Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: Theoretical framework and two-dimensional computations

A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the...

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Published in:	Computer methods in applied mechanics and engineering Vol. 197; no. 9; pp. 1056 - 1079
Main Authors:	Scovazzi, G., Love, E., Shashkov, M.J.
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 01.02.2008 Elsevier
Subjects:	Computational techniques Exact sciences and technology Fluid dynamics Fracture mechanics (crack, fatigue, damage...) Fundamental areas of phenomenology (including applications) General theory Hourglass instability Lagrangian shock hydrodynamics Mathematical methods in physics Physics Physics of gases Physics of gases, plasmas and electric discharges Solid mechanics Stabilization Structural and continuum mechanics Variational multi-scale method Variational multi-scale method Lagrangian shock hydrodynamics Stabilization Hourglass instability Predictor-corrector methods Lagrangian Gas dynamics Clausius Duhem inequality Mechanical strength Shear strength Hydrodynamics Momentum Entropy Finite element method Equations of state Lagrange interpolation Multiscale method Variational calculus Modelling Strength of materials Numerical stability
ISSN:	0045-7825, 1879-2138
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Abstract	A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the form of a pressure correction. With respect to broadly accepted hourglass control approaches, the novelty of the method resides in its residual-based character. The stabilizing residual has a definite physical significance, since it embeds a discrete form of the Clausius–Duhem inequality. Effectively, the proposed stabilization samples the production of entropy to counter numerical instabilities. The proposed technique is applied to materials with no shear strength (e.g., fluids), for which there exists a caloric equation of state, and extensions to the case of materials with shear strength (e.g., solids) are also envisioned. The stabilization operator is incorporated into a mid-point, predictor/multi-corrector time integration algorithm, which conserves mass, momentum and total energy. Encouraging numerical results in the context of compressible gas dynamics confirm the potential of the method.
AbstractList	A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the form of a pressure correction. With respect to broadly accepted hourglass control approaches, the novelty of the method resides in its residual-based character. The stabilizing residual has a definite physical significance, since it embeds a discrete form of the Clausius-Duhem inequality. Effectively, the proposed stabilization samples the production of entropy to counter numerical instabilities. The proposed technique is applied to materials with no shear strength (e.g., fluids), for which there exists a caloric equation of state, and extensions to the case of materials with shear strength (e.g., solids) are also envisioned. The stabilization operator is incorporated into a mid-point, predictor/multi-corrector time integration algorithm, which conserves mass, momentum and total energy. Encouraging numerical results in the context of compressible gas dynamics confirm the potential of the method.
Author	Scovazzi, G. Shashkov, M.J. Love, E.
Author_xml	– sequence: 1 givenname: G. surname: Scovazzi fullname: Scovazzi, G. email: gscovaz@sandia.gov organization: 1431 Computational Shock- and Multi-Physics Department, Sandia National Laboratories, P.O. Box 5800, MS 1319, Albuquerque, NM 87185-1319, USA – sequence: 2 givenname: E. surname: Love fullname: Love, E. organization: 1431 Computational Shock- and Multi-Physics Department, Sandia National Laboratories, P.O. Box 5800, MS 1319, Albuquerque, NM 87185-1319, USA – sequence: 3 givenname: M.J. surname: Shashkov fullname: Shashkov, M.J. organization: Theoretical Division, Group T-7, MS B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
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Keywords	Variational multi-scale method Lagrangian shock hydrodynamics Stabilization Hourglass instability Predictor-corrector methods Lagrangian Gas dynamics Clausius Duhem inequality Mechanical strength Shear strength Hydrodynamics Momentum Entropy Finite element method Equations of state Lagrange interpolation Multiscale method Variational calculus Modelling Strength of materials Numerical stability
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Snippet	A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type)...
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SubjectTerms	Computational techniques Exact sciences and technology Fluid dynamics Fracture mechanics (crack, fatigue, damage...) Fundamental areas of phenomenology (including applications) General theory Hourglass instability Lagrangian shock hydrodynamics Mathematical methods in physics Physics Physics of gases Physics of gases, plasmas and electric discharges Solid mechanics Stabilization Structural and continuum mechanics Variational multi-scale method
Title	Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: Theoretical framework and two-dimensional computations
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