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A multi-resolution SPH method for fluid-structure interactions
Chi Zhang, Massoud Rezavand, Xiangyu Hu
TL;DR
FSI simulations are challenging because moving and deforming interfaces complicate mesh-based approaches. This paper develops a multi-resolution SPH method using different spatial and temporal resolutions, momentum-conserving coupling, and time-averaged solid-particle quantities. Numerical examples report improved computational efficiency with reasonable accuracy and robustness, including bio-mechanical applications.
Problem
FSI simulations are highly challenging, while mesh-based methods can encounter difficulties with moving fluid-structure interfaces and large deformation.
Method
The method applies different spatial resolutions and time steps to fluid and structure, using position-based Verlet integration and time-averaged solid-particle velocity and acceleration for coupling.
Results
The numerical examples demonstrate considerably improved computational efficiency, robustness, and reasonable accuracy, including benchmark and bio-mechanical cases.
Takeaways & Limitations
Multi-resolution SPH can produce solid dynamics at higher resolution while using coarser fluid resolution when sufficient, supporting efficient FSI simulation.
Abstract
from arXiv · showhide
In this paper, we present a multi-resolution smoothed particle hydrodynamics (SPH) method for modeling fluid-structure interaction (FSI) problems. By introducing different smoothing lengths and time steps, the spatio-temporal discretization is applied with different resolutions for fluid and structure. To ensure momentum conservation at the fluid-structure coupling, a position-based Verlet time integration scheme is introduced. Furthermore, the time-averaged velocity and acceleration of solid particles are introduced to enhance force matching in the fluid and solid equations. A set of numerical examples including several bio-mechanical problems are considered to demonstrate the efficiency, accuracy and robustness of the present method. A open-source code for all the examples is also provided.
1. Introduction
FSI simulations are challenging because moving, deforming interfaces complicate mesh-based methods, while conventional single-resolution SPH can be computationally expensive. The paper proposes multi-resolution SPH with different spatial and temporal resolutions for fluid and structure, validated on benchmark and bio-mechanical examples.
- FSI is challenging, particularly when fluid-structure interfaces undergo large deformation and displacement.
- Mesh-based FEM and IBM approaches face difficulties with convective terms, moving meshes, and weakly coupled force and kinematic transfer.IBM uses overlapping Lagrangian and Eulerian meshes, with solid forces distributed to the fluid mesh through a kernel function.
- Lagrangian particle methods such as SPH, MPS, and DEM are effective for material interfaces, violent events, and large interface displacement.
- Single-resolution SPH applies uniform particle spacing and one time step, which can be expensive when thin or finely resolved structures require small time steps.
- The proposed method uses different particle resolutions and time steps for fluid and structure to improve computational efficiency while retaining higher structural resolution.It combines updated Lagrangian fluid dynamics, total Lagrangian solid dynamics, and a position-based Verlet scheme for momentum conservation at coupling.
- Time-averaged solid-particle velocity and acceleration improve force matching while the structure is updated several times during one fluid time step.The method is validated on flow-induced vibration, dam-break flow with an elastic gate, venous valves, and fish flapping.
2. Governing equations and preliminary work
The section formulates fluid and solid dynamics and their SPH discretizations, including fluid–structure interaction forces and momentum-conserving coupling. It also describes weak compressibility, constitutive models, and time-step considerations.
- Governing equations and discretization: The method begins with governing equations for fluid and solid dynamics, followed by unified SPH discretizations.The fluid uses mass and momentum conservation, while the solid is formulated in updated or total Lagrangian coordinates.
- Fluid dynamics: Weakly compressible fluid flow is closed with a Tait equation of state relating pressure to density.The artificial sound speed controls compressibility, with cF = 10vmax limiting density variations to about 1%.
- Solid dynamics: Solid dynamics assume an elastic, weakly compressible material and support both linear isotropic elasticity and Neo-Hookean nonlinear behavior.The Neo-Hookean model is used for materials undergoing large deformations.
- Fluid–structure coupling: At the fluid–structure interface, the solid acts as a moving boundary, no-slip is imposed, and pressure and viscous interaction forces are applied to both phases.Solid discretization uses a correction matrix and inter-particle averaged stress, while fluid forces include contributions from neighboring solid particles.
- Time integration and conservation: The anti-symmetric kernel derivative ensures momentum conservation for each interacting fluid–solid particle pair.A single time step is commonly used for force matching, but it can reduce efficiency when the solid sound speed greatly exceeds the fluid sound speed.
3. Multi-resolution method
The method uses different spatial and temporal resolutions for fluid and solid equations to improve computational efficiency, while addressing coupling inconsistencies with position-based Verlet integration and time-averaged solid quantities.
- 3.1. Multi-resolution discretization: Different particle spacings, smoothing lengths, and time steps discretize the fluid and solid equations at separate spatio-temporal resolutions.This multi-resolution discretization is introduced to increase computational efficiency.
- 3.2. Coupling issues: Different time steps can violate momentum conservation and create force mismatch at the fluid-structure coupling.Solid-particle velocity and acceleration may take different updated values after each solid time step.
- 3.3. Position-based Verlet scheme: A position-based Verlet time integration scheme is proposed to address momentum-conservation problems caused by multi-resolution coupling.The scheme is used for integrating the fluid and solid equations.
- 3.1. Multi-resolution discretization: The fluid uses hF for fluid-structure interaction force calculations, with hF ≥ hS to ensure neighboring fluid particles can be identified for solid particles.Here hF and hS denote the fluid and solid smoothing lengths, respectively.
- 3.3. Position-based Verlet scheme: Time-averaged solid-particle velocity and acceleration are introduced during a fluid acoustic time step to address force-calculation mismatch.These averaged quantities are used in calculating fluid-structure interaction forces.
- 3.3. Position-based Verlet scheme: Velocity is updated once per fluid acoustic time step, making fluid and solid momentum-equation marching exactly consistent and enabling strict momentum conservation.The velocity-based Verlet scheme does not guarantee momentum conservation under multiple time steps, whereas the position-based scheme is presented as doing so.
4. Numerical examples
The numerical examples validate the multi-resolution SPH method on flow-induced vibration and dam-break benchmarks, then demonstrate it on venous valve and fish-like flapping problems. Across these cases, the method shows agreement with reference results while reducing computational cost through differing spatial and temporal resolutions.
- Validation and applications: The study first validates accuracy and computational efficiency using benchmarks involving large structural deformation from flow-induced vibration or time-dependent water pressure.The benchmarks are followed by bio-mechanical applications involving a venous valve and passive flapping of a fish-like body.
- Flow-induced vibration: The beam reaches self-sustained periodic oscillation beyond dimensionless time 50, with a 2:1 horizontal-to-vertical frequency ratio and a Lissajous trajectory.The reported frequency and amplitude agree with previous computational results; the literature frequency is 0.19 and amplitude ranges from 0.78 to 0.92.
- Flow-induced vibration: A resolution ratio dpF/dpS = 2.0 with multiple time steps yields a speedup of 5.2 without notable loss of numerical accuracy.The converged amplitude and frequency are 0.855 and 0.189, respectively, although the y-direction amplitude is slightly overestimated.
- Dam-break flow: Multiple time steps provide especially high computational performance when a large Poisson ratio produces a small solid time step.Increasing spatial resolution slightly decreases the maximum horizontal deformation, consistently with the single-resolution results.
- Bio-mechanical applications: The venous-valve example exhibits self-sustained large-amplitude leaflet oscillations whose period remains tied to the pulsatile inflow as Womersley number increases.The simplified geometry nevertheless captures the essential flapping behavior of valve leaflets interacting with pulsatile blood flow.
5. Concluding remarks
The paper presents a multi-resolution SPH method for fluid-structure interaction that uses different spatio-temporal resolutions for fluid and structure. Its coupling mechanisms support momentum conservation, force matching, computational efficiency, robustness, and accuracy in numerical and biomechanical examples.
- The method applies different spatio-temporal resolutions to the fluid and structure in fluid-structure interaction simulations.
- The method demonstrates considerably improved computational efficiency while retaining reasonable accuracy and robustness in extensive numerical examples.
- A position-based Verlet scheme ensures momentum conservation during fluid-structure coupling with multiple time steps.
- Time-averaged solid-particle velocity and acceleration enhance force matching between the fluid and solid equations.
- Biomechanical applications include a venous valve and passive flapping of a fish-like body, where good performance indicates potential for future practical applications.
Appendix A : Velocity profile of the pulsative flow
The appendix describes a pulsative-flow velocity-profile setup and outlines the multi-resolution SPH algorithm used for fluid-structure simulations. The procedure initializes particles and structure geometry, advances fluid and solid equations with position-based Verlet integration, and updates coupling quantities between steps.
- The pulsative flow is driven by a periodic pressure gradient, with pressure amplitude A and oscillation period T defining the forcing.
- For the left-ventricle pulsative-flow model, the parameters are A0 = 2500N/m3 and T0 = 0.6s.
- Velocity profiles are reported during a typical period for Womersley numbers Wo = 1.0, Wo = 5.0, and Wo = 10.0.
- Algorithm 1 initializes the simulation, computes fluid and structure quantities, and iterates updates until the termination condition is satisfied.
- Fluid and solid equations are integrated with position-based Verlet, while pressure and viscous forces and time-averaged solid kinematics support coupling updates.
21 end
The paper concludes its computational procedure and reports benchmark setups and efficiency studies for flow-induced vibration and dam-break flow through an elastic gate. Computational timings are evaluated under specified parallelized simulation conditions.
- The multi-resolution SPH simulation terminates after the prescribed termination condition is reached.
- Tables 1 and 2 describe the flexible-beam flow-induced-vibration setup and its computational efficiency.
- CPU wall-clock time is evaluated for shared-memory parallelized computations and for simulation up to dimensionless time 0.4.
- Table 3 reports a convergence study for flow-induced vibration of a beam attached to a cylinder.
- Tables 4 and 5 present dam-break flow through an elastic gate, including case setups and computational efficiency.