The 100x Smaller Matrix That Could Turbocharge Offshore Wind Design

The Fastest Route to a Better Wave Energy Converter

Ten times faster. A hundred times smaller. Those are the numbers that matter in a new paper from researchers at Cornell University and the University of Michigan — and they have nothing to do with a new supercomputer or a machine-learning shortcut. They come from dusting off a mathematical technique that has existed in the literature since 1981, reformulating it rigorously, and finally comparing it head-to-head with the software the offshore engineering world actually uses.

The stakes are higher than they might first appear. The marine economy — offshore wind, wave energy converters (WECs), aquaculture platforms, floating hydrogen production — is growing fast, and every structure in it must survive ocean waves. Computing how waves push and pull on a floating body is called hydrodynamic analysis, and it is the foundation of structural design. The problem is that doing it well is slow. The standard approach, the boundary element method (BEM), discretizes a structure's surface into thousands of tiny panels and solves a large numerical system. That works for a single design. But if you want to optimize a design — to search through thousands of possible shapes for the one that absorbs the most wave energy or costs the least to build — running BEM thousands of times becomes a serious bottleneck.

The method this paper rehabilitates, called the matched eigenfunction expansion method (MEEM), sidesteps much of that cost by exploiting the mathematical structure of the problem itself.

The Science

MEEM was introduced by Yeung in 1981 to solve a problem that purely analytical methods could not: computing wave forces on a truncated cylinder — one whose bottom floats freely, not resting on the sea floor. The trick is to divide the fluid around the structure into regions (the water underneath each cylindrical ring, and the open ocean surrounding everything), write down analytical expressions for the wave potential in each region using what are called eigenfunctions — standing wave patterns that naturally satisfy the governing equations — and then numerically enforce the rule that pressure and fluid velocity must match continuously at every boundary between regions. The result is a system of linear equations $\mathbf{A}\vec{x}=\vec{b}$, where the unknowns $\vec{x}$ are the amplitudes of those standing-wave patterns (the eigencoefficients), and solving it gives the complete fluid motion everywhere (Bimali et al., 2026).

What makes this semi-analytical approach fast is what makes it mathematically elegant: the matrix $\mathbf{A}$ has a block bi-diagonal structure

. Each block ${\mathbf{A}}_m$ handles only the boundary between two neighboring fluid regions, and adjacent blocks do not overlap. For a structure with $M$ internal regions, the total matrix size scales as $N_T=N^{i_1}+{\sum }_{m=2}^{M}2N^{i_m}+N^e$, where each $N$ is a small integer representing how many eigenfunction terms are kept per region. BEM, by contrast, requires a mesh whose size scales with the surface area of the structure — and even a simple cylinder can require thousands of panel elements.

The team at Cornell and Michigan, led by Rebecca McCabe with co-authors Yinghui Bimali, Collin Treacy, Kapil Khanal, En Lo, and Maha Haji, built on preliminary work to develop OpenFLASH (Flexible Library for Analytical & Semi-analytical Hydrodynamics), an open-source Python package that implements this unified MEEM framework. Their benchmark opponent is Capytaine, the widely-used open-source BEM solver that represents the current community standard.

The key hydrodynamic quantities the method computes are: added mass $A_{pq}(\omega )$ — the extra inertia a body appears to have because it must accelerate the surrounding fluid when it moves; radiation damping $B_{pq}(\omega )$ — energy radiated away as outgoing waves; and excitation coefficients — how hard incoming waves push the structure. Together, these coefficients fully characterize how a floating body responds to ocean waves and are the inputs every offshore engineer needs for structural analysis and power output prediction.

Validation against Capytaine for a CorPower-like wave energy converter (a real commercial WEC design) showed MEEM reproduces the added mass, radiation damping, excitation magnitude, and excitation phase accurately across a wide range of wave frequencies — establishing that the semi-analytical foundation is sound before the speed and geometry tests begin.

What They Found

Convergence: How Fast Does the Method Settle?

The first major result concerns convergence — how quickly the computed forces stop changing as you include more eigenfunction terms. This is a subtlety that matters enormously in practice, because adding more terms costs more compute time. The paper documents, for the first time systematically, how convergence depends on the specific geometry of the problem.

MEEM Convergence: Error vs. Number of Eigenfunction Terms

Label	Value
N=5	38
N=10	18
N=20	7.5
N=40	3.2
N=60	2.1
N=80	1.4
N=100	1

Illustrative of the exponential convergence law e^(−αN + β) documented for a three-region configuration. Source: Bimali et al. (2026), Fig. 6.

The finding is that convergence follows a clean exponential law: the error in added mass and damping decreases as $e^{-\alpha N+\beta }$, where $N$ is the number of terms kept, and $\alpha$ and $\beta$ are geometry-dependent parameters the team characterizes explicitly (Bimali et al., 2026). Crucially, achieving 2% accuracy requires only a modest $N$ — and reaching that 2% threshold takes roughly ten times less wall-clock time than Capytaine needs to reach the same accuracy

.

The matrix involved is roughly one hundred times smaller by element count. For context: BEM meshes for moderately complex offshore structures can run to tens of thousands of degrees of freedom. MEEM achieves comparable accuracy with matrices in the hundreds. That is not a marginal improvement — it is a qualitative change in what becomes computationally feasible.

Slanted Geometries: Can MEEM Handle the Real World?

The most practically important question is whether MEEM, which is derived for perfect cylinders, can handle the slanted, tapered, and conical surfaces that appear on real offshore structures. The CorPower WEC, the RM3 float, the WaveBot — none of these are simple vertical cylinders. They have curved or angled sides.

The approach MEEM takes is geometric discretization: approximate a slanted surface as a staircase of many thin cylindrical rings. More rings means a better approximation of the true slope. The paper rigorously characterizes the tradeoff between the number of discretization steps, the steepness of the slant angle, and the resulting error in the computed forces.

Error in Hydrodynamic Coefficients vs. Slant Angle (MEEM vs. Capytaine)

Label	Value
0°	0.5 %
5°	1.8 %
10°	3.4 %
15°	4.9 %

MEEM approximates slanted surfaces as staircase discretizations of cylindrical rings. Source: Bimali et al. (2026), Section 4.

The result is encouraging. For slant angles up to 15 degrees from vertical — which covers a large fraction of practical offshore body shapes — MEEM stays within 5% of Capytaine's results, even with a manageable number of staircase steps. Steeper angles require more discretization steps to maintain accuracy, and the paper provides explicit guidance on how to choose the discretization resolution for a target error level. This is the first time such guidance has been documented in the literature, removing a major uncertainty about the method's applicability to non-cylindrical shapes.

The Speed Advantage in Numbers

MEEM vs. Capytaine: Speed and Matrix Size Comparison

Label	Value
Capytaine Runtime	100 relative
MEEM Runtime	10 relative
Capytaine Matrix Size	100 relative
MEEM Matrix Size	1 relative

MEEM achieves 2% convergence ~10× faster with a matrix ~100× smaller than Capytaine. Source: Bimali et al. (2026), Section 5.

The computational benchmark is the clearest statement of MEEM's utility. Across a range of test geometries, MEEM consistently reaches its converged answer faster than Capytaine — not by a few percent, but by roughly an order of magnitude. The matrix it solves is two orders of magnitude smaller. For single evaluations, both methods are fast enough. But for an optimization loop that might call the hydrodynamic solver thousands of times — varying the radius, draft, and taper of a WEC float to find the optimal design — that factor-of-ten speed advantage compounds into the difference between a study that is feasible in hours versus one that takes weeks.

Why This Changes Things

The offshore renewable energy sector is at an inflection point. Wave energy, in particular, has struggled to reach commercial scale partly because the design space is enormous and the physics is complex. A wave energy converter's power output depends sensitively on its shape — how its added mass and damping coefficients interact with the wave spectrum at a given deployment site. Finding the best shape requires running many hydrodynamic evaluations, and the cost of those evaluations has historically constrained designers to exploring only a handful of candidate geometries.

What MEEM offers is not just speed — it is tractability. When a hydrodynamic evaluation takes ten times less time, you can run ten times more design iterations within the same budget. When the method is packaged in an open-source Python library, it is accessible to the graduate student running optimization studies at 2 AM, not just the engineering firm with a commercial BEM license.

The block bi-diagonal structure of the MEEM matrix $\mathbf{A}$

also has a deeper implication: it is sparse, meaning most of its entries are zero. Sparse matrices can be solved with specialized algorithms that are faster still, and they can be analyzed symbolically. The paper shows that each sub-matrix ${\mathbf{A}}_m$ decomposes into an element-wise (Hadamard) product of purely radial and purely vertical components: ${\mathbf{A}}_m={\mathbf{A}}_{m,r}\odot {\mathbf{A}}_{m,z}$. That factorization is not just mathematically pleasing — it reveals the physical structure of the problem and opens doors to analytical sensitivity analysis and gradient-based optimization that are much harder to achieve with BEM.

It is also worth naming what MEEM does not replace. BEM remains the right tool for geometries that are genuinely irregular — structures with overhangs, non-axisymmetric shapes, or complex multi-body configurations that cannot be approximated as concentric rings. MEEM is not a universal solver. It is a precision instrument for a specific but commercially important class of shapes: the vertical axisymmetric bodies that dominate point-absorber wave energy converters, spar buoys, and many floating wind foundations. For that class, the paper argues convincingly, MEEM deserves a central place in the engineer's toolkit rather than being confined to academic literature.

The Haskind relation — a classical result in linear hydrodynamics that lets you compute wave excitation forces from the radiation problem alone, without explicitly solving the diffraction problem — is used throughout to reduce the computational scope (Bimali et al., 2026). This is a reminder that MEEM's speed advantage is partly about algorithmic cleverness and partly about theoretical depth: the method is fast because it is analytical, and analytical methods carry mathematical structure that purely numerical methods discard.

What's Next

The paper is explicit about its boundaries. The current framework handles heave — vertical motion — for axisymmetric bodies. Surge and pitch, the horizontal and rocking motions that matter equally for mooring design and structural fatigue, are not yet included. Extending MEEM to those degrees of freedom is the logical next step.

The framework also currently requires body profiles to be continuous and radially monotonic — meaning the radius at any depth can only increase or decrease steadily, never doubling back. That excludes concave shapes, reentrant profiles, and geometries where a body's bottom intersects a fluid region boundary. These are not exotic edge cases; some WEC designs deliberately use concave hulls to tune their hydrodynamic response. The mathematics for handling these cases is more involved, but the block matrix structure established in this paper provides a scaffolding to build on.

There is also the question of irregular frequencies — a known numerical artifact of BEM that does not afflict MEEM — and the behavior at very low and very high wave frequencies, which the paper characterizes analytically through asymptotic expansions. Understanding these limits helps engineers avoid numerical pitfalls and trust their results across the full frequency range relevant to ocean engineering practice.

Perhaps most importantly, the release of OpenFLASH as an open-source package turns this from a paper result into a community resource. The history of scientific software suggests that when a capable, well-documented tool becomes freely available, its use expands far beyond what its creators anticipated. Researchers who previously used slow BEM solvers not because they preferred them, but because no better option was packaged and accessible, now have an alternative.

The ocean is not getting calmer. Offshore structures are getting more numerous and more varied in shape as the energy transition accelerates. The tools we use to design them need to keep pace — not just in raw computing power, but in mathematical intelligence. A method that exploits the physics of the problem rather than brute-forcing it with mesh elements is exactly the kind of foundational work that enables the next generation of designs. The fact that it has been sitting in the literature since 1981, waiting to be properly characterized and packaged, makes this paper both a contribution and a small rebuke to the field's tendency to reach for the familiar.