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SIGGRAPH 2012 Course FEM Simulation of 3D Deformable Solids
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SIGGRAPH 2012 Course FEM Simulation of 3D Deformable Solids: A practitioner’s guide to theory, discretization and model reduction.
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SIGGRAPH 2012 Course
FEM Simulation of 3D Deformable Solids: A practitioner’s
guide to theory, discretization and model reduction.
Part One:
The classical FEM method and discretization methodology
Eftychios D. Sifakis
University of Wisconsin-Madison
Version 1.0 [10 July 2012]
For the latest version of this document, please see:
http://www.femdefo.org/
Contents
1 Preface 3
2 Elasticity in three dimensions 5
2.1 Deformation map and deformation gradient . . . . . . . . . . . . . . 5
2.2 Strain energy and hyperelasticity . . . . . . . . . . . . . . . . . . . . 8
2.3 Force and traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 The First Piola-Kirchhoff stress tensor . . . . . . . . . . . . . . . . . 11
3 Constitutive models of materials 14
3.1 Strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 St. Venant-Kirchhoff model . . . . . . . . . . . . . . . . . . . . . . . 18
1
2 CONTENTS
3.4 Corotated linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Isotropic materials and invariants . . . . . . . . . . . . . . . . . . . . 20
3.6 Neohookean elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Discretization 25
4.1 Energy and force discretization . . . . . . . . . . . . . . . . . . . . . 25
4.2 Linear tetrahedral elements . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Force differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 An implicit time integration scheme . . . . . . . . . . . . . . . . . . 32
Chapter 1
Preface
Simulation of deformable elastic solids has evolved into a popular tool for visual
effects, games and interactive virtual environments. The Finite Element Method
has been very popular in this context, especially in applications that can benefit
from its versatility in representing elastic bodies with intricate geometric features
and diverse material properties. Techniques for solids simulation that have been
broadly used in graphics draw upon a rich, decades-long literature in Galerkin
methods, discrete elliptic PDEs and continuum mechanics theory. As many of
these techniques originated in theoretical and engineering disciplines other than
graphics and visual computing, it may be somewhat challenging for a practitioner
with modest theoretical exposure or familiarity with these fields to navigate some
of the most established mechanical engineering or computational physics reference
textbooks, especially if their goal is to acquire a high-level understanding of the basic
tools needed for implementing a simulation system. This document aims to provide
a concise, yet lightweight synopsis of the relevant theory, with adequate attention
to implementation details from the perspective of a graphics developer. Most of the
material referenced in these notes resulted from the author’s long and rewarding
interactions with graduate students at Stanford, UCLA and UW-Madison, as well
as the experience of the graduate class “Introduction to physics-based modeling and
simulation” offered at the University of Wisconsin.
This document assumes minimal to no exposure to continuum mechanics or
finite element discretizations. However, a particular flavor of calculus background
is presumed, including:
• Familiarity with functions of several variables, partial derivaties, volume and
surface integrals.
• Some exposure to numerical techniques for solving linear systems of equa-
tions, and the Newton-Raphson method for finding approximate solutions to
nonlinear problems.
3
4 CHAPTER 1. PREFACE
• A good understanding of linear algebra, including concepts such as vectors,
matrices and (higher-order) tensors. Familiarity with determinants, eigen-
value problems and the Singular Value Decomposition is also assumed.
• Although many of the proofs and derivations are treated as optional reading,
the majority of them make heavy use of differentials (linearized tensors) and
reference complex differentiation concepts (such as the derivative of a matrix
function with respect to a matrix argument).
As a supplement to the present introduction to FEM methods for deformable
solids simulation, the following textbooks are highly recommended:
J. Bonet and R. Wood, Nonlinear continuum mechanics for Finite Element
Analysis, (2nd ed.), Cambridge University Press
O. Gonzalez and A. Stuart, A first course in Continuum Mechanics, Cam-
bridge University Press
T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Ele-
ment Analysis, Dover Publications
T. Belytschko, W. Lui and B. Moran, Nonlinear Finite Elements for Con-
tinua and Structures, Wiley
J. Simmonds, A Brief on Tensor Analysis, (2nd ed.), Springer-Verlag
G. Golub and C. van Loan, Matrix Computations, (3rd ed.), Johns Hopkins
University Press
J. Demmel, Applied Numerical Linear Algebra, SIAM
Text in shaded boxes presents theoretical proofs, provides examples or provides
further insight on the preceding topics. This content can be treated as optional
reading, and omitting it should not compromise the understanding of subsequent
topics.
Chapter 2
Elasticity in three dimensions
In this chapter we focus on three-dimensional elastic bodies deforming in space, and
discuss how we can formulate quantitative descriptions for the deformed shape of
an object and the forces resulting from it. To a certain extent, these formulations
are analogous to similar concepts from mass-spring systems, or deformable elastic
strands. However, since a volumetric body is able to alter its shape in more complex
ways than, for example, a one-dimensional elastic strand, many concepts that may
be familiar from simpler mechanical systems will need to be extended and become
more expressive. For the time being, and until chapter 4, we will not concern
ourselves with discretization issues. Our discussion will focus on the continuous
phenomenon of elastic deformation, as if we had infinite resolution at our disposal.
2.1 Deformation map and deformation gradient
Our initial objective is to provide a concise mathematical description of the defor-
mation that an elastic body has sustained. This formulation will lay the foundation
for appropriate representations of other physical properties such as force and en-
ergy. We begin by placing the undeformed elastic object in a coordinate system,
and denote by Ω the volumetric domain occupied by the object. This domain will
be referred to as the reference (or undeformed) configuration, and we follow the con-
vention that capital letters
~
X ∈ Ω are used when referring to individual material
points in this undeformed shape. Note that the precise position and orientation of
the undeformed elastic body within the reference space is not important and can be
chosen at will, as long as the shape of the object corresponds to a rest configuration.
When the object undergoes deformation, every material point
~
X is being dis-
placed to a new deformed location as seen in figure 2.1 (top) which is, by convention,
denoted by a lowercase variable ~x. The relation between each material point and its
respective deformed location is captured by the deformation function
~
φ : R
3
→ R
3
which maps every material point
~
X to its respective deformed location ~x =
~
φ(
~
X).
5
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