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SPH pressure forces are equal-opposite and central
Statement
For the system in Fixed-width regularized conservative SPH particle system, set g_ij=∇W_ε(q_i−q_j) and define the force contribution F_ij=−m_im_j(A_i+A_j)g_ij. Radial evenness gives g_ji=−g_ij and g_ii=0, so for i≠j, F_ji=−F_ij; also F_ij is parallel to q_i−q_j (the zero vector is allowed).
Facts & Assumptions
Given: A state in the positive-density domain, positive masses, and a radial C² kernel.
Fixed-width regularized conservative SPH particle system: A_i=p_i/ρ_i² and the acceleration sum is −Σ_jm_j(A_i+A_j)∇W_ε(q_i−q_j).
Scaled symmetric compactly supported kernel family: W_ε is differentiable and radial, hence even.
The chain rule for total derivatives: : the derivative of W_ε composed with r↦−r is obtained by the chain rule.
Proof
Since W_ε(−r)=W_ε(r), differentiating the identity in r using [F3] gives ∇W_ε(−r)=−∇W_ε(r). The scalar coefficient m_im_j(A_i+A_j) is symmetric in i,j, so F_ji=−F_ij.
If r≠0 and u·r=0, then |r+tu|²=|r|²+t²|u|², so the derivative at t=0 of W_ε(r+tu), which is w_ε(|r+tu|), is zero by the chain rule [F3]. Thus ∇W_ε(r) is orthogonal to every u orthogonal to r and lies in span(r). At r=0, differentiable evenness gives ∇W_ε(0)=0 by [F3]. Hence each F_ij is central.
Depends on
Used by
- A conservative positive-pressure SPH system can close a particle gap Counterexample
- Two SPH particles receive equal and opposite pressure forces Example
- Linear and angular momentum are conserved by semidiscrete SPH Proposition
- The matched internal-energy equation conserves total semidiscrete energy Proposition
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)