Computational Fluid-Structure Interaction_ Methods and Applications ( PDFDrive )

Governing Equations of Fluid and Structural Mechanics

Overview of Fluid-Structure Interaction (FSI)

Fluid-Structure Interaction (FSI) involves complex interactions between fluid dynamics and structural mechanics. The governing equations for these disciplines are typically formulated as partial differential equations, which are completed by appropriate boundary conditions and constitutive models. The study of FSI emphasizes the strong and weak forms of these equations. In particular, a three-dimensional solid representation is analyzed alongside discussions on behaviors of thin structures. Furthermore, computational methods like Arbitrary Lagrangian–Eulerian (ALE) are introduced, which are essential for simulations involving deformable or moving domains in fluid mechanics.

1. Governing Equations of Fluid Mechanics

1.1. Navier–Stokes Equations of Incompressible Flows

The dynamics of incompressible flows are primarily described by the Navier–Stokes equations, which consist of local equations detailing how fluid flows behave under various forces.

• Momentum balance equation: \frac{\partial(\rho u)}{\partial t} +
  abla \cdot (\rho u \otimes u - \sigma) - \rho f = 0

• Continuity (mass) equation:
  abla \cdot u = 0

1.1.1. Strong Form of the Navier–Stokes Equations

The strong form represents the fluid's behavior mathematically within a defined spatial domain ( ( \Omega_t \in \mathbb{R}^{nsd} )). Key variables include:

• Density (( \rho ))

• Velocity (( u ))

• External force (( f ))

• Stress tensor (( \sigma(u, p) = -pI + 2\mu \epsilon(u) )), capturing the internal forces within the fluid. The momentum equation underlines the local balance of linear momentum over the fluid domain, specifically underscoring the aspect of incompressibility through the divergence-free condition of the velocity field.

1.1.2. Weak Form of the Navier–Stokes Equations

Transitioning to the weak form necessitates defining trial and test function sets, resulting in: Find (( u )) and (( p )) such that: \int_{\Omega_t} w \cdot \left( \rho \frac{\partial u}{\partial t} + u \cdot
abla u - f \right) d\Omega + \int_{\Omega_t} \epsilon(w) : \sigma(u, p) d\Omega + \int_{(\Gamma_t)_h} w \cdot h d\Gamma + \int{\Omega_t} q
abla \cdot u d\Omega = 0

2. Governing Equations of Structural Mechanics

2.1. Kinematics

In structural mechanics, the kinematics aspect involves defining material domain representations and mappings from reference configurations to current states. The displacement of any point in the structure is defined as (( y(X, t) )). Basic relationships include:

• Velocity and acceleration, which can be derived from the displacement function.

2.2. Conservation of Mass

The principle of conservation of mass plays a critical role in structural mechanics, defining structural density through the formula (( \rho J = \text{constant} )), where ( J ) represents the determinant of the deformation gradient, thus ensuring mass consistency within the structure's domain.

2.3. Principle of Virtual Work

This principle articulates the balance of virtual work contributed internally by the structure against external work done. It leads to a variational formulation predicated on displacements, serving as a foundation for the development of finite element methods.

2.4. Weak Form of Structural Mechanics Equations

The weak formulation is derived under restrictions that reflect the equilibrium state and balance of forces, allowing for the application of numerical methods in structural analysis.

3. ALE and Space-Time Methods

3.1. Interface-Tracking and Interface-Capturing Techniques

These techniques are essential for managing the differences between moving and stationary mesh methods in fluid domains, highlighting trade-offs related to computational efficiency, accuracy, and complexity during simulations.

3.2. ALE Methods

The Arbitrary Lagrangian–Eulerian methods advance the understanding of temporal and spatial dependencies, employing versatile reference frames to foster numerical stability and accommodate changes in the fluid domain.

3.3. Space-Time Methods

Space-time methods are designed to manage fluid behavior across varying time frames effectively. These methods enhance fluid-structure interaction modeling, particularly during movement or dynamic changes in the system.

4. Finite Element Method (FEM) Elements

4.1. Finite Element Function Spaces

The finite element approach involves discretization and the assembly of governing equations based on shape functions derived from the physical geometries of the system being analyzed.

4.2. Stability and Convergence Analysis

A crucial aspect of this analysis involves addressing potential oscillatory behaviors, ensuring stability through appropriate parameter selection and thorough mathematical modeling, particularly in fluid dynamic contexts.

Key Points from each Section:

1. Develops and elucidates formulations for fluid mechanics within the FSI context, detailing the strength of equations and boundary conditions.

2. Establishes kinematic variables, conservation principles and virtual work formulations in structural mechanics designed for finite element applications.

3. ALE and space-time methods underscore significant adaptations in mesh and interface strategies aiding numerical stability while effectively managing computational loads in dynamic settings.

Summary

The governing equations outlined above establish the foundational framework for simulations analyzing fluid-structure interactions. The applied techniques promote flexibility, addressing the inherent complexities and dynamic behaviors inherent within computational mechanics, crucial for both theoretical analysis and practical applications in engineering disciplines.