Introduction to Reduced Order Modeling (ROM) for Structural Analysis

A Low-Dimensional Approach for Accelerating Simulations

In finite element analysis (FEM), the number of degrees of freedom can easily reach millions or even tens of millions.

This becomes particularly challenging in applications such as:

Parameter studies

• Topology optimization
• Inverse problems
• Real-time simulation
• Surrogate modeling

In these scenarios, the same or similar simulations must be solved repeatedly, making computational cost a major bottleneck. One of the most important approaches for overcoming this challenge is Reduced Order Modeling (ROM) . ROM is a technique that Compresses a large-scale FEM problem into a much smaller system using only the most important deformation modes.

Original Structural System

In linear static analysis, we typically solve:

Ku=F

where

• K \in \mathbb{R}^{N \times N} : stiffness matrix
• u \in \mathbb{R}^{N} : displacement vector
• F \in \mathbb{R}^{N} : force vector
• N: total number of degrees of freedom

In FEM, the stiffness matrix K is usually sparse.

Basic Idea of ROM

In ROM, the displacement vector is approximated using a small number of basis vectors:

u \approx \Phi q

where:

• \Phi \in \mathbb{R}^{N \times r} : reduced basis matrix
• q \in \mathbb{R}^{r} : reduced coordinates
• r \ll N

This can also be written as:

u \approx q_1 \phi_1 + q_2 \phi_2 + \cdots + q_r \phi_r

The core idea of ROM is:

Even though FEM models are extremely high-dimensional, the actual deformation behavior often lies in a much lower-dimensional subspace.

Galerkin Projection

Starting from:

Ku=F

we substitute:

u = \Phi q

to obtain:

K \Phi q = F

Applying Galerkin projection by multiplying both sides by ( \Phi^T ):

\Phi^T K \Phi q = \Phi^T F

We then define:

K_r = \Phi^T K \Phi\\
F_r = \Phi^T F

which leads to the reduced system:

K_r q = F_r

Why Is ROM Fast?

Suppose:

• Original system size: ( N = 10^6 )
• Reduced dimension: ( r = 50 )

The problem size is reduced from:

10^6 \times 10^6

to:

50 \times 50

This reduction dramatically decreases computational cost.

Constructing the Reduced Basis (POD)

One of the most widely used ROM techniques is Proper Orthogonal Decomposition (POD)

Snapshot Collection

First, multiple simulations are performed under different conditions:

\Phi =
\begin{bmatrix}
U_1 & U_2 & \cdots & U_r
\end{bmatrix}

These vectors form the reduced basis.

Can ROM Be Used with Sparse Matrices?

Yes — ROM is highly compatible with sparse FEM systems.

The original stiffness matrix (K) remains sparse throughout the process.

For example:

Kr = Phi.T @ (K @ Phi)

is typically computed using sparse matrix multiplication.

Here:

• (K @ \Phi): sparse matrix multiplication
• (\Phi^T): projection into the reduced space

How Is the Reduced Basis Actually Computed?

The reduced basis matrix ( \Phi ) is typically constructed from simulation snapshots using Singular Value Decomposition (SVD).

Suppose we collect (m) displacement snapshots:

u_1,\ u_2,\ \cdots,\ u_m

Each snapshot is a high-dimensional FEM solution:

u_i \in \mathbb{R}^N

We assemble them into a snapshot matrix:

X =
\begin{bmatrix}
u_1 & u_2 & \cdots & u_m
\end{bmatrix}
\in \mathbb{R}^{N \times m}

Next, we apply Singular Value Decomposition (SVD):

X = U \Sigma V^T

where:

• U \in \mathbb{R}^{N \times m}: left singular vectors
• \Sigma: singular value matrix
• V \in \mathbb{R}^{m \times m}: right singular vectors

The columns of U represent orthogonal deformation modes extracted from the dataset.

The singular values in \Sigma indicate the importance of each mode.

We then select the first r dominant modes:

\Phi =
\begin{bmatrix}
U_1 & U_2 & \cdots & U_r
\end{bmatrix}

This matrix becomes the reduced basis used in ROM.

Applications Where ROM Is Effective

ROM is particularly powerful for:

Digital twins

• Parameter studies
• Many-query problems
• Topology optimization
• Inverse analysis
• Uncertainty quantification (UQ)
• Real-time control