Axisymmetric Contact SPH for Impact Engineering

• Axisymmetric Contact SPH is a particle-based discretization method that uses local Riemann star states to enforce conservation in compressible multi-material flows.
• The method represents material as toroidal volumes in cylindrical coordinates, directly incorporating 1/r source terms to balance mass, momentum, and energy.
• Enhancements like TKC and MUSCL reconstruction improve interface sharpness and stability, leading to accurate shock capturing in impact engineering simulations.

to=arxiv_search 彩神争霸怎么样්ඩjson {"query":"(Rublev, 8 Jul 2025)"}Japgolly Axisymmetric Contact Smoothed Particle Hydrodynamics (CSPH) is an axisymmetric, particle-based discretization for compressible multi-material flow and impact engineering in which pairwise contact, formulated through local Riemann star states, replaces the pairwise averaging characteristic of classical SPH while retaining a conservative structure. In the formulation developed in “A family of conservative axisymmetric contact SPH schemes for impact engineering applications,” particles represent toroidal material volumes in the $(r,z)$ half-plane, the axisymmetric $1/r$ terms are incorporated directly into the discretization, and a one-parameter family of conservative schemes is constructed so that mass, linear momentum, and total energy remain consistent with the underlying axisymmetric PDEs (Rublev, 8 Jul 2025).

1. Governing equations in axisymmetric compressible flow

The formulation is posed in cylindrical coordinates $(r,z)$ under axial symmetry, with no $\theta$-dependence. For compressible media, the conservative continuity equation is

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$

The paper uses the volumetric strain $\varepsilon=-\ln(\rho/\rho_0)$ and writes

$$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$

with $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$. For a perfect fluid in the pressure-only case,

$$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$

and the internal-energy evolution is

$$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$

A central feature of the axisymmetric setting is that the geometric source terms proportional to $1/r$0 are not optional corrections but part of the conservation problem itself. The conservative total-energy form therefore requires a discretization in which those source terms are balanced consistently. This is the main distinction between axisymmetric CSPH and Cartesian CSPH, where no analogous geometric singularity appears (Rublev, 8 Jul 2025).

The formulation is extended to elastic solids through the stress decomposition

$1/r$1

with axisymmetric momentum equation

$1/r$2

where

$1/r$3

This solid formulation is required for impact problems in which elastic or elastoplastic strength cannot be neglected.

2. Particle representation and conservative axisymmetric discretization

In axisymmetric CSPH, each particle represents a toroidal material volume rather than a Cartesian point mass. The particle volume is

$1/r$4

and the cross-sectional area is

$1/r$5

Field interpolation in the $1/r$6–$1/r$7 half-plane uses the axisymmetric weight $1/r$8: $1/r$9

$(r,z)$0

The conservative construction replaces classical SPH pairwise averages by contact, or Riemann, star states while preserving exact conservation through antisymmetric pairwise forms. The family of conservative schemes is parameterized by a radial function $(r,z)$1. Two conditions guarantee conservation: $(r,z)$2 for pairwise symmetry of momentum, and

$(r,z)$3

for energy source balance. A symmetric form

$(r,z)$4

with $(r,z)$5, $(r,z)$6, and $(r,z)$7, satisfies both conditions. Three conservative choices emerge—geometric, arithmetic, and harmonic means—and the paper recommends the harmonic-mean scheme for robustness (Rublev, 8 Jul 2025).

For fluids, the acoustic contact problem between particles $(r,z)$8 and $(r,z)$9 yields

$\theta$0

where $\theta$1 is the acoustic impedance and $\theta$2 is the radial projection along the pair direction. For the recommended harmonic-mean axisymmetric CSPH scheme, the fluid equations become

$\theta$3

$\theta$4

$\theta$5

The formulation also admits a thermodynamic interpretation. For the acoustic Riemann solver, the entropy production satisfies

$\theta$6

which the paper uses to demonstrate thermodynamic consistency.

3. Contact treatment for multi-material fluids and solids

The defining element of CSPH is the use of local pairwise contact states at interfaces, including interfaces between different materials. In solids, the contact problem is formulated in a local $\theta$7 frame, where $\theta$8 is aligned with the line of centers and $\theta$9 is transverse to it. The transformation matrices are

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$0

The acoustic solid contact uses longitudinal and transverse sound speeds, $$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$1 and $$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$2, to construct star velocity and stress components. For example,

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$3

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$4

Similar formulas are used for the transverse components. In the harmonic-mean solid scheme, the momentum and total-energy equations are written in terms of these transformed star states and then mapped back to $$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$5 coordinates (Rublev, 8 Jul 2025).

The constitutive update for deviatoric stress is based on the axisymmetric strain-rate deviator and rotation: $$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$6

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$7

$$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$8

With shear modulus $$\partial_t \rho + \partial_r(\rho u_r) + \partial_z(\rho u_z) + \frac{\rho u_r}{r} = 0.$$9, the linear elasticity update is

$\varepsilon=-\ln(\rho/\rho_0)$0

This contact-centered construction serves two roles simultaneously. First, it enforces conservative flux exchange across material interfaces through antisymmetric pairwise terms. Second, it mitigates oscillations at contact discontinuities without recourse to the artificial-viscosity tuning typical of older SPH variants. In the Sedov calculations, the paper additionally uses a Roe-type approximation with a diffusion limiter to further stabilize shocks.

4. Consistency enhancement: TKC and MUSCL reconstruction

The conservative core can be augmented by two accuracy-improving devices: Total Kernel Correction (TKC) and MUSCL reconstruction. TKC is introduced to restore consistency, specifically linear completeness, and to reduce diffusion. It is defined through the local moment matrix

$\varepsilon=-\ln(\rho/\rho_0)$1

and raw kernel gradients are replaced by corrected ones using $\varepsilon=-\ln(\rho/\rho_0)$2. In the axisymmetric setting, the correction preserves the conservative structure while respecting the $\varepsilon=-\ln(\rho/\rho_0)$3 source-term consistency (Rublev, 8 Jul 2025).

With TKC, the continuity equation becomes

$\varepsilon=-\ln(\rho/\rho_0)$4

where

$\varepsilon=-\ln(\rho/\rho_0)$5

The corrected momentum equation uses

$\varepsilon=-\ln(\rho/\rho_0)$6

and star states $\varepsilon=-\ln(\rho/\rho_0)$7 and $\varepsilon=-\ln(\rho/\rho_0)$8 are computed using corrected projections.

MUSCL reconstruction provides second-order, piecewise linear interface states: $\varepsilon=-\ln(\rho/\rho_0)$9 where $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$0 is a slope and $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$1 is a limiter such as minmod or van Leer. These reconstructed left and right states are then fed into the pairwise Riemann solver. The paper states that, in practice, MUSCL reconstruction significantly sharpens shocks and contacts.

The combination of TKC and MUSCL is especially important in axisymmetric calculations because the reduced dimensionality does not itself guarantee high fidelity. A plausible implication is that conservative fluxes suppress nonphysical energy drift, while TKC and MUSCL address separate issues of completeness and interface sharpness.

5. Axis treatment, stabilization, and material modeling

Near the symmetry axis $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$2, the kernel support is truncated and the geometric singularity becomes numerically delicate. To avoid singular behavior, the method uses ghost particles reflected across the axis and temporarily switches to Parshikov’s original axisymmetric contact SPH form for particles within $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$3. The velocities and stresses of ghost particles are mirrored consistently. The paper notes that this local treatment slightly breaks global conservation but yields better accuracy near the axis. It also states that the overall scheme remains markedly more conservative than earlier axisymmetric CSPH (Rublev, 8 Jul 2025).

Shock capturing relies on Riemann-based dissipation rather than artificial viscosity, and TKC is used to reduce numerical diffusion. Time integration is first-order explicit Euler. A standard CFL condition is applied; the paper does not specify a value, but gives the typical constraints

$$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$4

and, for pairwise interactions,

$$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$5

with $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$6 a Courant factor.

The material models employed in the test and application problems include several standard high-pressure and high-strain-rate EOS and constitutive laws. For fluid tests, the ideal gas EOS is

$$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$7

For sand, the Mie–Grüneisen EOS is

$$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$8

with parameters $$\frac{d\varepsilon}{dt} = \frac{1}{r}\frac{\partial}{\partial r}(r U^r) + \frac{\partial U^z}{\partial z} = \boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r},$$9, $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$0, $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$1, and $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$2. The explosive products are modeled with a JWL EOS and Lee–Tarver detonation kinetics. The sand barrier strength is modeled with JH-2, including $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$3, $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$4, $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$5, $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$6, tensile strength $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$7, and shear modulus $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$8. Taylor bar impact calculations use Johnson–Cook plasticity.

The reported implementation practice also includes Wendland C2 kernels in Eulerian tests, smoothing-length factor $\boldsymbol{\nabla}=(\partial_r,\partial_z)^T$9–$$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$0, particle splitting and merging to maintain resolution, and particle shifting to improve packing. The paper warns that merging can lose kinetic energy unless corrected and states that it opts to avoid artificial heating, accepting a small energy deviation.

6. Verification, validation, and computational significance

The verification program spans canonical compressible-flow and impact benchmarks as well as an application-scale blast-mitigation problem. In the cylindrical Sod test, the initial discontinuity is at $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$1 with left state $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$2, $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$3 and right state $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$4, $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$5, the gas is initially at rest, the Wendland C2 kernel is used with $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$6, and the solution is evaluated at $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$7. The paper reports that TKC-MUSCL-SPH reproduces shock, contact, and rarefaction structures consistently with reference solutions, with sharp fronts and correct wave speeds (Rublev, 8 Jul 2025).

In the axisymmetric Sedov point explosion, the domain is $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$8, $$\frac{d\mathbf{U}}{dt} = -\frac{1}{\rho}\boldsymbol{\nabla}P,$$9 with $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$0, energy deposited in a sphere of radius $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$1, $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$2 inside and $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$3 elsewhere, and total energy $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$4 at approximately $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$5. The analytical shock radius is

$$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$6

Using TKC-CSPH, $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$7, particle splitting and merging, particle shifting, and a Roe-type contact with limiter, the simulation preserves spherical symmetry in the axisymmetric representation and reproduces the shock radius, front strength, and maximum compression of approximately $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$8.

The Taylor bar calculations cover OFHC copper, Armco iron, and 4340 steel under multiple initial lengths and impact velocities. The paper reports that the axisymmetric MUSCL-SPH results match reported final length ratios $$\frac{de}{dt} = -\frac{P}{\rho}\left(\boldsymbol{\nabla}\cdot\mathbf{U} + \frac{U^r}{r}\right).$$9 with differences within a few percent. It also provides a direct performance comparison between axisymmetric and full 3D Cartesian MUSCL-SPH. For a steel bar with $1/r$00, $1/r$01, and $1/r$02 up to $1/r$03, the 3D calculation uses approximately $1/r$04 million particles and takes approximately $1/r$05 h $1/r$06 min on $1/r$07 cores, whereas the axisymmetric calculation uses approximately $1/r$08k particles and takes approximately $1/r$09 min on $1/r$10 cores. The paper interprets this as a significant speedup and memory reduction while retaining fidelity.

The application study concerns shock wave weakening by a breakaway sand barrier. Experiments compare explosions in air with explosions inside protective sand cylinders of specified geometry, including rings and a capped cylinder. The numerical model uses ideal gas for air, JWL plus Lee–Tarver for the explosive, JH-2 plus Mie–Grüneisen for the sand barrier, and a rigid steel chamber. The axisymmetric TKC-CSPH simulations reproduce barrier breakup, the transfer of momentum from detonation products to sand fragments, and the attenuated air shock. Peak overpressures at multiple gauges match experiments reasonably, with typical differences of $1/r$11–$1/r$12, and the attenuation effect is captured.

These results define the method’s practical significance. Axisymmetric CSPH reduces dimensionality and computational cost for axially symmetric problems, but it does not merely repackage Cartesian CSPH: it requires careful treatment of $1/r$13 source terms, near-axis support truncation, and interface-consistent conservative fluxes. The paper’s comparative discussion emphasizes that the new family of schemes differs from prior axisymmetric CSPH by enforcing conservation through the choice of $1/r$14 and antisymmetric pairwise forms, thereby eliminating numerical heating as demonstrated in the cylindrical Verney problem, while remaining aligned with Godunov-type SPH in its use of Riemann solvers and conceptually related to correction-based variants such as CRKSPH through its use of TKC with $1/r$15 weighting.