A unified hyperbolic formulation for viscous fluids and elastoplastic solids

We discuss a unified flow theory which in a single system of hyperbolic partial differential equations (PDEs) can describe the two main branches of continuum mechanics, fluid dynamics, and solid dynamics. The fundamental difference from the classical continuum models, such as the Navier-Stokes for example, is that the finite length scale of the continuum particles is not ignored but kept in the model in order to semi-explicitly describe the essence of any flows, that is the process of continuum particles rearrangements. To allow the continuum particle rearrangements, we admit the deformability of particle which is described by the distortion field. The ability of media to flow is characterized by the strain dissipation time which is a characteristic time necessary for a continuum particle to rearrange with one of its neighboring particles. It is shown that the continuum particle length scale is intimately connected with the dissipation time. The governing equations are represented by a system of first order hyperbolic PDEs with source terms modeling the dissipation due to particle rearrangements. Numerical examples justifying the reliability of the proposed approach are demonstrated.Comment: 6 figure

Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences between the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences.Comment: 14 figure

A numerical scheme for non-Newtonian fluids and plastic solids under the GPR model

A method for modeling non-Newtonian fluids (dilatants and pseudoplastics) by a power law under the Godunov-Peshkov-Romenski model is presented, along with a new numerical scheme for solving this system. The scheme is also modified to solve the corresponding system for power-law elastoplastic solids. The scheme is based on a temporal operator splitting, with the homogeneous system solved using a finite volume method based on a WENO reconstruction, and the temporal ODEs solved using an analytical approximate solution. The method is found to perform favorably against problems with known exact solutions, and numerical solutions published in the open literature. It is simple to implement, and to the best of the authors’ knowledge it is currently the only method for solving this modified version of the GPR model.EPSRC Centre for Doctoral Training in Computational Methods for Materials Science under grant EP/L015552/

A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner

A new framework for the computational simulation of problems arising in continuum me- chanics is presented. It is unified in the sense that it can describe all three major phases of matter within the same set of equations. It is able to represent inviscid fluids, Newtonian and non-Newtonian viscous fluids, elastic and plastic solids, and reactive species. These materials are presented with a variety of equations of state, and there is a clear methodology for extending the framework to more exotic materials using other constitutive equations. It is capable of accurately modeling interfaces between regions occupied by different phases, and by the vacuum. The problem of impact-induced detonation in an elastoplastic confiner is one that incorpo- rates the whole range of aforementioned material types, representing a challenge to existing frameworks. This new framework is shown to accurately and efficiently solve this problem. The framework comprises a modification and extension of the Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, along with a new set of operator-splitting-based numerical solvers to allow for the efficient solution of the problems that it is put to, and a new Riemann ghost fluid method for accurate simulation of material interfaces. In addition to this work, novel mathematical analyses of the structure of the GPR equations - and the numerical methods currently used to solve them - are presented in this study. This new framework presents a range of benefits: the conceptual work required to implement a computational simulation involving many different components is greatly reduced, saving time and allowing for greater specialization of computational techniques. This has the po- tential to streamline development of simulation software by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required, for example in the treatment of interfaces in multimaterial problems.Financial support from the EPSRC Centre for Doctoral Training in Computational Methods for Materials Science under grant EP/L015552/

First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory

Motivated by numerically solving the Einstein field equations, we derive a first-order reduction of the second-order $f(T)$-teleparallel gravity field equations in the pure-tetrad formulation (no spin connection). We then restrict our attention to the teleparallel equivalent of general relativity (TEGR) and propose a 3+1 decomposition of the governing equations that can be used in a computational code. We demonstrate that for the matter-free space-time the obtained system of first-order equations is equivalent to the tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen and therefore also admits a symmetric hyperbolic formulation. The structure of the 3+1 equations resembles a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity

Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization)

On thermodynamically compatible finite volume schemes for continuum mechanics

In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the \textit{entropy inequality} as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper \textit{An interesting class of quasilinear systems} of 1961 \textit{exactly} at the discrete level. All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations