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Smoothed Particle Hydrodynamics

Simulates fluids with particles that estimate local density and pressure.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Simulation
Difficulty: Advanced
Family: Particle Simulation
Tier: Specialized
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 4
Code example: Not ready
Visualization: Ready
Aliases: SPH

Overview

SPH treats fluid as particles and uses smoothing kernels to approximate continuous fluid quantities.

Core Concepts

simulated statefluid stateparticle interactionphysical stateparticle simulationstate updatetime stepnumerical stability

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n)

Average

O(n k)

Worst

O(n^2)

Space

O(n)

When to Use

Use Smoothed Particle Hydrodynamics when you need to find neighboring particles.
Use it for fluid simulation with inputs like particles.
Use it when approximate behavior over time is more useful than a closed-form answer.
Use it when the Particle Simulation tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when the time step, grid size, or sample count cannot support stable results.
Avoid it when O(n^2) worst-case behavior is too costly for the expected input size.
Avoid it when the required output is not updated particle positions and velocities.

What It Does

Finds neighboring particles
Computes density, pressure, and forces from kernels

Use Cases

Fluid simulation
Splash effects
Astrophysics models

Inputs

Particles
Kernel radius
Fluid parameters

Outputs

Updated particle positions and velocities

Key Formulas (1)

Density estimate

density_i = ∑(mass_j · W(distance(p_i, p_j), h))

density_i = sum(mass_j * W(distance(p_i, p_j), h))

Estimates fluid density from nearby particles using a smoothing kernel.

Pseudocode

1.Find neighbors for each particle
2.Estimate density from nearby particles
3.Compute pressure and viscosity forces
4.Integrate velocities and positions
5.Apply boundary handling

Canonical Example

Input: position = 0, velocity = 2, time step = 0.5
Output: next simulated state
Explanation: Smoothed Particle Hydrodynamics: The algorithm advances the system by applying its update rule over a small step.

Implementation Notes

Keep the input contract explicit: Particles; Kernel radius; Fluid parameters.
Use this step as the implementation spine: Find neighbors for each particle.
Then preserve the next state change: Estimate density from nearby particles.
Track units and invariants such as mass, energy, or count where they matter.

Common Pitfalls

Efficient neighbor search is essential for large SPH simulations.
Using a time step that is convenient for rendering but unstable for the simulated system.
Returning updated particle positions and velocities without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show particles flowing while neighbor kernels shape pressure forces.

Step 1 of 5: Find neighbors for each particle

fluid-particles

Notes

Smoothed Particle Hydrodynamics operates in O(n k) time and O(n) space within the Particle Simulation family. Key governing relation: density_i = sum(mass_j * W(distance(p_i, p_j), h)). Often compared with Position Based Dynamics and N-Body Simulation.