On a certain class of one step temporal integration methods for standard dissipative continua

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ORIGINAL PAPER

On a certain class of one step temporal integration methods for standard dissipative continua Sebastian Stark1 Received: 25 February 2020 / Accepted: 18 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract A class of isothermal dissipative continua being often referred to as “standard dissipative” is considered. The initial boundary value problem describing the behavior of these continua can be conveniently formulated in terms of a Helmholtz free energy functional, a dissipation functional, a power functional, and, possibly, a Lagrangian multiplier functional. In order to obtain (approximate) solutions for the initial boundary value problem, temporal and spatial discretization are necessary in most cases. In the present contribution, certain approaches for temporal discretization are discussed. In particular, different straightforward to apply first and second order accurate one step schemes are investigated; and the properties of these schemes are demonstrated by applying them to a diffusion problem. Keywords Standard dissipative continua · One step temporal integration · Rate of Convergence · Finite element method

1 Introduction Many problems in the field of continuum mechanics can be formulated or reformulated as variational principles, such that their solution furnishes an extremum or a saddle point of some suitably defined functional. This approach is particularly appealing because it usually provides with a rather compact and elegant statement of the problem, which clearly exposes its structure, and thereby facilitates the mathematical analysis as well as the development of discretization and numerical solution procedures. The latter has e.g. been demonstrated by Hackl and Fischer [14], Ortiz and Stainier [23] and Simo and Honein [27] in the context of plasticity and viscoplasticity, by Yang et al. [30] in the context of thermomechanically coupled problems, by Miehe and co-workers in the context of Cahn–Hilliard type diffusion [19,20] and Darcy type fluid transport in porous media [21], and by Carstensen et al. [9] and Ortiz and Repetto [22] in the context of micro-structure formation. The present work is concerned with a certain class of isothermal, dissipative continua, the behavior of which can

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Sebastian Stark [email protected]

be described by a single incremental1 potential being the sum of (i) the rate of the Helmholtz free energy, (ii) a dissipation functional accounting for dissipative effects, (iii) the negative of a power functional incorporating the influence of the environment, and, possibly, (iv) a Lagrangian multiplier functional incorporating constraints. Following the literature, such a system is referred to as “standard dissipative”, see e.g. [11,15,18]. The associated incremental variational principle has been exploited in various contexts, with the contributions [19–21] of Miehe and co-workers being of particular relevance below. In the latter works, the incremental variational problem is generally first formulated in the spacetime-