Eulerian/Lagrangian Sharp Interface Schemes for Multimaterials
We present multimaterial simulations using both Eulerian and Lagrangian schemes. The methods employed are based on classical Godunov-like methods that are adapted to treat the case of interfaces separating different materials. In the models considered, th
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Abstract We present multimaterial simulations using both Eulerian and Lagrangian schemes. The methods employed are based on classical Godunov-like methods that are adapted to treat the case of interfaces separating different materials. In the models considered, the gas, liquids, or elastic materials are described by specific constitutive laws, but the governing equations are the same. Examples of gas–gas and gas–elastic material interactions in one- and two-spatial dimensions are presented.
1 Introduction Physical and engineering problems that involve several materials are ubiquitous in nature and in applications. The main contributions in the direction of simulating these phenomena go back to Godunov (1978) for the model and Favrie et al. (2009) for numerical simulations. However, the numerical scheme presented in that paper is relatively complicated and has the disadvantage that the interface is diffused over a certain number of grid points. We propose a simple second-order accurate method to recover a sharp interface description keeping the solution stable and nonoscillating. This scheme can be adapted to both Eulerian and Lagrangian frameworks. Y. Gorsse A. Iollo IMB and Inria, Bordeaux, France e-mail: [email protected] A. Iollo e-mail: [email protected] T. Milcent (&) I2M and Arts et Metiers Paristech, Bordeaux, France e-mail: [email protected]
Y. Zhou et al. (eds.), Fluid-Structure-Sound Interactions and Control, Lecture Notes in Mechanical Engineering, DOI: 10.1007/978-3-642-40371-2_12, Springer-Verlag Berlin Heidelberg 2014
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2 Eulerian Model The conservative form of elastic media equations in the Eulerian framework are 8 q þ divx ðquÞ ¼ 0 > > < t ðquÞt þ divx ðqu u rÞ ¼ 0 ð1Þ > ðqeÞt þ divx ðqeu rT uÞ ¼ 0 > : ðrx Y Þt þr x ðu rx Y Þ ¼ 0 The unknowns are the density qðx; tÞ, the velocity uðx; tÞ, the total energy per unit mass eðx; tÞ; and the backward characteristics of the problem Yðx; tÞ: Here, rðx; tÞ is the Cauchy stress tensor in the physical domain.
3 Lagrangian Model The counterpart of these equations in the lagrangian framework are ðq0 Xt Þt divn ðTÞ ¼ 0 ðq0 eÞt divn ðTT Xt Þ ¼ 0
ð2Þ
The unknowns are the velocity Xt ðn; tÞ (the direct characteristics of the problem are Xðn; tÞ and the total energy per unit mass eðn; tÞ: Here, Tðn; tÞ is the first Piola–Kirchhoff stress tensor in the reference domain and q0 is the initial density.
4 Constitutive Law To close the system, a constitutive law is chosen: s c1 exp cv q 1 2 p1 v 2Þ þ þ ðTrðBÞ e ¼ e juj ¼ 2 q0 c1 q
ð3Þ
is the modified left Cauchy–Green tensor which where sðx; tÞ is the entropy and B depends on rx Y: The constants cv ; c; p1 ; v characterize a given material. The two first terms of (3) represent a stiffened gas and the third one represents a NeoHookean elastic solid. The stress tensors r and T are then derived from this constitutive law as a function of the problem unknowns.
Eulerian/Lagrangian Sharp Interface Schemes for Multimate
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