Adaptation of Signorini’s problem for interactive
simulations
Christian Duriez, Claude Andriot
CEA LIST
SCRI
Fontenay-aux-Roses, France
christian.duriez@cea.fr
Abderrahmane Kheddar
AIST/ISI-CNRS
Joint Japanese-French Robotics Laboratory (JRL)
Tsukuba, Japan
abderrahmane.kheddar@aist.go.jp
Abstract—Having a representation of an object and its behavior
before a rough physical model is available is an option that is
being taken up more and more widely in industry. Advances
in virtual prototyping, which is becoming interactive thanks
to real-time computations, are opening up new prospects for
this type of simulation in many areas. Modeling deformable
bodies may be necessary for several tasks such as mounting /
dismounting snap-in tasks or when cables are considered in the
virtual environment. The physical model of deformable material
behavior and the geometric level of detail can’t be simplified too
much. To reach a “physical realism”, in the haptic perception,
contact forces between objects must be integrated correctly.
Signorini has proposed a physical model of contacts between
a deformable object and its rigid environment. This model is
extended to solve for contacts between two deformable objects.
This paper presents resolution of the Signorini’s problem applied
on deformable objects in an interactive simulation context. The
deformation behavior is modeled by finite element method.
I. INTRODUCTION
In this introduction, the advances in virtual prototyping are
described especially with the example of the CEA approach.
Challenging works on deformable models in this area are
presented.
A. From design to maintenance
Virtual prototyping used in industrial environments allows
reductions in development times, product industrialization
costs, and associated training and maintenance costs. Available
computation’ speed capabilities enable to generate virtual
environments representing manufacturing systems close to real
and make possible to operate in an interactive mode.
Virtual prototyping is actually only the front end of a
product life management (PLM) process taking on board all
constraints related to manufacturing (robotics, ergonomics),
use and maintenance (training). All the computer-aided design
(CAD) majors such as Catia (from Dassault Syst`emes) or
PTC (from Parametric Technology Corporation) are opened to
this functionality. There is a need from all buoyant industrial
sectors, such as automotive, aerospace, transportation, energy,
but equally from architecture, cultural and medical sectors.
Presently, automotive and aerospace manufacturers still widely
use physical mockups, in order for instance, to “prototype” assembly
or maintenance operations. However, as a visualization
tool, such mockups do have two major drawbacks:
² they are expensive, and
² often obsolete since they do not always represent the
latest version of the product.
This is why these industries are increasingly trying to substitute
them by numerical (“digital”) mockups based on best of
what may offer virtual/augmented reality techniques allowing
different degrees of immersion of the designer/user within the
virtual mockups, and increasing the design performance and
efficiency.
B. Interactive simulation with haptic interfaces
As an example, consider automotive industry in assembly.
Currently, assembly-line designers use parts made by stereolithography
to test part assembly and the associated tooling.
Users consider that use of such current simulation software
programs as Catia or Robcad is too complex for this type
of operation. In the near future, designers will be able to
use, on the one hand, stereoscopic visualization systems to
prototype their lines, and, on the other hand, haptic interfaces
coupled with real-time physical simulation software programs
(calculation of collisions and contact forces) to prototype
tasks. The goal? To make virtual prototyping tools usable by
professionals in industry rather than by specialists, and with no
need, for instance, to program a single line of code. Assembly
and/or maintenance operation are the major functions of virtual
prototyping with haptic feedback.
C. Deformable objects
A main composing part of a virtual reality platform is the
real-time physical simulator. It computes, thanks to describing
database and external stimuli, the behavior of the virtual world
and the rendering of this behavior output signals. Modeling
deformable bodies is necessary for several tasks such as
mounting / disassembling and snap-in operations of flexible
parts. Haptic interaction allows operators to experience these
tasks in a realistic situation manner. Computing deforming
objects’ behavior in real-time simulation is a challenging issue.
Models are based on continuum mechanics governing laws
(i.e. methods that drive links between the geometric strain and
the internal stress of the deformed material [9] [15]).
Modeling equations are not possible to solve analytically
for the general object’s shape. Nevertheless mathematically
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