Nonlinear Real-time Finite Element Analysis using CUDA

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1 Nonlinear Real-time Finite Element Analysis using CUDA Biomechanics section KU Leuven

2 Introduction Biomechanics & surgery application What are we trying to do? Why is FE relevant to us? FE and parallelization, TLED Which formulation is better and why? Basic Nonlinear Implicit Total Lagrangian Explicit Dynamic Results, use case Conclusion 3/20/2013 2/21

Intro Biomechanics Help DURING surgery, need fast computation of stresses/damage - Can t influence surgery workflow - tool: FE, generally very slow we use

3 Intro Biomechanics Help DURING surgery, need fast computation of stresses/damage - Can t influence surgery workflow - tool: FE, generally very slow we use CUDA Finite Element Analysis Speed and accuracy is what we are going for. Not embarrassingly parallel Heartport Endoarctic clamp, Redwood, USA 3/20/2013 3/21

4 Introduction to Finite Elements Time integration: To solve: Implicit Iterative schemes (Newton-Raphson) Explicit e.g. central difference method Convergence: Energy has stabilized t=0 t=1 t=2 Total and Updated Lagrangian 3/20/2013 4/21

5 Basic nonlinear implicit FE, example on elasticity Implicit, nonlinear: precomputed cheap cheap Pros: less timesteps unconditionally stable insensitive to incompressbility Impose B.C.s cheap cheap cheap expensive Newton-Raphson iteration Cons: Large matrix inversion More memory intensive More computationally intensive Little is precomputed cheap cheap *D=stress/strain matrix *B=strain/displacement matrix 3/20/2013 5/21

6 Total Lagrangian Explicit Dynamic precompute(); computeforces(); Pros: lots of precomputation no K matrix assembly no K inversion elements independent much less memory required Cons: many more timesteps addit. time to converge conditionally stable sensitive to incompressibility updatedisplacements(); doloading(); *neo-hookean material *d is damage 3/20/2013 6/21

7 TLED serial, iteration phase Get forces() 8 Update Displacements() 9 dodisplacementloading() b d c e f = precomputed 3/20/2013 7/21

8 TLED parallel, iteration phase Get forces( ) Update Displacements( ) doloading( ) x y scatter gather filter x y b d c e f Element granularity a.x a.y b.x b.y c.x c.y d.x d.y 1.x 1.y 2.x 2.y... Nodal granularity b.x b,y c.x c,y New: scatter, gather & filter. One more summation step.

9 Forces load & store maxconnections (m) nodes F.x F.y a b c d a.x 0 a.y a.x 0 b.x c.x 0 b.x b.x 0 c.x 0 d.x 0 a.x 0 c.x 0 d.x d.x a.y 0 b.y c.y 0 b.y b.y 0 c.y 0 d.y 0 a.y 0 c.y 0 d.y d.y b 6 F[4 * 1 i [me].x] = a; F[4 * 2 i [me].y] = a; F[4 * 4 i [me].z] = a; d c 3 a b c d 5 F[4 * 4 i [me].x] = b; F[4 * 2 i [me].y] = b; F[4 * 3 i [me].z] = b; F[4 * 4 i [me].x] = c; F[4 * 3 i [me].y] = c; F[4 * 5 i [me].z] = c; F[4 * 6 i [me].x] = d; F[4 * 4 i [me].y] = d; F[4 * 4 i [me].z] = d; Force[m * mynode [1] index[me].x] = f1; Force[m * mynode [2] index[me].y] = f2; Force[m * mynode [3] index[me].z] = f3; 3/20/2013 9/21

mm. Abaqus/Standard solution Cuda/TLED: displacement (mm) damage(-) Parameter Value μ 1006.7 MPa κ 50e3 MPa γ 0.

Hookean Damage Neo Hookean Damage Neo Hookean Damage 1.8135 1.8119 1.8125 1.8109 1.8163 1.

10 mm. Abaqus/Standard solution Cuda/TLED: displacement (mm) damage(-) Parameter Value μ MPa κ 50e3 MPa γ 0.9 [-] m 1000 MPa Method Material Model Displacement of node x[mm] Abaqus explicit Abaqus implicit CUDA TLED Neo Hookean Damage Neo Hookean Damage Neo Hookean Damage % deviation Calculation time 0.21% 0.21% 4min 4.6min 400ms 500ms 81ms 87ms *xeon e6520 vs Tesla C2075(in single precision) 3/20/ /21

11 Code in action: human aorta model slices Mimicking endoclamp balloon expansion. Loading is 0-500mmHg sine wave. Simulation at 14 fps ~70ms per solution, 1000 timesteps each. Material is Neo-Hookean with E=3000Pa, v=0,49. Small enough to fit all element/node data into registers. 3/20/ /21

12 Thanks! 3/20/ /21

13 TLED Precomputation phase Initialize and compute Mass matrix Shape function derivatives CDM coefficients Explicit integration scheme Central Difference Method No need for optimization/iterative methods Covers nonlinear behavior in an intuitive way given small enough timestep is used. diagonalized Add mass proportional damping according to The Dynamic Relaxation algorithm *CDM = central difference method 3/20/ /21

14 Accuracy considerations Single precision Some increments < macheps Total error after 1000ts <1mm = single precision warranted Accumulation of error(cuda vs Matlab) d=0 Loading curve / macheps d=3 Evolution of error(single vs double) 3/20/ /21

3-noded linear triangular element artery damage single

15 Endoclamp case study mm Hyperelastic constitutive model Plane strain formulation Neo-hookean matrix material of 3-noded linear triangular element artery damage single point Gaussian integration mm damage 0<t<Τ 3/20/ /21

Parallelizing a Real-Time 3D Finite Element Algorithm using CUDA: Limitations, Challenges and Opportunities.

Parallelizing a Real-Time 3D Finite Element Algorithm using CUDA: Limitations, Challenges and Opportunities. Vukasin Strbac Biomechanics section KU Leuven 2/21 The Finite Element Method and the GPU Basically