A Topological-Geometrical Pipeline for 3D Cracking-like Phenomena

Jérémy Riffet

, Nicolas Castagne

, Emmanuelle Darles

and Annie Luciani

Univ. Grenoble Alpes, Grenoble INP, Institute of Engineering, ICA, 38000 Grenoble, France

Univ. Poitiers, CNRS, XLIM, UMR 7252, F-86000 Poitiers, France

ACROE, Grenoble, France

Keywords: Computer Animation and Simulation, Fractures, Splits, Cracks, Tears, Physics-based Models, Geometry-free

Approaches, Masses-interaction Networks, Motion Coating, Adjacency Graphs, Topological Modelling.

Abstract: Animation of one-to-many phenomena (fractures, tears, breaks, cracks…) is challenging. This article builds

over recent works that proposed a 3-stages modelling and simulation pipeline, made of a cascade of models:

geometry-free physical model → explicit modelling of the evolving topology → geometrical model. On the

Physics’ side, in the framework of masses-interactions network modelling, the article extends the recent

Splitting-MAT method, where the physical splits occur onto the material points, toward 3 dimensional volume

models. Downstream, it introduces a topo-geometrical pipeline adapted to this upstream split-on-the-masses

property. Experiments, and analysis of the complexity of the topo-geometrical part, show that, while offering

constructible and manageable means, separating Physical, Topological and Geometrical aspects in the 3-

stages pipeline enables a rich variety of one-to-many dynamics, with good efficiency.

1 INTRODUCTION

Fracture, tearing, breaking, cracking, or more

generally one-to-many visual dynamics featuring

topological transformations are attractive, but their

modelling and simulation are challenging. Research

has recently worked on two categories of approaches.

The first approach focuses on extending

“geometry-based” physical methods, such as Finite

Elements, Mass-Spring Meshes or Diffuse Elements

methods, toward fracturing, tearing, breaking, etc. In

such cases, the physical algorithms are embedded into

a geometrical mesh. When a topological

transformation occurs, both the geometrical and

physical models need to be co-transformed with one

another. Managing globally such geometrico-

physical remeshing process is difficult. (Muguercia et

al., 2014) and (Frerichs et al., 2015) offer surveys of

recently proposed solutions and models. In parallel

with the present work, one can also note that it has

recently been proposed to root on an explicit

representation of the topological aspects, with the aim

to manage with more control and systematism the

physico-geometrical remeshing issues (Carter et al.,

2000; Meseure et al., 2010; Fléchon et al., 2013;

Paulus et al., 2015).

In the second category of approaches, to which

this article relates, modelling and computation of the

splittable dynamics root on “geometry-free” (or

“mesh-free”, or “morphology-free”) physics-based

methods, such as: masses-interactions networks

modelling (Jund et al., 2012; Luciani and Godard,

1997), meshless techniques (Zhuang et al., 2012;

Steinemann et al., 2009; Pauly et al., 2005), smoothed

particles hydro-dynamics (Chen et al., 2013), frame-

based simulation (Manteaux et al., 2015), and others.

In these cases, the physical model is not tied to a

pre-existing geometrical mesh, and does not express

matter contiguity. This eases the modelling and

computing of the physical state changes: the physical

model can be thought of, computed, without the

burden of geometrical aspects.

Anyhow, when working with Geometry-free

physics-based approaches, the dynamics is most often

generated as a set of moving points. Consequently, a

geometrical model must be set up to visualize such

punctual movements. In the case of one-to-many

phenomena, the physical state changes must also

control downstream topological transformations

within the geometrical model.

To address this issue, (Zhuang et al., 2012) build

over the principles of level sets. (Pauly et al., 2005)

propose employing small elliptical surfaces called

Riffet, J., Castagne, N., Darles, E. and Luciani, A.

A Topological-Geometrical Pipeline for 3D Cracking-like Phenomena.

DOI: 10.5220/0006552702070214

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 1: GRAPP, pages

207-214

ISBN: 978-989-758-287-5

207

Figure 1: A cascading pipeline for the modelling and

simulation of one-to-many phenomena.

surfels. (Steinemann et al., 2009) and (Chen et al.,

2013) employ an adjacency graph set up over the

moving particles.

Another recent proposal consists in introducing a

model explicitly dedicated to the topological aspects.

Besides first experiments in (Darles et al., 2011),

(Jund et al., 2012) and (luciani et al, 2014) position a

global modelling and simulation pipeline, made of a

cascade of submodels, from upstream to downstream

(Figure 1): a physics-based model → a topological

model → a geometrical model. Employing a central

topological model helps managing efficiently,

systematically and formally the topological

transformation, under control of the physics-based

punctual movements. Also, segmenting the complete

model in layers of submodels makes it possible to tie

each of the models to a clearly delimited part of the

final visual phenomena: splittable dynamics thanks to

the physical model; management and transformations

of the relations thanks to the topological model;

geometry thanks to the geometrical model.

Recently, in the context of Geometry-free

physics-based approaches, (Kalantari et al., 2014)

extended masses-interactions networks modelling, by

introducing the Splitting MAT system. This system

allows the splits to occur directly on the material

points, though without any computational overhead.

However, Splitting MAT have not yet been employed

within the global Physics → Topology → Geometry

pipeline.

This article extends the original pipeline proposed

in (Jund et al., 2012) and related articles to the cases

where Splitting-MAT (Kalantari et al., 2014) are

employed on the upstream Physics’ side. Our

contributions are:

1/ On the physics-based side, we extend the

Splitting-MAT method to the case of splittable three-

dimensional volume models.

2/ Downstream, we introduce a topological-

geometrical coating pipeline adapted to the splitting-

MAT principles in 3D. We explain how the upstream

split-on-the-masses property eases setting up the

topo-geometrical process downstream.

3/ We present various experiments on fractures,

breaks and tears phenomena. We show that the

pipeline allows modelling and simulating a range of

fracturing/cracking/tearing effects with good

efficiency, possibly at interactive frame-rate.

2 THE TOPO-GEOMETRICAL

PIPELINE

This section presents the proposed

physical→topological→geometrical pipeline built

over the Splitting MAT method. Section 2.1

summarizes the principles of the Splitting MAT

physics-based system, and discusses the upstream

physical model with Splitting MAT. Section 2.2

covers the data exported from the physics toward the

topological, then geometrical stages. Section 2.3

provides details on the G-Map topological system we

employ. Section 2.3 covers the topo-geometrical

pipeline downstream: as originally proposed in (Jund

et al. 2012), we present successively the

Construction, Association, Modification and

Affectation steps.

For more clarity, throughout this section,

explanations are first based on an exemplary 2D

model, which is globally summarized on Figure 3. In

each paragraph, we briefly explain how the pipeline

can be evolved to 3D volume models. The

experimental models presented in section 3 are all 3D

volume models.

2.1 Physical Model with Splitting MAT

The Splitting-MAT methodology extends the

possibilities of masses-interactions networks

modelling in regards to one-to-many phenomena

(fractures, breaks, tears…), by enabling the splits to

occur directly onto the MATerial points – the masses

of the network. The method has a fully constant

algorithmic complexity, and guarantees by

construction the stability of the physics, no matter

how the model evolves (break, tear, split…) during

simulation. (Kalantari et al., 2014) provides details on

the system and its stable computing cost.

Modelling with splitting–MAT starts by defining

the smallest possible physical entities, corresponding

to the fully-split state, by interconnecting some

masses with physical interactions. Then, masses of

various entities are united into Mass-unions. A Mass-

union is created by tying its masses one to another

with Duplets. As long as a duplet remains active, the

2 tied masses remain in the same Mass-union, and

will keep the same exact behaviour: same position,

same speed. Each Duplet is associated with a Sensor

which, when triggered, inactivates the Duplet.

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Depending on the emerging phenomena, duplets may

be progressively inactivated, leading to Mass-unions’

splits. Hence, masses may progressively gain their

autonomy, possibly down to the fully-split state.

When working with 2D models, each elementary

physical entity could be made of 3 or 4 masses. Figure

3 (a, b, c, left side) explain how the physical model

could be built while modelling, and split during

simulation, in the case of a very simple exemplary 2D

model made of only 4 elementary physical entities, 13

masses, and 5 Mass-unions.

In this work, on the physics-based side, we extend

the Splitting-MAT method to 3D volume models –

where only surfacic models where presented in

(Kalantari et al, 2014). To work in 3D, each

elementary physical entity can be made of 8 masses,

and various interactions, that form a hexahedral

physical entity (Figure 4a). Then, up to 8 masses

taken from 8 different smallest physical entities are

tied into initial Mass Unions, by using a minimum of

7 duplets (Figure 4b). During simulation, the splitting

process is then exactly the same as in 2D, even though

each mass has a 3D position and velocity.

2.2 Data Passed to the

Topo-geometric Stage

In our proposed pipeline, only phenomenological data

generated from the physics-based model are passed to

the topo-geometrical level. Two categories are

considered in this work.

The first consists in the punctual movements of

each of the masses of the physical model, sampled at

the physics’ simulation frequency, no matter they are

gathered in Mass-unions or not. Each of these

punctual movements is called an evolution function

(Luciani et al., 2014).

The second consists in the state of the Mass-

unions. This data stream is new as compared to

(Luciani et al., 2014), which did not employ Splitting

MAT. The Mass-unions’ state data stream is event-

based: whenever a Mass-union splits in the physics,

the indexes of the masses forming the newly created

Mass-unions are passed to the downstream model

along with the date of the event.

2.3 Adjacency Graphs Fundamentals

In order to handle easily and formally topological

constructs and modifications in large volume sets, we

employ a structure that stores cells adjacencies. There

exist many graph models in the literature, such as for

example half-edges graphs.

Figure 2: Two darts sewed by α

link form an edge (black

lines). Several edges sewed by α

links (red curves lines)

form a face. Several faces sewed by α

links (green large

dotted lines) form a volume. Several volumes sewed by α

links (blue thin dotted lines) form one connected

component.

In this work, we employ the generalized map

formalism (Lienhart, 1994). Each cell in dimension N

(N>0) is created by sewing different N-1-D cells to

obtain a N-D cell. Hence, a 1D cell (topological

vertex) is created by sewing 0D elementary cells

called “darts”. Sew operations are mathematically

defined as bijective functions 

(with i the dimension

of the sew operation) called involutions. The N-D

topological cells are the nodes of the adjacency graph,

and the involutions represent its edges. Topological

cells, such as topological vertices, edges or faces, then

correspond to a set of darts that are sewed with each

other’s with chosen involutions, called orbit (Figure

2).

Interestingly, the system enables finding any

adjacency relations quickly and automatically, by

using simple graph scanning. Additionally, it builds

on generic principles to guarantee consistency and

coherency of the topology during construction, and

whenever performing any topological modification.

2.4 Topo-geometrical Pipeline

Downstream Physics, to finally obtain a visible

evolving geometry, the proposed topo-geometrical

pipeline roots on 4 steps: in first, during the modelling

stage, construction and association steps. Secondly,

during the simulation stage, modification and

affectation steps.

2.4.1 Construction

The first step, called Construction, consists in

building an initial topology, which will be the core of

the entire topo-geometrical pipeline and will be

evolved during simulation.

As compared to (Kalantari et al, 2014), employing

Splitting MAT in the Physics makes it possible to

build a simple base topology. This is a important

advantage as compared to previous works on the 3

stages pipeline that did not employ Splitting-MATs.

A Topological-Geometrical Pipeline for 3D Cracking-like Phenomena

209

This base topology can be obtained simply by

employing a building process similar to the physical

model’s building process.

Figure 3a shows how the base initial topology

obtained in the case of simple use-case 2D model. It

is made 4 topological polygonal faces, sewed with

each other to form a single large polygonal surface.

Hence, each 2D elementary physical entity

corresponds to a polygonal face.

When working with 3D volume models, the

building of the base initial topology follows the same

process. Though, instead of leading to polygonal

faces, it results in tetrahedral topological volumes,

sewed in α

. Each of these volumes corresponds to an

elementary entity in the physical model. Noticeably,

besides the base topology discussed in this article, it

would be equally possible to experiment with other

topologies (e.g. refined).

2.4.2 Association

Association consists in bijectively associating each

evolution function to one or several elements in the

topological structure (darts, orbits, etc.).

Employing Splitting-MATs on the physical level

enables a cunning association strategy, as compared

to previous works without Splitting MATs. This

association is achieved by traversing the topological

structure in the same order the physical model was

built. During this scan, the evolutions functions are

associated one after another to a selected orbit in the

topological model.

In 2D, the proposed association roots on the

notion of face’s corner, topologically defined as the

set of darts in the orbit< α

> of a chosen dart. Figure

3b illustrate the resulting Association in the case of

the simple 2D exemplary model. Each evolution

function (each moving material point) is associated

with a single face’s corner orbit in the topological

structure. For example, the topological vertex in the

centre of the topological model is made of 4 face’s

corners, each one gathering 2 darts. These 4 face’s

corners are associated to the 4 corresponding

evolution functions output from the physics model. It

should be reminded that, since masses will keep the

same exact position as long as they remain in the same

Mass-Union, the corresponding face’s corners will be

associated downstream to this unique position, until a

split occurs in the Physics.

The proposed association strategy extends rather

simply to 3D, by considering the notion of volume’s

corner, instead of face’s corner. A volume’s corner is

topologically defined as the set of darts in the orbit<

> of a chosen dart. Hence, as we use tetrahedral

volumes in the Experiment section, we simply have

to associate each evolution function with a single

volume’s corner (instead of a face’s corner).

In G-Map data structure, storing associations in

the topology is achieved by storing the evolution

function’s index in the orbit. Constant time access is

achieved from any dart of the orbit to the value (i.e

position) of the associated evolution function.

2.4.3 Modification

During simulation, the Modification step consists in

progressively transforming the topological model, to

implement topologically fractures and splits under

control of the Physics. As compared to (Luciani et al.,

2014), employing splitting MATs upstream enables a

more direct control of the topological

transformations, thanks to the events received

whenever a Mass-union splits in the Physics. In the

case of our 2D simple exemplary model, the

modification process would simply consist in

unsewing from each other’s the face’s corners which

ids are not any more in the same Mass-union (figure

3c). In this process, the GMap’s implementation

automatically maintains the consistency of the

topology, so avoid non-manifold topology. As a

consequence, in 2D, unsewing two face’s corners

ultimately, and automatically, results in separating

topological edges.

When working with 3D volume models, a Mass-

union splits results in separating the two

corresponding volume’s corners (instead of face’s

corners). Even in the third dimension, the GMap’s

implementation ensures the consistency of the

topology. Consequently, unsewing the 2 volume’s

corners automatically unsews all the α

links between

the adjacent darts of these corners. This results in

separating faces of two adjacent topological volumes.

We can retrieve in constant time the involutions to

unsew in response to a split event received from the

physical model by the simple correspondences

between the upstream physical masses and the

topological face’s corners in 2D (or volume’s corners

in 3D) of the base topology.

2.4.4 Affectation and Geometrical Model

During simulation, Affectation, consists in setting up

on each rendering step a visible geometry, by

embedding geometrically the topological map. With

this step, the one-to-many phenomena are finally

shaped to the eye.

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Figure 3. Simple exemplary 2D model illustrating the topo-geometrical Pipeline’s Steps. (a) Construction process. Left

figure: elementary physical entities, made of masses (yellow circles) linked by interactions (dot lines), and Mass-Unions. In

the centre of the model, 3 duplets (red dot ellipses) initially gather 4 masses (yellow circles) taken from 4 different elementary

physical entities into a single Mass-union (black ellipses). As long as these 3 duplets remain active, these 4 masses will share

the same exact positions while computing. Central figure: while Mass-unions remain unsplit, all the masses gathered in each

Mass-union behave as a single mass (orange circle). Right figure: the construction of the base topological structure inspired

by the physics network. (b) Association. Association is based on the order of mass’s placements (numbers) in the physics

model. (c) Modification. In case the duplet between the mass 5 and 8 is inactivated by its Sensor, then the Mass-union splits

into two Mass-unions: some masses previously tied are now separated. This information are used in the topological model to

unsew in α

the two associated faces’ corners (face’s corner 2 with 11, and face’s corner 5 with 8). (d) and (e) Affectation

step. Two embedded geometry obtained from the topological structure in the course of the simulation. The first is the

geometry of the topological model before the split, and the second is the geometry of the topologic model after the split. Only

the edges that are free in α

(red large lines) are rendered in the geometric model.

The position given to each geometrical vertex can

be the current value of the evolution function

associated to any of the sewed face’s corner forming

the topological vertex.

To exemplify, in the case of our 2D simple model,

in figure 3d before the split and 4e after the split, we

choose the following geometrical embedding: each

topological vertex becomes a geometrical vertex,

each unsewed topological edge (without α

involution, in red) becomes a geometrical edge, etc.

Hence, topological faces sewed with each other’s are

rendered as a single connected geometrical face.

When working with 3D volume models, the

process is similar. We just have to take into account

the third dimension’s topological elements: the

topological volumes. When several topological

volumes are sewed, only the unsewed faces are

rendered, so as to form a single large connected

A Topological-Geometrical Pipeline for 3D Cracking-like Phenomena

211

(a)

(b)

Figure 4: (a) Elementary physical tetrahedral entities with,

in blue, constraint viscoelastic interactions. (b) The set of 7

duplets, that initially gathers 8 masses taken from 8

different elementary entities into a single Mass-union. The

experimental model gathers 70x70x10 elementary physical

entities.

component. This means that several sewed

topological volumes in α

are rendered as a single

geometrical volume. This is made very easy thanks to

the topological model: the faces of such connected

components are simply those which α

involutions are

free. More complex geometrical embedding are

possible, for example, one could also choose to

embed another, possibly more complex, geometry,

such as for example: adding geometrical vertices on

the centre of each geometrical face; creating

geometrical vertices in the centre of each topological

volume; etc.

3 EXPERIMENTS AND

COMPLEXITY

For the following experiments, the Physics network

upstream is made of 70x70x10 hexahedral basic

physical entities. Each entity, corresponding to the

fully-split state, is made of 8 masses, and 16

interactions, forming a hexahedral physical entity

(Figure 4a). Then, these entities are tied to each

other’s with Duplets (Figure 4b), so as to form a large

3D physics-based splittable block. To trigger duplet’s

inactivation, we employ distance Sensors mounted

between two masses chosen in adjacent entities

outside the Mass-union. In this article, from an

experiment to another, only the physical parameters

values, and the initial state, are modified.

On the topological side, the base topological

model is made of 70x70x10 topological cubic

volumes that are sewed with each other’s to form a

single large parallelepiped block (Figure 5).

The two animations shown on Figure 6a and

Figure 6b were both achieved with the same exact

physical, topological and geometrical models. The

physical model stands for a slightly deformable

matter, in which Sensors’ thresholds are chosen non-

homogeneous, so as to spread various cracking lines

Figure 5: The 3D volume topological model is inspired

from the building process employed in the physics’ model:

70x70x10 topological cubic volumes are sewed to each

other’s.

in the matter. In between the two animations, only the

initial state of the physical model differs. Depending

on this initial state, cracks emerge and propagate at

various places in the simulation.

The example shown on Figure 7 illustrates

possibility of thoroughly different effects with the

pipeline. In the physical model, the parameters are set

to achieve a very deformable matter behaviour – like

a thin sheet or soft body. Then, the Sensors’

thresholds controlling Duplets inactivation are

chosen inhomogeneous over the model: the distance

thresholds are made smaller along a vertical line close

to the centre of the model. Hence, we favour a chosen

tear propagation line in the model.

The videos associated with this paper

(http://147.171.151.195:8080/fbsharing/LTWswZVk)

provide other examples illustrating further variability

in the obtainable behaviours and renderings

In all these experiments, the physical model is run

at 1050Hz: employing Splitting MAT leads to a fully

stable complexity, no matter the occurring splits. The

topo-geometrical part of the pipeline is run at 50 Hz,

and we achieved interactive framerate for all our

experiments on a standard PC.

The algorithmic complexity of the Modification

step starts by analysing the received Splitting-Mat

events. For each newly created sub-Mass-union, a

local scan of orbit<α1 , α2> is performed to determine

which volume’s corners should be unsewed from

each other’s. The complexity is O(n*m*(n+2k)), with

n the number of masses in the new Mass-union, m the

number of faces incident on the volume’s corner, and

k the number of darts in each visited face. The number

of split events to process on each step is the number

of splits that occurred in the physics since the last

execution of the topo-geometrical pipeline. Hence, in

case many Mass-unions have split in the physics, the

duration of the topological Modification step might

be penalized. However, separating the physics from

the geometry implies that this might only impact the

stability of the visual framerate, but not at all the

consistency of the physical simulation. As for it, the

memory complexity of the topological model is

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212

(a)

(b)

Figure 6: Emergence of tears in the matter. The same exact physical, topological and geometrical models are used for a) and

b). Depending on the initial state of the physical model, various tears will emerge.

(a)

(b)

(c)

Figure 7: Tearing of a very deformable thin matter bloc. A tearing line is favoured by employing lower Sensor threshold in

the centre of the physical model. (a) Direct representation of the physical model, by using a sphere on each mass. (b)

Representation of the topological model, in which sewed and unsewed faces are rendered with distinct colours, allowing to

pursue the evolving topological modifications. (c) Rendering of the finally obtained geometrical model.

constant throughout simulation, thanks to the use of

the GMaps formalism: the number of darts is never

changed, even when topology is modified.

4 CONCLUSIONS

This paper proposed new contributions to the today’s

stream of research that envisages employing explicit

representations of the topological aspects to root the

coating processes over geometry free physics-based

methods. The general pipeline is made of cascading

models: physical → topological → geometrical.

On the physical side, we extended the Splitting-

MAT methodology, which enables the split to occur

onto the material elements, to 3D volume models.

We then introduced downstream a topologico-

geometrical pipeline adapted to this property of the

upstream physical model. As compared to previous

works on the 3-stages pipeline, which did not employ

Splitting MAT, the split-on-the-masses property

allows building rather simply a base topological

model, and further enables a cunning handling of the

topological transformations of this base structure

during simulation, under control of the Physics.

The obtained results exhibit precision in both the

dynamics and the visual (geometrical) aspects.

Hence, the Splitting MAT-powered 3-stages pipeline,

while being manageable, does not limit the richness

of the desired dynamics (dynamics of the fracture and

splits, propagation, etc.). Finally, a theoretical and

practical measure of the complexity of the

topologico-geometrical part shows that the pipeline

competes with integrated approaches and qualifies for

real time implementations.

In the future, we plan to creatively experiment

with more diverse constructs. Indeed, handling

physics (dynamics) / topology (spatial relations) /

A Topological-Geometrical Pipeline for 3D Cracking-like Phenomena

213

geometry (final image) in 3 clearly separated

cascading models introduces manageability in the

modelling processes, and variability in the

observation of the physically generated dynamics.

Hence, besides employing the base topology and

simple geometry, as presented in this article, the three

models may indeed drastically differ in their

structure, and in their complexity. We plan to build

over this advantage to experiment with varied

physical models upstream, and varied topological

constructs and geometrical renderings for each of

them: non-regular physical models, refined

topologies, diverse geometrical embedding, etc.

Rooting on formal approaches on both the physical

and topological sides will ease such future

explorations, we assume.

AKNOWLEDGMENTS

We thank to Saman Kalantari for the core 3D volumic

algorithms of the Splitting MAT system. Thanks to

Dr Lilian Aveneau for his help for rendering

animation. ACROE and ICA are supported by the

French Ministry of Culture.

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