Animating Highly Viscous Liquids with Particle System and Implicit Surface Modeling
Animating Highly Viscous Liquids
with Particle System and Implicit Surface Modeling
Kwansik Kim
University of California,Santa Cruz
July11,2000
Abstract
Physically based animations of variety of substances offers automatic generation of realistic simulations of many natural behavior of the substances in real world.Various
deformable models,collision detection and responses techniques are available for simu-
lating behaviors of substances with wide rage of physical properties which includes non
viscous?uids,elastic and viscoelastic solids.We present a technique to simulate the de-
formations of soft substances by handling large scale deformation with a particle system
and local viscoelasticity with implicit surface layer around each particle.Collisions are
handled by either fusion with the collided substances or repulsive action and friction with
other objects.In simulating highly viscous liquids,however,the”stickiness”of the sub-
stances should be considered to be able to animate the liquids realistically.It would be
almost impossible to accurately simulate this gluing behavior of viscous liquids.In this
paper,we are attempting to approximate this behavior with a relatively simple collision
response model and elastic model using implicit surface.This raises needs for new formu-
lation of collision detection and its responses which based on the collision history of the
substances because the deformation of the viscous liquid depends on its collision traces.
We extend models based on particles system and present collision responses techniques
based on implicit surface formulations.
1Introduction
Physically based animation is a continuing effort to simulate the behaviors of physical real world.Not only it adds striking realism to the static image synthesis but its simulation can
be used in many engineering applications.Many deformable models have been developed and available for simulating various natural phenomena including deformations of solids and ?uids.This paper presents a technique that simulates the behavior of highly viscous liquids.
Some?uid dynamics are simulated with a set of partial differential equations that approx-imate shallow water equations[9].Array of hydrolic columns are used to simulate a volume of”splashing”?uids[10]by computing?ows between columns and their hydrolic pressures. These models are suitable for less viscous?uids,like water,in very limited situation and not general enough to be used for highly viscous liquids.
In[15],they introduced elasticity theory to the deformable models and further extend the model that can handle viscoelasticity,plasticity and fracture[14].This model provides gen-eral frame work for simulating substances ranging from perfectly elastic solids to viscoelastic substances.The history of deformation should be considered to simulate plasticity and thus their model is consist of two layers;inelastic reference component which describes the mo-tion and general shape according to the applied force,and elastic layers that handles elasticity between the current shape and the reference shape.Because they tried to make general model for a wide range of substances and the lattice is used for descretizing equations,their model is restricted to the structured substances.
Other approaches are models based on particle systems[1,5,16].Modeling wide range of deformations including viscoelasticity,plasticity and handling collision,separation and fusion are done by simulating corresponding interactions between particles.Because they display iso-surface between particles they can give some anomalies due to incorrect collision detec-tions between particle and other objects.
In[8],precise modeling of collision between?exible solids are formulated as implicit surfaces.They present deformable models,which are de?ned as implicit surfaces around skeletons using?eld functions that de?ne the solids and its propagation area.The implicit de?nition of exact contact surface between two solid and their collision responses,which are deformations of implicitly de?ned solids,are formulated as an additional term in the?eld functions for the implicitly de?ned solid based on techniques by[3,4].Using these formu-lations,techniques for animating very soft substances are introduced[7,6].They use particle system for large scale deformations computing interaction between particles and the union of viscoelastic layer over particles de?nes the soft substances.Like other particle based models, they model viscoelasticity,fusion and separation by interaction between particles and they also solved problems of constant volume deformation[11]and unwanted blending[17].
A problem in animating highly viscous liquids,however,is that they often show high stickiness,which means the collided substances have tendencies to be glued at the collision point without fusion with other objects and this behavior is critical in simulating highly vis-cous”liquids”realistically.Once the gluey liquids collide with objects they will leave the trace of following the surface of the objects.This characteristic makes handling collision a
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Figure1:Modeling highly viscous liquids.
harder problem because the collision traces affects future collision responses and deforma-tion.It would be a gland challenge to run accurate simulation of this gluing behavior.We are attempting to?nd a relatively simple model that approximate the simulation reasonably. Therefore we need different formulation of collision detections and responses which based on collision history.This presentation addresses this problem and raises needs for new model of collision detection and its responses based on collision history by extending implicit surface technique.Our method is based on[6]and suggests that the collision trace of gluey liquids should be stored as a curve and the implicit deformation of each particle be done with the trace curve using techniques in[4].
2Overview
Particle systems[12,13]can be used to simulate a wide range of interesting behaviors.Parti-cles are objects that have mass,position,and respond to forces but have no spatial volume.The highly viscous substances are modeled as implicit surfaces of?eld functions around the con-nected graph between particles as a skeleton[4,3].The contact surfaces and the corresponding deformations are formulated as implicit de?nitions by the techniques in[8].The collision with other viscous liquids will be resulted in fusion[6].The animation of highly viscous liquids are implemented using these techniques(?gure1).The large scale inelastic or viscous movements and deformations are modeled with a particle system and local deformations due to collisions are formulated using implicit surface modeling techniques.The shape and displaying param-eters for interacting with other rigid solids are computed by de?ning deformations terms.The local deformation shapes are controlled by attenuation terms in the scope of in?uence of the reference point.These local deformation formulations are realized in ray tracing algorithm.
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We raise ideas of different kind of collision response when the soft viscous liquids collide with foreign objects.Each substance has its stickiness constant by which the attracting force is computed at the point of collision.If the attracting force is bigger than other forces the particle will follow the surface of the collided object.The liquid keeps colliding with the surface,so to speak,as long as the attracting force win over other force.We call this set of all collision points as collision history and save them in the data structure for the viscous liquid de?nition.With this collision history,we suggests to deform the related part of the skeleton for the liquid volume and thus the entire volume should be deformed correspondingly.As the collision history accumulated,the traces of collision can serve as reference components for the viscoelastic deformation of the liquid volume.
3Deformable Model
Implicit formulation models for?exible solids were introduced in[8,6].Gascuel[8]presented the implicit surface model for?exible and rigid bodies as well as formulations for interaction between rigid bodies and?exible bodies.Desbrun et al[6]introduced a hybrid model of particle system and implicit surface model for animating soft surfaces.We are using their model to formulate animation of viscous liquid interacting with rigid environments.
3.1Implicit Solids and Surfaces
An implicit solid is a set of?eld functions and the surface of the solid is a iso surface of the sum of?eld functions.An implicit surface generated by a set of skeletons,
with associated?eld functions,is de?ned by:
where
Each?eld function has a set of reference objects which can be any type of primitives,such as points,curves,parametric surface or volumes1.Therefore,implicit modeling can be used to design various kind of solid models.
An implicit model can be easily deformed locally by introducing a well designed defor-mation function term in the formulation,
which will be described in next sections.
3.2Elastic Modeling with Field Functions
The set of points P satisfying(where is the?eld functions)is used to de?ne a surface.Assuming the set of points?xed,we can use variations of?eld function values(e.g. gradients)around P in order to model physical properties such as stiffness.The applied force during a displacement of P from to is:
To express linear and non-linear elasticity in radial direction,using stiffness constant,fol-lowing condition is reported to be enough[8].
N is a normal direction of the?eld function and is a gradient vector at the point Y. From equation(),we get deformation function term which is which makes an equilibrium with interacting elastic solids.
(1)
where is a radial function term.We use equation(1)to de?ne general correspondence between deformation and forces characterizing implicit deformable solid.We use the equation to derive deformation term while interacting with other rigid or deformable solid.2From the equations above,the de?nition of stiffness in the elastic model is derived[8].
The stiffness is modeled with a function of which is the distance from point P from the closest point on the implicit surface.We use this formulation to control deformation term due to collision with rigid and elastic solid.We used the following?eld function given in[8] which satis?es conditions and formulation explained in this section.
Interpenetration zone i i
j i Solids after deformation
S 0000000000000000000011111111111111111111Figure 2:Modeling contacts consists of different deformation zones,interpenetration zone and propagation zone.
The ?eld functions currently implemented are parameterized by a scope of in?uence R,a
thickness value and a stiffness value k and
,.Constants a,b and c we used for linear or non-linear stiffness are
given in [8].
3.3Interactions between Implicit Models
Gascuel described formulations between rigid solids and soft solids as well as soft solids with soft solids [8].Here we describe interactions between soft solids and rigid solids,which are implemented,after describing general idea of modeling contacts between implicit models.3
3.3.1Modeling contact and Interpenetration
Our aim is to model deformations when two or more implicit solids contact or interpenetrates each other.As described in previous section,each ?eld function has the thickness and the scope of in?uence.Figure 2shows the various zones between two elastic solids.In “interpen-etration zone”,the values of both ?eld function are less than 1or user given constant.In this zone,precise contacting surface should be computed for each solid using elastic model.We call the intersection of scopes of in?uence of interacting solids as “propagation area”.In this area each solid should be deformed while maintaining underline elastic models like stiffness.
Suppose that two objects and are interacting locally.What we need is each deforma-tion term and to be added to their respective?eld function(represents the action of object j on object i).Then the objects will simply de?ned in this area by:
In the interpenetration area,g terms should be negative and in points P where.The local contact surface generated must?t with local rigidities of colliding objects.A natural choice is
The veri?cation of this deformation terms for contacting surface as well as local rigidities and correspondence between forces and deformations are derived in[8]
3.3.2Deformations in“propagation areas”
The deformation in the propagation area,however,are modeled locally rather than interacting with other solids.The force correspondences of deformation are considered only in interpen-etration area.Therefore deformation terms in the propagation area are geometric only.We can provide useful parameters such as thickness of propagation area to control shape of the deformation in the area.We are providing two basic control parameter for controlling’s propagation?eld terms(due to collision with:
A thickness value giving the size of offset where deformations propagate around an
interpenetration zone.
An“attenuation value”giving the ratio between the maximal value desired for
and the current maximal compression term in the interpenetration area.
There are couple of conditions that must be satis?ed geometrically.For example,the prop-agation?eld term must be positive in the propagation area and should preserve continuity and the derivatives must become zero out side of thickness range.The resulting“attenuation”function should be a function of distance from arbitrary point P to the closest point on the,the surface of colliding solid.
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a k a w x = d(P,P )0
k,a 0,w (x)
Figure 3:Attenuation function with control parameters de?ning the propagation ?eld terms.
where
,is the maximal propagation value,which is the size of the bulge.We are using the piecewise function given in [8]which satisfy all conditions 4.4The paper has typos which we found while implementing the function and con?rmed by the author.
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4Particle System
Previous sections described external elastic layer of our system for animating highly viscous liquids.As in[16,6],we use particle system of inelastic reference components to simulate the motion and large scale deformation while implicit elastic models gives shape of the liquid and local deformation due to collisions.
4.1Inelasticity
We consider attraction/repulsion forces and?uid friction forces depending on local particle density in addition to basic implementation of Newtonian physics.The forces applied by particle on particle are:
5Ray Tracing Liquid Objects
All implicit surface modeling for elastic shape control described in earlier section is primar-ily issues of visualizing resulting animation.We can either polygonalize the resulting shape deformation[2]or compute ray traced image.We chose to ray trace them for higher quality images and better controls of displaying options for multiple objects in a scene.
To the author’s knowledge,no decent method for analytically calculating intersection point of a ray with implicit surface of multiple non-linear functions that interacts with other func-tions.Note that the attenuation term and deformation term for each reference point is a function of the distance from the closest point and the point where the function is evaluated. Moreover we have multiple reference points and and interacts with other objects due to collisions.
As in[8,6],we use a binary search to?nd a point P on the ray that satis?es
where n is the total number of reference points for liquid object and rigid object.For each reference point,we?rst intersect with its bounding volume and step through the bounded ray until we?nd a point P where.Then we perform binary search between last two points until we?nd the correct intersection point within given tolerance.
As mentioned in the beginning of the section,because the implicit surface modeling con-tributes geometrically only in our implementation and particle system is used for viscous be-haviors and collision responses,all the formulations for implicit surfaces are realized in the ray tracing algorithm while evaluating the sum of all related function value,depending on position P relative to distances from reference points and other in?uence scopes.
6Animation Process
We provide a fast way of previewing animation sequence by using simple primitives for refer-ence points and rigid https://www.wendangku.net/doc/cd8110327.html,er can control the animation by easy-to-understand control-ling parameters described in earlier sections,such as stiffness,thickness,scope of in?uence and linear or non-linear options for the elastic area or mass,gravity and stickiness for particle system and collision response or shape controls of attenuation function.A typical process of creating an animation is,
Set the position and parameters for liquid objects and rigid objects.
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Figure5:Snapshots of animation sequences.
Set the control parameters for particle system and elastic model control for liquid ob-jects.
Preview large scale moving and deformation by animating with simple primitives while controlling animation parameters such as time step size.
Snapshot a ray traced images of key positions for resulting local deformations of liquid objects.
Generate animation with ray traced image for each frame of the entire animation se-quence.
Figure4shows the preview window and ray tracing window in our system.
7Implementation
The system is written in C++with GL library on SGI workstations.We relied on fast drawing capabilities for previewing the animation.The system is written in object orient style and thus,if programmer wants to add new rigid or implicit surface objects,they can be easily plugged in by de?ning only a few number of virtual functions de?ned in the abstract classes for general deformable objects.The system generates SGI image?le format for the?nal animation sequence.Figure5shows snapshots of animation sequences.
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8Conclusion and Future Research
A hybrid system of particle system for viscous behavior and implicit surface modeling for ex-ternal elastic area deformation are presented.An collision response method for highly viscous and gluey liquids is presented and a collision history model is suggested to add realism to the animation in the future.
Among the problems we have not solved,unwanted blending makes most noticeable ar-tifacts in the animation.Before the scope of in?uence reaches the other objects,the other attenuation functions can generate unwanted blending and deformations.Another less notice-able artifact is that the implicit modeling does not consider the constant volume deformation and thus the volume of liquids can be serverly increased or decreased unexpectedly.The solu-tions for both problems can be found in[6].We are also planning to implement more precise and optimized ray tracer.
The collision history issues will be implemented in the future.We believe the idea will solve the problems of lacking sense of gravity and unnatural graininess in the external shape deformations of liquids,especially when relatively small number of reference points are used. We also plan to adopt scripted user interface like Tcl/Tk for more modulization of code and independence from graphical user interface and speci?c hardware.
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