[{"type":"journal_article","publication":"Computational Mechanics","abstract":[{"text":"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.","lang":"eng"}],"status":"public","_id":"52233","user_id":"335","department":[{"_id":"154"},{"_id":"321"}],"keyword":["Applied Mathematics","Computational Mathematics","Computational Theory and Mathematics","Mechanical Engineering","Ocean Engineering","Computational Mechanics"],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0178-7675","1432-0924"]},"quality_controlled":"1","year":"2024","citation":{"short":"R. Mahnken, H. Westermann, Computational Mechanics (2024).","mla":"Mahnken, Rolf, and Hendrik Westermann. “Construction of A-Stable Explicit Last-Stage Diagonal Implicit Runge–Kutta (ELDIRK) Methods.” <i>Computational Mechanics</i>, Springer Science and Business Media LLC, 2024, doi:<a href=\"https://doi.org/10.1007/s00466-024-02442-y\">10.1007/s00466-024-02442-y</a>.","bibtex":"@article{Mahnken_Westermann_2024, title={Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods}, DOI={<a href=\"https://doi.org/10.1007/s00466-024-02442-y\">10.1007/s00466-024-02442-y</a>}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Mahnken, Rolf and Westermann, Hendrik}, year={2024} }","apa":"Mahnken, R., &#38; Westermann, H. (2024). Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods. <i>Computational Mechanics</i>. <a href=\"https://doi.org/10.1007/s00466-024-02442-y\">https://doi.org/10.1007/s00466-024-02442-y</a>","ama":"Mahnken R, Westermann H. Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods. <i>Computational Mechanics</i>. Published online 2024. doi:<a href=\"https://doi.org/10.1007/s00466-024-02442-y\">10.1007/s00466-024-02442-y</a>","chicago":"Mahnken, Rolf, and Hendrik Westermann. “Construction of A-Stable Explicit Last-Stage Diagonal Implicit Runge–Kutta (ELDIRK) Methods.” <i>Computational Mechanics</i>, 2024. <a href=\"https://doi.org/10.1007/s00466-024-02442-y\">https://doi.org/10.1007/s00466-024-02442-y</a>.","ieee":"R. Mahnken and H. Westermann, “Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods,” <i>Computational Mechanics</i>, 2024, doi: <a href=\"https://doi.org/10.1007/s00466-024-02442-y\">10.1007/s00466-024-02442-y</a>."},"date_updated":"2024-03-19T12:14:07Z","publisher":"Springer Science and Business Media LLC","date_created":"2024-03-03T13:23:28Z","author":[{"id":"335","full_name":"Mahnken, Rolf","last_name":"Mahnken","first_name":"Rolf"},{"full_name":"Westermann, Hendrik","id":"60816","orcid":"0000-0002-5034-9708","last_name":"Westermann","first_name":"Hendrik"}],"title":"Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods","doi":"10.1007/s00466-024-02442-y"},{"publication":"Computational Mechanics","type":"journal_article","status":"public","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n          <jats:p>In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into an autoencoder architecture. This method’s integration of parametric space information significantly reduces the amount of training data needed to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 <jats:inline-formula>\r\n              <jats:alternatives>\r\n                <jats:tex-math>$$\\times $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mo>×</mml:mo>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n            </jats:inline-formula> 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 <jats:inline-formula>\r\n              <jats:alternatives>\r\n                <jats:tex-math>$$\\times $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mo>×</mml:mo>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n            </jats:inline-formula> 101 grid using a newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and an interpolation approach as an upscaling technique. Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on representative test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high-fidelity while preserving critical details often lost in traditional upscaling methods, such as sharp interfaces features lost in the context of interpolation approaches.</jats:p>"}],"department":[{"_id":"952"},{"_id":"321"}],"user_id":"85414","_id":"62767","language":[{"iso":"eng"}],"issue":"4","quality_controlled":"1","publication_identifier":{"issn":["0178-7675","1432-0924"]},"publication_status":"published","intvolume":"        75","page":"1377-1406","citation":{"chicago":"Najafi Koopas, Rasoul, Shahed Rezaei, Natalie Rauter, Richard Ostwald, and Rolf Lammering. “Introducing a Microstructure-Embedded Autoencoder Approach for Reconstructing High-Resolution Solution Field Data from a Reduced Parametric Space.” <i>Computational Mechanics</i> 75, no. 4 (2024): 1377–1406. <a href=\"https://doi.org/10.1007/s00466-024-02568-z\">https://doi.org/10.1007/s00466-024-02568-z</a>.","ieee":"R. Najafi Koopas, S. Rezaei, N. Rauter, R. Ostwald, and R. Lammering, “Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space,” <i>Computational Mechanics</i>, vol. 75, no. 4, pp. 1377–1406, 2024, doi: <a href=\"https://doi.org/10.1007/s00466-024-02568-z\">10.1007/s00466-024-02568-z</a>.","ama":"Najafi Koopas R, Rezaei S, Rauter N, Ostwald R, Lammering R. Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space. <i>Computational Mechanics</i>. 2024;75(4):1377-1406. doi:<a href=\"https://doi.org/10.1007/s00466-024-02568-z\">10.1007/s00466-024-02568-z</a>","apa":"Najafi Koopas, R., Rezaei, S., Rauter, N., Ostwald, R., &#38; Lammering, R. (2024). Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space. <i>Computational Mechanics</i>, <i>75</i>(4), 1377–1406. <a href=\"https://doi.org/10.1007/s00466-024-02568-z\">https://doi.org/10.1007/s00466-024-02568-z</a>","short":"R. Najafi Koopas, S. Rezaei, N. Rauter, R. Ostwald, R. Lammering, Computational Mechanics 75 (2024) 1377–1406.","bibtex":"@article{Najafi Koopas_Rezaei_Rauter_Ostwald_Lammering_2024, title={Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space}, volume={75}, DOI={<a href=\"https://doi.org/10.1007/s00466-024-02568-z\">10.1007/s00466-024-02568-z</a>}, number={4}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Najafi Koopas, Rasoul and Rezaei, Shahed and Rauter, Natalie and Ostwald, Richard and Lammering, Rolf}, year={2024}, pages={1377–1406} }","mla":"Najafi Koopas, Rasoul, et al. “Introducing a Microstructure-Embedded Autoencoder Approach for Reconstructing High-Resolution Solution Field Data from a Reduced Parametric Space.” <i>Computational Mechanics</i>, vol. 75, no. 4, Springer Science and Business Media LLC, 2024, pp. 1377–406, doi:<a href=\"https://doi.org/10.1007/s00466-024-02568-z\">10.1007/s00466-024-02568-z</a>."},"year":"2024","volume":75,"author":[{"first_name":"Rasoul","full_name":"Najafi Koopas, Rasoul","last_name":"Najafi Koopas"},{"first_name":"Shahed","full_name":"Rezaei, Shahed","last_name":"Rezaei"},{"first_name":"Natalie","full_name":"Rauter, Natalie","last_name":"Rauter"},{"first_name":"Richard","full_name":"Ostwald, Richard","id":"106876","orcid":"0000-0003-2147-8444","last_name":"Ostwald"},{"first_name":"Rolf","full_name":"Lammering, Rolf","last_name":"Lammering"}],"date_created":"2025-12-03T12:37:08Z","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-03T12:51:26Z","doi":"10.1007/s00466-024-02568-z","title":"Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space"},{"publication":"Computational Mechanics","type":"journal_article","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>Three 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 <jats:italic>Explicit Last</jats:italic> stage in the general Butcher array of <jats:italic>Diagonal Implicit Runge–Kutta</jats:italic> (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.\r\n</jats:p>"}],"status":"public","_id":"45757","department":[{"_id":"9"},{"_id":"154"},{"_id":"321"}],"user_id":"335","keyword":["Applied Mathematics","Computational Mathematics","Computational Theory and Mathematics","Mechanical Engineering","Ocean Engineering","Computational Mechanics"],"language":[{"iso":"eng"}],"quality_controlled":"1","publication_identifier":{"issn":["0178-7675","1432-0924"]},"publication_status":"published","year":"2023","citation":{"ama":"Mahnken R. Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation. <i>Computational Mechanics</i>. Published online 2023. doi:<a href=\"https://doi.org/10.1007/s00466-023-02347-2\">10.1007/s00466-023-02347-2</a>","chicago":"Mahnken, Rolf. “Derivation of Third Order Runge–Kutta Methods (ELDIRK) by Embedding of Lower Order Implicit Time Integration Schemes for Local and Global Error Estimation.” <i>Computational Mechanics</i>, 2023. <a href=\"https://doi.org/10.1007/s00466-023-02347-2\">https://doi.org/10.1007/s00466-023-02347-2</a>.","ieee":"R. Mahnken, “Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation,” <i>Computational Mechanics</i>, 2023, doi: <a href=\"https://doi.org/10.1007/s00466-023-02347-2\">10.1007/s00466-023-02347-2</a>.","bibtex":"@article{Mahnken_2023, 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={<a href=\"https://doi.org/10.1007/s00466-023-02347-2\">10.1007/s00466-023-02347-2</a>}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Mahnken, Rolf}, year={2023} }","mla":"Mahnken, Rolf. “Derivation of Third Order Runge–Kutta Methods (ELDIRK) by Embedding of Lower Order Implicit Time Integration Schemes for Local and Global Error Estimation.” <i>Computational Mechanics</i>, Springer Science and Business Media LLC, 2023, doi:<a href=\"https://doi.org/10.1007/s00466-023-02347-2\">10.1007/s00466-023-02347-2</a>.","short":"R. Mahnken, Computational Mechanics (2023).","apa":"Mahnken, R. (2023). Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation. <i>Computational Mechanics</i>. <a href=\"https://doi.org/10.1007/s00466-023-02347-2\">https://doi.org/10.1007/s00466-023-02347-2</a>"},"date_updated":"2023-06-23T06:48:42Z","publisher":"Springer Science and Business Media LLC","author":[{"last_name":"Mahnken","full_name":"Mahnken, Rolf","id":"335","first_name":"Rolf"}],"date_created":"2023-06-23T06:47:36Z","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"},{"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Computational Mathematics","Computational Theory and Mathematics","Mechanical Engineering","Ocean Engineering","Computational Mechanics"],"department":[{"_id":"9"},{"_id":"154"},{"_id":"321"}],"user_id":"335","_id":"30655","status":"public","publication":"Computational Mechanics","type":"journal_article","doi":"10.1007/s00466-021-02117-y","title":"Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua","volume":69,"date_created":"2022-03-28T13:23:17Z","author":[{"first_name":"Xiaozhe","last_name":"Ju","full_name":"Ju, Xiaozhe"},{"first_name":"Rolf","id":"335","full_name":"Mahnken, Rolf","last_name":"Mahnken"},{"first_name":"Yangjian","full_name":"Xu, Yangjian","last_name":"Xu"},{"first_name":"Lihua","full_name":"Liang, Lihua","last_name":"Liang"}],"publisher":"Springer Science and Business Media LLC","date_updated":"2023-01-24T13:10:56Z","page":"847-863","intvolume":"        69","citation":{"ama":"Ju X, Mahnken R, Xu Y, Liang L. Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua. <i>Computational Mechanics</i>. 2022;69(3):847-863. doi:<a href=\"https://doi.org/10.1007/s00466-021-02117-y\">10.1007/s00466-021-02117-y</a>","ieee":"X. Ju, R. Mahnken, Y. Xu, and L. Liang, “Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua,” <i>Computational Mechanics</i>, vol. 69, no. 3, pp. 847–863, 2022, doi: <a href=\"https://doi.org/10.1007/s00466-021-02117-y\">10.1007/s00466-021-02117-y</a>.","chicago":"Ju, Xiaozhe, Rolf Mahnken, Yangjian Xu, and Lihua Liang. “Goal-Oriented Error Estimation and h-Adaptive Finite Elements for Hyperelastic Micromorphic Continua.” <i>Computational Mechanics</i> 69, no. 3 (2022): 847–63. <a href=\"https://doi.org/10.1007/s00466-021-02117-y\">https://doi.org/10.1007/s00466-021-02117-y</a>.","mla":"Ju, Xiaozhe, et al. “Goal-Oriented Error Estimation and h-Adaptive Finite Elements for Hyperelastic Micromorphic Continua.” <i>Computational Mechanics</i>, vol. 69, no. 3, Springer Science and Business Media LLC, 2022, pp. 847–63, doi:<a href=\"https://doi.org/10.1007/s00466-021-02117-y\">10.1007/s00466-021-02117-y</a>.","bibtex":"@article{Ju_Mahnken_Xu_Liang_2022, title={Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua}, volume={69}, DOI={<a href=\"https://doi.org/10.1007/s00466-021-02117-y\">10.1007/s00466-021-02117-y</a>}, number={3}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Ju, Xiaozhe and Mahnken, Rolf and Xu, Yangjian and Liang, Lihua}, year={2022}, pages={847–863} }","short":"X. Ju, R. Mahnken, Y. Xu, L. Liang, Computational Mechanics 69 (2022) 847–863.","apa":"Ju, X., Mahnken, R., Xu, Y., &#38; Liang, L. (2022). Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua. <i>Computational Mechanics</i>, <i>69</i>(3), 847–863. <a href=\"https://doi.org/10.1007/s00466-021-02117-y\">https://doi.org/10.1007/s00466-021-02117-y</a>"},"year":"2022","issue":"3","publication_identifier":{"issn":["0178-7675","1432-0924"]},"quality_controlled":"1","publication_status":"published"},{"year":"2019","citation":{"ama":"Ostwald R, Kuhl E, Menzel A. On the implementation of finite deformation gradient-enhanced damage models. <i>Computational Mechanics</i>. 2019;64(3):847-877. doi:<a href=\"https://doi.org/10.1007/s00466-019-01684-5\">10.1007/s00466-019-01684-5</a>","ieee":"R. Ostwald, E. Kuhl, and A. Menzel, “On the implementation of finite deformation gradient-enhanced damage models,” <i>Computational Mechanics</i>, vol. 64, no. 3, pp. 847–877, 2019, doi: <a href=\"https://doi.org/10.1007/s00466-019-01684-5\">10.1007/s00466-019-01684-5</a>.","chicago":"Ostwald, Richard, Ellen Kuhl, and Andreas Menzel. “On the Implementation of Finite Deformation Gradient-Enhanced Damage Models.” <i>Computational Mechanics</i> 64, no. 3 (2019): 847–77. <a href=\"https://doi.org/10.1007/s00466-019-01684-5\">https://doi.org/10.1007/s00466-019-01684-5</a>.","bibtex":"@article{Ostwald_Kuhl_Menzel_2019, title={On the implementation of finite deformation gradient-enhanced damage models}, volume={64}, DOI={<a href=\"https://doi.org/10.1007/s00466-019-01684-5\">10.1007/s00466-019-01684-5</a>}, number={3}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Ostwald, Richard and Kuhl, Ellen and Menzel, Andreas}, year={2019}, pages={847–877} }","short":"R. Ostwald, E. Kuhl, A. Menzel, Computational Mechanics 64 (2019) 847–877.","mla":"Ostwald, Richard, et al. “On the Implementation of Finite Deformation Gradient-Enhanced Damage Models.” <i>Computational Mechanics</i>, vol. 64, no. 3, Springer Science and Business Media LLC, 2019, pp. 847–77, doi:<a href=\"https://doi.org/10.1007/s00466-019-01684-5\">10.1007/s00466-019-01684-5</a>.","apa":"Ostwald, R., Kuhl, E., &#38; Menzel, A. (2019). On the implementation of finite deformation gradient-enhanced damage models. <i>Computational Mechanics</i>, <i>64</i>(3), 847–877. <a href=\"https://doi.org/10.1007/s00466-019-01684-5\">https://doi.org/10.1007/s00466-019-01684-5</a>"},"page":"847-877","intvolume":"        64","publication_status":"published","quality_controlled":"1","publication_identifier":{"issn":["0178-7675","1432-0924"]},"issue":"3","title":"On the implementation of finite deformation gradient-enhanced damage models","doi":"10.1007/s00466-019-01684-5","date_updated":"2025-12-03T13:03:51Z","publisher":"Springer Science and Business Media LLC","date_created":"2025-12-03T13:02:41Z","author":[{"last_name":"Ostwald","orcid":"0000-0003-2147-8444","id":"106876","full_name":"Ostwald, Richard","first_name":"Richard"},{"first_name":"Ellen","last_name":"Kuhl","full_name":"Kuhl, Ellen"},{"full_name":"Menzel, Andreas","last_name":"Menzel","first_name":"Andreas"}],"volume":64,"status":"public","type":"journal_article","publication":"Computational Mechanics","language":[{"iso":"eng"}],"_id":"62779","user_id":"85414","department":[{"_id":"952"},{"_id":"321"}]},{"status":"public","type":"journal_article","publication":"Computational Mechanics","language":[{"iso":"eng"}],"user_id":"75","_id":"19308","citation":{"ama":"Caylak I, Penner E, Dridger A, Mahnken R. Stochastic hyperelastic modeling considering dependency of material parameters. <i>Computational Mechanics</i>. Published online 2018:1273-1285. doi:<a href=\"https://doi.org/10.1007/s00466-018-1563-z\">10.1007/s00466-018-1563-z</a>","chicago":"Caylak, Ismail, Eduard Penner, Alex Dridger, and Rolf Mahnken. “Stochastic Hyperelastic Modeling Considering Dependency of Material Parameters.” <i>Computational Mechanics</i>, 2018, 1273–85. <a href=\"https://doi.org/10.1007/s00466-018-1563-z\">https://doi.org/10.1007/s00466-018-1563-z</a>.","ieee":"I. Caylak, E. Penner, A. Dridger, and R. Mahnken, “Stochastic hyperelastic modeling considering dependency of material parameters,” <i>Computational Mechanics</i>, pp. 1273–1285, 2018, doi: <a href=\"https://doi.org/10.1007/s00466-018-1563-z\">10.1007/s00466-018-1563-z</a>.","short":"I. Caylak, E. Penner, A. Dridger, R. Mahnken, Computational Mechanics (2018) 1273–1285.","mla":"Caylak, Ismail, et al. “Stochastic Hyperelastic Modeling Considering Dependency of Material Parameters.” <i>Computational Mechanics</i>, 2018, pp. 1273–85, doi:<a href=\"https://doi.org/10.1007/s00466-018-1563-z\">10.1007/s00466-018-1563-z</a>.","bibtex":"@article{Caylak_Penner_Dridger_Mahnken_2018, title={Stochastic hyperelastic modeling considering dependency of material parameters}, DOI={<a href=\"https://doi.org/10.1007/s00466-018-1563-z\">10.1007/s00466-018-1563-z</a>}, journal={Computational Mechanics}, author={Caylak, Ismail and Penner, Eduard and Dridger, Alex and Mahnken, Rolf}, year={2018}, pages={1273–1285} }","apa":"Caylak, I., Penner, E., Dridger, A., &#38; Mahnken, R. (2018). Stochastic hyperelastic modeling considering dependency of material parameters. <i>Computational Mechanics</i>, 1273–1285. <a href=\"https://doi.org/10.1007/s00466-018-1563-z\">https://doi.org/10.1007/s00466-018-1563-z</a>"},"page":"1273-1285","year":"2018","publication_status":"published","publication_identifier":{"issn":["0178-7675","1432-0924"]},"doi":"10.1007/s00466-018-1563-z","title":"Stochastic hyperelastic modeling considering dependency of material parameters","author":[{"first_name":"Ismail","id":"75","full_name":"Caylak, Ismail","last_name":"Caylak"},{"first_name":"Eduard","last_name":"Penner","id":"27973","full_name":"Penner, Eduard"},{"first_name":"Alex","full_name":"Dridger, Alex","last_name":"Dridger"},{"last_name":"Mahnken","full_name":"Mahnken, Rolf","first_name":"Rolf"}],"date_created":"2020-09-11T10:05:32Z","date_updated":"2022-07-27T07:05:15Z"},{"language":[{"iso":"eng"}],"_id":"24696","department":[{"_id":"9"},{"_id":"154"},{"_id":"321"}],"user_id":"335","status":"public","publication":"Computational Mechanics","type":"journal_article","title":"Parameter identification for rubber materials with artificial spatially distributed data","doi":"10.1007/s00466-015-1175-9","date_updated":"2023-01-24T14:33:33Z","author":[{"full_name":"Nörenberg, Nicole","last_name":"Nörenberg","first_name":"Nicole"},{"first_name":"Rolf","last_name":"Mahnken","id":"335","full_name":"Mahnken, Rolf"}],"date_created":"2021-09-20T11:00:26Z","year":"2015","page":"353-370","citation":{"bibtex":"@article{Nörenberg_Mahnken_2015, title={Parameter identification for rubber materials with artificial spatially distributed data}, DOI={<a href=\"https://doi.org/10.1007/s00466-015-1175-9\">10.1007/s00466-015-1175-9</a>}, journal={Computational Mechanics}, author={Nörenberg, Nicole and Mahnken, Rolf}, year={2015}, pages={353–370} }","mla":"Nörenberg, Nicole, and Rolf Mahnken. “Parameter Identification for Rubber Materials with Artificial Spatially Distributed Data.” <i>Computational Mechanics</i>, 2015, pp. 353–70, doi:<a href=\"https://doi.org/10.1007/s00466-015-1175-9\">10.1007/s00466-015-1175-9</a>.","short":"N. Nörenberg, R. Mahnken, Computational Mechanics (2015) 353–370.","apa":"Nörenberg, N., &#38; Mahnken, R. (2015). Parameter identification for rubber materials with artificial spatially distributed data. <i>Computational Mechanics</i>, 353–370. <a href=\"https://doi.org/10.1007/s00466-015-1175-9\">https://doi.org/10.1007/s00466-015-1175-9</a>","chicago":"Nörenberg, Nicole, and Rolf Mahnken. “Parameter Identification for Rubber Materials with Artificial Spatially Distributed Data.” <i>Computational Mechanics</i>, 2015, 353–70. <a href=\"https://doi.org/10.1007/s00466-015-1175-9\">https://doi.org/10.1007/s00466-015-1175-9</a>.","ieee":"N. Nörenberg and R. Mahnken, “Parameter identification for rubber materials with artificial spatially distributed data,” <i>Computational Mechanics</i>, pp. 353–370, 2015, doi: <a href=\"https://doi.org/10.1007/s00466-015-1175-9\">10.1007/s00466-015-1175-9</a>.","ama":"Nörenberg N, Mahnken R. Parameter identification for rubber materials with artificial spatially distributed data. <i>Computational Mechanics</i>. Published online 2015:353-370. doi:<a href=\"https://doi.org/10.1007/s00466-015-1175-9\">10.1007/s00466-015-1175-9</a>"},"publication_identifier":{"issn":["0178-7675","1432-0924"]},"quality_controlled":"1","publication_status":"published"},{"title":"A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems","publisher":"Springer Science and Business Media LLC","date_created":"2023-05-31T12:26:42Z","year":"2008","quality_controlled":"1","issue":"5","keyword":["Applied Mathematics","Computational Mathematics","Computational Theory and Mathematics","Mechanical Engineering","Ocean Engineering","Computational Mechanics"],"language":[{"iso":"eng"}],"publication":"Computational Mechanics","doi":"10.1007/bf00350355","date_updated":"2023-05-31T12:27:11Z","author":[{"first_name":"Rolf","last_name":"Mahnken","id":"335","full_name":"Mahnken, Rolf"}],"volume":15,"citation":{"ama":"Mahnken R. A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems. <i>Computational Mechanics</i>. 2008;15(5):408-425. doi:<a href=\"https://doi.org/10.1007/bf00350355\">10.1007/bf00350355</a>","ieee":"R. Mahnken, “A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems,” <i>Computational Mechanics</i>, vol. 15, no. 5, pp. 408–425, 2008, doi: <a href=\"https://doi.org/10.1007/bf00350355\">10.1007/bf00350355</a>.","chicago":"Mahnken, Rolf. “A Newton-Multigrid Algrithm for Elasto-Plastic/Viscoplastic Problems.” <i>Computational Mechanics</i> 15, no. 5 (2008): 408–25. <a href=\"https://doi.org/10.1007/bf00350355\">https://doi.org/10.1007/bf00350355</a>.","bibtex":"@article{Mahnken_2008, title={A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems}, volume={15}, DOI={<a href=\"https://doi.org/10.1007/bf00350355\">10.1007/bf00350355</a>}, number={5}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Mahnken, Rolf}, year={2008}, pages={408–425} }","short":"R. 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Numerical simulation of interface debonding with a combined damage/friction constitutive model. <i>Computational Mechanics</i>, <i>25</i>(5), 456–467. <a href=\"https://doi.org/10.1007/s004660050493\">https://doi.org/10.1007/s004660050493</a>","bibtex":"@article{Döbert_Mahnken_Stein_2002, title={Numerical simulation of interface debonding with a combined damage/friction constitutive model}, volume={25}, DOI={<a href=\"https://doi.org/10.1007/s004660050493\">10.1007/s004660050493</a>}, number={5}, journal={Computational Mechanics}, publisher={Springer Science and Business Media LLC}, author={Döbert, C. and Mahnken, Rolf and Stein, E.}, year={2002}, pages={456–467} }","short":"C. Döbert, R. Mahnken, E. 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