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A 2.5D NURBS-Trace Infinite-Element Method for Moving-Load Wave Propagation and Soil--Structure Interaction in Semi-Infinite Ground

This paper presents a 2.5D NURBS-trace infinite-element method (NBIEM) that combines isogeometric analysis for the near field with tensor-product NURBS and exponential radial functions for the far field to efficiently and accurately model moving-load wave propagation and soil-structure interaction in semi-infinite, heterogeneous geotechnical media without requiring radial quadrature or artificial boundaries.

Original authors: Yanhui Zhong, Hao Hong, Bei Zhang, Quansheng Zang, Hussein Rappel, Stéphane P. A. Bordas

Published 2026-07-15

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Yanhui Zhong, Hao Hong, Bei Zhang, Quansheng Zang, Hussein Rappel, Stéphane P. A. Bordas

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 listen to a train zooming past a field, but the field is actually an endless ocean of soil that stretches forever in every direction. If you tried to build a computer model of this entire infinite ocean to see how the ground shakes, your computer would likely melt. It's like trying to count every single grain of sand on a beach just to see how one footstep makes the sand ripple.

This paper introduces a clever new trick called the 2.5D NURBS-Trace Infinite-Element Method (or NBIEM for short) to solve this "infinite ocean" problem without needing a supercomputer the size of a city.

The Magic Trick: 2.5D

First, let's talk about the "2.5D" part. Usually, to model a train moving down a track, you need a full 3D model. But if the ground and the track look the same all the way down the line (like a long, straight tunnel or a straight road), you don't need to model the whole 3D world. You only need to model a single slice of the ground (the cross-section) and then use a mathematical "magic spell" (a Fourier transform) to imagine how that slice repeats itself as the train zooms by. This turns a massive 3D puzzle into a series of smaller, manageable 2D puzzles.

The Problem: The Endless Edge

Even with this 2D slice, there's a catch. The ground doesn't stop; it goes on forever. In old computer models, scientists would just chop the ground off at a certain distance and pretend it ends there. But that's like cutting off the end of a trampoline; the waves would bounce back off your fake cut-off edge and mess up the results.

To fix this, they used "infinite elements." Think of these as special, magical extensions attached to the edge of your model that know how to let waves escape into the void without bouncing back.

The New Solution: NURBS-Trace

The authors' big innovation is how they connect the "near field" (the ground right under the train) to this "infinite field" (the rest of the world).

The Old Way: Imagine the near field is made of Lego bricks, and the infinite field is made of Play-Doh. To connect them, you have to squish the Play-Doh to fit the Lego, which often creates gaps or wobbly connections.
The New Way (NBIEM): The authors use a technique called Isogeometric Analysis. Instead of Lego and Play-Doh, they use the same smooth, flexible material (mathematical curves called NURBS) for both the near field and the infinite field.

They call this a "NURBS-trace" method. It's like having a single, seamless ribbon that flows from the train tracks out into infinity. Because the ribbon is the same material on both sides, they don't need to force-fit them together. The connection is perfect, smooth, and mathematically exact.

The "No-Quadrature" Shortcut

Here is the really cool part. Usually, when calculating how these infinite waves behave, computers have to do millions of tiny, repetitive math steps (called numerical quadrature) to guess the answer. It's slow and can get messy.

The authors found a way to write down the answer for the infinite part using a closed-form formula. Think of it like this: instead of counting every single step a runner takes to get to the finish line, they found a single equation that tells you exactly how long the race takes. This means their computer doesn't have to chop the infinite ground into tiny pieces or do millions of guesses. It just plugs the numbers into the formula, and boom—the answer is there. This makes the method incredibly fast.

What They Tested (and What They Didn't)

The team didn't just dream this up; they put it through the wringer in a series of simulations:

The "Speed Test": They simulated trains moving at different speeds:
- Sub-Rayleigh: Slower than the ground's natural vibration speed (like a car driving on a bumpy road).
- Super-shear: Faster than the shear waves but slower than the compression waves (a bit like a sonic boom starting to form).
- Super-compressional: Faster than all ground waves (a true sonic boom in the soil).
- The Result: In all these cases, their new method matched the known "perfect" math solutions almost exactly, even for the stress and strain (how much the ground is being squeezed or stretched).
The "Comparison": They compared their new method against two other popular ways of doing this:
- FEM/IEM: The standard, older way.
- IGA/SBIGA: A more complex, newer way.
- The Result: Their new method was much faster than the standard way and much faster than the complex newer way (which took 168 seconds to solve a problem their method solved in 0.023 seconds!). It was also more accurate than the standard way, especially at higher frequencies.
Real-World Scenarios: They simulated:
- Layered Ground: Like a cake with different flavors of soil.
- Buried Tunnels: Like the Zhengzhou Metro Line 1. They checked how deep the tunnel needs to be to stop vibrations from reaching the surface. They found that once the tunnel is about 12 to 14 meters deep, the vibrations above it drop off significantly, and going deeper gives less and less benefit.
- Soft Layers: They tested what happens if there's a soft, squishy layer of soil under the road. They found that treating the soil to a depth of about 1.2 to 1.5 meters helps the most; going deeper than that doesn't help much more.

What They Explicitly Rule Out

The paper is very clear about what this method is not:

It is not a method that requires you to know the "perfect" answer beforehand to tune your settings. They created a "reference-free" workflow that works without needing an analytical solution to check against.
It is not a method that relies on "Perfectly Matched Layers" (PML), which are thick, artificial absorbing layers that need careful tuning. Their method uses the infinite elements directly attached to the boundary.
It is not a method that suggests the "calibrated" version (which uses extra complex math to tweak the decay) is always necessary. They found that the simpler "all-S" version (using just one type of wave info) works great for most engineering needs, and the complex calibration is only a bonus for specific low-frequency cases.

How Sure Are They?

The authors are very confident in their simulations. They have proved through their own computer models that:

The method works for different train speeds.
It is faster and accurate compared to existing methods.
It handles complex shapes (like curved tunnels and layered soil) well.

However, they are careful to say these are simulations. They haven't yet compared their results to real-world measurements from a real train track (though they suggest that would be a great next step). They also note that their "calibrated" settings were derived from a specific benchmark and might need adjustment if the ground conditions change drastically.

The Bottom Line

This paper presents a new, super-fast, and smooth way to model how the ground shakes when a train or heavy truck drives by. By using a seamless mathematical ribbon to connect the ground near the road to the infinite ground far away, and by using a "shortcut formula" to calculate the infinite part, they can solve these problems much faster than before. It's a tool that engineers can use to design better tracks and tunnels, ensuring that vibrations don't bother the people living nearby, all without needing a computer that costs more than a small country's GDP.

Technical Summary: A 2.5D NURBS-Trace Infinite-Element Method for Moving-Load Wave Propagation

Problem Statement
Ground vibration induced by moving loads (e.g., high-speed trains, heavy traffic) is a critical concern in transportation geotechnics. Predicting these vibrations requires modeling wave propagation in semi-infinite, unbounded ground domains. While the response is inherently three-dimensional, systems with geometry and material properties invariant along the traveling direction (longitudinal $z$ -axis) can be efficiently analyzed using a 2.5D formulation. This approach reduces the problem to a sequence of 2D cross-sectional problems in the wavenumber–frequency domain via Fourier transformation. However, accurately modeling the unbounded exterior domain while satisfying radiation conditions and maintaining geometric consistency with complex interfaces (curved tunnels, layered soils) remains a challenge. Existing methods, such as Finite/Infinite Element (FEM/IEM) coupling, Boundary Element Methods (BEM), and Scaled Boundary Finite Element Methods (SBFEM), often face trade-offs between computational cost, geometric flexibility, and the need for fundamental solutions or artificial absorbing layers.

Methodology
The paper proposes a 2.5D Non-Uniform Rational B-spline (NURBS)-trace Infinite-Element Method (NBIEM). This formulation couples a bounded near-field domain with a semi-infinite exterior domain using a unified isogeometric framework.

2.5D Reduction: The governing equations of motion for linear viscoelastic media are transformed into the frequency-wavenumber domain. The moving load is treated as a harmonic excitation where the temporal frequency $\omega$ and longitudinal wavenumber $k$ are coupled via the load speed $c_{load}$ ( $k = (\omega - \omega_0)/c_{load}$ ).
Bounded Near-Field (IGA): The near-field cross-section is discretized using Isogeometric Analysis (IGA). Both the geometry and the displacement field are represented by the same NURBS basis functions, allowing for exact representation of curved boundaries and high-order continuity.
Semi-Infinite Exterior (NURBS-Trace Infinite Elements): The exterior domain is modeled using infinite elements constructed as tensor products of:
- The boundary NURBS trace (shared with the near-field discretization).
- Admissible radial functions ( $P_q$ ) representing outgoing or evanescent waves (e.g., exponential decay).
- Crucially, the near-field and far-field share the same discrete NURBS trace space and boundary control-point degrees of freedom. This enforces displacement continuity directly without projection, mortar variables, or interface transfer operations.
Closed-Form Integration: For selected exponential radial functions, the radial moments required for the far-field stiffness and mass matrices are evaluated in closed form. This eliminates the need for finite radial cutoffs or numerical radial quadrature, ensuring exact satisfaction of the radiation condition within the chosen functional space.
Radiation Condition Handling: The method employs an "all-S" realization where S-wave-informed radial functions are assigned to all boundary segments. The decay parameters are determined by the exterior material properties and geometric scales. To ensure robustness without analytical references, the authors introduce two sensitivity indicators: $I_\beta$ (sensitivity to radial decay parameters) and $I_D$ (sensitivity to boundary placement).

Key Contributions
The paper makes three primary contributions:

Formulation: Development of a three-component 2.5D Galerkin NURBS-trace infinite-element formulation. It uniquely couples the bounded IGA domain and the semi-infinite exterior through an identical discrete NURBS trace, preserving full longitudinal-wavenumber-dependent strain and inertia coupling among all three displacement components.
Analytical Efficiency: Evaluation of far-field stiffness and mass contributions via closed-form radial moments for selected exponential radial functions. This avoids finite radial truncation and numerical integration in the radial direction.
Systematic Characterization: Comprehensive verification and assessment of the method, including:
- Validation against closed-form half-space solutions across sub-Rayleigh, super-shear, and super-compressional regimes.
- Development of reference-free sensitivity indicators ( $I_\beta$ and $I_D$ ) to screen boundary adequacy.
- Direct comparison with classical FEM/IEM and IGA/Scaled-Boundary IGA (IGA/SBIGA) regarding accuracy, phase fidelity, and computational cost.

Results

Accuracy: The NBIEM reproduces displacement and stress frequency-response functions with high accuracy, matching analytical solutions for both sub-Rayleigh and super-compressional moving loads. Stress results (derivative-dependent fields) confirm the method's ability to capture gradients, benefiting from the higher continuity of NURBS bases.
Comparison with Alternatives:
- vs. FEM/IEM: NBIEM achieves comparable or superior displacement and phase accuracy with significantly fewer degrees of freedom. It is approximately one order of magnitude faster in wall-clock time for moderate-to-large meshes due to the elimination of scaled-boundary dynamic stiffness construction.
- vs. IGA/SBIGA: NBIEM achieves similar accuracy to IGA/SBIGA in the intermediate-to-high frequency range but at a substantially lower computational cost. Unlike IGA/SBIGA, it does not require solving a scaled-boundary dynamic stiffness system, avoiding the associated high assembly costs.
Engineering Applications: The method successfully models:
- Layered Ground: Handling material interfaces and stiffness contrasts.
- Track-Subgrade Systems: Analyzing embankment vs. cutting configurations with multi-patch geometries, revealing speed-dependent amplification at deeper interfaces.
- Buried Structures: Simulating metro tunnels with varying cover depths, demonstrating how increased burial depth attenuates vibration transmission to the soil above the tunnel crown.
Sensitivity: The "all-S" default realization was found to be robust. The sensitivity indicators effectively identified cases requiring boundary enlargement (e.g., low frequencies or super-shear speeds), ensuring solution reliability without analytical benchmarks.

Significance and Claims
The paper claims that the NBIEM provides a geometrically consistent and computationally efficient framework for 2.5D wave propagation and soil-structure interaction in semi-infinite domains. Its significance lies in:

Geometric Consistency: By using NURBS traces for both near and far fields, it handles complex curved interfaces and multi-patch configurations naturally, avoiding the geometric approximation errors common in low-order FEM.
Computational Efficiency: It offers a favorable accuracy-cost balance, particularly for repeated frequency-domain analyses, by avoiding the expensive dynamic stiffness construction of SBFEM and the fundamental solution dependencies of BEM.
Robustness: The proposed sensitivity indicators allow for a reference-free, automated screening of boundary settings, making the method practical for engineering applications where analytical solutions are unavailable.

The authors position NBIEM not as a replacement for all existing methods but as a distinct, efficient alternative that bridges the gap between the geometric flexibility of isogeometric analysis and the unbounded domain treatment of infinite elements, specifically tailored for moving-load problems in geotechnical engineering.