A variational formulation of thermomechanical constitutive update for hyperbolic conservation laws

Heuzé, Thomas; Stainier, Laurent

doi:10.1016/j.cma.2022.114893

Cited by 4 publications

(4 citation statements)

References 79 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A last example involving large strains pertains to a multi‐holed elementary rectangular domain submitted to some impact loading, which is extracted from Reference 50. Figure 9 shows the computational domain and associated boundary conditions.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…According to the particular chosen set of conservation laws (5), the Jacobian determinant J may either be computed as the determinant of the deformation gradient (i.e., J = det F), or updated via the solution of an additional conservation law written on it, 50,51 especially when quasi-incompressible responses 51 or hydrodynamic equation of state 25 are described. The simplest expression one can consider for the volumetric part W H (J) is a quadratic one…”

Section: Constitutive Modelmentioning

confidence: 99%

“…It is clear that the moment matrix A(X) (50) of size m × m should be invertible wherever the MLS shape functions are to be evaluated. On the one hand, a necessary condition for the moment matrix to be invertible is that N p ≥ m. On the other hand, pathological arrangements of material points leading to a singular matrix should be avoided, especially if N p = m (see Reference 11 for more details on it), hence the locations of material points within the grid cell Ω e may not be completely arbitrary.…”

Section: Forward Mapping Particle To Grid Via a Moving Least Square A...mentioning

confidence: 99%

See 2 more Smart Citations

ADER discontinuous Galerkin Material Point Method

Lakiss,

Heuzé,

Tannous

et al. 2023

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

The first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

show abstract

Section: Numerical Examplesmentioning

confidence: 99%

Section: Constitutive Modelmentioning

confidence: 99%

Section: Forward Mapping Particle To Grid Via a Moving Least Square A...mentioning

confidence: 99%

See 1 more Smart Citation

ADER discontinuous Galerkin Material Point Method

Lakiss,

Heuzé,

Tannous

et al. 2023

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…In order to investigate shear band propagation, Eremin et al [30] used microstructure based finite-difference method to model plastic flow in low carbon steel. Heuze and Stainier [31] created a variational framework which will let hyperbolic conservation laws work with thermo-hyperelastic-viscoplastic constitutive equations [32]. Aided with a high-order and pressure oscillating-free scheme, Eulerian description of conservation laws as well as a hyper-elastic framework were also used to understanding multi-material elastic-plastic flow [33,34].…”

Section: Introductionmentioning

confidence: 99%

Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

Yang¹,

Lowe²

2023

Preprint

View full text Add to dashboard Cite

This paper reports a theoretical and numerical framework to model nonlinear waves in elastic-plastic solids. Formulated in the Eulerian frame, the governing equations employed include the continuity equation, the momentum equation, and an elastic-plastic constitutive relation. The complete governing equations are a set of first-order, fully coupled partial differential equations with source terms. The primary unknowns are velocities and deviatoric stresses. By casting the governing equations into a vector-matrix form, we derive the eigenvalues of the Jacobian matrix to show the wave speeds. The eigenvalues are also used to calculate the Courant number for numerical stability. The model equations are solved using the Space-Time Conservation Element and Solution Element (CESE) method. The approach is validated by comparing our numerical results to an analytical solution for the special case of longitudinal wave motion.

show abstract