Spatial Adaptive Refinement (SAR)

• Spatial Adaptive Refinement (SAR) is an adaptive resolution strategy in SPH that refines particles near free surfaces by reversibly splitting and merging based on a mass-ratio criterion.
• The method integrates a variable smoothing length, particle shifting, and an arc-gap search for free-surface detection to ensure numerical accuracy and uniform particle distribution.
• Validation cases including dam-breaks and water entry demonstrate that SAR can reduce computational costs by up to 95% while maintaining free-surface flow fidelity.

Spatial Adaptive Refinement (SAR) denotes an adaptive-resolution strategy within smoothed particle hydrodynamics with adaptive spatial resolution (SPH-ASR) for simulating free surface flows. In this formulation, particle spacing varies in space and time according to the distance to the free surface: particles near the free surface are split for refinement, whereas particles away from the free surface are merged for coarsening, with the objective of reducing computational demands while maintaining numerical accuracy (Yang et al., 2020). The method couples reversible numerical particle splitting and merging with a variable smoothing length formalism, a particle-shifting technique for improving particle distribution, and an angle-gap search for free-surface identification.

1. Definition and core formulation

In SPH-ASR, SAR extends standard Smoothed Particle Hydrodynamics (SPH) by allowing particle spacing, and hence resolution, to vary in space and time, driven by the distance to the free surface (Yang et al., 2020). The refinement mechanism is local: particles close to the free surface are “split” to increase local resolution, while particles away from the free surface are “merged” to reduce computational cost. A key feature is that this splitting and merging is reversible and follows a simple mass-ratio criterion tied to a reference particle spacing that grows with distance from the free surface.

The method partitions the fluid domain into bands indexed by $k = 0,1,2,\ldots$, measured in band-widths

$\Delta s_k = C_r^k \,\Delta s_0,$

where $\Delta s_0$ is the finest spacing and $C_r > 1$ is the adaptive ratio between adjacent bands. A particle at distance $d$ from the free surface falls into band

$$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$

with $K$ the kernel-width-to-spacing ratio, for example $K \approx 2$.

For each particle $i$, the reference mass is defined as

$m_{r,i} = \rho_i\,(\Delta s_k)^d,$

where $\Delta s_k = C_r^k \,\Delta s_0,$0 is the spatial dimension, with $\Delta s_k = C_r^k \,\Delta s_0,$1 in the reported formulation. The corresponding mass ratio is

$\Delta s_k = C_r^k \,\Delta s_0,$2

This mass ratio determines adaptation. The splitting criterion is

$\Delta s_k = C_r^k \,\Delta s_0,$3

with $\Delta s_k = C_r^k \,\Delta s_0,$4, and the merging criterion is

$\Delta s_k = C_r^k \,\Delta s_0,$5

with $\Delta s_k = C_r^k \,\Delta s_0,$6.

This construction makes SAR a distance-to-interface-driven adaptive discretization scheme rather than a globally remeshed one. The reported method is specifically organized around free-surface localization, rather than a generic error estimator.

2. Refinement and coarsening operations

The one-step SAR update is described procedurally. For each fluid particle $\Delta s_k = C_r^k \,\Delta s_0,$7, the method computes the distance to the free surface via free-surface detection, assigns the band index $\Delta s_k = C_r^k \,\Delta s_0,$8, computes $\Delta s_k = C_r^k \,\Delta s_0,$9 and $\Delta s_0$0, evaluates $\Delta s_0$1, and then applies either splitting or merging according to the threshold tests (Yang et al., 2020).

The splitting operation $\Delta s_0$2 is defined for a particle $\Delta s_0$3 with mass $\Delta s_0$4, velocity $\Delta s_0$5, and position $\Delta s_0$6. The nearest neighbor $\Delta s_0$7 is chosen, and a unit vector $\Delta s_0$8 perpendicular to $\Delta s_0$9 is constructed. Two child particles $C_r > 1$0 and $C_r > 1$1 are then created with

$C_r > 1$2

$C_r > 1$3

$C_r > 1$4

where $C_r > 1$5 is a splitting-distance factor. Each child is assigned the parent’s smoothing length $C_r > 1$6, and the parent particle is removed.

The merging operation $C_r > 1$7 is defined using particle $C_r > 1$8 and its nearest neighbor $C_r > 1$9, chosen so that both satisfy $d$0. The new mass is

$d$1

The new momentum, expressed as velocity, is updated by

$d$2

and the new position is the center of mass,

$d$3

The new smoothing length is taken as $d$4, or an average, and particle $d$5 is deleted.

The described implementation makes clear that SAR in SPH-ASR is not only a resolution-selection rule but also a concrete particle-transformation mechanism. A common misconception is that adaptive refinement in particle methods necessarily implies arbitrary multichild subdivision. In the reported formulation, splitting currently doubles only into two particles; the paper identifies more general tree-based splits such as $d$6 or $d$7 as future possibilities rather than present components.

3. Variable smoothing length and particle shifting

To maintain kernel consistency across varying particle spacings, each particle’s smoothing length $d$8 is updated every time step using the Hernquist and Katz formalism:

$d$9

with

$$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$0

and

$$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$1

Here $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$2 is the actual number of neighbors of particle $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$3 at step $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$4, $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$5 is the target reference neighbor count, and $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$6 is the spatial dimension, with $$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$7 in the reported cases. Bounds are enforced so that

$$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$8

where

$$k = \left\lfloor \frac{d}{K\,\Delta s_k} \right\rfloor,$$9

The SPH sums employ the symmetrized kernel gradient

$K$0

Particle shifting is applied to counteract disorder in the interior. For non-free-surface particles, a diffusion-like shift is used:

$K$1

with

$K$2

$K$3

$K$4

and

$K$5

This shift is applied only to particles whose neighbors are all identified as interior, not free surface, in order to avoid spurious diffusion at open-surface regions (Yang et al., 2020).

Within the method’s internal logic, variable $K$6 and shifting are not ancillary. They are explicitly introduced to stabilize resolution transitions, maintain kernel consistency across nonuniform spacing, and improve particle distribution in the bulk.

4. Free-surface detection and its role in adaptation

Free-surface detection is central to SAR because the adaptive rule is driven by the distance to the free surface. The reported method uses an arc-gap search. For each candidate particle $K$7, neighbors are collected within a reference radius

$K$8

with $K$9. The angle $K \approx 2$0 of the vector $K \approx 2$1 with respect to a fixed direction, the $K \approx 2$2-axis, is computed for each neighbor. The neighbors are sorted by ascending $K \approx 2$3, and the circular arc gaps

$K \approx 2$4

are scanned with wrap-around. If

$K \approx 2$5

the particle is marked as a free-surface particle, with an example value $K \approx 2$6 (Yang et al., 2020).

This search determines which particles are considered near the open interface and therefore governs the band assignment that ultimately controls splitting and merging. The method’s overall adaptive behavior is therefore tightly coupled to geometric free-surface identification rather than to pressure-based, density-based, or residual-based criteria.

A plausible implication is that the quality of the arc-gap classification affects both efficiency and accuracy, because SAR concentrates the finest spacing in regions that the search identifies as free-surface-adjacent. The paper correspondingly lists the critical angle $K \approx 2$7 among the user-set parameters.

5. Validation, performance, and representative cases

The SPH-ASR method was validated by simulating various free surface flows, and the results were compared to those obtained using SPH with uniform spatial resolution (USR) and experimental data (Yang et al., 2020). Seven test cases are reported to demonstrate SAR’s accuracy and efficiency relative to USR and experiments. Across all cases, the key metrics are particle count, CPU time, free-surface profiles, pressure histories, and object-penetration depths.

For a dam-break on a dry bed, the geometry is specified by $K \approx 2$8, $K \approx 2$9, $i$0, and $i$1. The USR resolution uses $i$2 and approximately $i$3 particles. The ASR range is $i$4 with a time-varying particle count in the range $i$5. CPU time is reduced by approximately $i$6. The free-surface evolution and pressure at wall gauge $i$7 agree within $i$8 of experiment and USR.

For a dam-break on a wet bed, the geometry is $i$9, $m_{r,i} = \rho_i\,(\Delta s_k)^d,$0, $m_{r,i} = \rho_i\,(\Delta s_k)^d,$1, and $m_{r,i} = \rho_i\,(\Delta s_k)^d,$2, with a gate rising at $m_{r,i} = \rho_i\,(\Delta s_k)^d,$3. The USR case uses approximately $m_{r,i} = \rho_i\,(\Delta s_k)^d,$4 particles. The ASR case uses $m_{r,i} = \rho_i\,(\Delta s_k)^d,$5 and approximately $m_{r,i} = \rho_i\,(\Delta s_k)^d,$6 particles. CPU time is reduced by approximately $m_{r,i} = \rho_i\,(\Delta s_k)^d,$7. Pressure and free-surface shape match USR and experiment to within $m_{r,i} = \rho_i\,(\Delta s_k)^d,$8.

For water entry of a slender body, the body size is $m_{r,i} = \rho_i\,(\Delta s_k)^d,$9, the entry speed is $\Delta s_k = C_r^k \,\Delta s_0,$00, and $\Delta s_k = C_r^k \,\Delta s_0,$01. The USR case with $\Delta s_k = C_r^k \,\Delta s_0,$02 uses approximately $\Delta s_k = C_r^k \,\Delta s_0,$03 particles and approximately $\Delta s_k = C_r^k \,\Delta s_0,$04 of CPU time. The ASR case with $\Delta s_k = C_r^k \,\Delta s_0,$05 uses approximately $\Delta s_k = C_r^k \,\Delta s_0,$06 particles and approximately $\Delta s_k = C_r^k \,\Delta s_0,$07 of CPU time. Splash, cavity evolution, and closure times agree with USR, while computational cost is approximately $\Delta s_k = C_r^k \,\Delta s_0,$08 lower.

Across all cases, computational effort is reduced by $\Delta s_k = C_r^k \,\Delta s_0,$09 to $\Delta s_k = C_r^k \,\Delta s_0,$10 depending on resolution contrast and refinement-area fraction. The reported evidence therefore places SAR as a targeted efficiency mechanism for free-surface SPH in cases where the dynamically important region is concentrated near the interface.

6. Parameters, limitations, and prospective extensions

The SAR approach in SPH-ASR is reported to adapt resolution successfully in response to free-surface motions while preserving accuracy and lowering computational cost (Yang et al., 2020). The mass-ratio thresholds $\Delta s_k = C_r^k \,\Delta s_0,$11, the band ratio $\Delta s_k = C_r^k \,\Delta s_0,$12, and the critical angle $\Delta s_k = C_r^k \,\Delta s_0,$13 are user-set parameters, and modest tuning of $\Delta s_k = C_r^k \,\Delta s_0,$14 is stated to suffice for a wide range of free-surface flows. Particle shifting with variable $\Delta s_k = C_r^k \,\Delta s_0,$15 is reported to improve interior uniformity without corrupting open boundaries.

The principal limitation explicitly identified is that splitting currently doubles only into two particles. More general tree-based splits, such as $\Delta s_k = C_r^k \,\Delta s_0,$16 or $\Delta s_k = C_r^k \,\Delta s_0,$17, are proposed as ways to yield smoother transitions. The paper also identifies automated parameter selection, extension to three-phase or thermal flows, and rigorous error-norm studies as future work.

These limitations clarify the scope of the reported SAR formulation. It is a specific adaptive strategy for free-surface SPH, not an exhaustive refinement framework. Its adaptation criterion, resolution bands, smoothing-length update, and free-surface search are tightly integrated, and the future directions indicate where broader generality and stronger a priori assessment remain open.