Self-Attraction and Loading (SAL)

• Self Attraction and Loading (SAL) is a framework addressing self-gravitational and self-weight effects to improve ocean tide modeling and structural grid-shell optimization.
• It employs spherical convolution and harmonic techniques to correct amplitude and phase errors in tidal simulations, reducing RMSE and eliminating spectral artifacts.
• In structural design, SAL leverages convex SOCP methods to optimize minimum-material grid-shells by incorporating self-weight as pure axial loads for robust, efficient structures.

Self Attraction and Loading (SAL) encompasses a set of physical corrections critical in models across geophysical fluid dynamics and structural optimization, specifically addressing, respectively, (i) the feedback of self-gravity and elastic deformation for ocean tides and (ii) structural design under self-weight. In ocean tide modeling, SAL comprises the self-gravitational attraction of the tide's mass, the elastic deformation ("loading") of the solid Earth resulting from this mass, and the feedback of these deformations on Earth’s gravity field. In structural engineering, SAL formulates the minimum-material topologies for grid-shells under their own design-dependent self-weight, ensuring purely axial load-carrying networks. The following sections review central concepts, formulations, numerical schemes, and state-of-the-art computational methods for SAL, guided by recent advances in both ocean modeling (Chen et al., 1 Feb 2026) and optimal structural design (Fairclough et al., 13 May 2025).

1. Physical Content and Mathematical Formalism

Ocean Tide Modeling

SAL in global tide models accounts for the coupled effects of self-gravity and loading induced by the redistribution of ocean water. A departure of the sea-surface elevation, $\eta(\theta,\phi)$, from equilibrium generates a surface load:

$\sigma(\theta,\phi) = \rho_w g \eta(\theta,\phi)$

where $\rho_w$ is seawater density and $g$ is gravitational acceleration. This surface mass deflects the solid Earth elastically, further altering its gravity field. The total perturbation in the gravitational potential at the Earth's surface is expressed as:

$$\delta U(\theta,\phi) = \delta U_\text{tide} + \delta U_\text{load} + \delta U_\text{self-grav}$$

SAL is essential for accuracy; neglecting it introduces amplitude errors up to $\sim 20\%$ and phase errors in dominant tidal constituents (Chen et al., 1 Feb 2026).

Structural Grid-Shells

In grid-shell design, SAL refers to the explicit inclusion of self-weight in the equilibrium and optimization, seeking minimum-material, axial-force (compression- or tension-only) networks. Each candidate member is treated as a catenary sustaining its own distributed weight purely axially—eliminating the need for lumped weight terms. This treatment embeds self-weight directly into convex equilibrium and optimization relations (Fairclough et al., 13 May 2025).

2. Classical Spherical Harmonic Approach (Geophysical SAL)

Traditionally, ocean SAL computations use a truncated spherical harmonic expansion. Both the load $\sigma$ and the gravity potential perturbation $\delta U$ are decomposed:

$$\sigma(\theta,\phi) = \sum_{n=0}^\infty \sum_{m=-n}^n \sigma_n^m Y_n^m(\theta,\phi)$$

$$\delta U(\theta,\phi) = \sum_{n=0}^\infty \sum_{m=-n}^n \delta U_n^m Y_n^m(\theta,\phi)$$

The harmonic coefficients are related through the degree-$n$ load Love numbers, $k_n$ and $h_n$, as:

$$\delta U_n^m = \frac{3G}{2n+1}\frac{R_E}{g}\rho_w(1 + k_n - h_n)\ \sigma_n^m$$

where $G$ is the gravitational constant and $R_E$ is Earth's mean radius.

To obtain the SAL-corrected potential for use in model momentum equations, the spherical harmonic coefficients are manipulated using these parameterizations, with summations truncated at degree $N$ (commonly $N\approx 40$). For practical implementation, modified load Love numbers $(1+k'_n-h'_n)$ are preferred, as they capture composite loading effects.

A significant limitation of this approach is the Gibbs phenomenon: truncation-induced spurious oscillations at sharp sea-land gradients, manifesting as non-physical "ringing" particularly along coasts, with overshoots up to $\sim 9\%$ of the discontinuity (Chen et al., 1 Feb 2026).

3. Spherical Convolution Formulation and Computational Techniques

To address the spectral truncation artifacts, a convolution-based method for SAL was introduced (Chen et al., 1 Feb 2026). The SAL potential at surface point $\mathbf x$ (on $S^2$) is represented as a spherical convolution:

$$\delta U(\mathbf x) = \int_{S^2} G_\text{SAL}(\mathbf x \cdot \mathbf y) \ \sigma(\mathbf y)\ d\Omega(\mathbf y)$$

$G_\text{SAL}(\mu)$ is a rotationally-invariant Green’s function kernel, expanded as a Legendre series:

$$G_\text{SAL}(\mu) = \sum_{n=0}^\infty \lambda_n P_n(\mu), \qquad \lambda_n = \frac{3 \rho_w (1+k'_n-h'_n)}{4\pi \rho_e}$$

This construction avoids explicit truncation and thereby eliminates Gibbs oscillations.

Chen et al. derive closed-form empirical asymptotics for the Love numbers: $k'_n \approx a_1/n$, $h'_n \approx b_0 + b_1/n$, to produce an explicit $G_\text{SAL}$ kernel with

$$G_\text{SAL}(\mu) \approx \frac{3\rho_w}{4\pi \rho_e} \left[ \frac{1-b_0}{\sqrt{2-2\mu}} + (a_1-b_1) \ln\left( \sqrt{2-2\mu} + 2-2\mu \right) \right]$$

The resulting acceleration, $\nabla \delta U(\mathbf x)$, is evaluated as a surface-weighted sum, where the gradient is taken analytically.

Implementation in ocean models such as MOM6 discretizes the sphere via a C-grid and evaluates the convolution using the Cubed-Sphere Fast Multipole Method (CSFMM) to reduce computational complexity from $O(N^2)$ to $O(N)$, employing barycentric Lagrange interpolation and Chebyshev proxy points. The self-term ($j=i$) is omitted to regularize the singularity.

4. Validation and Comparison: Tidal and Structural Applications

Ocean Tides

Validation against TPXO9 satellite-altimetry–derived tides reveals substantial improvements. On a $0.36^\circ$ grid, the traditional (degree-40) spherical harmonic SAL yields a deep-ocean RMSE of 8.10 cm (phase error 6.80 cm); convolution-based SAL lowers these to 5.82 cm (4.33 cm phase). On a $0.08^\circ$ grid, convolution SAL reduces the deep-ocean RMSE from 3.99 cm to 3.64 cm and the global RMSE from 6.68 cm to 5.93 cm. Wavenumber spectra demonstrate steeper roll-off for the acceleration field, indicating spurious high-frequency ringing is eliminated (Chen et al., 1 Feb 2026).

Structural Grid-Shells

In SAL-based grid-shell optimization, simultaneous topology and elevation optimization is formulated as a convex second-order cone program (SOCP):

$$\begin{aligned} \min_{\mathbf s,\mathbf q_A,\mathbf q_B} & \quad \frac{1}{\rho g}\mathbf 1^T(\mathbf q_A+\mathbf q_B) \ \text{s.t.} &\quad \mathbf B\mathbf s = \mathbf f_{xy} \ &\quad \mathbf D_A\mathbf q_A + \mathbf D_B\mathbf q_B = \mathbf f_z \ &\quad (\sin l_i q_{A,i} + \cos l_i s_i)(\sin l_i q_{B,i} + \cos l_i s_i) \geq s_i^2,\quad s_i \geq 0 \end{aligned}$$

Key results include:

• For a barrel vault, as self-weight $\rho g$ increases, the optimal rise and material volume both increase sharply.
• For a square grid with distributed load, increased self-weight leads from flat shells to pronounced barrel shapes; eventually, the optimal solution under strong upward loading becomes a “counterweight” nodal mass distribution.
• The method efficiently handles complex ground structures, including multiply-connected or self-intersecting domains, yielding global optima in seconds.

Convexity guarantees global optimality and efficient solution by modern SOCP solvers; "member-adding" (column generation) further enhances speed for large ground structures (Fairclough et al., 13 May 2025).

5. Practical Implementation and Computational Efficiency

Geophysical Models

The convolutional SAL approach is implemented in MOM6 with the following salient features:

• The convolution operates directly on the sphere and adapts naturally to varying grid resolution and coastal clustering.
• CSFMM amortizes the $O(N^2)$ evaluation cost to $O(N)$, supporting high-resolution global grids.
• Strong scaling studies show the convolution approach matches or surpasses traditional spherical harmonic SAL for $>$1000 parallel ranks, with SAL computations constituting $1\!-\!2\,\%$ of a time step at scale (Chen et al., 1 Feb 2026).

Structural Optimization

For grid-shells, formulation and solution steps are:

1. Discretize the horizontal plan and construct the ground structure of all potential bars.
2. Construct incidence matrices and calculate bar projection lengths.
3. Formulate and solve the SOCP using platforms such as CVXPY and solvers like MOSEK or ECOS/SCS.
4. Retrieve dual variables to compute nodal elevations, reconstruct bar catenaries, and postprocess global optimal topology.
5. For large-scale problems, employ member-adding to keep the active set minimal and maintain computational tractability.

Performance benchmarks show a two to three order-of-magnitude advantage over 3D truss-based topological optimization at similar discretizations (Fairclough et al., 13 May 2025).

6. Limitations and Prospects

Oceanographic Modeling

The convolution SAL retains full spectral content, eliminating Gibbs-induced artifacts and delivering smoother, more physically consistent tidal accelerations. Remaining challenges include:

• Extension of Green’s function kernels for Earth models with laterally varying elasticity.
• Incorporation of finite-speed loading signal propagation.
• Adaptation to regional domain boundaries and spatially varying or time-dependent Green’s functions, especially for non-global simulations.

Structural Optimization

SAL for grid-shells enables robust exploration of topology and geometry as functions of self-weight and material strength ratio ($\rho g /\sigma$), material system, boundary topology, and load configuration. Current limitations pertain primarily to:

• Incorporation of buckling sensitivity and imperfect geometry via additional conic constraints.
• Multi-load case or dynamic and robustness extensions.

A plausible implication is that integration with more generalized material models and imperfection sensitivity analyses will further broaden the applicability of SAL-based optimization (Chen et al., 1 Feb 2026, Fairclough et al., 13 May 2025).

7. Summary and Comparative Table

SAL is a foundational correction in both global tidal modeling and minimally-massed structural design. In both settings, state-of-the-art methods have superseded classical approaches in accuracy and computational efficiency due to advances in mathematical formulation (spherical convolution; convex SOCP) and fast numerical methods.

Application Domain	Classical Approach	Modern SAL Formulation
Ocean tides	Spherical harmonics (trunc.)	Spherical convolution (full spectrum)
Grid-shells	3D truss SOCP (lumped weight)	Convex SOCP w/ self-weight as catenaries

The central innovation across both domains is the embedding of self-attraction and load effects directly and efficiently into the governing equations, optimized over the full configuration space while preserving physical and numerical fidelity (Chen et al., 1 Feb 2026, Fairclough et al., 13 May 2025).