Energy-based mathematical methods for reactive multiphase flows

**Speaker:** Christian Wieners (KIT)

**Date:** November 19, 2020

**Abstract:** We present a new numerical method for dynamic fracture at small strains which is based on a discontinuous Galerkin approximation of a first order formulation for elastic waves and where the fracture is approximated by a phase field driven by a stress based fracture criterion.

The staggered algorithm in time combines the implicit midpoint rule for the wave propagation followed by an implicit Euler step for the phase field evolution. Then, driven by a stress based fracture criterion, the material is degradated, and the waves are reflected at the diffusive interface.

Then method is evaluated in detail in one dimension, and then we demonstrate in a 2D application the fracture evolution with multiple fractures initiated by reflections.

This is joint work with Kerstin Weinberg, Siegen.

**Speaker:** Miroslav Grmela (Polytechnique Montréal)

**Date: **November 26, 2020

**Abstract:** Multiscale thermodynamics is a theory of relations among levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated on microscopic, mesoscopic, and macroscopic levels. In this talk we formulate the multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels, see [1] for more details.

[1] M. Grmela: Multiscale Thermodynamics. arxiv-preprint 2020.

**Speaker:** Martin Kružík (Czech Technical University)

**Date: **December 10, 2020

**Abstract:** We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove the existence of energy minimizing equilibrium states and Γ-convergence of diffuse-interface approximations to the sharp-interface limit.

This is a joint work with D. Grandi, E. Mainini and U. Stefanelli.

**Speaker:** Denis Matignon (ISAE)

**Date: **January 14, 2021

**Abstract: **The numerical simulation of complex open multiphysics systems in Computational Science and Engineering is a challenging topic. Based on energy exchanges, the port-Hamiltonian formalism aims at describing physics in a structured manner. One of the major interests of this approach is its versatility, allowing for coupling and interconnection that preserve this structure.

We propose a Finite Element based technique for the structure-preserving discretization of a large class of port-Hamiltonian systems (pHs). Assuming a partitioned structure of the system associated to an integration-by-parts formula, it is possible to derive a consistent weak-formulation sharing the main features of the original boundary-controlled PDE. This allows using Galerkin approximations to obtain finite-dimensional systems that mimic the properties of the original distributed ones; these systems are either ODEs or Differential Algebraic Equations (DAEs). Moreover, the Partitioned Finite Element Method producing sparse matrices enables the use of dedicated algorithms in scientific computing. Indeed, this method can be easily implemented using well-established and robust libraries.

This strategy is illustrated by means of physically motivated PDEs with boundary control and observation, either in 2D or in 3D, both linear and non-linear: acoustic waves, Mindlin and Kirchhoff plates, heat equation, Shallow Water equation, Maxwell’s equation.

Interactive Jupyter notebooks are available, relying on the FEniCS open-source software. Advanced applications include multiphysics problems, e.g. fluid-structure interactions, thermoelasticity, and modular modelling of complex systems, e.g. multibody dynamics.

[1] Cardoso-Ribeiro, Flávio Luiz and Matignon, Denis and Lefèvre, Laurent. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. (2020) IMA Journal of Mathematical Control and Information.

https://doi.org/10.1093/imamci/dnaa038

**Speaker**: Pierre-Etienne Druet (WIAS)

**Date**: January 21, 2021

**Abstract**: This talk investigates the asymptotic behaviour of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy, that is valid in stable subregions of the phase diagram. In the incompressible limit where the average volume becomes independent of pressure, we are confronted with two problems:

(i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition, which in many cases violates empirical evidence.

(ii) As a consequence of the second law of thermodynamics, the incompressible limit implies that the average volume becomes independent of temperature as well. Most applications, however, reveals the non-appropriateness of this property. A more thorough analysis reveals that this conclusion is absent under a proper scale limit.

According to our mathematical treatment the free energy as a functions of temperature and the partial densities tends to a limit in the sense of epi– or Gamma–convergence. Within this setting we discuss and solve the first problem. The second problem will be treated by considering the asymptotic behaviour of both a general inequality relating thermal expansion and compressibility and an equation describing the pressure and temperature dependence of the specific internal energy.

This research is joined work with D.Bothe (TU Darmstadt) and W.Dreyer (WIAS Berlin).

**Speaker**: Markus Hütter (Uni Eindhoven)

**Date**: January 28, 2021

**Abstract**: In general, in every system with dissipation there are also fluctuations; the fluctuation-dissipation theorem is a key result of nonequilibrium statistical mechanics, which is encoded e.g. in the GENERIC framework [1,2,3]. Correspondingly, also complex fluids, which contain the relaxation of the microstructure (e.g. polymer chains), must show behavior with fluctuations. Fluctuations are relevant particularly for the behavior on small scales, where they are not averaged out, e.g. in micro-/nano-fluidics and microrheology.

To date, models for complex fluids do not include fluctuations systematically. In our recent work, we have extended conformation-tensor based models for complex fluids by including thermal fluctuations in a thermodynamically consistent way [4]. Furthermore, this work has been complemented by establishing a link between the fluctuating conformation-tensor models and the underlying description of polymer chains as bead-spring dumbbells [5]. To achieve this link, we follow two routes: (1) stochastic calculus, followed by averaging (direct approach); (2) nonequilibrium statistical mechanics, which includes the derivation of the coarse-grained relaxation tensor, the fluctuations, and the thermodynamic potential (particularly entropy). Denoting the number of polymer chains by N, it is shown that a finite (rather than infinite) value for N gives rise to a correction to the conventional entropy for the conformation tensor and leads to the fluctuations that vanish only in the thermodynamic limit (N to infinity). Finally, some subtleties are pointed out related to the appearance of the so-called spurious drift in the stochastic differential equation for the conformation tensor.

**References**:

[1] M. Grmela, H.C. Öttinger, Phys. Rev. E 56(6) (1997) 6620–6632 (http://dx.doi.org/10.1103/PhysRevE.56.6620)

[2] H.C. Öttinger, M. Grmela, Phys. Rev. E 56(6) (1997) 6633–6655 (http://dx.doi.org/10.1103/PhysRevE.56.6633)

[3] H.C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, 2005.

[4] M. Hütter, M.A. Hulsen, P.D. Anderson, J. Non-Newtonian Fluid Mech. 256 (2018) 42–56 (https://doi.org/10.1016/j.jnnfm.2018.02.012)

[5] M. Hütter, P.D. Olmsted, D.J. Read, Eur. Phys. J. E 43 (2020) 71 (http://dx.doi.org/10.1140/epje/i2020-11999-x)

**Speaker**: Eberhard Bänsch (Uni Erlangen)

**Date**: February 4, 2021

**Abstract**: We present a mathematical model for convective transport in nanofluids including thermophoretic effects that is very similar to the celebrated model of Buongiorno. Our model, however, is thermodynamically consistent and consequently an energy estimate can be shown.

We show existence of weak solutions for the time-dependent as well as stationary problem. Moreover, an efficient numerical scheme for the time dependent problem is proposed.

For the stationary system, regularity and subsequently optimal error estimates for finite element approximations can be shown under some smallness assumptions.