用于完全不可压缩超弹性固体流固耦合的有限元浸入边界法的体积稳定混合格式
A Volumetrically Stabilized Mixed Formulation of the Finite Element Immersed Boundary Method for Fluid Structure Interaction with Fully Incompressible Hyperelastic Solids
浏览论文内容
中文总结 AI 辅助
本研究提出一种体积稳定的混合格式，改进了带分布式拉格朗日乘子的有限元浸入边界法，可实现完全不可压缩超弹性固体的流固耦合，消除体积不稳定并精准重现下落圆盘终端速度至误差1%。
中文摘要 AI 辅助
带有分布式拉格朗日乘子的有限元浸入边界法（FE IBM）是流固耦合（FSI）领域极具吸引力的框架，它将不可压缩流体的欧拉描述与浸入固体的拉格朗日描述耦合在两个独立、非协调的网格上，避免了贴体任意拉格朗日欧拉法所需的昂贵重网格操作。然而，当浸入固体被建模为完全不可压缩超弹性材料时，对固体偏应力的直接有限元离散无法满足拉格朗日不可压缩约束，导致计算结构出现虚假体积不稳定和锁死现象。本研究提出一种分布式拉格朗日乘子FE IBM的混合格式，以恢复体积稳定性。基于近不可压缩超弹性理论，我们用膨胀应变能导出的体积项增强固体应力，并引入额外的固体压力场（弱形式施加），作为拉格朗日不可压缩约束J=1的拉格朗日乘子。所得格式采用有限元进行空间离散，无条件稳定的半隐式格式进行时间离散，并在基于libMesh有限元库构建的GRINS多物理场框架中实现为可复用的浸入边界物理内核。该方法通过三个FSI基准测试验证：椭圆位移厚环恢复平衡、径向拉伸的不可压缩环、粘性流体中重力下落的圆盘。结果表明，该混合格式消除了未稳定格式中观察到的体积失效，且能将下落圆盘的解析终端速度重现至误差1%以内。
英文摘要
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%.
