@article{52233, abstract = {{ELDIRK methods are defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge-Kutta 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{45757, abstract = {{AbstractThree prominent low order implicit time integration schemes are the first order implicit Euler-method, the second order trapezoidal rule and the second order Ellsiepen method. Its advantages are stability and comparatively low computational cost, however, they require the solution of a nonlinear system of equations. This paper presents a general approach for the construction of third order Runge–Kutta methods by embedding the above mentioned implicit schemes into the class of ELDIRK-methods. These will be defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge–Kutta (DIRK) methods, with the consequence, that no additional system of equations must be solved. The main results—valid also for non-linear ordinary differential equations—are as follows: Two extra function calculations are required in order to embed the implicit Euler-method and one extra function calculation is required for the trapezoidal-rule and the Ellsiepen method, in order to obtain the third order properties, respectively. Two numerical examples are concerned with a parachute with viscous damping and a two-dimensional laser beam simulation. Here, we verify the higher order convergence behaviours of the proposed new ELDIRK-methods, and its successful performances for asymptotically exact global error estimation of so-called reversed embedded RK-method are shown. }}, author = {{Mahnken, Rolf}}, 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 = {{{Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation}}}, doi = {{10.1007/s00466-023-02347-2}}, 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{34257, abstract = {{Galvanic corrosion is a destructive process between dissimilar metals. The present paper presents a constructed numerical test case to simulate galvanic corrosion of two dissimilar metals. This test case is used to study the accuracy of different implementations to track the dissolving anode boundary. One technique is to numerically simulate a mesh displacement based on the prescribed displacement at the anode boundary. The second method is to adjust only the boundary elements. Re-meshing after a certain number of time steps is applied to both implementations. They produce similar results for an electrical and electrochemical field problem. This work shows that mesh smoothing does not result in higher accuracy when modeling a moving anode front. Adjusting only the boundary elements is sufficient when frequent re-meshing is used.}}, author = {{Harzheim, Sven and Hofmann, Martin and Wallmersperger, Thomas}}, issn = {{0001-5970}}, journal = {{Acta Mechanica}}, keywords = {{Mechanical Engineering, Computational Mechanics}}, number = {{11}}, pages = {{4427--4439}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Comparison of two mesh-moving techniques for finite element simulations of galvanic corrosion}}}, doi = {{10.1007/s00707-022-03326-z}}, volume = {{233}}, year = {{2022}}, } @article{30655, author = {{Ju, Xiaozhe and Mahnken, Rolf and Xu, Yangjian and Liang, Lihua}}, issn = {{0178-7675}}, journal = {{Computational Mechanics}}, keywords = {{Applied Mathematics, Computational Mathematics, Computational Theory and Mathematics, Mechanical Engineering, Ocean Engineering, Computational Mechanics}}, number = {{3}}, pages = {{847--863}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua}}}, doi = {{10.1007/s00466-021-02117-y}}, volume = {{69}}, year = {{2022}}, } @article{30657, author = {{Henkes, Alexander and Wessels, Henning 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 = {{{Physics informed neural networks for continuum micromechanics}}}, doi = {{10.1016/j.cma.2022.114790}}, volume = {{393}}, year = {{2022}}, } @article{32592, author = {{Ju, X. and Mahnken, Rolf and Xu, Y. and Liang, L.}}, 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 = {{{NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems}}}, doi = {{10.1016/j.cma.2022.115199}}, volume = {{398}}, year = {{2022}}, } @article{33801, author = {{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 = {{{New low order Runge–Kutta schemes for asymptotically exact global error estimation of embedded methods without order reduction}}}, doi = {{10.1016/j.cma.2022.115553}}, volume = {{401}}, year = {{2022}}, } @article{45420, author = {{Mahnken, Rolf and Steinmann, P.}}, issn = {{0044-2267}}, journal = {{ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik}}, keywords = {{Applied Mathematics, Computational Mechanics}}, number = {{S1}}, pages = {{161--164}}, publisher = {{Wiley}}, title = {{{Parameter Identification of Constitutive Models for Fluidsaturated Porous Media within an FE-Setting}}}, doi = {{10.1002/zamm.20000801341}}, volume = {{80}}, year = {{2011}}, } @article{45421, author = {{Döbert, C. and Mahnken, Rolf and Stein, E.}}, issn = {{0044-2267}}, journal = {{ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik}}, keywords = {{Applied Mathematics, Computational Mechanics}}, number = {{S2}}, pages = {{467--468}}, publisher = {{Wiley}}, title = {{{Numerical Simulation of Interface Debonding during Fibre Pull-Out Tests}}}, doi = {{10.1002/zamm.200008014105}}, volume = {{80}}, year = {{2011}}, } @article{45419, author = {{Mahnken, Rolf}}, issn = {{0044-2267}}, journal = {{ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik}}, keywords = {{Applied Mathematics, Computational Mechanics}}, number = {{S2}}, pages = {{481--482}}, publisher = {{Wiley}}, title = {{{Improvements for finite-element implementation of the Gurson Model}}}, doi = {{10.1002/zamm.200008014112}}, volume = {{80}}, year = {{2011}}, } @article{45431, author = {{Mahnken, Rolf}}, issn = {{0178-7675}}, journal = {{Computational Mechanics}}, keywords = {{Applied Mathematics, Computational Mathematics, Computational Theory and Mathematics, Mechanical Engineering, Ocean Engineering, Computational Mechanics}}, number = {{5}}, pages = {{408--425}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems}}}, doi = {{10.1007/bf00350355}}, volume = {{15}}, year = {{2008}}, } @article{45307, author = {{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}}, number = {{39}}, pages = {{5057--5080}}, publisher = {{Elsevier BV}}, title = {{{Strength difference in compression and tension and pressure dependence of yielding in elasto-plasticity}}}, doi = {{10.1016/s0045-7825(00)00364-9}}, volume = {{190}}, year = {{2002}}, } @article{45413, author = {{Mahnken, Rolf and Steinmann, P.}}, issn = {{0363-9061}}, journal = {{International Journal for Numerical and Analytical Methods in Geomechanics}}, keywords = {{Mechanics of Materials, Geotechnical Engineering and Engineering Geology, General Materials Science, Computational Mechanics}}, number = {{5}}, pages = {{415--434}}, publisher = {{Wiley}}, title = {{{A finite element algorithm for parameter identification of material models for fluid saturated porous media}}}, doi = {{10.1002/nag.136}}, volume = {{25}}, year = {{2002}}, } @article{45412, author = {{Mahnken, Rolf and Kohlmeier, M.}}, 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}}, number = {{32-33}}, pages = {{4259--4278}}, publisher = {{Elsevier BV}}, title = {{{Finite element simulation for rock salt with dilatancy boundary coupled to fluid permeation}}}, doi = {{10.1016/s0045-7825(00)00317-0}}, volume = {{190}}, year = {{2002}}, } @article{45417, author = {{Döbert, C. and Mahnken, Rolf and Stein, E.}}, issn = {{0178-7675}}, journal = {{Computational Mechanics}}, keywords = {{Applied Mathematics, Computational Mathematics, Computational Theory and Mathematics, Mechanical Engineering, Ocean Engineering, Computational Mechanics}}, number = {{5}}, pages = {{456--467}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Numerical simulation of interface debonding with a combined damage/friction constitutive model}}}, doi = {{10.1007/s004660050493}}, volume = {{25}}, year = {{2002}}, } @article{45429, author = {{Mahnken, Rolf and Stein, Erwin}}, 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}}, number = {{3-4}}, pages = {{225--258}}, publisher = {{Elsevier BV}}, title = {{{A unified approach for parameter identification of inelastic material models in the frame of the finite element method}}}, doi = {{10.1016/0045-7825(96)00991-7}}, volume = {{136}}, year = {{2002}}, } @article{45428, author = {{Mahnken, Rolf and Stein, Erwin}}, 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}}, number = {{1-2}}, pages = {{17--39}}, publisher = {{Elsevier BV}}, title = {{{Parameter identification for finite deformation elasto-plasticity in principal directions}}}, doi = {{10.1016/s0045-7825(97)00008-x}}, volume = {{147}}, year = {{2002}}, }