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A hybrid s-version isogeometric strategy for dynamic crack propagation in 2D and 3D problems

This paper proposes a hybrid s-version isogeometric analysis (hS-IGA) strategy that combines B-spline global discretization with Lagrange-based local meshes to accurately and efficiently evaluate dynamic fracture quantities, significantly reducing integration costs while eliminating coupling discontinuities in both 2D and 3D dynamic crack propagation problems.

Original authors: Tianyu He, Kosei Kurosaki, Naoki Morita, Naoto Mitsume, Kazuki Shibanuma

Published 2026-08-07

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CC BY 4.0

Original authors: Tianyu He, Kosei Kurosaki, Naoki Morita, Naoto Mitsume, Kazuki Shibanuma

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 trying to predict how a crack will race through a piece of glass or steel when it gets hit. This isn't just about watching a crack grow; it's about understanding the invisible, intense forces right at the very tip of that crack. In the world of engineering, this is called fracture mechanics. To do this, scientists use powerful computer simulations. Think of these simulations like a digital map of the material. To get the answer right, the map needs to be incredibly detailed right where the crack is, because that's where the action is. But the rest of the object might be huge and boringly uniform. If you make the entire map super-detailed, your computer would take forever to crunch the numbers, like trying to count every single grain of sand on a beach just to measure a single pebble.

To solve this, engineers use a clever trick called the "s-method." Imagine you have a big, low-resolution photo of a landscape (the whole structure). Then, you take a tiny, high-resolution magnifying glass (a fine mesh) and stick it right over the crack. The computer tries to blend these two views together. The problem is, the "low-res" part of the photo is made of blocky, pixelated squares. When the computer tries to blend the smooth, high-res view with the blocky view, the edges get jagged and messy. To fix the mess, the computer has to cut the blocky squares into even tinier pieces to get the math right, which makes the calculation slow and heavy again. This paper introduces a new way to smooth out those edges without losing the detail, making the whole process much faster and more accurate.

The researchers, led by Tianyu He and Kazuki Shibanuma, propose a new strategy they call "hybrid s-version isogeometric analysis," or hS-IGA for short. Think of their idea as upgrading the "low-res" background photo from blocky pixels to smooth, flowing curves, while keeping the "high-res" magnifying glass exactly the same. In the old way, the background was built using standard building blocks (Lagrange basis functions) that only connect at their corners, creating sharp, discontinuous edges. The new method swaps these blocks for smooth, mathematical curves (B-splines) for the background. These curves flow seamlessly from one section to the next, like a smooth river instead of a staircase.

Why does this matter? Because when the smooth background meets the detailed magnifying glass, the computer doesn't have to do all that extra, messy cutting and pasting to make the math work. The paper shows that by using these smooth curves for the big picture and keeping the standard blocks for the crack area, the computer can calculate the forces at the crack tip much more efficiently. In their tests, this new method reduced the number of calculation steps needed by about 81% in 2D simulations and a staggering 95.6% in 3D simulations compared to the old method.

The team didn't just guess this would work; they put it to the test. They simulated cracks in flat plates and 3D solids, both stationary and moving at high speeds. They checked if the new method could accurately predict two critical things: the "Dynamic Stress Intensity Factor" (a measure of how hard the crack is being pushed) and the "local stress" (the actual pressure right in front of the crack). The results showed that the new hS-IGA method was just as accurate as the old, slower methods, and even slightly better at avoiding errors near the edges of the magnifying glass.

Crucially, the paper argues against the idea that you need to make everything smooth and high-tech to get good results. They found that while the background needs to be smooth to avoid math headaches, the area right around the crack actually works best with the standard, blocky approach. This is because the crack tip is a sharp, jagged point where the material tears apart, and the standard blocks handle that "tearing" behavior naturally. If you tried to use the smooth curves for the crack itself, it would be like trying to draw a sharp tear with a soft, flowing pen—it just doesn't fit the physics of the break.

So, the main finding is that a "hybrid" approach is the sweet spot: smooth curves for the big, quiet parts of the structure to speed things up, and standard blocks for the chaotic, tearing part of the crack to keep it accurate. The authors suggest that this strategy could be a game-changer for analyzing how materials fail, especially in complex 3D structures where the old methods get bogged down by the sheer number of calculations required. While the current study focused on simple, linear cracks in steel-like materials, the authors hint that this efficient method could eventually help tackle even tougher problems, like how metals bend and break under extreme stress. For now, it stands as a verified, highly efficient tool for simulating dynamic crack propagation, proving that sometimes, the best way to solve a complex problem is to mix the smooth with the sharp.

Technical Summary: A Hybrid s-Version of Isogeometric Analysis for Dynamic Crack Propagation

Problem Statement
Accurate simulation of dynamic crack propagation requires the reliable evaluation of near-crack fracture quantities, specifically the Dynamic Stress Intensity Factor (DSIF) and local stress ahead of the crack front. These quantities are governed by mechanical fields that exhibit high gradients, necessitating extremely fine spatial resolution near the crack tip. However, structural dimensions are typically orders of magnitude larger than the required refinement zone, creating a severe scale-gap problem. While the conventional s-version of the finite element method (s-method) addresses this by superimposing a refined local mesh onto a coarse global mesh, it suffers from a significant computational bottleneck. In the conventional Lagrange-based s-method, the global approximation employs basis functions with only $C^0$ -continuity across element boundaries. When a local element overlaps multiple global elements, the derivatives of the global basis functions become discontinuous within the local element. This discontinuity renders standard Gauss quadrature inefficient for coupling integrals, necessitating recursive subdivision of the integration domain. This subdivision drastically increases the number of integration points and matrix assembly costs, often offsetting the efficiency gains of the global-local strategy.

Methodology
The paper proposes a Hybrid s-version of Isogeometric Analysis (hS-IGA) strategy to resolve the integration bottleneck while preserving the advantages of the s-method. The core innovation lies in a hybrid discretisation approach:

Global Mesh: The global domain is discretised using B-spline basis functions (Isogeometric Analysis). These functions possess higher-order continuity ( $C^{p-1}$ for degree $p$ with simple knots), ensuring that their derivatives remain continuous across element boundaries.
Local Mesh: The local crack domain retains the standard Lagrange-based finite element discretisation. This choice is motivated by the need to naturally represent the dynamically propagating crack tip/front as a point of $C^0$ -continuity (but not $C^1$ ) and to facilitate standard post-processing procedures for fracture parameters and nodal force release techniques.

By employing B-splines in the global mesh, the discontinuities in the coupling integrands caused by global element boundaries are eliminated. Consequently, the coupling terms between the global and local domains can be integrated accurately using standard Gauss quadrature without the need for recursive subdivision. The crack propagation is modelled using a nodal force release technique on the local Lagrange mesh, with the local mesh updating dynamically to track the crack front.

Key Contributions

Formulation of hS-IGA: A novel framework that integrates IGA into the global approximation of the s-method while maintaining a Lagrange-based local mesh. This hybrid approach specifically targets the continuity-related integration bottleneck of the conventional s-method.
Elimination of Recursive Subdivision: The strategy enables accurate coupling integration using standard quadrature rules, removing the computational overhead associated with subdividing integration domains in the conventional s-method.
Preservation of Local Crack Representation: By retaining Lagrange elements locally, the method maintains the straightforward representation of crack surfaces and the established procedures for nodal force release and local stress evaluation, which are critical for dynamic fracture analysis.

Results
The proposed strategy was verified through two-dimensional (straight crack) and three-dimensional (circular crack) benchmarks under both stationary and dynamic loading conditions.

Accuracy: The hS-IGA strategy demonstrated accuracy comparable to both the standard Finite Element Method (FEM) and the conventional s-method. It accurately evaluated DSIF and local stress fields, with errors generally remaining below 1.5% for DSIF and 4% for local stress across various crack velocities and mesh configurations.
Computational Efficiency: The most significant finding was the drastic reduction in computational cost associated with coupling integration.
- In 2D dynamic benchmarks, the number of integration points required was reduced by approximately 81% compared to the conventional s-method.
- In 3D dynamic benchmarks, the reduction was even more pronounced, reaching approximately 95.6% relative to the conventional s-method.
- The method also reduced the number of integration points by approximately 98% in 2D stationary and 99% in 3D stationary benchmarks compared to the conventional approach.
Robustness: The strategy maintained stability and accuracy across a wide range of crack velocities (200–1500 m/s) and different global-to-local mesh size ratios, showing no sensitivity to the global mesh density for local stress evaluation near the crack tip, a known issue in the conventional s-method.

Significance
The paper claims that the hS-IGA framework provides an accurate and efficient global-local strategy for dynamic crack propagation analyses where reliable evaluation of near-crack fracture quantities is essential. Its primary significance lies in resolving the principal computational bottleneck of the conventional s-method—the excessive number of integration points required for coupling—without sacrificing the method's inherent advantages in handling multiscale problems. The authors suggest that while the current study is limited to planar linear-elastic problems, the concept is promising for more advanced applications, such as three-dimensional elastoplastic fracture analyses, where the computational burden of repeated coupling integration is even more critical.

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