Subzonal Pressure Method in High-Order SGH

• Subzonal Pressure Method is a stiffness-based hourglass control technique in SGH that enriches the cell-centered pressure field with subzonal corrections.
• It subdivides each cell into subzones, enforcing mass conservation to compute local density variations that yield an anti-hourglass force stabilizing mesh deformations.
• The method generalizes the classical Q1-P0 formulation (m=1) to high-order settings, providing a unified, variational approach that improves robustness and accuracy.

Searching arXiv for the cited SGH and related papers. The subzonal pressure method is a stiffness-based hourglass control technique in staggered Lagrangian hydrodynamics (SGH) that introduces subzonal pressure information to detect and penalize kinematic modes invisible to a purely cell-centered thermodynamic discretization. In the formulation developed for high-order SGH, the physical pressure field $p_h\in Q^{m-1}$ is enriched to a higher-order field $\tilde p\in Q^m$ at $(m+1)^2$ Gauss points, and the anti-hourglass force is defined as the difference between the momentum right-hand side evaluated with $\tilde p$ and that evaluated with $p_h$ (Sun et al., 9 Sep 2025). Within this framework, the classical $Q^1$-$P^0$ subzonal pressure method emerges as the special case $m=1$, so the method is presented not as an ad hoc correction but as a unified polynomial and quadrature-based construction (Sun et al., 9 Sep 2025).

1. SGH setting and the origin of subzonal pressure

In SGH, the unknowns are staggered between kinematic and thermodynamic locations. Positions and velocities $\vec u$ are nodal or vertex-based, while density $\rho$, pressure $\tilde p\in Q^m$0, and specific internal energy $\tilde p\in Q^m$1 are cell- or zone-based. The mesh moves with the material, and the momentum equation

$\tilde p\in Q^m$2

is discretized with nodal test functions, whereas the energy equation is discretized in terms of zone-centered thermodynamic degrees of freedom (Sun et al., 9 Sep 2025).

In the high-order SGH framework, velocities and positions are approximated in a continuous tensor-product polynomial space $\tilde p\in Q^m$3 on each element $\tilde p\in Q^m$4, while thermodynamic variables are approximated in a discontinuous space $\tilde p\in Q^m$5. For an element $\tilde p\in Q^m$6,

$\tilde p\in Q^m$7

with $\tilde p\in Q^m$8 and $\tilde p\in Q^m$9 (Sun et al., 9 Sep 2025).

The need for subzonal pressure arises from hourglass modes. In low-order SGH, particularly $(m+1)^2$0-$(m+1)^2$1, non-physical mesh distortions can leave cell volumes and cell-averaged quantities unchanged, making them invisible to a discretization that only senses cell-centered thermodynamic fields. These modes can grow in time and destroy mesh quality. Classical countermeasures are viscosity-based filters and stiffness-based filters; the subzonal pressure method belongs to the latter class (Sun et al., 9 Sep 2025).

2. Classical $(m+1)^2$2-$(m+1)^2$3 formulation

In classical $(m+1)^2$4-$(m+1)^2$5 SGH, the velocity field is bilinear on each quadrilateral zone and the pressure is constant per zone. A single pressure degree of freedom per zone is not enough to constrain all non-trivial $(m+1)^2$6 kinematic modes. The central idea of the subzonal pressure method is therefore to introduce multiple pressure-like degrees of freedom inside each cell (Sun et al., 9 Sep 2025).

The method is described by four conceptual steps. First, the cell is subdivided into subzones or subvolumes. Second, subzonal mass conservation is enforced so that each subzone carries its own density $(m+1)^2$7, computed from its initial mass and current subzonal volume. Third, subzonal pressures are defined through an equation of state using these subzonal densities. Fourth, a correction force is added, driven by the differences between subzonal pressures and the cell-averaged pressure. This force acts like a stiffness resisting hourglass patterns that change subzonal volumes without changing the cell average (Sun et al., 9 Sep 2025).

The classical interpretation is geometric. In Caramana et al. (1998), the subzonal volumes are the areas enclosed by the cell center, a node, and the two adjacent edge midpoints. The high-order treatment re-expresses the same idea in quadrature language rather than by explicit geometric subdivision (Sun et al., 9 Sep 2025).

The operative mechanism is selective sensitivity. The cell-averaged pressure $(m+1)^2$8 does not detect hourglass modes, but the subzonal density derived from subzonal mass conservation does detect local volume changes. Consequently, the induced subzonal pressure differs from the cell pressure, and the resulting force is zero for uniform modes and non-zero for hourglass distortions. This is the essence of the classical method as recast in the paper’s notation (Sun et al., 9 Sep 2025).

3. Quadrature-based reformulation and pressure enrichment

The high-order paper rewrites the method through the semi-discrete momentum equation

$(m+1)^2$9

For the actual pressure field $\tilde p$0, the right-hand side is evaluated with an $\tilde p$1-point Gauss–Legendre rule: $\tilde p$2 If a higher-order pressure $\tilde p$3 is constructed with additional internal degrees of freedom, the corresponding term is evaluated with a $\tilde p$4-point Gauss–Legendre rule: $\tilde p$5

The hourglass force is then defined as

$\tilde p$6

with

$\tilde p$7

This gives a stiffness-based subzonal pressure correction in a general polynomial and quadrature framework (Sun et al., 9 Sep 2025).

The enrichment is not arbitrary. The pressure increment is defined by

$\tilde p$8

where $\tilde p$9 is the sound speed and $p_h$0 is a subzonal density difference at the $p_h$1 quadrature points. The mass-conserving subzonal density is obtained from

$p_h$2

while the lower-order density $p_h$3 and sound speed are interpolated from the $p_h$4 quadrature points through an interpolation matrix $p_h$5 (Sun et al., 9 Sep 2025).

The density variation is thus

$p_h$6

The resulting $p_h$7 supplies precisely the missing sensitivity to subcell deformation (Sun et al., 9 Sep 2025).

4. Recovery of the classical method as the $p_h$8 case

For $p_h$9, the high-order construction reduces to the classical $Q^1$0-$Q^1$1 setting. Kinematics are $Q^1$2 on a quadrilateral with four nodes, thermodynamics are $Q^1$3, the physical pressure uses one Gauss point, and the enriched pressure uses four Gauss points, namely the standard $Q^1$4 Gauss points at $Q^1$5 (Sun et al., 9 Sep 2025).

In this case, the interpolation matrix is

$Q^1$6

so the constant cell-centered density and pressure are simply replicated to the four subzones. The enriched density satisfies

$Q^1$7

and the subzonal density variation is

$Q^1$8

The corresponding pressure variation is

$Q^1$9

which yields the nodal hourglass force

$P^0$0

The structure is identified as identical to the subzonal pressure stiffness in Caramana’s method, with one stated difference: in the reformulated version, the subzonal volume is represented by $P^0$1 at Gauss points, whereas in the classical construction it is represented by explicit geometric subzones. The paper states that the classical $P^0$2-$P^0$3 subzonal pressure method is therefore recovered as the $P^0$4 instance of the general enrichment $P^0$5 with subzonal mass conservation and $P^0$6 (Sun et al., 9 Sep 2025).

This identification is conceptually significant because it places the classical algorithm inside a general variational and quadrature framework. A plausible implication is that the method’s low-order success can be interpreted as a special case of polynomial pressure enrichment rather than solely as a geometric subcell construction.

5. Variational role, conservation properties, and algorithmic placement

In this formulation, subzonal density and pressure are not primary unknowns advanced in time. Instead, $P^0$7 is recomputed each time step through local mass conservation, $P^0$8 is obtained through the equation of state and the linearization $P^0$9, and subzonal pressure enters the discrete momentum equation only through the hourglass force (Sun et al., 9 Sep 2025). The auxiliary status of these quantities is central: they do not introduce global degrees of freedom and do not alter the conservation structure of the base SGH scheme.

Mass conservation is enforced in the $m=1$0 thermodynamic space using $m=1$1-point quadrature, while the enriched density also satisfies mass conservation pointwise at the $m=1$2 quadrature points. The latter does not change total mass. Momentum conservation is retained because the hourglass force is derived from a pressure divergence term, the mass matrix is diagonal via Gauss–Lobatto quadrature, and nodal forces are assembled locally and divided by nodal mass (Sun et al., 9 Sep 2025).

The energy equation explicitly accounts for the work of the hourglass contribution. The paper writes

$m=1$3

and in discrete form

$m=1$4

where $m=1$5 includes both hourglass and viscosity forces (Sun et al., 9 Sep 2025). The stated purpose is to ensure that the work done by the hourglass force in the momentum equation is exactly accounted for in the energy equation at the discrete level.

The algorithmic workflow places subzonal pressure after the base thermodynamic update and before force assembly. Within a time step, the method uses diagonal nodal masses, updates positions and Jacobians, evaluates thermodynamic variables at $m=1$6 points, constructs $m=1$7 and $m=1$8 at $m=1$9 points, assembles hourglass and viscous forces, updates velocity, and then updates internal energy using the discrete work term (Sun et al., 9 Sep 2025). Because subzonal mass conservation is local per quadrature point, no global solve is required.

6. Interaction with artificial viscosity and numerical behavior

The paper treats artificial viscosity and subzonal pressure as separate force contributions added to the same momentum equation. The viscous momentum equation is

$\vec u$0

with $\vec u$1 taken either as the velocity gradient or its symmetrized form, and the discretization is written in a compact form that reuses $\vec u$2 at $\vec u$3 quadrature points (Sun et al., 9 Sep 2025). The viscosity coefficient is

$\vec u$4

or, with vorticity correction,

$\vec u$5

The two stabilizing mechanisms are written as

$\vec u$6

and

$\vec u$7

There are no mixed or cross terms; they simply add, and the same higher-order quadrature and geometric data are reused (Sun et al., 9 Sep 2025).

At the same time, the paper emphasizes a shared sensitivity: both hourglass control and viscosity depend on $\vec u$8, either through subzonal mass conservation or through denominator terms in the viscous force. If $\vec u$9, both $\rho$0 and $\rho$1 can blow up. The choice of $\rho$2 quadrature for both terms is presented as a compromise between accurate damping of hourglass modes and acceptable robustness (Sun et al., 9 Sep 2025).

The reported numerical evidence covers smooth and shock-dominated regimes. In the 2D Taylor–Green vortex, with no artificial viscosity and hourglass control active, the observed convergence rates match theoretical expectations, hourglass control does not degrade accuracy, and $\rho$3-$\rho$4 achieves a given accuracy with far fewer degrees of freedom than $\rho$5-$\rho$6 (Sun et al., 9 Sep 2025). In the Noh problem, with both hourglass control and artificial viscosity enabled, density and pressure profiles match the analytical solution well, the mesh remains well-behaved, and the combined treatment yields stable, accurate solutions with strong symmetry (Sun et al., 9 Sep 2025).

For interface- and shear-dominated tests such as Dukowicz–Meltz and triple point, the findings reported are good shock capturing and interface representation, preserved mesh quality, and no visible hourglass patterns. In the triple point problem, $\rho$7-$\rho$8 shows more detailed vortex twisting than $\rho$9-$\tilde p\in Q^m$00 (Sun et al., 9 Sep 2025). In the Sedov blast test, the method is used specifically to diagnose hourglass motion: with artificial viscosity but no hourglass control, the mesh shows hourglass distortion away from the shock where viscosity is inactive, whereas activating hourglass control significantly improves mesh quality and density profiles (Sun et al., 9 Sep 2025).

These results directly address a common misconception that artificial viscosity alone is sufficient to control hourglass behavior. The paper’s evidence indicates that viscosity may be inactive precisely in regions where hourglass distortion develops, so the subzonal or enriched pressure method is crucial, especially away from strong shocks (Sun et al., 9 Sep 2025).

7. Scope, comparison, and limitations

Within the paper’s framework, the subzonal pressure method is defined by three ingredients: enrichment of $\tilde p\in Q^m$01 to $\tilde p\in Q^m$02 at $\tilde p\in Q^m$03 Gauss points, computation of subzonal density variation from mass-conserving $\tilde p\in Q^m$04 minus interpolated $\tilde p\in Q^m$05, and formation of the hourglass force as the difference between enriched and physical pressure contributions to the momentum equation (Sun et al., 9 Sep 2025). The method’s principal conceptual contribution is the reinterpretation of subzonal pressure as pressure enrichment.

The paper compares this stiffness-based approach with viscosity-based hourglass control. Viscosity-based control is described as constructed by adding artificial viscosity that targets hourglass modes via velocity filters, whereas stiffness-based control is built from subcell mass-conserving density and pressure variations. The stated advantages of stiffness-based control are that it naturally extends from classical SGH to high order, maintains high-order accuracy, works in tandem with existing artificial viscosity, and provides a clear physical interpretation through subzonal mass conservation and equation-of-state linearization. The stated disadvantages are the need for $\tilde p\in Q^m$06 quadrature, which increases cost, and greater sensitivity to mesh tangling and singular Jacobians (Sun et al., 9 Sep 2025).

The limitations are explicit. Stability issues remain for very low initial internal energy or very fine meshes, and these are linked to singular Jacobians and the sensitivity of both subzonal mass conservation and viscosity to $\tilde p\in Q^m$07 (Sun et al., 9 Sep 2025). This qualifies the method’s robustness and indicates that the gain in mode control comes with geometric sensitivity. A plausible implication is that, in practical high-order SGH implementations, subzonal pressure stabilization is most effective when combined with mesh-quality safeguards and careful time-step control rather than treated as a standalone remedy.

In summary, the subzonal pressure method, as formulated in high-order SGH, is a local stiffness-like operator induced by subcell mass-conserving density variations evaluated at higher-order quadrature points. It suppresses hourglass modes by restoring sensitivity to deformation patterns that leave cell averages unchanged, and the classical $\tilde p\in Q^m$08-$\tilde p\in Q^m$09 method is recovered exactly as its lowest-order instance (Sun et al., 9 Sep 2025).