The problem LEACS addresses arises directly from the push toward tightly coupled multiphysics simulation. Modern reactor analysis increasingly iterates Monte Carlo transport solvers such as OpenMC, Serpent2, MCNP, and RMC together with thermal-hydraulics, fuel performance, and structural mechanics codes. When Monte Carlo transport is coupled to a finite element thermal solver, the temperature and density fields it receives are not flat. Fuel pellets can exhibit centerline-to-surface temperature gradients of several hundred Kelvin, and coolant density can change by tens of percent across an axial span. Those gradients feed back into neutron transport through temperature-dependent cross sections, and capturing that feedback is precisely the point of running a coupled calculation in the first place. Yet conventional Monte Carlo treats cross sections as piecewise-constant along a particle’s flight path, an approximation that quietly discards exactly the physics the coupling was meant to resolve.

At the heart of the challenge is the collision distance sampling step. In a standard Monte Carlo particle history, the code calculates the macroscopic cross section of the current material, samples a distance to collision from an exponential distribution, and moves the particle to whichever comes first: the sampled collision, a cell boundary, or a cutoff distance. When cross sections vary continuously in space, the cumulative distribution function from which the collision distance is sampled depends on the optical depth accumulated along the particle’s actual trajectory through the temperature field. Evaluating that integral exactly is impractical, so every practical method must approximate it somehow, and each existing approximation imposes its own tradeoffs in accuracy, generality, or speed.

The dominant existing strategies fall into three broad camps. Delta tracking, originally devised by Woodcock in 1965, samples collisions against a majorant cross section and then accepts or rejects each pseudo-collision probabilistically. It handles complex geometry gracefully, which is why it remains popular for problems like TRISO particle fuels, but it struggles when a region contains very high cross sections, such as gadolinium absorbers, and constructing a true majorant is difficult when temperature-dependent Doppler broadening shifts resonance peaks in an energy-dependent way. Adaptive binning, recently implemented in the CARDINAL code via the MoabSkinner user object, regenerates the geometry into cells of roughly equal temperature and density, restoring the simple sampling formula at the cost of a much larger geometry. Its resolution is also tied to the multiphysics mesh, which cannot be subdivided below the element size, and tetrahedral meshes are inefficient for geometries like fuel pins that are meters long but only centimeters wide. Numerical estimation methods, dating back to work by Brown, Martin, and Greisheimer, fit polynomial functions to cross sections sampled along the flight path and integrate them with quadrature rules; they are fast, but representing functional expansions on unstructured, potentially deformed meshes while preserving orthogonality is nearly impossible in three dimensions.

LEACS, which stands for Lagrange Error-Adaptive Collision Sampling, takes a different route. Rather than converting the finite element solution into a lower-order representation, it accesses the native element shape functions directly during particle transport. When a neutron enters a material flagged for LEACS treatment, the method places sample points along the flight path, queries the finite element solution at each point to obtain the local temperature, uses windowed multipole formalism to Doppler-broaden the cross sections on the fly, fits piecewise quadratic functions to the resulting cross-section profile, and inverts the reconstructed cumulative distribution function to obtain the collision distance. The windowed multipole approach is itself notable: it broadens cross sections at arbitrary temperatures with far higher accuracy than linear interpolation between library temperatures, and recent advances evaluating the Faddeeva function as a rational fraction have made it extremely efficient.

The genuinely novel element is how LEACS chooses where to place those sample points. Because querying the finite element solution is expected to be the most expensive part of the transport calculation, every sample must earn its keep. The naive choice, equidistant spacing, wastes effort: for large optical depths, the most likely collision distances cluster near the particle’s initial position, yet equal spacing devotes as many points to the unimportant region near the boundary as to the critical region near the start. LEACS instead exploits the Lagrange error bound, a classical result from Taylor’s theorem that bounds the remainder of a polynomial approximation. By requiring the approximation error to be uniform across all intervals, the method derives an analytic spacing function that concentrates points where the curvature of the collision probability distribution demands them. Every sample location is therefore analytically optimal for reconstructing the true collision distance distribution, which is how LEACS aims to limit the slowdown that finite element sampling inevitably introduces.

To test the idea, the researchers built two pseudo-pin test cases spanning the extremes of neutron flight distance in a fuel pin: a 0.95-centimeter path emulating a purely radial crossing and a 364.85-centimeter path emulating a purely axial one. The material was uniform-density uranium-238, chosen because resonance absorption in U-238 is the dominant temperature feedback effect in a typical light water reactor, and one-dimensional thermal simulations in the MOOSE framework supplied realistic temperature profiles. Comparing the reconstructed cumulative distribution functions against a finely sampled control, the analysis showed that LEACS spacing consistently outperformed equidistant spacing, with the advantage growing dramatically for long flight paths. When neutron energies were sampled from a realistic pressurized water reactor pin-cell spectrum computed in OpenMC, the error in average collision distance dropped substantially, and the gains from Lagrange spacing over equidistant spacing increased as more sample points were used.

The practical payoff showed up most clearly in the spatial tallies. Using an AP1000-based pin cell with 1 percent mixed-oxide fuel, a composition chosen because it produces the largest Doppler effect in benchmark studies, and a C5G7 benchmark assembly for larger-scale cost testing, the team coupled OpenMC transport to MOOSE thermal solutions with volumetric heating derived from the peak AP1000 linear power rating. The results revealed a clear radial and axial variation in the neutron capture reaction rate that the constant-temperature treatment misses entirely. Strikingly, the root mean square difference between a three-sample LEACS run and a twenty-one-sample reference stayed below 0.35 percent, with average z-scores below 1.0, suggesting that as few as three sample points per flight segment are sufficient to capture the spatial impact of temperature gradients on the reaction rate distribution.

Computational cost remains the method’s principal hurdle. Default collision distance sampling takes roughly 0.068 microseconds per call; LEACS timing is orders of magnitude higher, with point location and inverse mapping within the finite element sampling routine dominating the bill. Slowdown factors depend strongly on the model: 4.7 to 9.7 for a two-dimensional pin cell, up to 16.4 for a three-dimensional pin cell with eleven samples, but only 1.3 to 2.1 for full assembly models, where the baseline calculation is already expensive and neutrons spend much of their history in regions that do not invoke LEACS. The authors note that a slowdown exceeding sixteen would likely be impractical for production work, while a factor of 1.3 may be acceptable in many applications. Future optimization will target the element search and inverse mapping algorithms, and planned extensions include density coupling, a capability that would matter for molten salt and boiling water reactors. The researchers foresee the method’s best home inside the CARDINAL coupling framework, where a direct head-to-head comparison with adaptive binning on both accuracy and cost would settle how this new chapter in Monte Carlo transport should be written.

Subject of Research: Error-adaptive collision sampling for Monte Carlo neutron transport coupled to finite element thermal solutions

Article Title: LEACS: lagrange error-adaptive collision sampling for multiphysics coupled Monte carlo transport

Article References: Kendrick, W. R., & Forget, B. (2026). LEACS: lagrange error-adaptive collision sampling for multiphysics coupled Monte carlo transport. Frontiers in Nuclear Engineering, 5, Article 1900800. https://doi.org/10.3389/fnuen.2026.1900800

Image Credits: AI Generated

DOI: 10.3389/fnuen.2026.1900800

Keywords: Monte Carlo transport, neutron transport, multiphysics coupling, LEACS, OpenMC, MOOSE, finite element methods, Doppler broadening, windowed multipole, reactor physics, collision sampling, nuclear engineering