On Verification and Validation of Spring Fabric Model

On Verification and Validation of Spring Fabric Model Zheng Gao, Qiangqiang Shi, Yiyang Yang, Bernard Moore, and Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University Stony Brook, NY 11794, 2015

Introduction Challenges in Parachute Simulation 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 Figure: Breathing motion of C9 parachute 2/14

Fabric Simulation Introduction Figure: fabric surface is modeled by spring mesh Numerical method Finite element method Spring mesh Advantages of spring mesh Conceptually simple Computationally efficient Criticism on spring mesh No relationship with continuum model 3/14

Introduction Relevant Work 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 [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 4/14

Spring Mesh Introduction The coordinates X i of the i-th spring vertex is decided by: m d2 X i dt 2 m i : mass of each point X i : position of each point dl ij : relative displacement e ij : unit vector from point i to point j k ij : tensile stiffness between point i and point j γ i : angular stiffness at point i N = j=1 η ij f ij + F external 5/14

Introduction Enhanced spring-mass model Traditional spring model The force formula: f ij = k ij dl ij e ij 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: f ij = k ij T 1 + k ij T 2 +(γ i T 1 dl im + γ j T 1 dl jm + γ i T 2 dl in T + γ 2 j dl jn )e ij Advantages: Include angular stiffness Derived from energy density function Reproduce physical properties X m T 1 X j T 2 X n X i 6/14

Numerical Results Numerical Setup 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 7/14

Numerical Results Convergence test for a swinging string Figure: Numerical error of length with time 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 Figure: Numerical error with mesh refinement Free 8/14

Numerical Results Convergence test for a vibrating membrane Figure: Numerical error of area with time Vibrating membrane with boundary fixed Calculate the Cauchy error of area and energy Estimate the convergence order Free Figure: Numerical error with mesh refinement Fixed 9/14

Numerical Results Measure Young s modulus and Poisson ratio Figure: Young s modulus E = stress/strain Measure Young s modulus E and Poisson ratios ν Stretching the fabric to different length Compare with theoretical solution Figure: Poisson Ratio ν = dε y /dε x 10/14

Numerical Results Does angular stiffness really matter? Reproduce Young s modulus E and Poisson ratio ν Model is limited to small strain ( 0.01) Figure: Inputs and outputs without angular stiffness Figure: Inputs and outputs with angular stiffness 11/14

Application GPU acceleration 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%) Pure CPU code CPU/GPU code Figure: Time cost for different mesh sizes 12/14

Application Coupling with fluid solver 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 Figure: streamline around C9 parachute with vent in x-z slides 13/14

Application 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. 14/15

Application Simulation of multi-chutes deployment 15/15

Thank you Acknowledgment 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. 16

Thank you for your kind attention