Physics-Based Simulation

Choice of Basis for Nonlinear Deformations

The quality of the reduced model is determined by the choice of the time-invariant basis matrix $\mathbf{U}\in {\mathbb{R}}^{3n\times r}$, which we assumed existance of in the previous chapters. This basis, which is pre-processed to be mass-orthonormal (${\mathbf{U}}^T\mathbf{MU}=\mathbf{I}$), must effectively span the space of expected nonlinear deformations.

Basis from Simulation Data (POD)

This data-driven approach, also known as Principal Orthogonal Directions (POD), constructs an optimal basis from pre-existing simulation data.

First, an offline, unreduced simulation is run to generate a set of $N$ deformation state vectors $\{{\mathbf{u}}_1,{\mathbf{u}}_2,...,{\mathbf{u}}_N\}$. Afterwards, the snapshots are assembled into a matrix $\mathbf{A}=[{\mathbf{u}}_1,{\mathbf{u}}_2,...,{\mathbf{u}}_N]$. A basis is extracted by performing a Singular Value Decomposition (SVD), typically with respect to the mass-weighted inner product $⟨\mathbf{x},\mathbf{y}{⟩}_{\mathbf{M}}={\mathbf{x}}^T\mathbf{My}$ (a technique known as mass-PCA). The columns of $\mathbf{U}$ from the decomposition $\mathbf{A}=\mathbf{U}\Sigma {\mathbf{V}}^T$ that correspond to the largest $r$ singular values form the basis $\mathbf{U}$!

While optimal for the training data, this method's primary drawback is the necessity of a slow, offline pre-simulation.

Basis from Modal Derivatives

This approach constructs a basis automatically by analyzing the system's fundamental nonlinear response, without requiring a non-reduced pre-simulation like POD.

The key insight is that linear modes are an incomplete basis. When a nonlinear system is excited in the direction of a linear mode, other deformations naturally co-appear due to nonlinear coupling. Modal derivatives are precisely these coupled deformations.

Mathematically, we analyze the static deformation $\mathbf{u}(\mathbf{p})$ resulting from a force applied along the linear modes ${\mathbf{U}}_{lin}$: ${\mathbf{f}}_{int}(\mathbf{u})=\mathbf{M}{\mathbf{U}}_{lin}\mathbf{\Lambda p}$, where $\mathbf{\Lambda }$ is the diagonal matrix of squared frequencies and $\mathbf{p}$ is a parameter vector. The solution $\mathbf{u}(\mathbf{p})$ can be expressed as second order Taylor series expansion around $\mathbf{p}=0$:

$$\mathbf{u}(\mathbf{p})=\sum _{i=1}^k{\psi }_i{p}_i+\frac{1}{2}\sum _{i=1}^k\sum _{j=1}^k{\mathbf{\Phi }}_{ij}{p}_i{p}_j+O(||\mathbf{p}|{|}^3).$$

This expansion reveals the system's response structure. The first-order derivatives, ${\psi }_i=\frac{\partial \mathbf{u}}{\partial p_i}{|}_{\mathbf{p}=0}$, are precisely the familiar linear vibration modes, representing the initial linear response to the applied force. The crucial nonlinear couplings are captured by the second-order derivatives, $${\mathbf{\Phi }}_{ij}=\frac{{\partial }^2\mathbf{u}}{\partial p_i\partial p_j}{|}_{\mathbf{p}=0},$$ which are known as the modal derivatives. Each ${\mathbf{\Phi }}_{ij}$ is a deformation vector that can be computed by solving a linear system involving the constant, rest-state stiffness matrix $\mathbf{K}$.

The full set of vectors $\{{\psi }_i,{\mathbf{\Phi }}_{ij}\}$ is typically too large. A compact, $r$-dimensional basis $\mathbf{U}$ is formed by collecting all linear modes and modal derivatives, scaling them to prioritize low-frequency contributions, and then using mass-PCA to extract the most significant combined shapes. This produces a general-purpose basis that captures essential nonlinear behavior without any prior knowledge of runtime forces.

For more information and the derivation please refer to the SIGGRAPH course notes [Sifakis & Barbic 2012].