Detecting Topological Changes in Dynamic Delaunay Triangulations - PowerPoint PPT Presentation

Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA GTC 2017 Speaker: Dr. Matteo Lulli Prof. M. Bernaschi and Prof. M. Sbragaglia May the 10th, 2017 Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 1 / 26

Outlook Motivation and Definitions 1 Physical system and Simulations 2 Algorithm Description 3 Physics Results 4 Conclusions and Perspectives 5 Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 2 / 26

Outline for section 1 Motivation and Definitions 1 Physical system and Simulations 2 Algorithm Description 3 Physics Results 4 Conclusions and Perspectives 5 Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 3 / 26

Motivation... Many systems are composed of a number of entities which dynamically interact with each other in real continuous space , i.e. not on a lattice... Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 4 / 26

Motivation... Many systems are composed of a number of entities which dynamically interact with each other in real continuous space , i.e. not on a lattice... Pedestrians on the street Birds in a flock Droplets in an emulsion Many others... Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 4 / 26

Motivation... Many systems are composed of a number of entities which dynamically interact with each other in real continuous space , i.e. not on a lattice... Pedestrians on the street Birds in a flock Droplets in an emulsion Many others... How to associate a topology? There are different ways to associate a topology to a given set of points: triangulations, quadrangulations... In the literature it is common to use Voronoi tesselation (dual to Delaunay) Delaunay triangulation (dual to Voronoi) Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 4 / 26

Motivation... Many systems are composed of a number of entities which dynamically interact with each other in real continuous space , i.e. not on a lattice... Pedestrians on the street Birds in a flock Droplets in an emulsion Many others... How to associate a topology? There are different ways to associate a topology to a given set of points: triangulations, quadrangulations... In the literature it is common to use Voronoi tesselation (dual to Delaunay) Delaunay triangulation (dual to Voronoi) Topology changes Many important properties on the dynamics of such sets of points can be studied by analyzing changes occurring in the topology during the time evolution of the system, e.g., topological changes are associated to stress redistribution in emulsions... Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 4 / 26

...and Definitions Commonly used Voronoi tesselation : given the set of points { z i } , the Voronoi cell associated to z i is a convex region of the plane defined as C i = { x ∈ R 2 | | x − z i | < | x − z k | ∀ k � = i } The simplest topology change is the T1 neighbours switching: one shared edge of two adjacent Voronoi cells degenerates in a point Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 5 / 26

...and Definitions Commonly used Voronoi tesselation : given the set of points { z i } , the Voronoi cell associated to z i is a convex region of the plane defined as C i = { x ∈ R 2 | | x − z i | < | x − z k | ∀ k � = i } The simplest topology change is the T1 neighbours switching: one shared edge of two adjacent Voronoi cells degenerates in a point We follow an alternative approach based on the dual of the Voronoi tesselation: the Delaunay triangulaton in which topology changes correspond to changes in the triangulation There can be more complex events in bulk and in the presence of boundaries Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 5 / 26

...and Definitions Commonly used Voronoi tesselation : given the set of points { z i } , the Voronoi cell associated to z i is a convex region of the plane defined as C i = { x ∈ R 2 | | x − z i | < | x − z k | ∀ k � = i } The simplest topology change is the T1 neighbours switching: one shared edge of two adjacent Voronoi cells degenerates in a point We follow an alternative approach based on the dual of the Voronoi tesselation: the Delaunay triangulaton in which topology changes correspond to changes in the triangulation There can be more complex events in bulk and in the presence of boundaries A. Corbetta PhD thesis, “Multiscale crowd dynamics: physical analysis, modeling and applications” Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 5 / 26

Our Approach We propose a unified approach for tracking topological changes occurring in the Delaunay triangulations of the points composing 2D systems We built up a fast CUDA library which can be integrated with already existing simulations or data analysis programs Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 6 / 26

Our Approach We propose a unified approach for tracking topological changes occurring in the Delaunay triangulations of the points composing 2D systems We built up a fast CUDA library which can be integrated with already existing simulations or data analysis programs Features Lattice based algorithm Library interface only needs a i) density field, ii) a threshold, iii) (optional) a map for irregular geometries Arbitrary obstacles geometry handling Periodic and non-periodic boundary conditions handled in both directions Negligible overhead on our Lattice Boltzmann simulations: check for topology changes every ∆ t = 10 2 steps on overall simulation time ∼ 10 7 steps Variable number of generating points... The implementation is composed by several elements that can be individually replaced according to the actual needs Our code can be downloaded from http://twin.iac.rm.cnr.it/dynvorcuda.tgz Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 6 / 26

Our Approach We propose a unified approach for tracking topological changes occurring in the Delaunay triangulations of the points composing 2D systems We built up a fast CUDA library which can be integrated with already existing simulations or data analysis programs Features Lattice based algorithm Library interface only needs a i) density field, ii) a threshold, iii) (optional) a map for irregular geometries Arbitrary obstacles geometry handling Periodic and non-periodic boundary conditions handled in both directions Negligible overhead on our Lattice Boltzmann simulations: check for topology changes every ∆ t = 10 2 steps on overall simulation time ∼ 10 7 steps Variable number of generating points... The implementation is composed by several elements that can be individually replaced according to the actual needs Our code can be downloaded from http://twin.iac.rm.cnr.it/dynvorcuda.tgz Current limitations Lattice based Triangulation relies on a non-compressed adjacency matrix Order ∼ 10 4 points Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 6 / 26

Outline for section 2 Motivation and Definitions 1 Physical system and Simulations 2 Algorithm Description 3 Physics Results 4 Conclusions and Perspectives 5 Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 7 / 26

What are emulsions? Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 8 / 26

What are emulsions? Ensembles of Droplets that do not coalesce - surfactants... Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 8 / 26

What are emulsions? Ensembles of Droplets that do not coalesce - surfactants... Display both solid and liquid features: yield stress σ Y Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 8 / 26

What are emulsions? Ensembles of Droplets that do not coalesce - surfactants... Display both solid and liquid features: yield stress σ Y Solid-like behaviour - elastic response: σ < σ Y γ n , Liquid-like behaviour - flow is non-Newtonian: σ > σ Y , σ ∝ ˙ n � = 1 Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 8 / 26

What are emulsions? Ensembles of Droplets that do not coalesce - surfactants... Display both solid and liquid features: yield stress σ Y Solid-like behaviour - elastic response: σ < σ Y γ n , Liquid-like behaviour - flow is non-Newtonian: σ > σ Y , σ ∝ ˙ n � = 1 Emulsions flow when the droplets configuration undergoes irreversible topological changes: plastic events Comput. Phys. Commun. 213 , 19 (2017) Detecting Topological Changes in Dynamic Delaunay Triangulations Using CUDA 8 / 26