Adaptive Particle Refinement

• Adaptive Particle Refinement is a collection of techniques that dynamically modify the local particle, node, or cell representation to concentrate resolution where geometry or solution structure demands it.
• These methods employ various strategies such as particle splitting, merging, nodal remeshing, and multilevel mesh hierarchies across domains like PFEM, SPH, PIC, and Bayesian inference.
• By refining regions with high error indicators and coarsening elsewhere, adaptive particle refinement balances computational efficiency with enhanced accuracy in simulation outcomes.

Adaptive particle refinement denotes a family of adaptive discretization strategies in which the local particle, node, or cell representation is modified during simulation or inference so that resolution is concentrated where geometry, solution structure, or posterior mass demands it. The term is not uniform across the literature: in the particle finite element method (PFEM) it often means insertion and deletion of Lagrangian mesh nodes; in smoothed particle hydrodynamics (SPH) it typically means particle splitting and merging with variable spacing or volume; in particle-in-cell (PIC) and hybrid PIC it is frequently realized through multilevel mesh hierarchies with level-local particle populations; and in Bayesian phase estimation it can mean adaptive refinement and merging of posterior grid cells that carry weighted particles or cell masses (Leyssens et al., 2024, Nealon et al., 2024, Innocenti et al., 2012, Tipireddy et al., 2020).

1. Conceptual scope and taxonomy

Across the recent literature, adaptive particle refinement is best understood as an umbrella term for methods that change local representational resolution during the computation rather than fixing a globally uniform particle or nodal spacing. In PFEM, particles are also mesh nodes, so mesh adaptation and particle refinement coincide. In SPH, particles remain the primary carriers of mass and state, and refinement is usually effected by splitting and merging. In PIC and hybrid PIC, the dominant pattern is often not free-form particle splitting everywhere, but rather level-aware particle populations attached to refined meshes. In posterior inference, the adaptive object may be a grid cell rather than a physical fluid particle, but the algorithmic role is analogous: refine where structure matters, coarsen where it does not (Leyssens et al., 23 Dec 2025, Yang et al., 2018, Aunai et al., 2022, Tipireddy et al., 2020).

The literature also draws a sharp distinction between true particle-count adaptivity and broader adaptive-resolution ideas. The $p$-adaptive mesh-free framework based on the Local Anisotropic Basis Function Method, for example, explicitly does not implement true particle/node refinement: the node set is fixed, while the local approximation order $p_i$ is increased or decreased dynamically (Feng et al., 26 Nov 2025). Likewise, mesh refinement around the witness beam in QuickPIC and adaptive mesh refinement in OPAL are particle-informed but remain primarily mesh-side adaptivity rather than particle splitting/coalescence (Su et al., 2024, Frey et al., 2018).

Context	Adaptive object	Typical mechanism
PFEM	Lagrangian nodes/particles	Delaunay refinement, edge collapse, surface/bulk node insertion
SPH	Particles	Splitting, merging, variable smoothing length, shifting
PIC / hybrid PIC	Level-local particle populations on refined meshes	Coarse-to-fine splitting, ghost particles, subcycled AMR
Bayesian inference / mesh-free methods	Posterior cells or local order	Grid refinement/merging or $p$-adaptivity

A common misconception is that adaptive particle refinement always refers to the same numerical operation. The surveyed work shows instead that the phrase spans at least four distinct technical meanings: particle-number adaptivity, nodal remeshing, multilevel particle ownership, and adaptive local approximation order.

2. Fundamental mechanisms

In PFEM, adaptive particle refinement is often formulated as quality-controlled maintenance of a moving Lagrangian point cloud. A representative 2D construction is based on constrained Delaunay triangulation and Chew’s first Delaunay refinement algorithm. The fluid satisfies the incompressible Navier–Stokes equations in Lagrangian form,

$$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$

and particle positions are updated by

$$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$

After fluid-domain detection by a variable-$\alpha$ shape criterion with

$\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$

the method coarsens internal edges that are too small, refines bad elements by circumcenter insertion, splits constrained edges when a circumcenter lies outside the domain, recovers the Delaunay property by edge swaps, interpolates velocity to new particles, solves the incompressible system, and then advects the particles (Leyssens et al., 2024).

In 3D PFEM, the same conceptual identification of particles with mesh nodes remains, but the geometry pipeline is separated into boundary and bulk stages. The boundary surface is refined first by longest-edge splitting whenever

$$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$

subject to a Split_allowed predicate, and the bulk is then refined by circumcenter insertion in a constrained Delaunay tetrahedralization. Coarsening is realized by filtering excessively close particles before the next triangulation. Domain reconstruction is no longer based on $\alpha$-shapes alone; instead, the advected boundary from the previous time step supplies an inside/outside oracle through the generalized winding number (Leyssens et al., 23 Dec 2025).

In compressible SPH, the most explicit recent formulations are volume based. The fundamental adaptive variable is the particle-associated volume

$V_i=\frac{m_i}{\rho_i},$

with a local target spacing $p_i$0 and target volume $p_i$1. Split and merge hysteresis are imposed through

$p_i$2

A split produces $p_i$3 offspring, each with volume $p_i$4, and linearly extrapolated density, velocity, and thermal energy. A merge is pairwise, with

$p_i$5

This formulation is explicitly designed for compressible flows, where mass alone is not an adequate proxy for spatial resolution (Villodi et al., 15 Apr 2025).

A different SPH construction assigns each particle a refinement level $p_i$6 based only on spatial position and uses

$p_i$7

Particles entering a finer zone are split, particles entering a coarser zone are merged, and adjacent refinement levels differ in mass by exactly a factor of two. The implementation in Phantom uses two-child splits, tangential placement relative to the refinement boundary, and a modified $p_i$8-d tree that returns leaf nodes with exactly two particles for efficient pairwise merging (Nealon et al., 2024).

In PIC and hybrid PIC, adaptive particle refinement is frequently realized through multilevel ownership rather than unrestricted dynamic split/merge everywhere. In the Multi Level Multi Domain method, each level is a fully functional PIC domain with its own fields and particles; particles are bound to their level of origin, prevented from transitioning to coarser levels, and refined-boundary particles are repopulated from coarse particles with a splitting technique. In 1D, each coarse particle is split into $p_i$9 refined particles with

$p$0

plus a level-dependent positional reconstruction (Innocenti et al., 2012). PHARE adopts a related philosophy: every AMR level carries its own fields and particle population, new refined levels are populated by splitting coarse particles so that the fine population preserves the assignment function, and particles are deleted when they leave a level rather than merged back (Aunai et al., 2022).

In Bayesian phase estimation, adaptive particle refinement becomes adaptive posterior support. Parameter space is divided into cells $p$1, each representing mass by

$p$2

Refinement is triggered by a midpoint-rule error indicator

$p$3

and cells are split when $p$4. Merging occurs when adjacent cell weights satisfy

$p$5

Here, the “number of particles” is data driven and posterior driven, rather than fixed in advance (Tipireddy et al., 2020).

3. Refinement indicators and region-definition strategies

A central divide in adaptive particle refinement is between geometric indicators and solution-based indicators. In PFEM for free-surface flows, the size field is prescribed geometrically, often by distance to the free surface. In the 2D Delaunay-refinement PFEM implementation, a KD-tree computes distance to the advected free surface and smaller target element sizes are imposed near that surface. The same local target scale also controls free-surface identification through the $p$6-shape criterion, which tightly couples mesh quality and geometry detection (Leyssens et al., 2024).

Multiphase SPH adopts an analogous geometric strategy. In adaptive-resolution SPH for liquid–gas flow, each particle is assigned a reference particle spacing that depends on its distance from the liquid–gas interface; this distance is represented discretely through particle bands of width $p$7, and adjacent bands satisfy

$p$8

Particles are then split or merged according to the ratio of actual mass to reference mass, where $p$9 in 2D (Yang et al., 2018). The free-surface SPH-ASR method follows the same basic logic: particles are split near the free surface, merged away from it, and a search algorithm identifies free-surface particles, while a particle-shifting technique with variable smoothing length improves particle distribution (Yang et al., 2020).

Block-based APR for SPH-FSI generalizes predefined refinement zones into a dynamic block array. Blocks are activated when a targeted characteristic exists within them, and the characteristic may be the presence of solid particles, high velocity magnitude, high vorticity, or free surface. Activated blocks are further partitioned into refinement and transition subdomains, so refinement can track moving structures, vortices, or interfaces with changing topology (Gao et al., 2022).

Recent compressible SPH work adds solution adaptivity through a shock sensor. The sensor is

$$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$0

with neighborhood spreading by

$$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$1

Particles receive the finest target spacing when

$$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$2

with $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$3 reported to work well generally. Transition bands are then built with a refinement ratio $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$4 (Villodi et al., 15 Apr 2025).

In Bayesian phase estimation, the indicator is posterior curvature rather than flow geometry. Refinement comes from the midpoint-rule estimate $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$5, while merging removes cells whose posterior mass has become negligible. In the $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$6-adaptive mesh-free framework, the indicator is instead the difference between two local Laplacian approximations,

$$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$7

and the local order is updated via an upper/lower threshold hysteresis between $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$8 and $$\rho \frac{D\mathbf u}{Dt} = -\nabla p + \mu \nabla^2 \mathbf u + \rho \mathbf g, \qquad \nabla \cdot \mathbf u = 0,$$9 (Tipireddy et al., 2020, Feng et al., 26 Nov 2025).

Particle-informed mesh refinement in beam and plasma PIC uses yet another class of indicators. OPAL-AMR supports six refinement policies: charge density, electrostatic potential, electric field, minimum particles per cell, maximum particles per cell, and particle momentum; neighboring-bunch studies use charge-density tagging with threshold $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$0 (Frey et al., 2018). QuickPIC, by contrast, refines around the witness beam and uses empirical beam-size constraints such as

$$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$1

with analogous relations in $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$2, so the adaptive box tracks the beam rather than an a posteriori field estimator (Su et al., 2024).

4. Stability, conservation, and interface management

Because particle refinement changes local support, neighbor topology, and data transfer patterns, stability hinges on transition management. In Delaunay-refinement PFEM, the main formal mesh-quality guarantee is that no angle smaller than $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$3 will be generated, inherited from 2D constrained Delaunay refinement. Practical termination is enforced by not inserting points for elements whose circumradius is smaller than a limit value, and geometric robustness is supported by the timestep restriction

$$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$4

which limits particle motion to about one local target length per step (Leyssens et al., 2024).

The compressible SPH split/merge formulas are conservative in a limited but explicit sense. The proposed splitting and merging conserve mass, volume, linear momentum, and thermal energy. With extrapolative offspring initialization, however, kinetic energy and angular momentum are not conserved; those can be recovered only by disabling extrapolation and copying the parent velocity to all offspring (Villodi et al., 15 Apr 2025). In the Phantom APR method, mass is perfectly conserved, and linear momentum and kinetic energy are conserved by the split/merge assignment, but the later relaxing stage can slightly affect angular momentum (Nealon et al., 2024).

Particle regularization is another recurring requirement. The multiphase adaptive-resolution SPH method does not rely on particle shifting near interfaces; instead it uses a close-particle repulsive force, symmetric variable-$$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$5 kernels, periodic density reinitialization by Shepard filtering every 20 time steps, and hysteresis between split and merge thresholds. Its improved smoothing-length update blends the neighbor-count correction with the average smoothing length of neighbors, damping abrupt $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$6-jumps between adjacent particles (Yang et al., 2018).

The BAPR and large-ratio APR literature for SPH-FSI addresses interface quality through inactive particles, transition zones, and explicit regularization. In the large-scale-ratio method, the transition zone is split into a regularized and a non-regularized sub-zone, and inactive particles are driven toward isotropic distributions by a modified background force while their pressure, velocity, acceleration, and density are obtained by Shepard interpolation. Free-surface particles are exempted from the normal component of this regularization through a surface-aware projection (Gao et al., 2022). BAPR uses temporary block boundary particles and iterative relaxation until

$$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$7

again to render newly generated particles isotropic (Gao et al., 2022).

In PIC, the critical interface issue is particle shape consistency across mesh levels. The MLMD method resolves this by fixing particle shape size on each level and never allowing a particle to migrate across levels as the same particle; refined-boundary particles are regenerated from coarse particles by splitting, which avoids the self-force associated with time-varying particle shape functions (Innocenti et al., 2012). PHARE reaches a similar goal by maintaining separate particle populations on each AMR level, using level ghost particles generated by splitting coarse particles and deleting particles on level exit rather than merging them back (Aunai et al., 2022).

5. Representative domains and empirical behavior

In free-surface PFEM, adaptive refinement mainly improves geometry capture and volume conservation. In the 3D vortex-in-a-box test, the no-refinement simulation loses $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$8 of its volume, whereas the refined simulation loses only $$\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{U}_i^t \Delta t.$$9. In the 2D PFEM vortex-in-a-box advection test, adaptive refinement tracks the filament down to mesh scale much better than a non-adaptive mesh with a similar total number of elements (about $\alpha$0); in the falling-drop benchmark, the most refined adaptive case uses about $\alpha$1 particles and shows smaller volume variation than literature results using around $\alpha$2 particles on a uniform mesh, while the dam-break dry-bed case maintains relative volume variation below about $\alpha$3 (Leyssens et al., 23 Dec 2025, Leyssens et al., 2024).

For multiphase SPH, interface-following resolution brings large cost reductions. In drop impact on a wet surface, adaptive gas refinement with $\alpha$4, $\alpha$5, and $\alpha$6 reproduces the crown shape well while reducing gas-particle counts and yielding reported speedups of about $\alpha$7, $\alpha$8, and $\alpha$9. In the deep-pool case, reported speedups reach about $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$0, $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$1, and $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$2, depending on $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$3 (Yang et al., 2018). In free-surface SPH-ASR, adaptive refinement near the moving free surface similarly reduces computational demands while maintaining numerical accuracy (Yang et al., 2020).

In compressible SPH, the Phantom APR paper reports sink-accretion errors of less than $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$4 and speedups of $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$5-$\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$6 across circumbinary disc, embedded-planet, and flyby problems, with storage requirements of $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$7–$\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$8 of the corresponding globally high-resolution runs (Nealon et al., 2024). The adaptive compressible SPH paper reports stronger problem-dependent gains: in the rotating square projectile, final particle count drops from $\alpha_e = \frac{R_e}{\alpha(\mathbf x_e)},$9 to $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$0, timestep count from $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$1 to $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$2, and runtime from $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$3 to $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$4 s, a $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$5 speedup; in the Apollo reentry capsule, runtime falls from $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$6 to $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$7 s, a $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$8 speedup (Villodi et al., 15 Apr 2025).

In SPH-FSI, block-based or large-ratio APR primarily targets moving structures and wakes. For the flapping beam behind a cylinder, BAPR with $$l(\tau_{ij})>\frac12\big(h(\mathbf v_m)+h(\mathbf v_n)\big),$$9 reports $\alpha$0 s per $\alpha$1 s simulated versus $\alpha$2 s for the uniform case on 1 MPI rank with 32 threads, while preserving beam-tip amplitude and period. For multi-body water entry with $\alpha$3, BAPR reports $\alpha$4 s versus $\alpha$5 s on 4 MPI tasks, and the depths nearly overlap with the uniform-resolution simulation (Gao et al., 2022). In the large-ratio APR method for hydroelastic impact and swimming, ratio 4 produces results consistent with ratio 2 while cutting cost by about $\alpha$6–$\alpha$7 relative to ratio 2 and by about an order of magnitude relative to uniform fine resolution in the tested cases (Gao et al., 2022).

In particle filtering and Bayesian inference, adaptive refinement is most valuable when posterior structure is multimodal. For $\alpha$8, the distribution of final grid-point counts over 1000 random simulations has median $\alpha$9, with 2.5 and 97.5 percentiles of $V_i=\frac{m_i}{\rho_i},$0 and $V_i=\frac{m_i}{\rho_i},$1. In the bimodal $V_i=\frac{m_i}{\rho_i},$2 regime, adaptive refinement still converges, whereas Liu–West resampling performs poorly even when evaluated by maximum likelihood; this is the paper’s main empirical argument that adaptive refinement preserves multimodal structure that moment-preserving particle filters can destroy (Tipireddy et al., 2020).

6. Limitations, ambiguities, and active directions

The recent literature makes clear that adaptive particle refinement is not yet a single mature methodology with uniform guarantees. A first fault line is definitional. Some papers use the term for true split/merge particle methods, some for node insertion/deletion in PFEM, some for level-local particles on adaptive meshes, and some only in a broad sense that includes fixed-node $V_i=\frac{m_i}{\rho_i},$3-adaptivity. This suggests that “adaptive particle refinement” is currently a family resemblance concept rather than a single algorithmic class (Feng et al., 26 Nov 2025, Su et al., 2024, Frey et al., 2018).

A second fault line concerns refinement criteria. Many practically successful methods remain geometric or heuristic rather than a posteriori error controlled. The 2D PFEM Delaunay-refinement method uses distance to the free surface rather than solution-based error indicators, and explicitly notes that such indicators are future work (Leyssens et al., 2024). The compressible SPH scheme uses heuristic thresholds such as $V_i=\frac{m_i}{\rho_i},$4 and $V_i=\frac{m_i}{\rho_i},$5 (Villodi et al., 15 Apr 2025). The Bayesian grid-refinement method depends on $V_i=\frac{m_i}{\rho_i},$6 and $V_i=\frac{m_i}{\rho_i},$7, with no automatic tuning rule (Tipireddy et al., 2020).

A third fault line concerns dimensionality and transfer operators. Several influential implementations are still 2D only or leave crucial transfer details underspecified. The 2D PFEM Delaunay-refinement method states that generalization to 3D is future work (Leyssens et al., 2024). The BAPR method is demonstrated only in 2D with two resolution levels (Gao et al., 2022). The 3D PFEM mesh-adaptation paper describes the geometry pipeline in detail but does not spell out a dedicated interpolation or $V_i=\frac{m_i}{\rho_i},$8-projection formula for newly inserted nodes (Leyssens et al., 23 Dec 2025).

Finally, conservation and interface artifacts remain active issues rather than solved details. Compressible SPH APR in Phantom exhibits a persistent density “blip” of order $V_i=\frac{m_i}{\rho_i},$9 at refinement interfaces (Nealon et al., 2024). QuickPIC explicitly states that it does not directly address self forces at the refinement boundary and instead mitigates them by multiple transition levels and boundary placement (Su et al., 2024). OPAL-AMR discusses spurious self-forces near coarse-fine interfaces and uses buffer cells as mitigation (Frey et al., 2018). Even when methods are conservative in mass or momentum, they often sacrifice exact angular-momentum, kinetic-energy, or flux consistency under split/merge or multilevel coupling (Villodi et al., 15 Apr 2025).

A plausible implication is that future work will continue to move in three directions: sharper problem-dependent indicators, more robust interface-transfer operators, and clearer separation between true particle adaptivity and broader adaptive-resolution frameworks. The present literature already shows that all three are numerically consequential, but none is yet standardized across domains.