Thermodynamically consistent time‐stepping algorithms for non‐linear thermomechanical systems

We present the basic theory for developing novel monolithic and staggered time-stepping algorithms for general non-linear, coupled, thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems they preserve the total energy and the entropy never decreases. Furthermore, if the governing equations of the problem have symmetries, the proposed integrators preserve them too. The formulation of such methods is based on two ideas: expressing the evolution equation in the so-called General Equations for Non-Equilibrium Reversible Irreversible Coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields. The new methods can be considered as an extension of the energy–momentum integration algorithms to coupled thermomechanical problems, to which they reduce in the purely Hamiltonian case. In the article, the new ideas are applied to a simple coupled problem: a double thermoelastic pendulum with symmetry. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  J. C. Simo,et al.  Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations☆ , 1996 .

[2]  Donald Greenspan,et al.  Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion , 1975 .

[3]  Francisco Armero,et al.  A new unconditionally stable fractional step method for non‐linear coupled thermomechanical problems , 1992 .

[4]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[5]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods , 2001 .

[6]  Peter Betsch,et al.  An energy consistent hybrid space‐time Galerkin method for nonlinear thermomechanical problems , 2006 .

[7]  Francisco Armero,et al.  Stable and time‐dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows , 1995 .

[8]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. I. Development of a general formalism , 1997 .

[9]  J. Marsden,et al.  Variational time integrators , 2004 .

[10]  Francisco Armero,et al.  Volume-preserving energy–momentum schemes for isochoric multiplicative plasticity , 2007 .

[11]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[12]  J. Marsden,et al.  Mechanical integrators derived from a discrete variational principle , 1997 .

[13]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

[14]  M Markus Hütter,et al.  Finite anisotropic elasticity and material frame indifference from a nonequilibrium thermodynamics perspective , 2008 .

[15]  T. Laursen,et al.  Energy consistent algorithms for dynamic finite deformation plasticity , 2002 .

[16]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics , 2001 .

[17]  J. C. Simo,et al.  Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations , 1994 .

[18]  J. C. Simo,et al.  On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry , 1996 .

[19]  O. Gonzalez.  Time integration and discrete Hamiltonian systems , 1996 .

[20]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[21]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[22]  J. Marsden,et al.  Symplectic-energy-momentum preserving variational integrators , 1999 .

[23]  Jerrold E. Marsden,et al.  The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates , 1988 .

[24]  J. C. Simo,et al.  How to render second order accurate time-stepping algorithms fourth order accurate while retaining the stability and conservation properties , 1994 .

[25]  M. Hütter,et al.  Thermodynamic considerations on non-isothermal finite anisotropic elasto-viscoplasticity , 2008 .

[26]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[27]  J. C. Simo,et al.  A new energy and momentum conserving algorithm for the non‐linear dynamics of shells , 1994 .

[28]  F. Armero,et al.  A new dissipative time-stepping algorithm for frictional contact problems: formulation and analysis , 1999 .