Fluid-Structure Interaction Method for Parachute Simulation Using Fabric Spring Model based on Rayleigh-Ritz Analysis

Xiaolin Li,

Zheng Gao, and Xiaolei Chen Department of Applied Math and Statistics

SUNY at Stony Brook

1

2

“Quantitative engineering analysis of parachutes and inflatables has been part of the routine design process since the days of World War II. But in most cases, the shear complexity in which their flexible structure interact both externally and internally with the surrounding air demands that empirical data be used to either validate or supplement such analysis. Advanced modeling embodied in the techniques of Computational Fluid Dynamics (CFD), Computational Structure Dynamics (CSD) and Fluid-Structure Interaction (FSI) has great potential for diminishing such reliance. But even though its application to aerodynamic decelerator system (ADS) has been under consideration for the past four decades, progress has been painfully slow and the results rarely integrated into today’s engineering design practice.” Jean Potvin et al. AIAA 2011-2501

The Need for Parachute FSI M&S

Three Regimes of Parachute Dynamics

3

1. Deployment 2. Inflation 3. Terminal Descent

Application to Parachute Deceleration System Collaboration with Edwards AFB and Natick Soldier System Center

Porosity modelling, low (left) and high (right) porosity parachutes. Fabric collision and turbulence modeling

A robust computational platform for parachute simulation with spring-mass model Work supported by Army Research Office under grants W911NF-14-1-0428 and W911NF-15-1-0403, PI: Xiaolin Li.

Multi-chutes, parallelization and inclusion of parachutists.

Modeling of parachute gores and reinforced borders

The top-left illustrate the gore-strengthened spring-mass model for parachute canopy. The other three plots show the canopy surface mesh of different types of parachutes using constrained Delaunay triangulation. The right plot is the inflated parachute using this model.

Lagrangian Equation

Apply Lagrangian equation

we can get the equations

• Model-I

• Model-II

here is the unit vector from point i to point j.

In our mode, we consider the force needed to bend the fabric is negligible

compared to the force needed to stretch it. We will see that Model-I contains

strong bending force and is not suitable for fabric modeling.

Eigen frequency

By a substitution we can get (omitting the prime)

or

Notice here and .

Eigen frequency of Model-I

In matrix form, the system is

where

Levy-Desplanques Theorem

If a matrix is strictly diagonally dominant, it is

nonsingular.

Definition: A matrix A is said to be diagonally

dominant if

𝑎𝑖𝑖 ≥ 𝑎𝑖𝑗𝑗≠𝑖

for all 𝑖

Proposition 2: The eigenvalues have an upper

bound, assuming that every mass point has at

most M neighbors.

Bound of Eigen frequency

Every eigenvalue of A lies within at least one of

the Gershgorin discs

Definition Let A be a complex matrix, with

entries . For , let be the

sum of the absolute values of the non-diagonal

entries in the ith row. Let be the closed

disc centered at with radius .Such a disc is

called a Gershgorin disc.

Gershgorin circle theorem

Proof: According to Gershgorin circle theorem,

all the eigenvalues of lie within the circles

, where

Bound of Eigen frequency

Then we can see

Therefore all eigenvalues of satisfy

Bound of Eigen frequency of Model-I

The numerical experiments show that the

minimum upper bound should be

Oscillatory Motion of Model-II

Proof: Rewrite the force as

where

and

Oscillatory Motion of Model-II

Oscillatory Motion of Model-II

Finally the force is

Oscillatory Motion of Model-II

It indicates that the force on a mass point

along each direction to its neighbors is

restoring, therefore the motion is oscillatory.

Conservation of Energy

Conservation of Energy

Oscillatory Motion of Fabric Spring Model

We analyzed the spectra of the oscillatory motion in the tangential direction of

the fabric surface and found that the frequencies of the oscillatory modes are

indeed bounded by

Numerical convergence of spring-mass model

The spring-mass elastic membrane model for both string chord and fabric surface is convergent numerically under computational mesh refinement. The left table shows the first order length convergence in the swing tests and the right table shows the area convergence in the drum tests.

Delingette’s variation of spring model (2011)

𝑊 𝑇𝑿0 = 1

2𝑘𝑖

𝑇𝑿0 𝑑𝑙𝑖2 + 𝛾𝑘

𝑇𝑿0𝑑𝑙𝑗𝑑𝑙𝑖𝑖≠𝑗

3

𝑖=1

𝑘𝑖

𝑇𝑿0 =𝑙𝑖0 2

2 cot2 𝛼𝑖 𝜆 + 𝜇 + 𝜇

8𝐴𝑿0

𝛾𝑘𝑇𝑿0 =

𝑙𝑖0𝑙𝑗0 2 cot 𝛼𝑖 cot 𝛼𝑗 𝜆 + 𝜇 − 𝜇

8𝐴𝑿0

𝑭𝑖𝑗 = 𝑘𝑖𝑗𝑇1 + 𝑘𝑖𝑗

𝑇2 𝑑𝑙𝑖𝑗 + 𝛾𝑖𝑇1𝑑𝑙𝑖𝑚 + 𝛾𝑗

𝑇1𝑑𝑙𝑗𝑚 + 𝛾𝑖𝑇2𝑑𝑙𝑖𝑛 + 𝛾𝑖

𝑇2𝑑𝑙𝑗𝑛 𝒆𝑖𝑗

= 𝑘 𝑖𝑗𝑑𝑙𝑖𝑗𝒆𝑖𝑗 + 𝛾 𝑖𝑗𝑑𝑙𝑖𝑗𝒆𝑖𝑗

Verification of Young’s modulus and Poisson ratio

The spring-mass model has excellent agreement with theoretical Young’s modulus and Poisson ratio when the strain is less than 0.1 (10%), very good agreement for strain below 0.2 (20%). The deviation increases as strain exceeds 0.2. This is sufficient for the simulation of parachute canopy.

Po

isson

ratio

You

ng’s m

od

ulu

s

Continuum Mechanics

• Linear Elastic Solids

– A continuum body with reference configuration 𝐵 is said to be a linear elastic solid:

• The first Piola-Kirchhoff stress is 𝑷 𝑿, 𝑡 = 𝑷 𝑭 𝑿, 𝑡 .

• The response function 𝑷 is of the form

𝑷 𝑭 = 𝑪 𝛻𝑿𝒖

where 𝛻𝑿𝒖 = 𝑭 − 𝑰 is the displacement gradient and 𝑪 is a given fourth-order tensor called elasticity tensor for the body.

• The tensor 𝑪 satisfies left and right minor symmetric condition.

– Implied properties:

• First Piola-Kirchhoff stress 𝑷 is symmetric, thus 𝑺 is not necessarily symmetric, balance law of angular momentum in general will not be satisfied.

• Right symmetry implies that 𝑷 𝑭 = 𝑪 𝑠𝑦𝑚 𝛻𝑿𝒖 , in general not

compatible with the frame-indifference.

Continuum Mechanics

– A Linear elastic solid is hyperelastic if the response function 𝑷 satisfieis

𝑷 𝑭 = 𝐷𝑊 𝑭 , for some function 𝑊:𝜈2 → 𝑅 called a strain energy density.

– (Condition for Linear Hyperelastic Solids) If the elasticity tensor 𝑪 has major symmetric, then a linear elastic solid is hyperbolic. And the energy density function is

𝑊 𝐹 =1

2𝛻𝑿𝒖:𝑪(𝛻𝑿𝒖)

where𝛻𝑿𝒖 = 𝑭 − 𝑰. Equivalently, by the minor symmetries of 𝐶

𝑊 𝐹 =1

2𝑬:𝑪(𝑬)

where 𝑬 = sym(𝛻𝑿𝒖).

The Impulse Method The idea of the impulse method is to separate the memory of external and Internal impulses on the mass points of the canopy

The action and reaction between fluid and canopy

Validation of parachute modeling and simulation

Here we present examples of validation study: (a) left plot, C-9 parachute breathing; (b) center, cross parachute inflation; (c) right, the comparison of inflation drag (the blue and red lines are drags recorded in simulation) with experiments.

Simulation of Angled Drop in Parachute Deployment

Our objective is to carry out predictive computational simulations on parachute malfunction during the inflation. This sequence of simulations feature the test of parachute forming an angle with the ambient fluid velocity during the deployment. The sequence of simulations are (from left to right) 𝛼 = 15°, 30°, 45°, 60° respectively. In the last simulation (𝛼 = 60°), the canopy is wrapped from inside out to form the canopy inversion, one of the dangerous malfunction of parachute inflation which may result in fatal consequence.

Simulation of multi-chutes deployment

Turbulence Modelling: RANS equation

• The velocity 𝐮 and pressure 𝑝 of a turbulent incompressible fluid is governed by

𝜕𝐮

𝜕𝑡+ 𝐮 ∙ 𝛻𝒖 = −

1

𝜌𝛻𝑝 + 𝛻 ∙ [ 𝜈 + 𝜈𝑇 𝛻𝐮 + 𝛻𝐮𝑇 ]

𝛻 ∙ 𝐮 = 0

• 𝜈𝑇 is the turbulent eddy viscosity emulating the effect of unresolved velocity fluctuations

• Specifying 𝜈𝑇 is the main task of RANS-based turbulence model

The 𝑘 − 휀 model

• 𝜈𝑇 can be specified by solving two additional equations about turbulent kinetic energy 𝑘 and the dissipation rate 휀:

• 𝜈𝑇 = 𝐶𝜇𝑘2/휀

𝜕𝑘

𝜕𝑡+ 𝛻 ∙ 𝑘𝐮 −

𝜈𝑇

𝜎𝐾+ 𝜈 𝛻𝑘 = 𝑃𝑘 − ε

𝜕𝜀 𝜕𝑡+ 𝛻 ∙ 휀 𝐮 −

𝜈𝑇

𝜎𝜀 + 𝜈 𝛻휀 =

𝜀 𝑘(𝐶1𝑃𝑘−𝐶2휀)

• Some improvements on the standard model:

– RNG model: resolve different scales of motion

Figure: viscosity in x-z slide Figure: vorticity magnitude in x-z slide

Numerical results with RNG model

Figure: streamline in x-z slide Figure: streamline in x-y slide

Background

• Porosity is a measure of the void spaces in a material between 0 and 1

• Permeability is the description of flow velocity through the material

Porosity Modelling

• Introduce penetration ratio 𝛾, 0 ≤ 𝛾 ≤ 1

– When 𝛾 = 0, no penetration, pure boundary case

– When 𝛾 = 1, full penetration, no boundary case

– When 0 < 𝛾 < 1, partial penetration, mixed boundary case

• Porosity Treatment → Boundary Condition Treatment

– Advection term: ghost point reconstruction

– Diffusion term: CONST_V_BOUNDARY

– Projection term: NEUMANN_BOUNDARY

Previous method Model the microstructure and solve the fluid equation at the pore level High computational cost, especially for parachute

Our method Consider the average aerodynamic motion of canopy surface Low computational cost, robust, easy to be coupled with the current fluid solver

[1] K. Takizawa and T.E. Tezduyar, Computational methods for parachute fluid-structure interactions, 2012

Fig. Takizawa 2012 [1] Fig. Tutt 2010 [2]

[2] Tutt B, Richard C, Roland S, Noetscher G, Development of parachute simulation techniques in LS-DYNA, 2010

Methodology

Porosity model

• The pressure drop is modeled with Ghost fluid method (GFM) based on Darcy’s law

[𝑝]Γ = 𝛼𝑢Γ ∙ 𝑛 + 𝛽 𝑢Γ ∙ 𝑛 𝑢Γ ∙ 𝑛

• Couple GFM with projection method – Adding a source term to the pressure (Poisson)

equation

– Not affect the symmetry of the coefficient matrix, easy to converge with KSP iterations

Fig. define the two domains for GFM

Fig. porous surface in the channel

Assumptions

• The canopy surface is extremely thin and the thickness can be ignored

• The microscopic structure is not considered; the permeability is modeled by applying a resistance on the air flow

h=25cm >> canopy thickness 0.1mm

14m

50m with 200 grid points

Mathematics Model

Resistance added on the interface Γ 𝜕𝑢

𝜕𝑡+ 𝑢 ∙ 𝛻𝑢 = 𝜇∆𝑢 − 𝛻𝑝 + 𝑓 + 𝑟Γ𝛿Γ

𝛻 ∙ 𝑢 = 0

Enforce the continuity of the velocity and pressure jump according to the Darcy’s law (laminar flow) or Ergun’s law (turbulence flow)

𝑝 = 𝑟Γ = 𝛼 𝑢 ∙ 𝑛 + 𝛽(𝑢 ∙ 𝑛)2 𝑢 = 0

𝛼 is viscous coefficient and 𝛽 is inertial coefficient

Numerical Method

• Projection method: decouple the velocity and pressure; apply jump condition for pressure Poisson equation

• Ghost fluid method: approximate the gradient and Laplacian of pressure with discontinuity across the interface. Different domain is determined by coating algorithm

2d benchmark test

2d parachute: permeability test

High porosity Medium porosity Low porosity

Y-velocity

𝛾 = 0. 5 𝛾 = 0. 25 𝛾 = 0

Vorticity

𝛾 = 0. 5 𝛾 = 0. 25 𝛾 = 0

3d Benchmark

Streamwise Velocity

Pressure

Validation: 3d fabric test

• According to the Ergun’s equation for turbulence flow 𝑝 = 𝛼 𝑢 ∙ 𝑛 + 𝛽(𝑢 ∙ 𝑛)2

Validation

Compare with experimental data [1]

[1] Air Force Flight Dynamics Laboratory Technical Report (AFFDL-TR-78-151), Recovery System Design Guide, June 1978

3d Parachute test

High porosity Low porosity

Parachutist/Cargo

• The parachutist/cargo is considered as a rigid body in the parachute system.

• The motion of the rigid body consists of two parts:

o the translational motion of the center of mass, which can be described by Newton’s second law.

𝐹 = 𝑚𝑎,𝑑2𝑋

𝑑𝑡2= 𝑎

o the rotational motion with respect to the center of mass which is governed by Euler’s equation of motion.

I1w 1 − w2w3 I2 − I3 = τ1I2w 2 −w3w1 I3 − I1 = τ2I3w 3 −w1w2 I1 − I2 = τ3

where 𝐼 = (𝐼1, 𝐼2, 𝐼3) is the moment of inertia, 𝑤 = (𝑤1, 𝑤, 𝑤3) is the angular velocity and

𝜏 = (𝜏1, 𝜏2, 𝜏3) is the outside torque.

Parachutist/Cargo (cont.)

• FronTier++ can generate different shapes of parachutist/cargo. For example, the following figures demonstrates the suspension line connection with a box (left), a ball (middle) and a human body (right).

Parachutist/Cargo Simulations

Box Ball Human

Parachutist/Cargo Simulations

Intruder T-10 T-11

Robust Treatment of Fabric Collision

Theory and Implementation

Procedure

• CGAL AABB Tree Algorithm

• Detect Proximity

• Detect Collision

• Impact Zone Algorithm

• Numerical Results

CGAL and AABB tree algorithm

• The axis aligned bounding box (AABB) tree algorithm can efficiently find the possible intersection or proximity between elements.

Basic Idea

• Carefully modify the average velocity between 𝑡𝑛 and 𝑡𝑛+1 to avoid any intersections

• The average velocity is calculated with: 𝑣 = (𝑥𝑛+1−𝑥𝑛)/𝑑𝑡

• A impulse method is used to change the velocity

• The new position is updated with the modified average velocity 𝑥𝑛+1 = 𝑥𝑛 + 𝑣 ∗ 𝑑𝑡

𝑥𝑛

Old 𝑥𝑛+1

New 𝑥𝑛+1

obstacle

Detect Proximity (no intersection)

• Most important and basic part

– Point to Triangle detection

– Edge to Edge detection

𝑥1

𝑥2

𝑥3

𝑥4

𝑥1 𝑥2

𝑥3 𝑥4

Add impulse

• Inelastic impulse

– Only apply when two points approach to each other

– Momentum conservation

• Elastic impulse

– Make sure the two points are separated by a distance

– Mimic the spring model: the smaller distance, larger impulse

Detect collision

• Deal with the case when intersection has already happened

• Find the time when the point and triangle are coplanar, then apply the impulse in the previous slides

• Iterate the method until no collision exist

• Solve one collision may involve new collisions

𝑥4

𝑥1 𝑥2

𝑥3

Impact zone algorithm

• Even if one collision is resolved, new collisions may occur. (Multiple collision)

• Impact Zone algorithm:

– Each point in its own list (impact zone)

– When two points collide, the lists containing them should be merged to form a larger impact zone

– The impact zone grows until collision free

– The impact zone is dealt like a rigid body, so no new collision exist within the impact zone

Implementation

• Base Class CollisionSolver

• Class CollisionSolver2d::CollisionSolver

• Class CollisionSolver3d::CollisionSolver

• Only two functions need to be called

STATE of the points

Union and find algorithm for impact zone method

Data structure for collision handling

Collision Handling

Simulation based folding

Fix at apex

Pull at load node

• Pre-processor method: • Can fold simple pattern very quickly but

cannot handle the complex folded shapes

• Simulation based approach: • Requires preparation and calculation time but

apply to any kind of folding pattern. Physical and realistic.

• Directly apply force to nodes or curves in any direction

Fig. before folding

Fig. after folding

Parachute inflation test

68

Introducing FronTier++

69

FronTier++ application: the software platform

1. It started from Glimm-McBryan’s front tracking code.

2. FronTier was SciDAC supported (ITAPS).

3. It is now modularized with a user API (FronTier++).

4. It is parallelized with MPI and GPU interface.

5. It implemented hyperbolic, parabolic, elliptic solvers.

6. It has incompressible fluid solver.

7. It has compressible fluid solver.

8. It has advection-diffusion solver.

9. It has phase transition solver.

10. It has fluid-structure solver.

11. It has fabric ODE system ODE solver.

12. It uses PETSc as the linear equation solver.

13. It generates output used by ViSit, paraview, geomview.

14. It is linked to HDF4 for 2D animation.

Some shortcomings: 1. The parallel load balancing and scaling is not perfect. 2. There are still bugs in geometry and topology handling. 3. Problems with the PETSc solver. 4. Others.

70

FronTier++ on Geometry Handling

Bifurcation of mesh Mesh deformation

Merging of mesh

Subgrid resolution

Geometry preservation

71

FronTier++ application: Rayleigh-Taylor instability

Chaotic mixing 3D Single bubble

Incompressible

Fluid instability

Compressible

Fluid instability

FronTier++ application: phase transition

Crystal formation Melting simulation

Erosion

73

FronTier++ application: other fluid problems

Richtmyer-Meshkov

Instability (implosion) Rayleigh instability

Jet simulation

74

FronTier++ application: fluid-structure interaction

Windmill simulation Bullet firing

Coffee mixing

75

FronTier++ application: fluid-structure interaction

76

FronTier++ application:cloud entrainment

Vorticity Number density Vapor concentration

Radius distribution

• We would like to thank Dr. Joseph Myers and Dr. Michael Kendra for providing the opportunity and to foster the communication between university faculty and army and Air Force scientists.

• Dr. Richard Charles is our Army scientific advisor.

• Thanks to Prof. Jean Potvin for providing us with important experimental data on verification and validation of the parachute model.

• This work is supported in part by the US Army Research Office under the award W911NF0910306, W911NF1410428 and the ARO-DURIP Grant W911NF1210357.

Acknowledgement