Validation of Fabric Model Zheng Gao (Stony Brook)

Zheng Gao, Qiangqiang Shi, Yiyang Yang, Bernard Moore, and Xiaolin Li

Department of Applied Mathematics and Statistics

Stony Brook University

Stony Brook, NY 11794

April 12, 2015

On Verification and Validation of Spring Fabric Model

Validation of Fabric Model Zheng Gao (Stony Brook)

• Computational fluid dynamics (CFD): turbulence, flow separation

• Computational structure dynamics (CSD): real fabric canopies

• Front tracking method and FronTier™ : interactions of moving interface and fluid

Challenges in Parachute Simulation

Figure: Breathing motion of C9 parachute

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Introduction

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Numerical method

– Finite element method

– Spring mesh

• Advantages of spring mesh

– Conceptually simple

– Computationally efficient

• Criticism on spring mesh– No relationship with continuum

model

Fabric Simulation

Figure: fabric surface is modeled by spring mesh

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Introduction

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Relevant work– Van Gelder showed that simple spring-mass models cannot

represent linear elastic membranes [1]

– Delingette proposed a revision of the spring-mass model that includes angular deformation [2]

– Our work is validating the model numerically and applying to the parachute simulation

Relevant Work

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Introduction

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[1] Gelder Van, Approximate simulation of elastic membranes by triangulated spring meshes, JGT (1998): 21-41.[2] Delingette Herve, Triangular springs for modeling nonlinear membranes, VCG, IEEE Transactions on 14.2 (2008): 329-341

Validation of Fabric Model Zheng Gao (Stony Brook)

• The coordinates 𝑋𝑖 of the 𝑖-th spring vertex is decided by:

𝑚𝑑2𝑋𝑖𝑑𝑡2= −

𝑗=1

𝑁

𝜂𝑖𝑗𝑓𝑖𝑗 + 𝐹𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙

Spring Mesh

𝑚𝑖: mass of each point

𝑋𝑖: position of each point

𝑑𝑙𝑖𝑗: relative displacement

𝑒𝑖𝑗: unit vector from point 𝑖 to point 𝑗

𝑘𝑖𝑗: tensile stiffness between point 𝑖 and point 𝑗

𝛾𝑖: angular stiffness at point 𝑖

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Introduction

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Validation of Fabric Model Zheng Gao (Stony Brook)

Enhanced spring-mass model

Traditional spring model• The force formula:𝒇𝒊𝒋 = 𝑘𝑖𝑗𝑑𝑙𝑖𝑗𝒆𝒊𝒋

• Disadvantage:• The spring force is

only proportional to the relative displacement from equilibrium distance

• No connection with continuum mechanics.

Enhanced spring model• The force formula:

𝒇𝒊𝒋 = 𝑘𝑖𝑗𝑇1 + 𝑘𝑖𝑗

𝑇2

+(𝛾𝑖𝑇1𝑑𝑙𝑖𝑚 + 𝛾𝑗

𝑇1𝑑𝑙𝑗𝑚 + 𝛾𝑖𝑇2𝑑𝑙𝑖𝑛

+ 𝛾𝑗𝑇2𝑑𝑙𝑗𝑛)𝒆𝒊𝒋

• Advantages:• Include angular stiffness • Derived from energy density

function• Reproduce physical

properties

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Introduction

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𝑻𝟏 𝑻𝟐𝑿𝒎 𝑿𝒏

𝑿𝒋

𝑿𝒊

Validation of Fabric Model Zheng Gao (Stony Brook)

• Fix total mass of string or fabric with different mesh size

• Gradually double the mesh size to test the convergence

• Input appropriate Young’s modulus and Poisson ratio

• Calculate the numerical Young’s modulus and Poisson ratio

Numerical Setup

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Numerical Results

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Validation of Fabric Model Zheng Gao (Stony Brook)

Convergence test for a swinging string

• The string is fixed on the one end• The Cauchy errors show nice

convergence of length and energy• The numerical result is convergent

at first order

Fixed

Free

Figure: Numerical error of length with time

Figure: Numerical error with mesh refinement

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Numerical Results

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Vibrating membrane with boundary fixed

• Calculate the Cauchy error of area and energy

• Estimate the convergence order

Convergence test for a vibrating membrane

Fixed

Free

Figure: Numerical error of area with time

Figure: Numerical error with mesh refinement

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Numerical Results

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Measure Young’s modulus 𝐸 and Poisson ratios 𝜈

• Stretching the fabric to different length

• Compare with theoretical solution

Measure Young’s modulus and Poisson ratio

𝜈 = 𝑑𝜖𝑦/𝑑𝜖𝑥

𝐸 = 𝑠𝑡𝑟𝑒𝑠𝑠/𝑠𝑡𝑟𝑎𝑖𝑛

Figure: Young’s modulus

Figure: Poisson Ratio

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Numerical Results

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Reproduce Young’s modulus 𝐸 and Poisson ratio 𝜈

• Model is limited to small strain (≤ 0.01)

Does angular stiffness really matter?

Figure: Inputs and outputs without angular stiffness

Figure: Inputs and outputs with angular stiffness

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Numerical Results

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Validation of Fabric Model Zheng Gao (Stony Brook)

• Parallel computing technique for acceleration (MPI and CUDA)

• Suitable for GPU computation

• Implementation with CUDA:– Copy velocity and position of each

points to device (0.4%-1.5%)

– Call GPU to solve ODE with 4-th order explicit Runge-Kutta method (95%-98%)

– Copy data back from device (0.8%-3%)

GPU acceleration

Figure: Time cost for different

mesh sizes

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Application

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Pure CPU code

CPU/GPU code

Validation of Fabric Model Zheng Gao (Stony Brook)

• Incompressible fluid solver– Fractional step method

– Second order accurate

• Turbulence modeling– High Reynolds number

– Realizable 𝑘 − 𝜀 model

• Fabric permeability– Coupling with Ergun’s law

Coupling with fluid solver

Figure: streamline around

C9 parachute with vent in

x-z slides

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Application

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Validation of Fabric Model Zheng Gao (Stony Brook)

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.

Angled Drop in Parachute Deployment

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Application

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Validation of Fabric Model Zheng Gao (Stony Brook)

Simulation of multi-chutes deployment

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Application

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Validation of Fabric Model Zheng Gao (Stony Brook)

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

• We would like to thank Dr. Richard Charles as our Army scientific advisor.

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

Acknowledgment

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Thank you

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Thank you for your kind attention