JST-SPH: a total Lagrangian, stabilised meshless methodology for mixed systems of conservation laws in nonlinear solid dynamics
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Abstract
The combination of linear finite elements space discretisation with Newmark
family time-integration schemes has been established as the de-facto
standard for numerical analysis of fast solid dynamics. However, this set-up
suffers from a series of drawbacks: mesh entanglements and elemental distortion
may compromise results of high strain simulations; numerical issues,
such as locking and spurious pressure oscillations, are likely to manifest;
and stresses usually reach a reduced order of accuracy than velocities.
Meshless methods are a relatively new family of discretisation techniques
that may offer a solution to problems of excessive distortion experienced by
linear finite elements. Amongst these new methodologies, smooth particles
hydrodynamics (SPH) is the simplest in concept and the most straightforward
to numerically implement. Yet, this simplicity is marred by some
shortcomings, namely (i) inconsistencies of the SPH approximation at or near
the boundaries of the domain; (ii) spurious hourglass-like modes caused by
the rank deficiency associated with nodal integration, and (iii) instabilities
arising when sustained internal stresses are predominantly tensile.
To deal with the aforementioned SPH-related issues, the following remedies
are hereby adopted, respectively: (i) corrections to the kernel functions
that are fundamental to SPH interpolation, improving consistency at and
near boundaries; (ii) a polyconvex mixed-type system based on a new set
of unknown variables (p, F, H and J) is used in place of the displacementbased
equation of motion; in this manner, stabilisation techniques from
computational fluid dynamics become available; (iii) the analysis is set in a
total Lagrangian reference framework.
Assuming polyconvex variables as the main unknowns of the set of first
order conservation laws helps to establish the existence and uniqueness
of analytical solutions. This is a key reassurance for a robust numerical
implementation of simulations. The resulting system of hyperbolic first
order conservation laws presents analogies to the Euler equations in fluid
dynamics. This allows the use of a well-proven stabilisation technique in
computational fluid dynamics, the Jameson Schmidt Turkel (JST) algorithm.
JST is very effective in damping numerical oscillations, and in capturing discontinuities
in the solution that would otherwise be impossible to represent.
Finally, we note that the JST-SPH scheme so defined is employed in a
battery of numerical tests, selected to check its accuracy, robustness, momentum
preservation capabilities, and its viability for solving larger scale,
industry-related problems