Physics-Based Simulation

Summary

The elastic potential energy $P_e$ is an integration of the strain energy density function $\Psi (\mathbf{F})$ at every local point in the solid domain. From the rigid null space, we derived an intuitive formulation of the strain energy density function, similar in structure to standard models like Neo-Hookean. Nonlinear elastic models are also rotation invariant, meaning any rotations applied after the deformation $\mathbf{F}$ do not change $\Psi$.

Linear elasticity features a quadratic energy density function and is specifically designed for infinitesimal strains $\mathbf{ϵ}$, lacking rigid modes in its null space. Yet, with the corresponding Lame Parameters $\mu$ and $\lambda$, it can accurately capture behaviors of small deformations observed in the real world. Standard elasticity models are required to be consistent with linear elasticity under small deformations.

This lecture focused on isotropic elasticity, where no special directions exist that make the material harder or easier to deform. Performing Polar SVD on $\mathbf{F}=\mathbf{U}\mathbf{\Sigma }\mathbf{V}$ allows us to rewrite $\Psi (\mathbf{F})$ of isotropic models using only principal stretches ${\sigma }_i={\mathbf{\Sigma }}_{ii}$.

Using the closest rotation ${\mathbf{R}}^n={\mathbf{U}}^n({\mathbf{V}}^n{)}^T$ to ${\mathbf{F}}^n$ in the last time step, we constructed a corotated linear elasticity to make linear elasticity rotation-aware while maintaining its simplicity. Simplifying further by retaining only the $\mu$-term enhances efficiency for visual computing.

Similar to how non-interpenetrations are enforced in IPC, the energy density function of Neo-Hookean acts as a barrier function, ensuring non-inversion ($\det (\mathbf{F})>0$). All other elasticity models introduced in this lecture are invertible, and they do not guarantee non-inversion.

In the next lecture, we will explore the derivatives of $\Psi (\mathbf{F})$ with respect to $\mathbf{F}$.