arXiv subjects

Evgeniy Romenski

Publications and source records attributed to Evgeniy Romenski.

At least 19 recordsLinked to original sources

arXiv2025

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

This paper presents a derivation of a first-order reduction and 3+1 decomposition of the teleparallel equivalent of general relativity (TEGR) in the pure-tetrad formulation (no spin connection). Our analysis demonstrates that in vacuum spacetimes, our 3+1 TEGR equations has the principal part of the differential operator equivalent to the one of tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen, and therefore the presented 3+1 decomposition of TEGR also admits a symmetric hyperbolic formulation, a desirable property for ensuring well-posedness of the initial value problem. Furthermore, the structure of the 3+1 equations possess a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity.

gr-qc↗

arXiv2024

A unified SHTC multiphase model of continuum mechanics

In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) equations, can describe the arbitrary number of phases that can be heat-conducting inviscid and viscous fluids, as well as elastoplastic solids. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the SHTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the SHTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging, we restrict our numerical examples to a simplified version of the model, focusing on the isentropic limit for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications for compressible multiphase problems. Notably, results are presented for multiphase flows in all the relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids.

math.NA↗

arXiv2023

Nonequilibrium model for compressible two-phase two-pressure flows with surface tension

In continuum thermodynamics, models of two-phase mixtures typically obey the condition of pressure equilibrium across interfaces between the phases. We propose a new non-equilibrium model beyond that condition, allowing for microinertia of the interfaces, surface tension, and different phase pressures. The model is formulated within the framework of Symmetric Hyperbolic Thermodynamically Compatible equations, and it possesses variational and Hamiltonian structures. Finally, via formal asymptotic analysis, we show how the pressure equilibrium is restored when fast degrees of freedom relax to their equilibrium values.

physics.flu-dyn↗

arXiv2023

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.

math.NA↗

arXiv2022

Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase flows

In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in \cite{Romenski2007,Romenski2009}. All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine-Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer-Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer-Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer-Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer-Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.

math.AP↗

arXiv2022

Comparison of the Symmetric Hyperbolic Thermodynamically Compatible framework with Hamiltonian mechanics of binary mixtures

How to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar but not identical, and are compared in this article. They are compared both analytically and numerically on several levels of description, varying in the amount of detail. The GENERIC equations, stemming from the Liouville equation, contain terms expressing self-advection of the relative velocity by itself, which lead to a vorticity-dependent diffusion matrix after a reduction. The SHTC equations, on the other hand, do not contain such terms. We also show how to formulate a theory of mixtures with two momenta and only one temperature that is compatible with the Liouville equation and possesses the Hamiltonian structure, including Jacobi identity.

physics.flu-dyn↗

arXiv2021

Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme

We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure of relaxation of tangential stresses similar to Maxwell's visco-elasticity theory. The model is formulated as a system of first-order hyperbolic partial differential equations with possibly stiff non-linear relaxation source terms. The computational strategy is based on a staggered semi-implicit scheme which can be applied in particular to low-Mach number flows as usually required for flows of non-Newtonian fluids. The applicability of the model and numerical scheme is demonstrated on a few standard benchmark test cases such as Couette, Hagen-Poiseuille, and lid-driven cavity flows. The numerical solution is compared with analytical or numerical solutions of the Navier-Stokes theory with the Herschel-Bulkley constitutive model for nonlinear viscosity.

physics.flu-dyn↗

arXiv2021

Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation

We derive and study a new hyperbolic two-phase model of a porous deformable medium saturated by a viscous fluid. The governing equations of the model are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems and by generalizing the unified hyperbolic model of continuum fluid and solid mechanics. Similarly to the unified model, the presented model takes into account the viscosity of the saturating fluid through a hyperbolic reformulation. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturated fluid. Using the presented nonlinear finite-strain SHTC model, the governing equations for the propagation of small-amplitude waves in a porous medium saturated with a viscous fluid are derived. As in the conventional Biot theory of porous media, three types of waves can be found: fast and slow compression waves and shear waves. It turns out that the shear wave attenuates rapidly due to the viscosity of the saturating fluid, and this wave is difficult to see in typical test cases. However, some test cases are presented in which shear waves can be observed in the vicinity of interfaces between regions with different porosity.

physics.flu-dyn↗

arXiv2020

A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics

We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homogeneous PDE system is endowed with two stationary linear differential constraints (involutions), which require the curl of distortion field and the curl of the thermal impulse to be zero for all times. In the stiff relaxation limit, the unified model tends asymptotically to the compressible Navier-Stokes equations. The new structure-preserving scheme presented in this paper can be proven to be exactly curl-free for the homogeneous part of the PDE system, i.e. in the absence of relaxation source terms. We furthermore prove that the scheme is quasi asymptotic preserving in the stiff relaxation limit, in the sense that the numerical scheme reduces to a consistent second-order accurate discretization of the compressible Navier-Stokes equations when the relaxation times tend to zero. Last but not least, the proposed scheme is suitable for the simulation of all Mach number flows thanks to its conservative formulation and the implicit discretization of the pressure terms.

physics.flu-dyn↗

arXiv2020

A unified first order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

Earthquake fault zones are more complex, both geometrically and rheologically, than an idealised infinitely thin plane embedded in linear elastic material. To incorporate nonlinear material behaviour, natural complexities, and multi-physics coupling within and outside of fault zones, here we present a first-order hyperbolic and thermodynamically compatible mathematical model for a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases. The fault geometry and secondary cracks are described via a scalar function $ξ\in [0,1]$ that indicates the local level of material damage. The model also permits the representation of arbitrarily complex geometries via a diffuse interface approach based on the solid volume fraction function $α\in [0,1]$. Neither of the two scalar fields $ξ$ and $α$ needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). The model shares common features with phase-field approaches but substantially extends them. We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material.

physics.geo-ph↗

arXiv2020

Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure

We are concerned with the numerical solution of a unified first order hyperbolic formulation of continuum mechanics that originates from the work of Godunov, Peshkov and Romenski (GPR model) and which is an extension of nonlinear hyperelasticity that is able to describe simultaneously nonlinear elasto-plastic solids at large strain, as well as viscous and ideal fluids. The proposed governing PDE system also contains the effect of heat conduction and can be shown to be symmetric and thermodynamically compatible. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes and material fatigue effects, by adding a new scalar variable to the governing PDE system. This extra parameter describes the material damage and is governed by an advection-reaction equation, where the stiff and highly nonlinear reaction mechanisms depend on the ratio of the local von Mises stress to the yield stress of the material. The stiff reaction mechanisms are integrated in time via an efficient exponential time integrator. Due to the multiple space-time scales, the model is solved on space-time adaptive Cartesian meshes using high order discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiting. A key feature of our new model is the use of a twofold diffuse interface approach that allows the cracks to form anywhere and at any time, independently of the chosen computational grid, without requiring that the geometry of the rupture fault be known a priori. We furthermore make use of a scalar volume fraction function that indicates whether a given point is inside the solid or outside, allowing the description of solids of arbitrarily complex shape.

math.NA↗

arXiv2020

Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for multiphase mixtures

A two-phase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a two-phase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They form a hyperbolic system of conservation equations that satisfy the fundamental laws of thermodynamics. Two types of phase interaction are introduced in the model: phase pressure relaxation to a common value and interfacial friction. Inelastic deformations also can be accounted for by source terms in the equation for distortion. The thus formulated model can be used for studying general compressible fluid flows in a deformable elastoplastic porous medium, and for modeling wave propagation in a saturated porous medium. Governing equations for small-amplitude wave propagation in a uniform porous medium saturated with a single fluid are derived. They form a first-order hyperbolic PDE system written in terms of stress and velocities and, like in Biot's model, predict three types of waves existing in real fluid-saturated porous media: fast and slow longitudinal waves and shear waves. For the numerical solution of these equations, an efficient numerical method based on a staggered-grid finite difference scheme is used. The results of solving some numerical test problems are presented and discussed

physics.flu-dyn↗

arXiv2020

A new continuum model for general relativistic viscous heat-conducting media

The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein's Special Relativity and the Euler-Lagrange structure of General Relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as the viscosity and heat conductivity are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach.

gr-qc↗

arXiv2019

A new causal general relativistic formulation for dissipative continuum fluid and solid mechanics and its solution with high-order ADER schemes

We present a unified causal general relativistic formulation of dissipative and non-dissipative continuum mechanics. The presented theory is the first general relativistic theory that can deal simultaneously with viscous fluids as well as irreversible deformations in solids and hence it also provides a fully covariant formulation of the Newtonian continuum mechanics in arbitrary curvilinear spacetimes. In such a formulation, the matter is considered as a Riemann-Cartan manifold with non-vanishing torsion and the main field of the theory being the non-holonomic basis tetrad field also called four-distortion field. Thanks to the variational nature of the governing equations, the theory is compatible with the variational structure of the Einstein field equations. Symmetric hyperbolic equations are the only admissible equations in our unified theory and thus, all perturbations propagate at finite speeds (even in the diffusive regime) and the Cauchy problem for the governing PDEs is locally well-posed for arbitrary and regular initial data which is very important for the numerical treatment of the presented model. Nevertheless, the numerical solution of the discussed hyperbolic equations is a challenging task because of the presence of the stiff algebraic source terms of relaxation type and non-conservative differential terms. Our numerical strategy is thus based on an advanced family of high-accuracy ADER Discontinuous Galerkin and Finite Volume methods which provides a very efficient framework for general relaxation hyperbolic PDE systems. An extensive range of numerical examples is presented demonstrating the applicability of our theory to relativistic flows of viscous fluids and deformation of solids in Minkowski and curved spacetimes.

gr-qc↗

arXiv2019

Continuum mechanics with torsion

This paper is an attempt to introduce methods and concepts of the Riemann-Cartan geometry largely used in such physical theories as general relativity, gauge theories, solid dynamics, etc. to fluid dynamics in general and to studying and modeling turbulence in particular. Thus, in order to account for the rotational degrees of freedom of the irregular dynamics of small scale vortexes, we further generalize our unified first-order hyperbolic formulation of continuum fluid and solid mechanics which treats the flowing medium as a Riemann-Cartan manifold with zero curvature but non-vanishing torsion. We associate the rotational degrees of freedom of the main field of our theory, the distortion field, to the dynamics of microscopic (unresolved) vortexes. The distortion field characterizes the deformation and rotation of the material elements and can be viewed as anholonomic basis triad with non-vanishing torsion. The torsion tensor is then used to characterize distortion's spin and is treated as an independent field with its own time evolution equation. This new governing equation has essentially the structure of the non-linear electrodynamics in a moving medium and can be viewed as a Yang-Mills-type gauge theory. The system is closed by providing an example of the total energy potential. The extended system describes not only irreversible dynamics (which raises the entropy) due to the viscosity or plasticity effect but it also has dispersive features which are due to the reversible energy exchange (which conserves the entropy) between micro and macro scales. Both the irreversible and dispersive processes are represented by relaxation-type algebraic source terms so that the overall system remains first-order hyperbolic. The turbulent state is then treated as an excitation of the equilibrium laminar state due to the non-linear interplay between dissipation and dispersion.

physics.flu-dyn↗

arXiv2018

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.

physics.flu-dyn↗

arXiv2017

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).

physics.class-ph↗

arXiv2017

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.

physics.flu-dyn↗