when calculating boundary integrals in multiple target points. We also introduce a multi-level look-up algorithm in order to select the additional quadrature nodes in vicinity of the pole of the Green-Oseen kernel. The expected convergence rate of the proposed algorithm is of order $\mathcal{O}(1/N^2)$ while the computational complexity is of order $\mathcal{O}$($N^2$ln$N$), where $N$ is the number of degrees of freedom used for surface discretization. Several numerical tests are presented to demonstrate the convergence and the efficiency of the method.

Alexander Farutin & Chaouqi Misbah. (2020). Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells. Numerical Mathematics: Theory, Methods and Applications. 7 (4). 413-434. doi:10.4208/nmtma.2014.1308si
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