skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A thermodynamically compatible splitting procedure in hyperelasticity

Abstract

A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations. In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simplemore » criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.« less

Authors:
; ;
Publication Date:
OSTI Identifier:
22314882
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 270; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY; ELASTICITY; ENTROPY; LIMITING VALUES; MATHEMATICAL MODELS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; SHEAR; SOLIDS; STORED ENERGY; STRESSES; TENSORS; VELOCITY

Citation Formats

Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr. A thermodynamically compatible splitting procedure in hyperelasticity. United States: N. p., 2014. Web. doi:10.1016/J.JCP.2014.03.051.
Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, & Ndanou, S., E-mail: serge.ndanou@univ-amu.fr. A thermodynamically compatible splitting procedure in hyperelasticity. United States. doi:10.1016/J.JCP.2014.03.051.
Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr. Fri . "A thermodynamically compatible splitting procedure in hyperelasticity". United States. doi:10.1016/J.JCP.2014.03.051.
@article{osti_22314882,
title = {A thermodynamically compatible splitting procedure in hyperelasticity},
author = {Favrie, N., E-mail: nicolas.favrie@univ-amu.fr and Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr},
abstractNote = {A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations. In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.},
doi = {10.1016/J.JCP.2014.03.051},
journal = {Journal of Computational Physics},
number = ,
volume = 270,
place = {United States},
year = {Fri Aug 01 00:00:00 EDT 2014},
month = {Fri Aug 01 00:00:00 EDT 2014}
}
  • An application of the theory of thermodynamically compatible hyperbolic systems to design a multiphase compressible flow models is discussed. With the use of such approach the governing equations are derived from the first principles, formulated in a divergent form and can be transformed to a symmetric hyperbolic system in the sense of Friedrichs. A usage of the proposed approach is described for the development of multiphase compressible fluid models, including two-phase flow models.
  • A field compatible calibration procedure for generating ppb levels of peroxyacetyl nitrate is presented. The procedure is based upon the condensed-phase synthesis of peroxyacetyl nitrate (PAN) in octane solution. These solutions show PAN loss of <4% month/sup -1/ when stored at -20/sup 0/C. The dilute PAN solutions are injected directly into known volumes of air in Tedlar bags to obtain ppb gas-phase concentrations. The addition of low levels of NO/sub 2/ into the bags stabilizes PAN in the gas phase.
  • Purpose: This study was designed to systematically investigate artifacts caused by interventional needles recommended for use in MRI, with focus on field strength, needle/mandrin type, orientation and sequence. Methods: Eight different MRI compatible needles were placed in porcine tissue and examined at 1.5 and 3.0 T with balanced-steady-state-free-precession (B-SSFP) and T1-weighted-spoiled-gradient-echo (T1-SPGR) sequences in different orientations to B{sub 0}. Artifact diameters with regards to the primary, inner, and secondary, outer artifacts were assessed and statistically evaluated. Results: The types and degree of artifacts varied considerably, especially between different mandrin types even for the same needles. Orientation of the needle inmore » the magnetic field was another main contributor to the artifact dimensions. Less important factors were the type of pulse sequence and field strength. Artifacts ranged from 0.7 mm (steel, 0 Degree-Sign , B-SSFP, 3.0 T, inner) to 71.4 mm (nitinol, 90 Degree-Sign , B-SSFP, 1.5 T, outer). Inner artifact diameters in B-SSFP were slightly larger (8.2 {+-} 5.7 mm) than those in T1-SPGR (7.6 {+-} 5.4 mm) and comparable between 1.5 and 3.0 T (e.g., 8.0 vs. 8.4 mm, B-SSFP). Conclusions: Although all were sold as 'MR compatible,' the artifacts differed greatly between needle types, and even more so for different mandrins. The results suggest an empirical approach to the needle choice based on lesion type and approach angle.« less