Controlling Anisotropy in Mass-Spring Systems

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DOI: 10.1007/978-3-7091-6344-3_9
Abstract
This paper presents a deformable model that offers control of the isotropy or anisotropy of elastic material, independently of the way the object is tiled into volume elements. The new model is as easy to implement and almost as efficient as mass-spring systems, from which it is derived. In addition to controlled anisotropy, it contrasts with those systems in its ability to model constant volume deformations. We illustrate the new model by animating objects tiled with tetrahedral and hexahedral meshes.
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Controlling Anisotropy
in Mass-Spring Systems
David Bourguignon and Marie-Paule Cani
iMAGIS-GRAVIR/IMAG-INRIA
iMAGIS is a joint research project of CNRS/INRIA/UJF/INPG
E-mail:
David.Bourguignon Marie-Paule.Cani @imag.fr
http://www-imagis.imag.fr/
Abstract. This paper presents a deformable model that offers control of the
isotropy or anisotropy of elastic material, independently of the way the object is
tiled into volume elements. The new model is as easy to implement and almost
as efficient as mass-spring systems, from which it is derived. In addition to con-
trolled anisotropy, it contrasts with those systems in its ability to model constant
volume deformations. We illustrate the new model by animating objects tiled
with tetrahedral and hexahedral meshes.
1 Introduction
Mass-spring systems have been extensively used in Computer Graphics over the last fif-
teen years, and are still very popular. Easier to implement and faster than finite element
methods, these systems allow animation of dynamic behaviors. They have been ap-
plied to the animation of inanimate bodies such as cloth or soft material [17, 10, 15, 5]
and to the animation of organic active bodies such as muscles in character anima-
tion [11, 1, 12].
Both isotropic and anisotropic elastic materials can be found among the objects
to animate. For instance, a rubber toy is isotropic, while most natural objects (animal
organs, plants) are strongly anisotropic, due to their fiber structure and/or the composite
materials they are made of. One of the main drawbacks of mass-spring systems is
that neither isotropic nor anisotropic materials can be generated and controlled easily.
Another problem is that most of the materials found in nature maintain a constant or
quasi-constant volume during deformations (this is well known for muscles, but also
holds for inanimate materials). Mass-spring models do not have this property.
1.1 Background
Animating an elastic object using a mass-spring system usually consists of discretizing
the object with a given 3D mesh, setting point masses on the mesh nodes and damped
springs on the mesh edges. Then, most implementations simply integrate point dynam-
ics equations for each mass from the set of applied forces due to the mesh deformation
at the previous time step [10].
Well-known advantages of mass-spring systems include their ability to generate
dynamic behaviors, while finite elements methods are generally used in the static case,
and their ability to handle both large displacements and large deformations.
Among the intrinsic limitations of mass-spring systems, one of the main problems is
parameter setting. Computing the masses in ordertoset up a homogeneousmaterial can
be done by computing each mass according to the volume of the Voronoi region around
it [4]. However, there is no easy solution for spring parameters. Since damped springs
are positioned along the edges of a given volume mesh, the geometrical and topological
structure of this mesh strongly influences the material behavior. A consequence of this
problem is that changing the mesh density during the simulation while maintaining the
same global mechanical properties is very difficult [7].
If all springs are set to the same stiffness, the mesh geometry may generate unde-
sired anisotropy, as shown in Fig. 1.a. The undesired behavior disappears when hexa-
hedral elements aligned with the forces directions are used (Fig. 1.b). Of course, if the
tiling of the object volume was computed from the triangulation of random uniformly-
distributed sample points, the unwanted anisotropy problem would tend to disappear
when the density of the mesh increases. However, using an extremely dense mesh
would reduce efficiency.
Approximating a desired behavior using a given mesh can be achieved, as in [4, 9],
by using optimization to tune individual spring stiffnesses. This technique could be
used, in theory, for generating both isotropic and anisotropic behaviors. However, due
to the large computational cost, this method has only been tested in the 2D case [4].
The most common approach to control the behavior of a mass-spring system, at
least along a few “directions of interest”, is to specifically design the mesh in order
to align springs on these specific directions, such as in Fig. 1.b. This was done for
instance in Miller’s “snakes and worms” models [11] and in the muscle modelofNg and
Fiume [12], where some of the springs were aligned with the muscle fibers and the rest
were set perpendicular to them. Unfortunately, manually creating such meshes would
be time consuming in the general case, where fiber directions generating anisotropy
vary in an arbitrary way inside the object. We are rather looking for an approach that
uses a 3D mesh obtained, for example, with a commercial meshing package (such as
GHS3D [16]) fed with a 3D surface mesh, and still displays the deformable model
behavior, with specified properties in specific directions.
shear spring
point mass
structural spring
a b c d
Fig. 1. Mass-spring systems drawbacks. At left, comparison between two meshes undergoing a
downward pull at their bottom end while their top end is fixed. We observe undesired anisotropy
in the tetrahedral mass-spring system (a), but not in the hexahedral mesh with springs aligned
in the gravity and pull force directions (b). At right, equilibrium state of a cantilever beam,
which left end is fixed, under force of gravity (c). All things being equal, the mass-spring system
considered (tetrahedral mesh) is unable to sustain flexion, as opposed to our model (Fig. 5.c).
The spring configurations used for tetrahedral and hexahedral meshes are given in (d).
1.2 Overview
This paper presents an alternative model to classical mass-spring systems that enables
one to specify isotropic or anisotropic properties of an elastic material, independently
from the 3D mesh used for sampling the object. The approach we use is still related
to mass-spring systems, in the sense that we animate point masses subject to applied
forces. However, the forces acting on each mass are derived from the anisotropic be-
havior specified for each of the volume elements that are adjacent to it.
Since there are no springs along the mesh edges, the geometry and topology of the
mesh do not restrict the simulated behavior. Moreover, constant volume deformations
can be obtained easily, by adding extra forces. We illustrate this on both tetrahedral and
hexahedral meshes. Our results show that computation time remains low, while more
controllable behaviors are achieved.
2 Modeling Anisotropy
Our aim is to specify the mechanical properties of the material independently from
the mesh geometry and topology. In usual mass-spring systems, internal forces acting
inside the material are approximated exclusivelyby forces acting along the edges of the
mesh (i.e. along the springs). This is the reason for the undesired anisotropy problem
described earlier, and for the difficulty in specifying desired anisotropic properties.
The basic idea of our method is to let the user define, everywhere in the object,
mechanical characteristics of the material along a given number of axes corresponding
to orientations of interest at each current location. All internal forces will be acting
along these axes instead of acting along the mesh edges. For instance, in the case of
organic materials such as muscles, one of the axes of interest should always correspond
to the local fiber orientation.
Since the object is tiled using a mesh, axes of interest and the associated mechanical
properties are specified at the barycenter of each volume element inside the mesh. We
currently use three orthogonal axes of interest. The possible use of a larger number of
axes will be discussed in Section 6.
2.1 General Scheme
During deformations of the material, the three axes of interest, of given initial orien-
tation, evolve with the volume element to which they belong. In order to be able to
know their position at each instant, we express the position of the intersection point of
one axis with one of the element faces as a linear combination of the positions of the
vertices defining the face. The corresponding interpolation coefficients are computed
for each face in the rest position (see Figures 2 and 4).
Given the position of the point masses of a volume element, we are thus able to
determine the coordinates of the six intersection points and consequently the three axes
that constitutes the local frame, up to the precision of our linear interpolation.
From the deformation of the local frame, we can deduce the resulting forces on
each intersection point. Then, for a given face, we can compute the force value on each
point mass belonging to this face by “inverse” interpolation of the force value at the
intersection point. The interpolation coefficients previously defined are therefore also
considered as weighting coefficients of the force on each point mass.
2.2 Forces Calculations
Damped springs with associated stiffness and damping coefficients are used to model
stretching characteristics along each axis of interest. In order to specify shearing prop-
erties, angular springs are added between each pair of axes. Rest lengths and rest angles
are pre-computed from the initial position of the object that defines its rest shape. The
equations we use for these springs are detailed below.
Axialdamped spring. The spring forces f
1
and f
2
between a pair of intersection points
1 and 2 at positions x
1
and x
2
with velocities v
1
and v
2
are
f
1
k
s
l
21
r k
d
˙
l
21
l
21
l
21
l
21
l
21
f
2
f
1
where l
21
x
1
x
2
, r is the rest length,
˙
l
21
v
1
v
2
is the time derivative of l
21
, ks
and kd are respectively the stiffness and damping constants.
Angular spring. The spring forces
f
1
f
2
and f
3
f
4
between two pairs of intersec-
tion points
1 2 and 3 4 are
f
1
k
s
l
21
l
43
l
21
l
43
c
l
43
l
43
f
2
f
1
f
3
k
s
l
21
l
43
l
21
l
43
c
l
21
l
21
f
4
f
3
where l
21
x
1
x
2
and l
43
x
3
x
4
, c is the cosine of the rest angle between l
21
and
l
43
, ks is the stiffness constant.
Here, two approximations are made: first, we assume a small variation of the angle
and take the variation of the angle’s cosine instead; second, we consider it sufficient
to use as unit vector the other vector of the pair, instead of a vector normal to the one
considered, in the plane where the angle is measured. These two approximations gave
good results in practice. Furthermore, we found no necessity to use damped angular
springs.
3 Application to Tetrahedral Meshes
Many objects in Computer Graphics are modeled using triangularsurface meshes. Gen-
erating a 3D mesh from such a description, using tools like GHS3D [16] yields to tetra-
hedral volume meshes. This section details our method in this case.
Fig. 2 depicts a tetrahedral element, with the associated frame defining the three
axes of interest. We express the position x
P
of point P as a function of the positions of
vertices A, B and C of the given face, using barycentric coordinates:
x
P
α x
A
βx
B
γ x
C
(e.g. if α
1 and β γ 0, we get x
P
x
A
). Therefore, a force f
P
applied to point P
is split into forces αf
P
, βf
P
and γf
P
, respectively applied on points A, B and C.
We can note that since the elementary volume has four faces, and since there are
three axes of interest defining six intersection points, two such points may lie on the
same face of the volume. This has not been problematicin practice, since forces applied
on mesh nodes are correctly weighted.
P
α
BA
C
γ
BA
C
β
Fig. 2. Tetrahedral element. A point mass is located at each vertex. A local frame is defined
at the barycenter of the element (left). Each axis is characterized by the barycentric coordinates
α, β and γ (with α
β γ 1) of its two intersection points (right, for a given face). These
coordinates are easily obtained using an area ratio.
3.1 Volume Preservation
Animating constant volume deformations with a classical mass-spring system is not
straightforward. For these systems, forces are only applied along the edges of each
volume element, while maintaining a constant volume basically requires adding radial
forces or displacements, as shown by Promayon et al. [14].
To simply ensure volume preservation, we propose a volume force formulation
adapted to tetrahedral volume element. It is loosely related to soft volume-preservation
constraint of Lee et al. [8].
Let us define x
B
the position of the barycenter of the tetrahedral element, with
x
B
1
4
3
i
0
x
i
where x
i
is the position of the ith vertex. Then, we define the force applied on the jth
vertex as
f
j
k
s
3
i
0
x
i
x
B
3
i
0
x
i
x
B
t 0
x
j
x
B
x
j
x
B
where k
s
is the constraint stiffness and
3
i
0
x
i
x
B
t 0
is the rest length of this “vol-
ume spring”. It was not necessary to add damping forces with this constraint.
This method gavesatisfactory results in pratice, since we get less than 1
5% volume
variation in our experiment (see Fig. 3), but results depend on the material parameters
chosen and the type of experiment conducted. In applications where these volume
variations are considered too high, volume preservation could be enforced directly as a
hard constraint like in Witkin’s work [20, 19].
4 Application to Hexahedral Meshes
The use of hexahedralmeshes is not as common as tetrahedral ones, since the geometry
they can define is more limited. However, these meshes may be useful for animating
objects modeled using voxels [2]. This kind of data, with information about material
characteristics specified in each voxel (possibly includinganisotropy), may be provided
by medical imaging applications.
Applying the general method presented in Section 2.1 to hexahedral meshes is
straightforward. Fig. 4 depicts an hexahedral element, with the associated frame defin-
ing the three axes of interest. We express the position x
P
of point P as a function of
-14
-12
-10
-8
-6
-4
-2
0
0 5 10 15 20 25
Volume relative variation (in %)
Time (in s)
Fig. 3. Volume preservation experiments using the same tetrahedral mesh lying on a table under
force of gravity. In our model, one axis of interest is set to the vertical direction (the direction of
application ofgravity) and thetwo others in horizontal directions. Parametersare chosen identical
along the 3 axes. The same stiffness and damping values are used in all experiments. Bottom
graph: our model without volume preservation forces. Middle graph: equivalent mass-spring
system. Top graph: our model with volume preservation forces.
the positions of vertices A, B, C and D of the given face, using bilinear interpolation
coordinates:
x
P
ζη x
A
1 ζ ηx
B
1 ζ 1 η x
C
ζ 1 η x
D
(e.g. if ζ
1 and η 1, we get x
P
x
A
). Therefore, a force f
P
applied to point P
is split into forces ζηf
P
,
1 ζ ηf
P
, 1 ζ 1 η f
P
and ζ 1 η f
P
, respectively
applied on points A, B, C and D.
Here, there is only one intersection point per face of the volume element. Since
the element has eight vertices, the system is under-constrained instead of being over-
constrained, as in the tetrahedral case. As a consequence, each elementary volume may
have several equilibrium states, corresponding to the same rest position of the three
axes of interest but to different positions of the vertices, if volume preservation forces
are not applied.
P
D C
BA
C
η
ζ
D
A B
Fig. 4. Hexahedral element. A point mass is located at each vertex. A local frame is defined
at the barycenter of the element (left). Each axis is characterized by the bilinear interpolation
coordinates ζ and η (with 0
ζ 1 and 0 η 1) of its two intersection points (right, for a
given face).
4.1 Volume Preservation
Given the characteristics of hexahedrongeometry, we use aslightly differentexpression
for volume preservationforces, while keeping the idea of employing a set of forces that
act in radial directions with respect to the volume element. This formulation is also
loosely related to soft volume-preservation constraint of Lee et al. [8].
Let us define x
B
as the position of the barycenter of the hexaedral element, with
x
B
1
8
7
i
0
x
i
where x
i
is the position of the ith vertex.
Then, we define the force applied on the jth vertex as
f
j
k
s
l l
t 0
k
d
˙
l
l
l
l
l
l x
j
x
B
˙
l
v
j
v
B
where v
j
and v
B
are respectively velocities of the jth vertex and barycenter,
˙
l is the time
derivative of l, ks and kd are respectively the stiffness and damping constants. This is
the classical formulation for a damped spring tying the jth vertex to the barycenter (see
Section 2.2).
5 Results
All the experiments presented in this section have been computed by setting point
masses to the same value. Thus, objects sampled using tetrahedral meshes are generally
heavier than those sampled using hexahedral meshes. Moreover, objects are slightly in-
homogeneous in the former case, since mesh nodes are not evenly distributed. Better
results would be obtained by computing the mass values according to the density of
the simulated material and to the volume of the Voronoi region associated with each
point mass, as was done by Deussen et al. [4]. However, we found the results quite
demonstrative as they are.
Numerical simulation of all experiments was achieved using Stoermer’s explicit
integration method [13] with no adaptive time step, and therefore might be improved.
Each figure depicts outer mesh edges and one of the three axes of interest inside
each elementary volume. In Fig. 6 this axis represents the orientation along which the
material is the stiffest.
5.1 Comparison with Mass-Spring Systems
The same experiments as in Fig. 1 are performed using our model instead of a classic
mass-spring system (see Fig. 5). Here, one axis of interest is set to the vertical direction
(the direction of application of gravity and pull forces) and the two others in horizontal
directions. The same stiffness and damping values are used in each direction.
5.2 Controlling Anisotropy
A set of experiments with different anisotropic behaviors is presented in Fig. 6. It is
interesting to notice that isotropic material can be modelled using a random orientation
for the stiffest axis in each volume element.
a b c
Fig. 5. Experiments similar to those of Fig. 1, but computed with our model. As expected, we
do not observe undesired anisotropy in both the tetrahedral (a), and the hexahedral (b) meshes.
With the same mesh and material parameters as in Fig. 1, our tetrahedral model is perfectly able
to sustain flexion, as shown by its equilibrium state (c).
5.3 Performance Issues
Our benchmarks are on an SGI O2 workstation with a MIPS R5000 CPU at 300 MHz
with 512 Mbytes of main memory. Experiments use tetrahedral and hexahedral meshes
lying on a table under force of gravity. Other conditions are similar to those of volume
preservation experiments (see caption of Fig. 3). Note that material stiffness strongly
influences computation time since we use an explicit integration method.
Maximum number of springs per element. For a classical mass-spring system, a
tetrahedral element has 6 structural springs along its edges, and an hexahedral element
has 12 structural springs along its edges plus 4 shear springs along its main diagonals.
We do not use bending springs between hexahedral elements, as in Chen’s work [2].
This has to be compared with 3 axial springs, 3 angular springs and 4 volume springs
(undamped), that gives approximately 10 springs for our tetrahedral element, and 3
axial springs, 3 angular springs and 8 volume springs, that gives 14 springs for our
hexahedral element.
We can conclude from the results displayed in Table 1 that simulating anisotropic
behavior and ensuring volume preservation are not very expensive in our model. These
properties make it suitable for interactiveapplications. However, the cost of our method
is directly related to the number of elements. Thus, unlike mass-spring systems, our
benchmark experiment using the tetrahedral mesh is slower than the one using the hex-
ahedral mesh.
Masses Elements Springs Sp./Elt. Time (in s)
Ms.-Sp. Sys. Tetra 222 804 1175 1.461 0.129
Hexa 216 125 1040 8.320 0.117
Our Model Tetra 222 804 8040 10 1.867
Hexa 216 125 1750 14 0.427
Table 1. Benchmarks results for classical mass-spring system and our model with tetrahedral
and hexahedral meshes. See explanations in the text concerning the estimated number of springs
per element in our model. Legend: Ms.: mass, Sp.: spring, Elt.: element, Time: time spent to
compute one second of animation, with a time step of 0.01 s.
a b c
d e f
Fig.6. Different anisotropic behaviors were obtained using the same tetrahedralmesh undergoing
a downward pull at its bottom end while its top end is xed. Anisotropy is tuned by changing
the stiffest direction in the material. This direction is: (a) horizontal (as a result, the material
tends to get thinner and longer), (b) diagonal (with angle of
π
4
, which constrains the material to
bend in this manner), (c) hemicircular (as a C shape, which causes a snake-like undulation of the
material), side (d) and top view (e), concentric helicoidal (the material successively twists and
untwists on itself) and finally (f) random (the material exhibits an isotropic behavior).
6 Conclusion and Future Work
We have presented an alternative formulation for mass-spring systems, where aniso-
tropy of a deformable volume is specified independently from the geometry of the un-
derlying mesh. There are no requirements for the mesh, that may be built from either
tetrahedral or hexahedralelements. Moreover, a method for generatingconstant volume
deformations is provided.
The new model stays very close to mass-springs systems, since it is as easy to
implement and almost as efficient in computation time. It also benefits from the ability
of mass-spring systems to animate large deformations and large displacements.
Further investigations are needed in order to validate our model. In particular, we
are planning to study the equivalent stiffness along orientations that do not correspond
to axes of interest. Once this is done, we may be able to generalize the method to
anisotropic material where more than three axes of interest are defined.
Other interesting possibilities arise by combining different volume element types to
obtain an hybrid mesh which better approximates the shape of the object; or by using
a b
Fig. 7. Two examples of complex anisotropic materials. In (a), angular cartographies of the
muscle fiber direction obtained on a human heart (at left, map of the azimuth angle, at right, map
of the elevation angle). In (b), a human liver with the main veinous system superimposed.
elements of different orders (linear vs quadratic interpolation, etc.) in the same mesh.
On the application side, we are currently working on human heart motion simu-
lation. This is a challenging problem since the heart is an active muscle of complex
geometry, where anisotropy (caused by muscle fibers varying directions, see Fig. 7.a)
plays an important role [6]. Important work has already been done to measure fiber
direction inside a human heart [18]. We plan to use this data for animating a full scale
organ. To do so, we will have to change our linear axial springs to non-linear active
axial springs, whose stiffness and rest length vary over time.
The human liver is also a good example of anisotropic material, although it has
been previously animated using isotropic elastic models [3]. In fact, it can be seen as a
composite material: the root-like structures of rather rigid vessels are embedded in the
liver tissue itself, which is a soft material (see Fig. 7.b).
Future work finally includes possible generalization to surface materials, such as
cloth. To do so, extra parameters controlling bending will have to be added to the
current volume model.
Acknowledgments
This work was performed within the framework of the joint incentive action “Beating Heart” of the research
groups ISIS, ALP and MSPC of the French National Center for Scientific Research (CNRS). It is partly
supported by the Lipha Sant´e Company, a subsidiary of the group MERCK KGaA. The human liver data
were kindly provided by the Epidaure project and IRCAD. We thank Franc¸ois Faure and Jacques Ohayon for
fruitful discussions; James Stewart and George Drettakis for rereading the paper. Finally, thanks to our first
reviewer for her/his helpful comments.
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Supplementary resources

  • ... This contribution uses the Lenghu fold-and-thrust belt, an example from the Qaidam basin, NE Tibetan Plateau ( Fig. 1) (e.g., Mao et al., 2016;M etivier et al., 1998;Wei et al., 2016;Wu et al., 2011;Yin et al., 2008b;Yin and Harrison, 2000), to demonstrate the bene- fits of three-dimensional geospatial models in providing new in- sights into our understanding, including fold geometry, fault zone architecture, and fault throw distribution. Geomechanical model- ling upon the three-dimensional geospatial models, using a Mass- Spring algorithm (e.g., Baraff and Witkin, 1998;Bourguignon and Cani, 2000;Provot, 1995;Terzopoulos et al., 1987), can also pro- vide an effective three-dimensional tool for advanced structural strain analysis. ...
    ... Here, we used the "Geomechanical Modelling" module within 3D Move of Midland Valley, which provides a workflow-managed three-dimensional restoration tool. The geomechanical modelling, using a Mass-Spring algorithm (e.g., Baraff and Witkin, 1998;Bourguignon and Cani, 2000;Provot, 1995;Terzopoulos et al., 1987), provides an effective three-dimensional tool for model validation and advanced structural strain analysis. The Mass-Spring approach is an established and extensively used technique in the discipline of computer graphics, and it is typically used for modelling real-time deformation of rigid and non-rigid bodies. ...
    ... The geospatial model was then restored in three dimensions to reveal the kinematic evolution of the Lenghu fold-and-thrust belt. Geomechanical modelling, using a Mass-Spring algorithm (e.g., Baraff and Witkin, 1998;Bourguignon and Cani, 2000;Provot, 1995;Terzopoulos et al., 1987), provided an effective three- dimensional tool for structural strain analysis, which was used to predict the strain distribution throughout the overall structure (e.g., normal faults with throws ranging from meters to tens of meters in the hanging-wall). The strain distribution predicted by geomechanical modelling was then compared with the natural normal faults observed in the fieldwork to validate the applicability of geomechanical modelling. ...
  • ... Adaptando a equação para o modelo, considera-se que a variação do ângulo entre dois eixos é Θ = α t jk − α 0 jk . Desta forma, as forças angulares são dadas por: LIU; ABD-ELMONIEM; PRINCE, 2009; HONG et al., 2006; BOURGUIGNON; CANI, 2000) e empiricamente foi a que apresentou melhor custo benefício, em termos de eficiência computacional, simplicidade de programação e funcionalidade(CAMPOS et al., 2013). Assim, a força total em cada ponto de interseção é dada pelo somatório da força ativa, das molas axiais e angulares, de amortecimento e de preservação de volume: ...
    ... Adaptando a equação para o modelo, considera-se que a variação do ângulo entre dois eixos é Θ = α t jk − α 0 jk . Desta forma, as forças angulares são dadas por: LIU; ABD-ELMONIEM; PRINCE, 2009; HONG et al., 2006; BOURGUIGNON; CANI, 2000) e empiricamente foi a que apresentou melhor custo benefício, em termos de eficiência computacional, simplicidade de programação e funcionalidade(CAMPOS et al., 2013). Assim, a força total em cada ponto de interseção é dada pelo somatório da força ativa, das molas axiais e angulares, de amortecimento e de preservação de volume: ...
    Thesis
    Full-text available
    This work proposes a computational heart model named FisioPacer, which aims to reproduce the electrical pulse propagation over the cardiac tissue and its mechanical deformation. In order to perform fast simulations, it was used a cellular automaton coupled with a mass-spring system. A genetic algorithm was also used to automatically adjusting model parameters, in order to reproduce \textit{in silico} experiments and a real left ventricle behavior. For the model validation, seventy two experiments were performed and its results were compared to another robust simulator, based on partial differential equations. The comparisons showed FisioPacer satisfactorily reproduced the tissue behavior, with up to 15000-fold improvement on computational time. Furthermore, a real patient left ventricle was simulated, with data obtained via MRI.
  • ... The parameter k T determines the magnitude of the active stress build- up in each cell of the model and defines the strength of the contractions occurring in the medium, see Figure 2E. The elasto-mechanical model consists of a mass-spring damper system with controllable, tunable linearly transverse muscle fiber anisotropy (Bourguignon and Cani, 2000;Christoph, 2015). Figure 2B illustrates the lattice structure of the mass-spring damper system. ...
    Article
    Full-text available
    Optical mapping is a high-resolution fluorescence imaging technique, which provides highly detailed visualizations of the electrophysiological wave phenomena, which trigger the beating of the heart. Recent advancements in optical mapping have demonstrated that the technique can now be performed with moving and contracting hearts and that motion and motion artifacts, once a major limitation, can now be overcome by numerically tracking and stabilizing the heart's motion. As a result, the optical measurement of electrical activity can be obtained from the moving heart surface in a co-moving frame of reference and motion artifacts can be reduced substantially. The aim of this study is to assess and validate the performance of a 2D marker-free motion tracking algorithm, which tracks motion and non-rigid deformations in video images. Because the tracking algorithm does not require markers to be attached to the tissue, it is necessary to verify that it accurately tracks the displacements of the cardiac tissue surface, which not only contracts and deforms, but also fluoresces and exhibits spatio-temporal physiology-related intensity changes. We used computer simulations to generate synthetic optical mapping videos, which show the contracting and fluorescing ventricular heart surface. The synthetic data reproduces experimental data as closely as possible and shows electrical waves propagating across the deforming tissue surface, as seen during voltage-sensitive imaging. We then tested the motion tracking and motion-stabilization algorithm on the synthetic as well as on experimental data. The motion tracking and motion-stabilization algorithm decreases motion artifacts approximately by 80% and achieves sub-pixel precision when tracking motion of 1–10 pixels (in a video image with 100 by 100 pixels), effectively inhibiting motion such that little residual motion remains after tracking and motion-stabilization. To demonstrate the performance of the algorithm, we present optical maps with a substantial reduction in motion artifacts showing action potential waves propagating across the moving and strongly deforming ventricular heart surface. The tracking algorithm reliably tracks motion if the tissue surface is illuminated homogeneously and shows sufficient contrast or texture which can be tracked or if the contrast is artificially or numerically enhanced. In this study, we also show how a reduction in dissociation-related motion artifacts can be quantified and linked to tracking precision. Our results can be used to advance optical mapping techniques, enabling them to image contracting hearts, with the ultimate goal of studying the mutual coupling of electrical and mechanical phenomena in healthy and diseased hearts.
  • ... The 3D geometry of faults and horizons, supplemented with averaged mechanical parameters, acted as input for the geomechanical model. Geomechanical modeling is based on the mass-spring algorithm, in which the surface is considered as a point mass array linked with springs following Hooke's law (Bourguignon and Cani, 2000). Each vertex displace- ment generates mesh deformation and a strain tensor that can be calculated. ...
  • Article
    Peridynamics is a formulation of the classical elastic theory that is targeted at simulating deformable objects with discontinuities, especially fractures. Till now, there are few studies that have been focused on how to model general hyperelastic materials with peridynamics. In this paper, we target at proposing a general strain energy function of hyperelastic materials for peridynamics. To get an intuitive model that can be easily controlled, we formulate the strain energy density function as a function parameterized by the dilatation and bond stretches, which can be decomposed into multiple one‐dimensional functions independently. To account for nonlinear material behaviors, we also propose a set of nonlinear basis functions to help design a nonlinear strain energy function more easily. For an anisotropic material, we additionally introduce an anisotropic kernel to control the elastic behavior for each bond independently. Experiments show that our model is flexible enough to approximately regenerate various hyperelastic materials in classical elastic theory, including St. Venant‐Kirchhoff and Neo‐Hookean materials.
  • Article
    In this paper we present a multi-modal framework for offline learning of generative models of object deformation under robotic pushing. The model is multi-modal in that it is based on integrating force and visual information. The framework consists of several sub-models that are independently calibrated from the same data. These component models can be sequenced to provide many-step prediction and classification. When presented with a test example–a robot finger pushing a deformable object made of an unidentified, but previously learned, material–the predictions of modules for different materials are compared so as to classify the unknown material. Our approach, which consists of offline learning and combination of multiple models, goes beyond previous techniques by enabling i) predictions over many steps, ii) learning of plastic and elastic deformation from real data, iii) prediction of forces experienced by the robot, iv) classification of materials from both force and visual data, v) prediction of object behaviour after contact by the robot terminates. While previous work on deformable object behaviour in robotics has offered one or two of these features none has offered a way to achieve them all, and none has offered classification from a generative model. We do so through separately learned models which can be combined in different ways for different purposes.
  • Chapter
    Die Arbeitssysteme von Morgen müssen sich mit neuartigen Problemlösungsstrategien den Herausforderungen der Zukunft stellen. Die Schlagwörter demografischer Wandel, Fachkräftemangel, Steigerung von Effizienz, Produktivität und Qualität sowie Produktion in Hochlohnländern erfordern in diesem Kontext gleichermaßen eine zügige Anpassung und eine stetige Entwicklung. Kapitel 2 „Arbeitssysteme der Zukunft“ befasst sich mit den verschiedensten Formen von Arbeitssystemen und den damit verbundenen gestiegenen Anforderungen. Durch die kontinuierliche Zunahme der Komplexität von Produkten steigt auch deren Individualität. Diese speziellen Herausforderungen, entsprechende Lösungsstrategien und Forschungsergebnisse werden im Folgenden beispielhaft anhand von Arbeitssystemen, die durch eine Interaktion zwischen Mensch und Roboter gekennzeichnet sind, von visuellen Assistenzsystemen und Bedienarbeitsplätzen sowie eines Kommissionier-Arbeitsplatzes vorgestellt.
  • Technical Report
    Full-text available
    Este trabalho propõe um modelo eletromecânico do tecido cardíaco, com o objetivo principal de não ser computacionalmente custoso. Para desenvolvê-lo foram utilizados autômatos celulares acoplados a um sistema massa-mola, a fim de reproduzir o fenômeno da propagação da eletricidade pelo coração que dispara a contração mecânica, além da preservação de volume e do efeito eletrotônico. As simulações foram comparadas com as de um modelo tradicional matematicamente robusto e os resultados foram promissores. Apesar de apresentar algum erro numérico, o modelo proposto foi capaz de reproduzir resultados aproximados, com a vantagem de ser entre 50 e 1900 vezes mais rápido. Também foram utilizadas técnicas de computação paralela para diminuir ainda mais o tempo de execução. Este modelo simplificado é a primeira etapa do desenvolvimento de um software cujo objetivo é auxiliar o implante de marcapassos ressicronizadores, o FisioPacer. A ideia é auxiliar no processo de tomada de decisão referente ao posicionamento do eletrodo no coração, pois isto é realizado durante o procedimento cirúrgico através de tentativa e erro. Ao se realizar este processo in silico antes da cirurgia, pode-se diminuir o tempo do procedimento e consequentemente reduzir a exposição do paciente a raios-x e infecções, além de aumentar a probabilidade de sucesso do tratamento. Para que o objetivo seja alcançado, ainda restam algumas etapas: 1) análise de sensibilidade dos parâmetros dos modelos, 2) ajuste de parâmetros para reproduzir o funcionamento do coração de um paciente específico e 3) simulação do posicionamento dos eletrodos de um marcapasso. O baixo tempo de execução obtido nesta primeira fase é de suma importância para que o FisioPacer seja capaz de computar as demais etapas em tempo hábil.
  • Conference Paper
    Full-text available
    We present the LeVen computational system for simulation of the electro-mechanical function of the left ventricle (LV) of mammalian heart on parallel computers. A macroscopic LV model incorporates state-of-the-art cellular models, Ekaterinburg--Oxford (EO) model or ten Tusscher--Noble--Noble--Panfilov (TNNP). Ventricular mechanical activity on the organ level is described by means of the modified mass-spring system method, while the cellular mechanics is based on the myofilament activity. An open architecture of the LeVen system is proposed, which allows one to add and use new mechanical and electrophysiological models. The parallelization is implemented using the OpenMP technology. The performance of the LeVen system on multicore CPUs is evaluated. Parallel LeVen implementation provides a significant speedup, but the scalability is limited.
  • Chapter
    In this work, a method to adapt a Mass-Spring system to the energy density function describing myocardial mechanics, by using continuum mechanics methods is presented. For the development of a computer model of the myocardial mechanics, a modified Mass-Spring system proposed by Bourguignon [1] is used. This system discretized the simulated object into voxels arranged in a hexahedral mesh. Suitable spring functions for this system were derived analytically from energy density function describing the hyperelastic properties of heart tissue. In this paper two different energy functions, proposed by Hunter et al. [2] and Guccione et al. [3] are used. To evaluate the passive mechanical behavior of the Mass-Spring system, different deformation experiments are simulated and the results are discussed.
  • Article
    This paper introduces an adaptive component to a mass-spring system as used in the modelling of cloth for computer animation. The new method introduces non-active points to the model which can adapt the shape of the cloth at inaccuracies. This improves on conventional uniform mass-spring systems by producing more visually pleasing results when simulating the drape of cloth over irregular objects. The computational cost of simulation is decreased by reducing the complexity of collision handling and enabling the use of coarser mass-spring networks.
  • Article
    Full-text available
    We argue that B-spline solids are effective primitives for the animation of physically-based deformable objects. After reviewing the mathematical formulation of B-spline solids, we describe how to quickly display and modulate their shapes. We apply our ideas to muscle modelling and provide techniques for initial shape definition and subsequent shape deformation. Data-fitting techniques are developed to build muscles from profile curves or from contour data taken from medical images. By applying a spring-mass model to the resulting B-spline solid, we have transformed a static model to a deformable one. The 3-D parameterization of the solid allows us to model microstructures within the solids such as fibre bundles in a muscle. B-spline solids are powerful and versatile deformable shape primitives that can be used in practical settings, such as the building-blocks of a muscle-based modeller and animation system for anatomical design.
  • A methodology is proposed for creating and animating computer generated characters which combines recent research advances in robotics, physically based modeling and geometric modeling. The control points of geometric modeling deformations are constrained by an underlying articulated robotics skeleton. These deformations are tailored by the animator and act as a muscle layer to provide automatic squash and stretch behavior of the surface geometry. A hierarchy of composite deformations provides the animator with a multi-layered approach to defining both local and global transition of the character's shape. The muscle deformations determine the resulting geometric surface of the character. This approach provides independent representation of articulation from surface geometry, supports higher level motion control based on various computational models, as well as a consistent, uniform character representation which can be tuned and tweaked by the animator to meet very precise expressive qualities. A prototype system (Critter) currently under development demonstrates research results towards layered construction of deformable animated characters.
  • Article
    We develop physically-based graphics models of non-rigid objects capable of heat conduction, thermoelasticity, melting and fluid-like behaviour in the molten state. These deformable models feature non-rigid dynamics governed by Lagrangian equations of motion and conductive heat transfer governed by the heat equation for non-homogeneous, non-isotropic media. In its solid state, the discretized model is an assembly of hexahedral finite elements in which thermoelastic units interconnect particles situated in a lattice. The stiffness of a thermoelastic unit decreases as its temperature increases, and the unit fuses when its temperature exceeds the melting point. The molten state of the model involves a molecular dynamics simulation in which ‘fluid’ particles that have broken free from the lattice interact through long-range attraction forces and short-range repulsion forces. We present a physically-based animation of a thermoelastic model in a simulated physical world populated by hot constraint surfaces.
  • Legless figures such as snakes and worms are modelled as mass-spring systems. Muscle contractions are simulated by animating the spring tensions. Directional friction due to the surface structure is included in the dynamic model and legless figure locomotion results. Various modes of locomotion are described.
  • Article
    Full-text available
    We describe a novel method for surgery simulation including a volumetric model built from medical images and an elastic modeling of the deformations. The physical model is based on elasticity theory which suitably links the shape of deformable bodies and the forces associated with the deformation. A real time computation of the deformation is possible thanks to a preprocessing of elementary deformations derived from a finite element method. This method has been implemented in a system including a force feedback device and a collision detection algorithm. The simulator works in real time with a high resolution liver model
  • Article
    This paper presents a method of constraining physically-based deformable objects. In this method, an object can be defined locally in terms of kinetic and dynamic (mass, position, speed), and physical parameters (compressibility, elasticity, motor functioning). Several problems are solved: constant volume deformation, displacement constraints (fixed or moving required positions), and real object modelling. An object is described by a set of mass points on its contour. The evolution algorithm runs in two phases dealing successively with forces and constraints (which are presented as reaction forces). The main contribution of the method is the control of object volumes during evolution. We define a function that explicitly gives the inside volume of an object in order to use it as a constraint. Thus, the volume can be kept exactly constant during deformation without using an iterative process, in opposition to lagrangian approaches. Some results are illustrated by examples at the end of the paper.
  • Article
    In this paper we present a method to obtain good approximations of deformable bodies with spring/mass systems. An iterative algorithm based on voronoi diagrams is used to get a good mass distribution. The elastic properties of the system are optimized by simulated annealing. Results are shown, and some applications are discussed.