1. Introduction

Mixed cells in arbitrary Lagrangian-Eulerian (ALE) or Eulerian methods contain interfaces between different materials or a mixture of materials. In the next section, we will not distinguish these two methods, considering that both methods solve the advection equation, including the vicinity of mixed cells. Most of these methods use a two-stage approximation of equations. The first stage considers gas dynamics or elastoplasticity equations without convective terms. The convective transfer comes into play at the second stage. Among many similar methods, we consider only the ALE methods that contain Lagrangian gas dynamics and elastoplasticity in the pure form, and the problem of mixed cells at that particular stage is the subject of research reported here. Note that mixed cells can be present even in purely

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Lagrangian techniques, and the problems related to their presence should also be addressed in this case.

Here, we will generally use the term "Lagrangian gas dynamics" (or simply "gas dynamics"), bearing in mind that, in the case of elastoplasticity, this will also involve equation terms related to the stress tensor deviator. Historically, several approaches to the problem of mixed cells in gas dynamics associated with materials distinction in such cells have been considered. In this chapter, we consider only the single-velocity model of matter. The major approach that has become predominant these days uses complete thermodynamic distinction of materials.<sup>1</sup> Next, we will use the term "material" meaning that, mathematically, an interface can also divide identical materials; moreover, one of the materials can be vacuum and/or a perfectly rigid body.

Thermodynamic parameters in gas dynamics include density, internal energy, and pressure. If other processes are modeled, the number of parameters increases; for example, for elastoplasticity, additional parameters will include components of the stress tensor deviator. In addition to thermodynamic parameters, volume fractions of constituent materials are introduced in each mixed cell that can be used to determine the geometric location of the interface inside a mixed cell, which is used in some models.2

This approach to materials identification allows one to model mixed cells containing not only contacting but also intermingled materials. When mixed cells are used for gas dynamics equations, additional closing relations are needed, which in fact define the interaction of materials inside the cell (the subcell interaction). Most of the known models manage with information about volume (or mass) fractions of the materials and their thermodynamic states [12–26]. Such models can be divided into two classes according to the number of computational stages involved.3

The first class of models is based on introducing closure models at a single stage, while the second class includes two-stage models, in which the second stage is in fact complementary to the first one and involves additional interaction between materials inside a mixed cell (so-called subcell interaction).

Next, we often use the terms "model" and "method" without distinction. One should note here that a method is understood to be an algorithm implemented in the form of a program and based on some physical model.

<sup>1</sup> Early in the development of Eulerian methods, a smaller number of parameters have been used to identify the materials; for example, in [1, 2], mass fractions of the materials and average energy of matter were employed. Accordingly, other closing relations were used, the required number of which in this case is plus one compared with the complete materials distinction. The models thus considered include the "isobar-isothermal" and the "isobar-isodQ" models, which, although successful in some respect, in the general case failed to deliver acceptable accuracy of results.

<sup>2</sup> The problem of identifying the contact location based on the material volume fractions is beyond the scope of this study; it is a separate problem discussed in dedicated studies (see, e.g. [1, 3–11]).

<sup>3</sup> Our classification and description of models is limited to the case of two materials in a cell, although many formulas mentioned in this chapter are also suitable for their larger number. For this reason, some models developed specifically for the case of several materials in a cell are left beyond the scope of our review.

Basic single-stage closure methods include the following:


Lagrangian techniques, and the problems related to their presence should also be addressed in

Here, we will generally use the term "Lagrangian gas dynamics" (or simply "gas dynamics"), bearing in mind that, in the case of elastoplasticity, this will also involve equation terms related to the stress tensor deviator. Historically, several approaches to the problem of mixed cells in gas dynamics associated with materials distinction in such cells have been considered. In this chapter, we consider only the single-velocity model of matter. The major approach that has become predominant these days uses complete thermodynamic distinction of materials.<sup>1</sup> Next, we will use the term "material" meaning that, mathematically, an interface can also divide identical materials; moreover, one of the materials can be vacuum and/or a perfectly rigid

Thermodynamic parameters in gas dynamics include density, internal energy, and pressure. If other processes are modeled, the number of parameters increases; for example, for elastoplasticity, additional parameters will include components of the stress tensor deviator. In addition to thermodynamic parameters, volume fractions of constituent materials are introduced in each mixed cell that can be used to determine the geometric location of the interface inside a mixed

This approach to materials identification allows one to model mixed cells containing not only contacting but also intermingled materials. When mixed cells are used for gas dynamics equations, additional closing relations are needed, which in fact define the interaction of materials inside the cell (the subcell interaction). Most of the known models manage with information about volume (or mass) fractions of the materials and their thermodynamic states [12–26]. Such models can be divided into two classes according to the number of computa-

The first class of models is based on introducing closure models at a single stage, while the second class includes two-stage models, in which the second stage is in fact complementary to the first one and involves additional interaction between materials inside a mixed cell

Next, we often use the terms "model" and "method" without distinction. One should note here that a method is understood to be an algorithm implemented in the form of a program

Early in the development of Eulerian methods, a smaller number of parameters have been used to identify the materials; for example, in [1, 2], mass fractions of the materials and average energy of matter were employed. Accordingly, other closing relations were used, the required number of which in this case is plus one compared with the complete materials distinction. The models thus considered include the "isobar-isothermal" and the "isobar-isodQ" models, which, although

The problem of identifying the contact location based on the material volume fractions is beyond the scope of this study;

Our classification and description of models is limited to the case of two materials in a cell, although many formulas mentioned in this chapter are also suitable for their larger number. For this reason, some models developed specifically for

successful in some respect, in the general case failed to deliver acceptable accuracy of results.

it is a separate problem discussed in dedicated studies (see, e.g. [1, 3–11]).

the case of several materials in a cell are left beyond the scope of our review.

this case.

70 Lagrangian Mechanics

body.

1

2

3

cell, which is used in some models.2

tional stages involved.3

(so-called subcell interaction).

and based on some physical model.


Note that these four models and methods developed on their basis are relaxation with respect to pressure. In a number of studies, nonrelaxation methods have been proposed, which use the following assumptions (models):


Two-stage models include the stage of subcell interactions between the materials in the nonequilibrium state; so the first stage here can only use models 5–7. This approach for closure models has been proposed independently in [20, 21]. The subcell pressure relaxation method in [20] is versatile and it is used in combination with models 5–7, denoted as the ∇ u-PR, Δp-PR and Δu-PR (pressure relaxation) methods.

All the above-mentioned methods do not employ the contact location inside a mixed cell. However, there are methods that make essential use of the information about the contact location. A method of this kind was first proposed in [21] and then developed in the "interface-aware subscale dynamics" IA-SSD method [24, 25] for the multimaterial case. It offers a two-stage model, the first stage of which employs the ∇ u model. At the second stage, driven by the materials' individual pressures, the interface between the materials moves normally to it. The interface is reconstructed based on the volume fractions, and its motion is accomplished based on the solution of an acoustic Riemann problem (model 3).

Let us point out one common feature (associated with the assumptions made in the models) of all the above closure methods. Velocity in the methods is normal to the interface (definite or imaginary) irrespective of interface location relative to the vector of velocity. In fact, they are isotropic in the sense that compression (expansion) ratios of materials are assumed to be equal in all directions. One can mention a number of other closure methods that employ algorithms similar to those used in the above-mentioned models [27–33]. This property of the methods is quite acceptable for most applications, but there are problems (see below), as applied to which it results in significant errors in simulations.

In [34], anisotropic closure methods, ACM-1 and ACM-2, are proposed, which are an extension of models 5–7. They possess all the advantages of methods 5–7, which are central to the EGAK code [35] when modeling flows, for which one can assume that they are isotropic, but have an important advantage when modeling more complex flows.

Apart from the basic closure method, mixed cells require additional relations to address the ways of pressure and artificial viscosity calculations for the whole cell and artificial viscosity calculations for the materials. Six approaches to calculate the artificial viscosity of materials are discussed in [36].
