Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.10217

A Volumetrically Stabilized Mixed Formulation of the Finite Element Immersed Boundary Method for Fluid Structure Interaction with Fully Incompressible Hyperelastic Solids

Abhishek Mishra·Paul T. Bauman
View original source ↗Read source document ↗Export metadata ↓

Abstract

The finite element immersed boundary method (FE IBM) with a distributed Lagrange multiplier is an attractive framework for fluid structure interaction (FSI) because it couples an Eulerian description of an incompressible fluid to a Lagrangian description of an immersed solid on two independent, nonconforming meshes, avoiding the costly remeshing required by body fitted arbitrary Lagrangian Eulerian methods. When the immersed solid is modelled as a fully incompressible hyperelastic material, however, a direct finite element discretization of the deviatoric solid stress fails to enforce the Lagrangian incompressibility constraint, and the computed structure exhibits spurious volumetric instabilities and locking. In this work we present a mixed formulation of the distributed Lagrange multiplier FEIBM that restores volumetric stability. Following the theory of nearly incompressible hyperelasticity, we augment the solid stress with a volumetric contribution derived from a dilatational strain energy and introduce an additional solid pressure field, enforced weakly, that plays the role of the Lagrange multiplier for the Lagrangian incompressibility constraint J = 1. The resulting formulation is discretized in space by finite elements and in time by an unconditionally stable semi implicit scheme, and is implemented as a reusable Immersed Boundary physics kernel within the GRINS multiphysics framework, built on the libMesh finite element library. The method is verified on three FSI benchmarks, an elliptically displaced thick ring returning to equilibrium, a radially stretched incompressible ring, and a disk falling under gravity in a viscous fluid, for which the mixed formulation removes the volumetric failure observed with the unstabilized formulation and reproduces the analytical terminal velocity of the falling disk to within 1%.

Explore related subjects

math.NAcs.NAphysics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Abhishek Mishra, Paul T. Bauman. 2026-08-10. A Volumetrically Stabilized Mixed Formulation of the Finite Element Immersed Boundary Method for Fluid Structure Interaction with Fully Incompressible Hyperelastic Solids. https://arxiv.org/abs/2608.10217

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗