Skeleton-stabilized ImmersoGeometric Analysis for incompressible viscous flow problems

TL;DR

This paper introduces a novel skeleton-stabilized immersed geometric analysis method for incompressible viscous flows, ensuring stability and high accuracy even in complex, highly cut immersed domains, with demonstrated applications in porous media flow.

Contribution

The paper presents a new stabilized finite element formulation that avoids stability issues in immersed flow problems and is computationally efficient with high-order accuracy.

Findings

Achieves oscillation-free, high-order convergence in 2D and 3D tests.

Remains stable even when most elements are cut, without interior cells.

Has a smaller algebraic stencil, improving computational efficiency.

Abstract

A Skeleton-stabilized ImmersoGeometric Analysis technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed formulation fits within the framework of the finite cell method, where essential boundary conditions are imposed weakly using a Nitsche-type method. The key idea of the proposed formulation is to stabilize the jumps of high-order derivatives of variables over the skeleton of the background mesh. The formulation allows the use of identical finite-dimensional spaces for the approximation of the pressure and velocity fields in immersed domains. The stability issues observed for inf-sup stable discretizations of immersed incompressible flow problems are avoided with this formulation. For B-spline basis functions of degree $k$ with highest regularity, only the derivative of order $k$ has to be controlled, which requires specification of only a single stabilization parameter for the pressure field. The Stokes and Navier-Stokes equations are studied numerically in two and three dimensions using various immersed test cases. Oscillation-free solutions and high-order optimal convergence rates can be obtained. The formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut, and hence it does not require the existence of interior cells. In terms of the sparsity pattern, the algebraic system has a considerably smaller stencil than counterpart approaches based on Lagrange basis functions. This important property makes the proposed skeleton-stabilized technique computationally practical. To demonstrate the stability and robustness of the method, we perform a simulation of fluid flow through a porous medium, of which the geometry is directly extracted from 3D $\mu{CT}$ scan data.