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arxiv: 2605.19730 · v2 · pith:YNDQSKMMnew · submitted 2026-05-19 · ⚛️ physics.ao-ph · physics.flu-dyn

Matrix structure and convergence behaviour of the matched eigenfunction method for computing heave wave forces on generalized concentric bodies

Yinghui Bimali , Rebecca McCabe , Collin Treacy , Kapil Khanal , En Lo , Maha Haji This is my paper

Pith reviewed 2026-05-20 01:45 UTC · model grok-4.3

classification ⚛️ physics.ao-ph physics.flu-dyn
keywords matched eigenfunction expansionhydrodynamic coefficientswave forcesconcentric cylindersslanted geometriesboundary element methodoffshore structuresconvergence behavior
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The pith

The matched eigenfunction expansion method computes heave wave forces on slanted concentric cylinders within 5 percent of boundary element results using matrices two orders of magnitude smaller.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unifying framework for the matched eigenfunction expansion method that models wave interactions with any number of fixed or heaving annular cylinders whose profiles stay continuous and radially monotonic. It examines the resulting block matrix structure, documents convergence rates for both vertical and slanted shapes, and directly compares accuracy and speed against the boundary element solver Capytaine. A sympathetic reader would care because traditional boundary element calculations create a computational bottleneck when sizing offshore structures, and a faster semi-analytical route would let designers evaluate more shapes in the same time.

Core claim

The central claim is that the matched eigenfunction expansion method supplies a single framework for an arbitrary number of fixed or heaving surface-piercing annular cylinders with continuous and radially monotonic profiles; numerical experiments then show that this framework approximates hydrodynamic coefficients of slanted geometries to within 5 percent of Capytaine even at 15 degrees from vertical, while reaching 2 percent convergence an order of magnitude faster than Capytaine and with a matrix size two orders of magnitude smaller.

What carries the argument

The block matrix structure obtained by dividing the fluid domain into annular regions and matching eigenfunction expansions of the velocity potential at each interface, which organizes the linear system solved for the added mass, radiation damping, and excitation coefficients.

If this is right

  • Hydrodynamic coefficients for a broad range of continuous radially monotonic shapes become computable with far less effort than boundary element methods.
  • Optimization loops that iterate over many offshore structure geometries become practical because each evaluation finishes an order of magnitude sooner.
  • Both fixed and heaving surface-piercing bodies can be treated inside the same matching procedure.
  • Convergence rates are now documented for slanted profiles, giving users a clear rule of thumb for choosing matrix size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The compact matrix size could let the method run inside larger coupled simulations that include multiple structures or time-varying waves.
  • A hybrid scheme that switches to boundary element methods only near sharp features might extend the approach to bodies that violate the monotonic-profile assumption.
  • Insights into the block structure may suggest similar semi-analytical reductions for related problems such as diffraction by arrays of cylinders or motion in irregular waves.

Load-bearing premise

The body profiles must stay continuous and radially monotonic so that the eigenfunction expansions in neighboring annular regions can be matched directly without extra interface conditions or geometric approximations.

What would settle it

A direct numerical comparison on a body whose profile contains a radial discontinuity or reversal would show whether the hydrodynamic coefficients diverge sharply from a high-resolution boundary element reference solution.

Figures

Figures reproduced from arXiv: 2605.19730 by Collin Treacy, En Lo, Kapil Khanal, Maha Haji, Rebecca McCabe, Yinghui Bimali.

Figure 1
Figure 1. Figure 1: Side view of concentric cylindrical bodies. 2.1. Linear Hydrodynamics and Eigenfunctions To model the fluid in the internal and external regions, linear potential flow theory is used. This implies small body motion, an inviscid, irrotational, and incompressible fluid, and a wave amplitude-to-wavelength ratio less than unity (Mei et al. 2005). With these 0 X0-3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Side view of taller and shorter region. will be defined as the taller t and shorter s fluid regions. When considering the vertical boundary dividing the two regions at 𝑟 = 𝑎, there are three different conditions to enforce: 1) the value of the velocity potentials at fluid-fluid boundaries are equal 𝜙 t (𝑎, 𝑧) = 𝜙 s (𝑎, 𝑧) for − ℎ ≤ 𝑧 ≤ −𝑑s , (2.2) 2) the radial fluid velocities at the fluid-fluid boundary … view at source ↗
Figure 3
Figure 3. Figure 3: Left: Comparison of the low frequency approximation, the infinite frequency limit, and standard MEEM (with 𝑁 𝑖𝑚 = 𝑁 𝑒 = 100) for the geometry described in Section 2.7. Right: A comparison of the first four 𝐶 𝑖2 1𝑛 over low frequencies, demonstrating that the behavior of 𝐶 𝑖2 10 eventually dominates as frequency approaches zero for both the real and imaginary parts. This trend is representative of the other… view at source ↗
Figure 4
Figure 4. Figure 4: shows the sparsity pattern of the A, B, and C matrices from Eq. 2.13 for an example configuration consisting of two bi-cylinder bodies with 𝑀 = 4, M1 = {1, 2}, M2 = {3, 4}, 𝑁 𝑖𝑚 = 𝑁 𝑒 = 10, and 𝑑𝑖+1 < 𝑑𝑖 . The A matrix features a block bi-diagonal structure. This can be observed by partitioning A into rectangular blocks such that each block encompasses the full height and half the blockwise width of the A𝑚… view at source ↗
Figure 5
Figure 5. Figure 5: Added mass, radiation damping, excitation magnitude, and excitation phase from MEEM and Capytaine for CorPower-like WEC without slanted portions. 3. Convergence of Hydrodynamic Coefficients Characterizing the convergence behavior of numerical solvers is crucial for ensuring accuracy while avoiding unnecessarily long runtimes. For BEM solvers, a mesh con￾vergence study is typically performed, where the numb… view at source ↗
Figure 6
Figure 6. Figure 6: Left: Added mass and damping calculated for a three body region configuration at 𝑁 𝑖1 = 𝑁 𝑖3 = 𝑁 𝑒 = 200 with region 𝑖2 heaving, for varying 𝑁 𝑖2 . Right: The data at left is transformed to the natural log of the associated error, and fitted to obtain error envelope parameters 𝛼, 𝛽 for each of added mass and damping. 𝜖 in the hydrodynamic coefficients solely based on the geometry and wave conditions being … view at source ↗
Figure 7
Figure 7. Figure 7: Flowchart of term count prediction. A geometry and desired error (green) are passed into the formula (blue) determined by the convergence study, producing a term count recommendation (orange). the region itself, such as its fluid height, radial width, or the ratio of its fluid height to those of its neighbors. Cumulative parameters are overall metrics combining the geometries of other regions, such as tota… view at source ↗
Figure 8
Figure 8. Figure 8: Process of choosing the fitting function for each dimensionless parameter (here ℎ−𝑑2 𝑎2−𝑎1 ), once identified. suggest functional forms for the relationship between the dimensionless parameters and fit parameters ( [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Randomly generated (described in Sec. 3.3) configurations with the target region heaving were fit with the product of the dependencies in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cross-sectional view of a slanted geometry and its equivalent discretized geometry. The close-up shown on the right-hand side shows how a slanted region can be approximated by a finite number of cylindrical rings. The unit vector normal to the horizontal part of all cylindrical rings is ˆ𝑛, while the unit vector normal to the true slanted body in the 𝑚th region is ˆ𝑛𝜁𝑚 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 11
Figure 11. Figure 11: Cross-sectional view of different discretization schemes for approximating a slanted geometry with cylindrical rings. The geometry of the cylindrical rings can be chosen such that their horizontal surfaces are a) within, b) partially within, or c) outside the true slanted body shape. 0 X0-25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the radiated velocity potential computed from MEEM and Capytaine. (a) and (c) show the real part of the potential from MEEM, (b) and (d) show the error in the real potential from MEEM relative to Capytaine, and (e) and (f) show the error in the potential along the slanted and stepped outlines. 0 X0-28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Convergence of the hydrodynamic coefficients of five cones of varying steepness calculated with MEEM’s step approximation (radius 10, ℎ = 50, 𝜔 = 1, 𝑁 𝑖𝑚 = 𝑁 𝑒 = 400). Subdivisions calculated ranged from 1 to 30. Error was calculated relative to values given by Capytaine. 0.6 0.8 1.0 Added Mass [kg] ×105 MEEM Capytaine Within 5% of Capytaine 0.8 1.0 1.2 Added Mass [kg] ×105 0.4 0.6 0.8 1.0 1.2 1.4 Frequen… view at source ↗
Figure 14
Figure 14. Figure 14: Left: Computed hydrodynamic coefficients for the CorPower-like WEC geometry with a slanted region (the slant intersects the vertical at 𝑑 = 7.13). MEEM approximates the slanted region using 30 subdivisions. Right: Geometry from left plots at 𝜔 = 1. In both cases, MEEM uses 𝑁 𝑖𝑚 = 𝑁 𝑒 = 400 and Capytaine has 5940 panels. not require evaluating Bessel functions with any arguments that were not already evalu… view at source ↗
Figure 15
Figure 15. Figure 15: Left: Distribution of function computation times for the geometry in [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: For the CorPower WEC geometry in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
read the original abstract

Structural survival of offshore structures is crucial for the growing marine economy. Calculating the added mass, radiation damping, and excitation coefficients to quantify wave loads with the traditional boundary element method (BEM) presents a computational bottleneck. The matched eigenfunction expansion method (MEEM), a long-known but rarely-used alternative, offers computational benefits due to its semi-analytical nature. However, previous work fails to directly compare its accuracy and computational performance with BEM, leaving the extent of its utility unknown. Furthermore, the geometry-dependent convergence for cylindrical and slanted geometries has not yet been documented, making the method's practicality for general geometries unclear. This paper presents a unifying MEEM framework for modeling an arbitrary number of fixed or heaving surface-piercing annular cylinders with continuous and radially-monotonic body profiles, and explores the method's block matrix structure, convergence behavior, ability to accurately approximate slanted geometries, and computational advantages over the BEM solver Capytaine. The numerical experiments show that MEEM can compute hydrodynamic coefficients of slanted geometries within 5% of Capytaine, even for angles as steep as 15 degrees from vertical. Finally, MEEM can achieve 2% convergence of its hydrodynamic coefficients an order of magnitude faster than Capytaine with a matrix size two orders of magnitude smaller, making it a computationally effective alternative to traditional BEM solvers. These contributions enable hydrodynamic analysis of a broad range of shapes with increased speed and confidence, paving the way for future optimization studies to yield improved designs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a unifying matched eigenfunction expansion method (MEEM) framework for an arbitrary number of fixed or heaving surface-piercing annular cylinders possessing continuous and radially-monotonic body profiles. It analyzes the resulting block-matrix structure, documents geometry-dependent convergence for both cylindrical and slanted cases, demonstrates approximation of slanted geometries, and reports computational comparisons against the BEM solver Capytaine, claiming 5% agreement on hydrodynamic coefficients for slants up to 15° from vertical together with order-of-magnitude speed-ups at 2% convergence using matrices two orders of magnitude smaller.

Significance. If the accuracy and performance claims survive clarification of the geometric-proxy issue, the work supplies a semi-analytical, matrix-structured alternative to BEM that could materially accelerate hydrodynamic-coefficient evaluation for generalized offshore geometries and thereby support subsequent optimization studies.

major comments (2)
  1. [Numerical experiments / slanted-geometry results] Numerical experiments (results section on slanted geometries): the headline claim that MEEM reproduces Capytaine coefficients within 5% for 15° slants rests on replacing the true linear slant with a continuous radially-monotonic profile for the MEEM computation. It is not stated whether the Capytaine reference solutions are obtained on the identical monotonic proxy or on the exact slanted geometry; without this clarification the reported tolerance conflates profile-approximation error with MEEM truncation and interface-matching error, weakening attribution of the observed accuracy to the method itself.
  2. [Convergence behavior and performance comparison] Convergence and performance subsection: the statements that 2% convergence is reached an order of magnitude faster with a matrix two orders of magnitude smaller lack explicit truncation criteria (number of retained eigenfunctions per region), definition of matrix size (total unknowns versus per-annulus blocks), Capytaine mesh resolution and convergence tolerance, and precise error-bar construction. These omissions make the quantitative speed-up claims difficult to reproduce or compare directly.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'generalized concentric bodies' is immediately qualified by the continuous radially-monotonic restriction; a single clarifying sentence would prevent readers from over-generalizing the scope.
  2. [Throughout] Notation: ensure that the block-matrix partitioning and the matching conditions at vertical interfaces are denoted consistently between the theoretical derivation and the numerical implementation sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help strengthen the clarity and reproducibility of the work. We address each major comment below and will incorporate revisions to resolve the identified ambiguities.

read point-by-point responses

  1. Referee: Numerical experiments (results section on slanted geometries): the headline claim that MEEM reproduces Capytaine coefficients within 5% for 15° slants rests on replacing the true linear slant with a continuous radially-monotonic profile for the MEEM computation. It is not stated whether the Capytaine reference solutions are obtained on the identical monotonic proxy or on the exact slanted geometry; without this clarification the reported tolerance conflates profile-approximation error with MEEM truncation and interface-matching error, weakening attribution of the observed accuracy to the method itself.

    Authors: We agree that explicit clarification is required. In the numerical experiments, Capytaine reference solutions were computed on the exact linear-slant geometry, while MEEM was applied to a continuous radially-monotonic profile chosen to approximate the slant. The reported 5% agreement therefore represents the combined effect of profile approximation and MEEM truncation/interface-matching error. Because the manuscript positions MEEM as a method for monotonic profiles that can approximate slanted bodies, this comparison is intentional; however, we will revise the text to state the distinction unambiguously so that readers can correctly attribute the observed accuracy. revision: yes

  2. Referee: Convergence and performance subsection: the statements that 2% convergence is reached an order of magnitude faster with a matrix two orders of magnitude smaller lack explicit truncation criteria (number of retained eigenfunctions per region), definition of matrix size (total unknowns versus per-annulus blocks), Capytaine mesh resolution and convergence tolerance, and precise error-bar construction. These omissions make the quantitative speed-up claims difficult to reproduce or compare directly.

    Authors: We accept that the current presentation lacks sufficient detail for direct reproduction. In the revised manuscript we will add: (i) the truncation criterion (number of retained eigenfunctions retained in each fluid region), (ii) an explicit definition of matrix size as the total number of unknowns in the assembled block system, (iii) the Capytaine mesh density and solver tolerance employed for the reference solutions, and (iv) the precise procedure used to construct the error bars shown in the convergence plots. These additions will allow readers to verify the reported order-of-magnitude speed-up at 2% convergence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; quantitative claims rest on external Capytaine benchmarks

full rationale

The paper defines a MEEM framework for continuous radially-monotonic annular bodies, derives the block-matrix structure and interface matching conditions from standard eigenfunction expansions in annular regions, and validates all accuracy (within 5% for 15° slants) and convergence (2% with order-of-magnitude speed-up) claims through direct numerical comparison to the independent Capytaine BEM solver. No fitted parameters are renamed as predictions, no self-citation chain is load-bearing for the central results, and the monotonic-profile assumption is stated explicitly as a modeling precondition rather than derived from the outputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard potential-flow assumptions and eigenfunction matching at cylindrical interfaces; no new free parameters, ad-hoc constants, or postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption The velocity potential in each fluid region can be expanded in a complete set of eigenfunctions that satisfy the linearized free-surface and bottom boundary conditions.
    Standard assumption in linear water-wave theory for axisymmetric geometries.

pith-pipeline@v0.9.0 · 5829 in / 1308 out tokens · 61272 ms · 2026-05-20T01:45:10.260961+00:00 · methodology

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