3  Deformable Models in Graphics

In graphical system we often want to simulate deformable objects. Simply put, deformable objects are ones that can change shape through time. To simulate such objects we need to describe how external forces on these objects translate into a deformation and how deformed objects generate forces that oppose external forces.Most of these methods have been thoroughly studied in context of continuum mechanics, but for our purposes we will only focus on parts relevant to physical simulation in graphics.

In these notes, we will first describe what a deformation is, then we will discuss a few ways of measuring the magnitude of the deformation, and then finally discuss how we can derive energies and forces from these deformations.

3.1 Describing Deformations

All motion can be separated into rigid-body displacement and deformation. The former is a combination of a rotation and a translation and does not affect the body shape \(B\). A deformation is “the change in the metric properties of a continuous body”, excluding rotation and translation. That is, a deformation describes how a “rest state” configuration \(\mathcal{K}_0(B)\) of a body \(B\) is transformed into a “deformed state” configuration \(\mathcal{K}_t(B)\) at timestep \(t\). Whereas in rigid-body motion the relative distances between particles \(p,q \in \mathcal{K}_0(B)\) are preserved, in a deformation they are not. We call the the function that maps a point \(X \in \mathcal{K}_0(B)\) to a point \(x \in \mathcal{K}_t(B)\) the deformation map \(\phi\):

\[x = \phi(X,t).\] This is a continuous map from the rest state to the deformed state, because reaby points in the rest state are mapped to nearby points in the deformed state (at least as long as the material is not tearing).

We classify deformations into two types: infinitesimal and finite. Without getting ahead of ourselves with all the notation, infinitesimal deformations are simply deformations where the displacements are much smaller than any relevant dimension of the body. In this setting we can approximate all deformations by linearizing around the point of interest. Finite deformations on the other hand are ones in which displacements are large compared to body dimensions. When dealing with finite deformations we sadly loose the privilidge to use linear approximations of deformations, rather we have to deal with non-linear deformations. In graphics we deal with the latter. You can see a comparison between finite and infitesimal formalations applied to large deformations.

Commonly, we represent objects as meshes of points connected by edges or faces. We describe all possible deformations of a single triangle as an affine map \(\textbf x = A \textbf X + t\).

3.2 Measuring Deformations: What is the “size” of a deformation?

More to come soon!

4 Refresher on Differential Equations

Ode. Function relating the variable and its derivatives:

\[F(t, y, y', y'', \dots) = 0.\] Most cases we can’t get closed-form solution for \(y\), instead we want an algorithm that gives us values of \(y\) at many \(t_k\) to approximate \(y\). We call such solvers numerical integrators.

If we have a vectored valued function \(\textbf y \in R^N\), then the solution to the initial value problem is just a path through \(\mathrm{R}^N\). Most often we try to work with the form of ODE solving for highest derivative:

\[y^{(k)} = F(t,y, y', \dots, y^{(k-1)}).\]

Reduction to 1st order. Given an explicit higher order ODE we can reduce it to first order (only 1st and 0th derivatives of \(y\)). We define a new family of unknown functions \(y_1 = y, y_2=y', ..., y_{k-1} = y^(k-2), F(x, y_{1}, y_{2},\dots, y_{k-1}) = y^{k}\) and rewrite our ODE as a system of simpler ODEs:

\[[y_{1}, y_{2}, \dots, y_{k-1}, f(t, y, y_{1}, \dots, y_{k-1})] = [y', y_{1}', \dots, y_{k-2}', y'_{k-1}]\] which is a single first order ODE in \(kN\) variables.

Autonomous vs non-autonomous. Sometimes we see \(t\) as an explicit parameter, and sometimes we do not. A differential equation is autonomous if it does not depend on the variable x, or in our example \(t\):

\[ y'(t) = f(y(t)) \text{ is autonomous}\] \[y'(t) = f(t, y(t)) \text{ is NOT autonomous}\]

We can switch from non-autonomous to autonomous with a conversion by relabeling variables.

Vector field picture. \(y(t)\) is a path of a point through the state space (this is \(y\) after reduction). \(f\) is a vector field in that state space, and tells particles where to go. So the process is an advection through the flow field.