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Comparison of model order reduction techniques with one-shot procedure for topology optimization for thermal applications

This paper proposes a model order reduction framework for thermal topology optimization that utilizes distinct reduced bases for forward and adjoint equations, demonstrating significant computational speedups of up to 16 times compared to high-fidelity workflows and 1.54 times over the one-shot method by leveraging solution snapshots and optimized linear solver stopping criteria.

Original authors: Luis Fernando Cusicanqui Lopez, Ramadan Krasniqi, Florian Feppon, Karl Meerbergen

Published 2026-04-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luis Fernando Cusicanqui Lopez, Ramadan Krasniqi, Florian Feppon, Karl Meerbergen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to design the perfect heat sink for a computer chip. Your goal is to arrange material inside a block so that heat flows away as efficiently as possible. You have two materials: a "super conductor" (like copper) that moves heat fast, and an "insulator" (like plastic) that blocks it. You want to use as little of the expensive copper as possible while keeping the chip cool.

This is a Topology Optimization problem. The computer tries millions of different arrangements to find the best one.

The Problem: The "Slow Motion" Bottleneck

To test a design, the computer has to solve a massive physics puzzle (the heat equation) for every single attempt.

The Old Way (High-Fidelity): Imagine trying to solve a giant jigsaw puzzle where every piece is a tiny, high-resolution photo. It's incredibly accurate, but it takes hours to finish just one puzzle. If you need to try 250 designs, you're waiting days or weeks.
The "One-Shot" Shortcut: Recently, researchers found a trick. Instead of finishing the puzzle, they just look at the first few pieces and guess the picture. It's fast (10x faster!), but sometimes the guess is a little off.

The Solution: The "Cheat Sheet" (Model Order Reduction)

This paper introduces a new strategy that combines the best of both worlds. Think of it as building a smart cheat sheet (a Reduced Order Model) based on the puzzles you've already solved.

Here is how the authors' method works, using simple analogies:

1. The Two-Track System (Forward & Adjoint)

In this design process, the computer runs two types of calculations simultaneously:

The Forward Track: "If I put copper here, how hot does the chip get?"
The Adjoint Track: "If I want to cool the chip, where should I move the copper?"

The authors realized these two tracks are different enough that they need two separate cheat sheets. One sheet learns from the "Forward" guesses, and the other learns from the "Adjoint" guesses. This prevents the computer from getting confused, which happened in older methods.

2. The "Smart" Stop Button (Stopping Criterion)

When the computer checks its cheat sheet, it needs to know: "Is this guess good enough, or do I need to go back to the slow, high-resolution puzzle?"

The Old Rule: The computer used a very strict ruler to measure the error. It was like checking if a sketch was perfect down to the pixel. This meant the computer often gave up on the cheat sheet and went back to the slow puzzle too often.
The New Rule: The authors found a better ruler. It's more forgiving and understands that for the "Adjoint" track, a slightly rougher sketch is actually fine. This allows the computer to trust the cheat sheet much more often, saving huge amounts of time.

3. The "Recycling" Trick

As the design changes, the computer needs to update its cheat sheet.

The Problem: If you just keep adding new pages to a notebook, it gets too heavy to carry. You have to throw away old pages. But if you throw away the wrong page, you lose important information.
The Fix: The authors created a special "recycling bin" algorithm. When they remove an old page, they mathematically ensure the remaining pages still cover the whole picture perfectly. It's like editing a video: you can cut out an old clip without making the rest of the movie look choppy.

The Results: Speed vs. Quality

The researchers tested this on a complex 3D heat sink design (think of a tiny, intricate metal tree inside a block).

Scenario A (Cheat Sheet + Slow Solver): When they used their new cheat sheet with the accurate (slow) solver, they finished the job 3 times faster than the standard method, with almost no loss in quality.
Scenario B (Cheat Sheet + One-Shot): This is the real winner. They combined the cheat sheet with the "One-Shot" shortcut.
- The "One-Shot" method alone was already 10x faster than the standard.
- By adding the cheat sheet on top of that, they got an extra 1.5x speed boost.
- Total Result: They achieved a design that looked and performed just like the super-accurate version, but in a fraction of the time.

The Bottom Line

Imagine you are trying to find the best route through a city.

Old Way: You drive every single street to measure traffic (Too slow).
One-Shot: You ask a local for a quick guess (Fast, but maybe wrong).
This Paper's Way: You ask the local for a guess, but you also keep a notebook of past traffic patterns. You use the notebook to correct the local's guess instantly. If the notebook says the guess is bad, then you drive the street to check.

By using this "notebook" (Model Order Reduction) and a smarter way to decide when to trust it, the authors made the design process significantly faster without sacrificing the quality of the final product. They turned a weeks-long calculation into a days-long one, making it much easier to design better electronics, engines, and cooling systems.

1. Problem Statement

Density-based Topology Optimization (TO) is a powerful method for generating optimized designs but faces a significant computational bottleneck when applied to large-scale 3D thermal problems. The process requires solving large linear systems for both the forward state equation (governing heat conduction) and the adjoint equation (for sensitivity analysis) at every optimization iteration.

The Challenge: Solving these systems to high accuracy using standard iterative solvers (like Preconditioned Conjugate Gradient with Multigrid) is computationally expensive, often dominating the total design cycle time.
The Trade-off: Existing acceleration techniques involve a trade-off between speed and accuracy. For instance, the "one-shot" method (solving equations with only one iteration) is fast but inaccurate, while full convergence is accurate but slow. The authors aim to bridge this gap by combining Model Order Reduction (MOR) with approximate solvers to achieve high speed without sacrificing design quality.

2. Methodology

The authors propose a hybrid framework that integrates Model Order Reduction (MOR) into the standard TO workflow. The core components of their methodology include:

A. Dual Reduced Basis Construction

Unlike previous methods that often use a single basis for both equations, this framework constructs two distinct reduced bases:

Forward Basis ( $V$ ): Spanning the state variable solutions.
Adjoint Basis ( $W$ ): Spanning the adjoint variable solutions.
These bases are updated incrementally using a Gram-Schmidt orthogonalization procedure. A key innovation is the proposed algorithm (Algorithm 1) for managing the basis size: when the maximum dimension is reached, the oldest vector is removed, and the remaining basis is re-orthogonalized to ensure the subspace accurately reconstructs the original snapshots, preventing information loss common in incremental approaches.

B. Adaptive Solver Strategy

The workflow employs an adaptive switching mechanism between the Reduced Order Model (MOR) and the Full Order Model (FOM):

MOR Step: Approximate solutions are projected onto the reduced bases.
Assessment: The accuracy of the MOR approximation is checked against a residual tolerance ( $\tau_{mor}$ ).
Fallback: If the MOR approximation fails the tolerance, the full linear system is solved using the high-fidelity solver (either a converged CG or a "one-shot" iteration), and the bases are updated with the new solution.

C. Improved Stopping Criterion

The paper identifies that standard stopping criteria for the Conjugate Gradient (CG) method (based on relative residual $\|r\|/\|b\|$ ) are too strict for adjoint equations in TO, leading to unnecessary iterations.

Proposed Solution: The authors adopt an alternative stopping criterion based on the work of Banoczi et al. (1998) and Arioli et al. (1992):
$w_2 = \frac{\|r^{(j)}\|}{\|b\| + \|A\|\|x^{(j)}\|} \leq \tau$
Benefit: This criterion provides a more uniform error bound that is less dependent on the right-hand side vector, allowing for more efficient use of reduced models, particularly for the adjoint system where the standard criterion often fails.

D. Integration with One-Shot Method

The authors investigate two strategies for generating the snapshots used to build the reduced bases:

High-Fidelity Snapshots: Using fully converged solutions.
One-Shot Snapshots: Using solutions from the "one-shot" method (a single iteration of the multigrid preconditioned CG). This allows the MOR to leverage the extreme speed of the one-shot method while correcting errors via the reduced model.

3. Key Contributions

Dual-Basis MOR Framework: A robust method for separately reducing forward and adjoint equations, allowing independent update frequencies and better handling of non-self-adjoint problems.
Robust Basis Update Algorithm: An incremental Gram-Schmidt procedure that correctly recovers the desired subspace when discarding old vectors, ensuring stability in long optimization runs.
Optimized Stopping Criterion: The demonstration that the alternative residual norm ( $w_2$ ) is superior for TO applications, specifically enabling effective MOR usage for adjoint equations where standard criteria are too restrictive.
Hybrid Acceleration Strategy: The novel combination of MOR with the one-shot method, proving that reduced bases can be effectively constructed from inaccurate, fast solutions to achieve further speedups.

4. Results

The methods were validated on 2D and 3D thermal topology optimization problems (heat sink design) using OpenFOAM.

2D Problem (Stopping Criterion Analysis)

Finding: Using the standard stopping criterion ( $w_1$ ) resulted in poor performance for the adjoint MOR because the error metric was too sensitive to the right-hand side.
Outcome: Switching to the proposed criterion ( $w_2$ ) allowed the adjoint reduced model to be used effectively, resulting in designs with identical quality to the standard method but with better computational efficiency.

3D Problem (Performance Comparison)

The authors compared four strategies:

Standard MGCG: Full convergence for all steps.
MOR + MGCG: MOR bases built from converged solutions.
One-Shot (MGCG_1): Single iteration per step.
MOR + One-Shot: MOR bases built from one-shot solutions.

Key Metrics:

MOR vs. Standard MGCG: Achieved a 3.5x speedup (walltime reduced from ~3722s to ~1061s) with final objective values within 4% of the high-fidelity solution.
One-Shot vs. Standard MGCG: The one-shot method alone provided a 10x speedup over the standard method.
MOR + One-Shot vs. One-Shot: Combining MOR with the one-shot method yielded an additional 1.53x speedup over the one-shot method alone.
- Total speedup over standard MGCG: ~16x.
- Final designs maintained qualitative similarity (tree-like structures) and thermal performance within 3% of the reference.
Gradient Accuracy: While the one-shot method showed higher directional accuracy in gradients, the MOR-enhanced versions maintained sufficient accuracy to converge to optimal designs without instability.

5. Significance

This paper demonstrates that Model Order Reduction is not just a replacement for high-fidelity solvers but a powerful accelerator that can be layered on top of other acceleration techniques like the one-shot method.

Scalability: The approach significantly reduces the computational cost of 3D thermal TO, making large-scale problems more tractable.
Efficiency: By using the alternative stopping criterion and dual bases, the method maximizes the utility of reduced models, avoiding the "bottleneck" of frequent full-order solves.
Practicality: The ability to build accurate reduced bases from "inaccurate" one-shot solutions suggests a pathway to even faster optimization cycles for complex physics problems where full convergence at every step is prohibitive.

In summary, the proposed framework achieves a 16-fold reduction in total simulation time compared to standard high-fidelity workflows while preserving the quality of the optimized thermal designs.