Stochastic Smoothed Particle Hydrodynamics for Stochastic Mechanics Problems

2026-05-07

AI Summary

This work addresses the challenge of modeling uncertainty in mechanical systems within the smoothed particle hydrodynamics (SPH) framework. It presents the first fully mesh-free stochastic SPH formulation by integrating orthogonal polynomial chaos expansion with SPH kernel functions. Random fields are discretized via Karhunen–Loève expansion, and stochastic partial differential equations are transformed into coupled systems of ordinary differential equations through Galerkin projection. Accurate enforcement of boundary conditions is achieved using a ghost particle method enhanced with a gradient correction matrix, while time integration employs a predictor–corrector scheme. Validated on uncertain Burgers’ equation and convection problems, the proposed method yields statistical moments in excellent agreement with Monte Carlo simulations, achieving up to three orders of magnitude reduction in computational cost.

Abstract

Smoothed Particle Hydrodynamics (SPH_ is a mesh-free Lagrangian method renowned for modeling large deformations and free-surface flows, yet classical formulations remain confined to deterministic systems. We introduce Stochastic SPH (S-SPH), which employs orthogonal Polynomial Chaos expansions to represent uncertainties in system parameters, forcing functions, and boundary or initial conditions, while spatial variation is captured via the SPH kernel. Random fields are discretized through Karhunen-Loève expansions, and a Galerkin projection in the polynomial basis transforms the underlying SPDE into a coupled system of ordinary differential equations governing the time evolution of expansion coefficients. To enforce Dirichlet and Neumann conditions in a mesh-free context, ghost-particle techniques augmented by a gradient-correction matrix are employed, and a predictor-corrector integration scheme ensures numerical stability. We validate S-SPH on benchmark problems, including one-dimensional advection with stochastic advection speed, inviscid Burgers' equations with random initial amplitudes, and two-dimensional Burgers' flows with uncertain Fourier-mode initial fields and viscosity, demonstrating excellent agreement with Monte Carlo simulation statistics of mean and variance. Remarkably, S-SPH achieves up to three orders of magnitude reduction in computational cost relative to direct sampling approaches. The proposed framework thus provides an efficient, accurate, and fully mesh-free methodology for uncertainty quantification in complex mechanics applications.

Problem

• Stochastic Mechanics
• Uncertainty Quantification
• Smoothed Particle Hydrodynamics
• Random Fields
• Mesh-free Methods

Innovation

• Stochastic SPH
• Polynomial Chaos Expansion
• Uncertainty Quantification
• Mesh-free Method
• Karhunen-Loève Expansion