D.L. Hicks

An analysis of smoothed particle hydrodynamics

SPH (Smoothed Particle Hydrodynamics) is a gridless Lagrangian technique which is appealing as a possible alternative to numerical techniques currently used to analyze high deformation impulsive loading events. In the present study, the SPH algorithm has been subjected to detailed testing and analysis to determine its applicability in the field of solid dynamics. An important result of the work is a rigorous von Neumann stability analysis which provides a simple criterion for the stability or instability of the method in terms of the stress state and the second derivative of the kernel function. Instability, which typically occurs only for solids in tension, results not from the numerical time integration algorithm, but because the SPH algorithm creates an effective stress with a negative modulus. The analysis provides insight into possible methods for removing the instability. Also, SPH has been coupled into the transient dynamics finite element code PRONTO, and a weighted residual derivation of the SPH equations has been obtained.

Conservative smoothing stabilizes discrete-numerical instabilities in SPH material dynamics computations

Analysis of Smoothed Particle Hydrodynamics reveals that SPH: (i) has an instability; (ii) cannot be stabilized with artificial viscosities; (iii) can be stabilized with conservative smoothing.

Lanczos' generalized derivative: Insights and Applications

Generalizations of the Lanczos derivative provide a basis for certain grid-free Finite Interpolation Methods (FIMs) which appear to have advantages over alternatives such as the standard Finite Difference Methods (FDMs) or Finite Element Methods (FEMs) for certain problems, e.g., hypervelocity impact problems in computational material dynamics.

SPH hydrocodes can be stabilized with shape-shifting

Grid free methods, such as SPH (Smoothed Particle Hydrodynamics), may, eventually, be more efficacious in their representations of material dynamics than the standard fixed grid methods. However, standard SPH (with stress = σ and interpolation weight = W) has instabilities when σW″ 0) and in tension (σ < 0). Conservative smoothing can control the SPH instabilities, but it may smooth out more of the short wave length structure than desired. SPH can also be stabilized by shifting the shape of W to change the sign of W″.

Continuum and discrete hydrodynamical models and globally well-posed problems

Two models for one-dimensional hydrodynamical motion are model C and model D. Model C is the standard, classical, continuum model. Well-posedness proofs for problems based on model C encounter extreme technical difficulties. There are dubious steps in the derivation of model C from first principles. Model D is a modification of model C which avoids these dubious steps. A mixed, initial-boundary-value problem based on model D is properly posed globally. The proof presented here is for the pressure obeying a generalization of the ideal-gas law and a viscous stress obeying a generalization of the Navier-Stokes form. No assumptions requiring the initial data to be small deviations from a constant state are required; the initial data are only required to be physically acceptable.

Initial-boundary value problems for some nonlinear conservation laws

We prove existence and uniqueness of a global weak solution to initial-boundary value problems with time dependent conditions to the equation under the assumption that P is Lipschitz continuous and α is positive, bounded, and measurable.

Weak solutions of initial-boundary value problems for class of nonlinear viscoelastic equations

Existence and uniqueness theorems are established for initials-boundary value problems corresponding to the equaction under assumptions on P and a that include as special cases the isentropic gas law P(v) = v−γ with γ>1 with α(v) = v −1 along with generalizations that allow for Non convex P. The smoothness assumptions made on P and α are much weaker then are usual in studying these problems and the initial data is only required to satisfy .

Introduction to computational continnum dynamics: A personal perspective

This is a survey of some open questions in the computational aspects of continuum (solid, liquid, gas, plasma, multiphase or multimaterial) dynamics.

The convergence of numerical solutions of hydrodynamical shock problems

Models C and D are mathematical formulations for the motion of materials. Model C is the classical, continuum model, and Model D is an alternative with some advantages. This paper proves a discretization of Model D converges when the material law is the ideal gas with a Navier-Stokes viscosity, the boundary data are given by the zero-velocity boundary conditions, and the initial data are physically acceptable. The discretization is conservative and entropy increasing. This numerical scheme can be used in the calculation of solutions to problems involving shock waves in hydrodynamical materials.

Eigenform error estimates of computational material dynamics with shocks. 1.1. Hooke's law materials

Abstract Eigenform error estimates are sharper than previous a priori error bounds for the von Neumann–Richtmyer (vNR) scheme. Theoretical results give insight into the experimental results on the test problems which include shocks. The vNR-RK4 scheme, a Runge–Kutta modification of the original vNR, is not only more accurate than the original vNR but also the eigenform error estimates are more accurate for vNR-RK4.