A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for solid dynamics

• Published in: Computer methods in applied mechanics and engineering 2019-08, Vol.353 (C), p.467-490
• Main Authors: Lieberman, Evan J., Liu, Xiaodong, Morgan, Nathaniel R., Luscher, Darby J., Burton, Donald E.
• Format: Article
• Language: English
• Subjects: Analytic solutions; Computer simulation; Conservation laws; Constitutive models; Deformation; Discontinuous Galerkin; ENGINEERING; Exact solutions; Galerkin method; Hydrodynamics; Lagrangian; Model accuracy; Plastic properties; Polynomials; Runge-Kutta method; Shocks; Solid dynamics; Stability; Strain; Stress distribution; Taylor series; Test procedures; Time integration; Two dimensional models; Velocity gradient; Yield criteria
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2019.05.006

We present a new multidimensional high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher-order elements. We also present new one-dimensional test problems that have an analytic solution corresponding to a hyperelastic–plastic wave. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated high-order Taylor series polynomials. The stress fields are represented using nodal quantities. The constitutive models used to calculate the deviatoric stress are either a hypoelastic–plastic, infinitesimal strain hyperelastic–plastic, or finite strain hyperelastic–plastic model. These constitutive models require new methods for calculating high-order polynomials for the velocity gradient and deformation gradient in an element. The plasticity associated with the strength model is determined using a radial return method with a J2 yield criterion and perfect plasticity. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. A diverse suite of 1D and 2D test problems are calculated. The new 1D piston test problems, which have analytic solutions for each elastic–plastic model, are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new Lagrangian DG method. 2D test problems are calculated to demonstrate the stability and robustness of the new Lagrangian DG method on multidimensional problems with high-order elements, which have faces that can bend. •New analytic solutions to hyperelastic–plastic piston problems are derived.•First high-order Lagrangian DG hydrodynamic method with a hypoelastic–plastic model.•Hyperelastic–plastic models generally more accurate than hypoelastic–plastic model.•The new Lagrangian DG method is accurate and stable for 2D solid dynamics problems.