@article{60124,
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  issn         = {{0020-7683}},
  journal      = {{International Journal of Solids and Structures}},
  publisher    = {{Elsevier BV}},
  title        = {{{Thermodynamically consistent phase-field modeling for polycrystalline multi-phase continua}}},
  doi          = {{10.1016/j.ijsolstr.2025.113465}},
  year         = {{2025}},
}

@article{52233,
  abstract     = {{ELDIRK methods are defined to have an <jats:italic>Explicit Last</jats:italic> stage in the general Butcher array of <jats:italic>Diagonal Implicit Runge-Kutta</jats:italic> methods, with the consequence, that no additional system of equations must be solved, compared to the embedded RK method. Two general formulations for second- and third-order ELDIRK methods have been obtained recently in Mahnken [21] with specific schemes,  e.g. for the embedded implicit Euler method, the embedded trapezoidal-rule and the embedded Ellsiepen method. In the first part of this paper, we investigate some general stability characteristics of ELDIRK methods, and it will be shown that the above specific RK schemes are not A-stable. Therefore, in the second part, the above-mentioned general formulations are used for further stability investigations, with the aim to construct new second- and third-order ELDIRK methods which simultaneously are A-stable. Two numerical examples are concerned with the curing for a thermosetting material and phase-field RVE modeling for crystallinity and orientation. The numerical results confirm the theoretical results on convergence order and stability.}},
  author       = {{Mahnken, Rolf and Westermann, Hendrik}},
  issn         = {{0178-7675}},
  journal      = {{Computational Mechanics}},
  keywords     = {{Applied Mathematics, Computational Mathematics, Computational Theory and Mathematics, Mechanical Engineering, Ocean Engineering, Computational Mechanics}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods}}},
  doi          = {{10.1007/s00466-024-02442-y}},
  year         = {{2024}},
}

@article{48464,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>Initial value problems can be solved efficiently by means of Runge–Kutta algorithms with adaptive step size control. Diagonally implicit Runge–Kutta (DIRK) methods are the most popular class among the diverse family of Runge–Kutta algorithms. In this paper, the novel class of low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods are explored, which combine implicit schemes with an additional explicit evaluation as an explicit last stage. ELDIRK Butcher tableaus are used to control embedded RK methods to obtain solutions of different orders. The lower‐order solution is obtained by classical implicit RK stages and the higher‐order solution is obtained by additional explicit evaluation. As a result, a significant reduction in computational cost is achieved by skipping the iterative solution of nonlinear systems for the additional step. The examination of the heat problem and the use of the innovative Butcher tableau in the finite‐element method are the main contributions of this work. Thus, it is possible to establish adaptive step size control for the new low‐order embedded methods based on an empirical method for error estimation. Two‐dimensional simulations are used to show an appropriate algorithm for the ELDIRK schemes. The new Runge–Kutta schemes' predictions of higher‐order convergence are confirmed, and their successful outcomes are illustrated.</jats:p>}},
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  issn         = {{1617-7061}},
  journal      = {{PAMM}},
  keywords     = {{Electrical and Electronic Engineering, Atomic and Molecular Physics, and Optics}},
  number       = {{2}},
  publisher    = {{Wiley}},
  title        = {{{Numerical investigations of new low‐order explicit last stage diagonal implicit Runge–Kutta schemes with the finite‐element method}}},
  doi          = {{10.1002/pamm.202300071}},
  volume       = {{23}},
  year         = {{2023}},
}

@article{48465,
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  issn         = {{0045-7825}},
  journal      = {{Computer Methods in Applied Mechanics and Engineering}},
  keywords     = {{Computer Science Applications, General Physics and Astronomy, Mechanical Engineering, Mechanics of Materials, Computational Mechanics}},
  publisher    = {{Elsevier BV}},
  title        = {{{On the accuracy, stability and computational efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK) for the adaptive solution of phase-field problems}}},
  doi          = {{10.1016/j.cma.2023.116545}},
  volume       = {{418}},
  year         = {{2023}},
}

@article{44891,
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  issn         = {{1617-7061}},
  journal      = {{PAMM}},
  keywords     = {{Electrical and Electronic Engineering, Atomic and Molecular Physics, and Optics}},
  number       = {{1}},
  publisher    = {{Wiley}},
  title        = {{{A thermodynamic framework for the phase‐field approach considering carbide precipitation during phase transformations}}},
  doi          = {{10.1002/pamm.202200080}},
  volume       = {{22}},
  year         = {{2023}},
}

@article{23794,
  author       = {{Westermann, Hendrik and Reitz, Alexander and Mahnken, Rolf and Schaper, Mirko and Grydin, Olexandr}},
  issn         = {{1611-3683}},
  journal      = {{steel research international}},
  title        = {{{Microstructure transformations in a press hardening steel during tailored thermo‐mechanical processing}}},
  doi          = {{10.1002/srin.202100346}},
  year         = {{2022}},
}

@article{29089,
  author       = {{Westermann, Hendrik and Reitz, Alexander and Mahnken, Rolf and Grydin, Olexandr and Schaper, Mirko}},
  issn         = {{1617-7061}},
  journal      = {{PAMM}},
  title        = {{{Constitutive modeling of viscoplasticity including phase transformations for graded thermo‐mechanical processing}}},
  doi          = {{10.1002/pamm.202100041}},
  year         = {{2021}},
}

@article{24384,
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  issn         = {{1617-7061}},
  journal      = {{PAMM}},
  title        = {{{Constitutive modeling of dynamic recrystallization coupled to viscoplasticity}}},
  doi          = {{10.1002/pamm.202000186}},
  year         = {{2021}},
}

@article{24390,
  author       = {{Mahnken, Rolf and Westermann, Hendrik}},
  issn         = {{0749-6419}},
  journal      = {{International Journal of Plasticity}},
  title        = {{{A non-equilibrium thermodynamic framework for viscoplasticity incorporating dynamic recrystallization at large strains}}},
  doi          = {{10.1016/j.ijplas.2021.102988}},
  year         = {{2021}},
}

@inproceedings{24388,
  author       = {{Westermann, Hendrik and Mahnken, Rolf}},
  booktitle    = {{14th WCCM-ECCOMAS Congress}},
  title        = {{{On the Thermodynamics of Dynamic Recrystallization for Viscoplasticity at Large Strains}}},
  doi          = {{10.23967/wccm-eccomas.2020.261}},
  year         = {{2021}},
}