GPU-based Biomechanical Simulations and Steering

Transcription

1 GPU-based Biomechanical Simulations and Steering Rüdiger Westermann Chair for Computer Graphics & Visualization Faculty of Informatics

2 GPU-based Biomechanical Simulations and Steering Focus on interactive physically-based simulation State-of-the-art report by Jun etal. 2015, EG 2014 A Survey of Physically Based Simulation of Cuts in Deformable Bodies Elastic body simulation Cutting for virtual surgery simulation Hip-joint implant planning Take away Accurate biomedical finite-element simulations can be steered on desktop PCs Topological changes for simulating cuts can be embedded into interactive environments Efficient extraction of smooth surfaces from hexahedral domains

3 GPU-based Biomechanical Simulations and Steering Applications: Simulation and cutting elastic bodies

4 GPU-based Biomechanical Simulations and Steering What s behind!? A finite element method (FEM) in combination with a geometric multigrid solver on Graphics Processing Units Model of linear elasticity Co-rotational formulation Hexahedral finite elements Topological changes of the simulation domain Interactive surface reconstruction and rendering

5 Biomechanical finite element simulations Model of linear elasticity Co-rotated strain B DB u e T e e K f u e : displacements f e : external forces B: strain matrix D: material law 8 j 1 T 0 0 ij j j j i 8 8 T T 0 0 RKijR uj fi RKij R pj pj j 1 j 1 Aˆ ij bˆ RK R p u p f, i 1,...,8 i Rotate displacements back into initial configuration Rotate forces from initial into current configuration R : Element rotation 0 : Undeformed vertex positions p j Linear strain Linear, co-rotated strain

6 Biomechanical finite element simulations Hexahedral finite elements Numerical stencil of regular shape (well parallelizable on the GPU) Parallel GPU voxelization to generate simulation model Target voxel grid k-buffer capturing fragments Depth k-buffer

7 Biomechanical finite element simulations Hexahedral finite elements Parallel GPU voxelization to generate simulation model grid in 25 ms

8 Biomechanical finite element simulations Matrix-free multigrid FEM Efficient GPU hierarchy construction Interpolation/restriction weights are implicitly given by vertex locations Level 0 Level 1 Level 2 I h 2h R A 2h h : Tri-linear interpolation h I2h T R A I 2h 2h h h h 2h Restriction is transpose of interpolation Galerkin-based coarsening Fine Grid Coarse Grid 1 1/2 1/2 1 1/2 1/2 Interpolation 1 1 1/2 1/2 1 1/2 1/2 Restriction 1

9 Biomechanical finite element simulations Performance (floating point operations) Model Simulation-Steps / second ,000 Elements 291,000 Vertices Each time step includes the re-assembly of the per-vertex equations (simulation level + coarse grids) for the co-rotational strain formulation and 2 multigrid V-cycles, each with 2 pre- and 1 post-smoothing Gauss-Seidel steps

10 Topological changes of the simulation domain Cutting and fracturing

11 Topological changes of the simulation domain Linked volume Face-adjacent elements are connected via links Cutting: links between elements are disconnected fine level Virtual cell 1 coarse level Virtual cell 2 Hierarchical representation of cuts

12 Adaptive simulation grid refinements Adaptive linked octree Cutting: refine local elements, then break links Regular 1:8 hexahedral decomposition No ill-shaped elements Initial octree Refined octree

13 Smooth boundaries for visualization and collision detection Smooth boundary surfaces from hexahedral simulation grids High resolution render surface bound to the simulation vertices When topological changes occur, new surface parts have to be extracted Using marching cubes Using splitting cubes Using dual contouring Surface reconstruction after cutting by different methods

14 Smooth boundaries for visualization and collision detection Smooth boundary surfaces from hexahedral simulation grids When topological changes occur, new surface parts have to be extracted Duplication of surface vertices when cuts occur

15 Smooth boundaries for visualization and collision detection Adaptive and smooth boundary surfaces tetrahedral grid 166,000 elements Hexahedral grid 141,000 elements

16 Convergence and scalability Convergence in complicated domains

17 Patient-specific hip-joint implant planning The implant should replicate the preoperative stress state Otherwise, stress shielding occurs Stresses are bypassed by the implant Adaptive bone remodeling leads to loosening of the implant A planning tool should allow selecting and positioning implants interactively, applying loads and simulating instantly the occurring stresses [Bergschmidt et al., 2009]

18 Patient-specific hip-joint implant planning Trabeculae are aligned along the principal stress directions (Wolff s law) Trabecular bone Cortical bone Cross section X-ray Principal stress directions

19 Patient-specific hip-joint implant planning Interactive workflow Interactive implant positioning Distance visualization for user navigation GPU multigrid FE simulation Linear elasticity model stress tensor field Computation of eigenvector field Integration along major stress direction GPU Rendering Visibility and optical attenuation Color encodes tension and compression

20 Patient-specific hip-joint implant planning

21 GPU-based Biomechanical Simulations and Steering Thank you for your attention