💻 computer science

Thermo-Structural Topology Optimization Considering Nonlinear Creep

This paper introduces a differentiable thermo-structural topology optimization framework using the Norton creep model and JAX-based automatic differentiation to minimize permanent deformation in high-temperature metallic components, demonstrating its efficacy through 2D benchmarks and a complex 3D graded material turbine blade design.

Original authors: Stefan Knapik, Aaditya Chandrasekhar, Deepak Sharma, Jian Cao, Changjie Sun, Wei Chen

Published 2026-07-28

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Stefan Knapik, Aaditya Chandrasekhar, Deepak Sharma, Jian Cao, Changjie Sun, Wei Chen

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 building a bridge, but instead of worrying about it snapping under a heavy truck today, you are worried about it slowly stretching out of shape over the next fifty years. This slow, permanent stretching is called creep. It happens when materials, especially metals, are subjected to constant stress while sitting in very hot environments, like inside a jet engine or a power plant. Even if the metal isn't hot enough to melt, the heat makes its atoms wobble and slide past each other, causing the part to slowly deform, like a piece of taffy left in the sun. If engineers don't account for this, the part might eventually fail, closing gaps where they shouldn't be or losing its shape entirely.

To stop this, engineers use a powerful computer tool called topology optimization. Think of this as a digital sculptor that figures out the perfect shape for a part by moving material around. Usually, these tools only look at how strong a shape is right now (elasticity). But for hot, long-lasting parts, that's not enough. You need a tool that can predict how the shape will change over time as it "creeps." This paper introduces a new, super-smart version of that digital sculptor. It combines the heat of the environment with the slow, time-dependent stretching of the metal to find shapes that stay strong and true for decades, rather than just for a moment.

The Digital Sculptor That Sees the Future

In the world of engineering, designing parts for jet engines is a bit like trying to build a house of cards in a wind tunnel. You need it to be light enough to fly, but strong enough to hold up under massive forces and scorching heat. The problem is that heat changes the rules. When metal gets hot and stays under pressure, it doesn't just bend; it slowly, permanently stretches. This is creep.

For a long time, engineers had to guess how to design for this. They would design a shape to be stiff today, hoping it wouldn't stretch too much tomorrow. But as the authors of this paper point out, that's a bit like trying to drive a car by only looking at the rearview mirror. You need to see where you're going.

The researchers, working at Northwestern University and GE Aerospace, built a new computer framework that acts like a time-traveling architect. They created a system that doesn't just ask, "Is this shape strong?" but asks, "How will this shape look and feel after 100 years of being hot and stressed?"

The Secret Sauce: A Differentiable Time Machine

The magic behind this new tool is a clever mix of physics and computer science. The team used a famous rule called the Norton model to describe how metals creep. Think of the Norton model as a recipe that tells you exactly how fast a specific metal will stretch based on how hot it is and how hard you are pulling on it. The hotter it gets, the faster it stretches, and the harder you pull, the faster it stretches even more.

To make this work in a design tool, they had to solve a tricky problem: calculating how changing the shape affects the creep over time. Usually, this requires doing complex math by hand, which is slow and prone to errors. Instead, the authors used a modern programming library called JAX. You can think of JAX as a super-powered calculator that can automatically figure out how every single tiny change in the design ripples through the entire simulation. It's like having a robot that can instantly tell you, "If I make this beam 1% thicker, the whole engine will stretch 5% less over 10 years."

They also used a method called backward Euler to step through time. Imagine watching a movie of the metal stretching, but instead of watching it frame-by-frame, the computer jumps ahead in big, safe steps, calculating the final state without getting lost in the details of every single second. This keeps the computer from getting overwhelmed, even when simulating a whole year of service life in just a few minutes.

What They Found: Shape Matters, But Heat Matters More

The team tested their new tool on several problems, and the results were surprising and instructive.

1. The "Stiff" Trap
First, they compared a design made with their new "creep-aware" tool against a traditional design made just to be stiff today. At first glance, the two shapes looked almost identical. But when they simulated a year of operation, the traditional design had stretched significantly, while the creep-aware design stayed nearly the same. The new tool had quietly rearranged the material to handle the long-term stress, proving that what looks good today might fail tomorrow.

2. The Heat Gradient Effect
Next, they looked at what happens when one side of a part is hot and the other is cold. In a uniform temperature, the shape stays symmetrical. But when they created a temperature gradient (hot on one side, cold on the other), the optimizer did something fascinating. It made the hot side of the structure much thicker and the cold side thinner. Why? Because the heat makes the metal on the hot side want to stretch much faster. The optimizer decided to "armor" the hot side with extra material to stop it from giving in, while letting the cool side get away with being thinner. It's like wearing a heavy winter coat on the side of the house facing the blizzard, but leaving the sunny side with just a light jacket.

3. Heavy Loads Change the Game
They also tested how the design changed when the load (the force pulling on the part) got heavier. Under light loads, the optimizer created intricate, web-like structures with many thin struts. But as the load increased, the design suddenly became blocky and solid, with fewer, much thicker beams. The reason? The math of creep is non-linear. A tiny bit of extra stress on a thin beam causes a massive amount of stretching. So, under heavy loads, the smartest move is to abandon the fancy, thin webs and just build thick, sturdy pillars to keep the stress low.

The Ultimate Challenge: The Graded Blade

The final test was the most ambitious: designing a 3D turbine blade. Instead of just moving material around, they let the computer mix two different metals: Inconel 625 (a super-strong, expensive alloy that resists creep) and Stainless Steel 316 (cheaper and more conductive, but less creep-resistant).

The goal was to figure out exactly where to put the expensive Inconel and where to use the cheaper steel. The result was a "graded" blade. The optimizer placed the expensive, creep-resistant Inconel right where the heat and stress were highest—the hot tip and the suction side of the blade. It left the cooler, less-stressed root and pressure side mostly made of the cheaper steel.

This approach was incredibly efficient. By grading the materials, they achieved the same performance as a blade made entirely of expensive Inconel, but using only 40% of the superalloy. It's like building a high-performance car where you only use the most expensive, high-tech tires on the wheels that actually touch the road, and standard tires everywhere else.

Why This Matters

This paper doesn't just offer a new way to draw shapes; it offers a new way to think about time and heat in engineering. The authors show that if you ignore the slow, creeping stretch of hot metals, you will design parts that look great on paper but fail in reality. By using their new, differentiable framework, engineers can now design components that are not only strong today but will stay true to their shape for decades.

The simulations suggest that for high-temperature applications, the best design isn't just about being stiff; it's about being smart about where you put your resources. Whether it's thickening a beam on the hot side or mixing in a super-alloy exactly where the heat is worst, the future of engineering is about designing for the long haul. And thanks to this new tool, we can finally see that future before we build it.

Technical Summary: Thermo-Structural Topology Optimization Considering Nonlinear Creep

Problem Statement

Metallic components operating at high temperatures, such as turbine blades and heat exchangers, are subject to creep: a time-dependent, permanent deformation that occurs under sustained loads even when stresses remain below the yield stress. While Topology Optimization (TO) has been successfully applied to static stiffness and stress-constrained designs, its application to creep remains limited. Existing creep-aware TO frameworks predominantly rely on linear viscoelastic models (e.g., Maxwell models). These models fail to capture the critical temperature dependence and nonlinear stress sensitivity inherent in metallic creep at elevated temperatures. Furthermore, prior work often neglects the coupling between transient thermal fields and nonlinear creep behavior, leading to designs that may be suboptimal for high-temperature service.

The core challenge addressed is the development of a design framework that can handle nonlinear, history-dependent, and thermomechanically coupled creep responses to minimize permanent deformation over a prescribed service life.

Methodology

The authors propose a differentiable thermo-structural TO framework implemented entirely within the JAX library, leveraging automatic differentiation (AD) for sensitivity analysis.

1. Constitutive Modeling

Creep Law: The framework utilizes the Norton power-law model to describe the secondary (steady-state) creep regime, which dominates long-term deformation in metals. The creep strain rate is defined as $\dot{\varepsilon}_{cr} = A(T) \sigma_{eq}^n \mathbf{N}$ , where the coefficient $A(T)$ follows an Arrhenius relationship ( $A(T) = A_0 \exp(-Q/RT)$ ) to capture temperature sensitivity.
Strain Decomposition: The total strain is additively decomposed into elastic, thermal, and inelastic creep components: $\varepsilon_{ij} = \varepsilon_{ij}^{el} + \varepsilon_{ij}^{cr} + \varepsilon_{ij}^{th}$ .
Time Integration: The transient response is solved using an implicit backward Euler scheme over discrete time steps representing the service life.

2. Numerical Implementation

Finite Element Analysis (FEA): The thermal problem is solved as a steady-state PDE first. The resulting temperature field drives the structural problem, which is solved as a transient, nonlinear equilibrium problem.
Differentiable Solver: The entire simulation pipeline (thermal solve, structural equilibrium, and local constitutive updates) is implemented in JAX.
- Global Equilibrium: Solved via Newton-Raphson.
- Local Constitutive Update: Solved via Newton-Raphson at each Gauss point to resolve the implicit creep residual.
- Sensitivity Analysis: Instead of manual derivation of adjoint equations, the framework uses reverse-mode automatic differentiation combined with the Implicit Function Theorem (IFT). This approach avoids unrolling iterative solver histories, significantly reducing memory requirements while providing machine-precise gradients.

3. Optimization Formulation

Objective: Minimize the creep compliance, defined as the work done by external loads acting through the accumulated creep displacement ( $J = \int_{\Gamma_t} \mathbf{u}_{cr} \cdot \bar{\mathbf{t}} \, d\Gamma$ ).
Constraints: A volume constraint limits the material usage.
Design Variables: Elemental pseudodensities are filtered and penalized (using RAMP) to drive the design toward a binary solid-void distribution.
Optimizer: The Method of Moving Asymptotes (MMA) is used for gradient-based updates.

Key Results and Experiments

The framework was validated and demonstrated through a series of numerical experiments:

Validation against Elastic Design:
- A 2D cantilever beam optimized for creep compliance was compared to one optimized for initial elastic compliance.
- Result: While topologies appeared visually similar, the creep-aware design exhibited significantly lower long-term permanent deformation. The optimizer reallocated material to suppress peak stress regions (where Norton flow is most aggressive), thickening members near fixed boundaries.
Pareto Trade-off (Volume Fraction):
- Varying the volume fraction constraint revealed a strongly convex trade-off between material usage and creep compliance.
- Result: Due to the nonlinear stress scaling ( $\sigma^n$ ), modest increases in material volume (reducing peak stress) yield disproportionately large reductions in creep compliance. Beyond a certain volume fraction, diminishing returns set in.
Thermal Coupling Effects:
- Experiments compared designs under uniform low/high temperatures versus a thermal gradient.
- Result: Under a thermal gradient, the optimizer produced an asymmetric topology, thickening members on the hot side and thinning them on the cool side. This "creep shielding" strategy exploits the fact that reducing stress in high-temperature regions (where the Arrhenius factor is large) is far more valuable than in cool regions. Uniform temperature shifts only rescaled the compliance without altering the topology.
Load Magnitude Sensitivity:
- Designs were optimized for varying load magnitudes (100, 200, 400 MPa).
- Result: Unlike elastic optimization where topology is often load-invariant, the creep-optimal topology depends qualitatively on the load magnitude. At high loads, the optimizer abandoned intricate truss-like substructures in favor of fewer, thicker load-bearing members to suppress peak stresses, as the marginal creep penalty of stress concentrations grows explosively with load.
3D Graded Alloy Turbine Blade:
- A challenge problem involving the compositional design of a turbine blade using a mixture of Inconel 625 (high creep resistance) and Stainless Steel 316 (high thermal conductivity).
- Result: Continuously graded designs strictly Pareto-dominated homogeneous blends. By concentrating the creep-resistant alloy (Inconel 625) specifically in the hot tip and high-stress suction side, the graded design achieved the same creep compliance as a homogeneous design using roughly 33% more of the expensive superalloy.

Significance and Claims

The paper claims the following contributions and significance:

First Coupled Framework: To the authors' knowledge, this is the first TO framework to couple a steady-state thermal field with a transient, nonlinear Norton creep model, capturing the Arrhenius temperature dependence of the creep coefficient.
Differentiable Simulation: The work demonstrates that implementing complex, history-dependent physics in a differentiable environment (JAX) eliminates the need for manual adjoint derivations. This enables efficient, exact sensitivity analysis for nonlinear multi-physics problems.
Design Insights: The study establishes that:
- Creep-aware designs differ qualitatively from elastic designs over long service horizons.
- Thermal gradients fundamentally reshape optimal material distribution, necessitating coupled thermo-structural analysis rather than isothermal approximations.
- The applied load magnitude is a critical design specification that alters the optimal topology, not merely a scaling parameter.
- These geometric insights extend to compositional design, where optimizing material distribution spatially can significantly reduce the usage of expensive high-performance alloys.

The authors note that the current structural model assumes small strains and linear elasticity, and the Norton model captures only secondary creep. Future work would require extending the framework to include plasticity, finite strains, and primary/tertiary creep models for a complete lifespan assessment.