On immersed boundary kernel functions: a constrained quadratic minimization perspective

TL;DR

This paper presents a new perspective on generating immersed boundary kernel functions by formulating it as a constrained quadratic minimization problem, enabling flexible and bounded kernel design on complex domains.

Contribution

It introduces a quadratic minimization framework for IB kernel generation, extending beyond analytical and MLS methods to include anisotropic and bounded kernels.

Findings

Quadratic minimization effectively generates IB kernels.

The approach allows for anisotropic and bounded kernel functions.

It broadens kernel design options beyond traditional methods.

Abstract

In the immersed boundary (IB) approach to fluid-structure interaction modeling, the coupling between the fluid and structure variables is mediated using a regularized version of Dirac delta function. In the IB literature, the regularized delta functions, also referred to IB kernel functions, are either derived analytically from a set of postulates or computed numerically using the moving least squares (MLS) approach. Whereas the analytical derivations typically assume a regular Cartesian grid, the MLS method is a meshless technique that can be used to generate kernel functions on complex domains and unstructured meshes. In this note we take a viewpoint that IB kernel generation, either analytically or via MLS, is a constrained quadratic minimization problem. The extremization of a constrained quadratic function is a broader concept than kernel generation, and there are well-established numerical optimization techniques to solve this problem. For example, we show that the constrained quadratic minimization technique can be used to generate one-sided (anisotropic) IB kernels and/or to bound their values.