---
_id: '52233'
abstract:
- lang: eng
  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.
author:
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
citation:
  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>
  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>
  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} }'
  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>.'
  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>.
  short: R. Mahnken, H. Westermann, Computational Mechanics (2024).
date_created: 2024-03-03T13:23:28Z
date_updated: 2024-03-19T12:14:07Z
department:
- _id: '154'
- _id: '321'
doi: 10.1007/s00466-024-02442-y
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta
  (ELDIRK) methods
type: journal_article
user_id: '335'
year: '2024'
...
---
_id: '62767'
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>"
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'
  last_name: Ostwald
  orcid: 0000-0003-2147-8444
- first_name: Rolf
  full_name: Lammering, Rolf
  last_name: Lammering
citation:
  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>
  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}
    }'
  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>.'
  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>.
  short: R. Najafi Koopas, S. Rezaei, N. Rauter, R. Ostwald, R. Lammering, Computational
    Mechanics 75 (2024) 1377–1406.
date_created: 2025-12-03T12:37:08Z
date_updated: 2025-12-03T12:51:26Z
department:
- _id: '952'
- _id: '321'
doi: 10.1007/s00466-024-02568-z
intvolume: '        75'
issue: '4'
language:
- iso: eng
page: 1377-1406
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Introducing a microstructure-embedded autoencoder approach for reconstructing
  high-resolution solution field data from a reduced parametric space
type: journal_article
user_id: '85414'
volume: 75
year: '2024'
...
---
_id: '45757'
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>"
author:
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
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>
  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>
  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} }'
  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>.'
  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).
date_created: 2023-06-23T06:47:36Z
date_updated: 2023-06-23T06:48:42Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1007/s00466-023-02347-2
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower
  order implicit time integration schemes for local and global error estimation
type: journal_article
user_id: '335'
year: '2023'
...
---
_id: '30655'
author:
- first_name: Xiaozhe
  full_name: Ju, Xiaozhe
  last_name: Ju
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Yangjian
  full_name: Xu, Yangjian
  last_name: Xu
- first_name: Lihua
  full_name: Liang, Lihua
  last_name: Liang
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>
  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>
  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} }'
  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>.'
  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>.'
  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>.
  short: X. Ju, R. Mahnken, Y. Xu, L. Liang, Computational Mechanics 69 (2022) 847–863.
date_created: 2022-03-28T13:23:17Z
date_updated: 2023-01-24T13:10:56Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1007/s00466-021-02117-y
intvolume: '        69'
issue: '3'
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
page: 847-863
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Goal-oriented error estimation and h-adaptive finite elements for hyperelastic
  micromorphic continua
type: journal_article
user_id: '335'
volume: 69
year: '2022'
...
---
_id: '62779'
author:
- first_name: Richard
  full_name: Ostwald, Richard
  id: '106876'
  last_name: Ostwald
  orcid: 0000-0003-2147-8444
- first_name: Ellen
  full_name: Kuhl, Ellen
  last_name: Kuhl
- first_name: Andreas
  full_name: Menzel, Andreas
  last_name: Menzel
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>
  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>
  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} }'
  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>.'
  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>.'
  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>.
  short: R. Ostwald, E. Kuhl, A. Menzel, Computational Mechanics 64 (2019) 847–877.
date_created: 2025-12-03T13:02:41Z
date_updated: 2025-12-03T13:03:51Z
department:
- _id: '952'
- _id: '321'
doi: 10.1007/s00466-019-01684-5
intvolume: '        64'
issue: '3'
language:
- iso: eng
page: 847-877
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: On the implementation of finite deformation gradient-enhanced damage models
type: journal_article
user_id: '85414'
volume: 64
year: '2019'
...
---
_id: '19308'
author:
- first_name: Ismail
  full_name: Caylak, Ismail
  id: '75'
  last_name: Caylak
- first_name: Eduard
  full_name: Penner, Eduard
  id: '27973'
  last_name: Penner
- first_name: Alex
  full_name: Dridger, Alex
  last_name: Dridger
- first_name: Rolf
  full_name: Mahnken, Rolf
  last_name: Mahnken
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>
  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>
  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} }'
  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>.'
  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>.
  short: I. Caylak, E. Penner, A. Dridger, R. Mahnken, Computational Mechanics (2018)
    1273–1285.
date_created: 2020-09-11T10:05:32Z
date_updated: 2022-07-27T07:05:15Z
doi: 10.1007/s00466-018-1563-z
language:
- iso: eng
page: 1273-1285
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
status: public
title: Stochastic hyperelastic modeling considering dependency of material parameters
type: journal_article
user_id: '75'
year: '2018'
...
---
_id: '24696'
author:
- first_name: Nicole
  full_name: Nörenberg, Nicole
  last_name: Nörenberg
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  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>
  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>
  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} }'
  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>.'
  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.
date_created: 2021-09-20T11:00:26Z
date_updated: 2023-01-24T14:33:33Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1007/s00466-015-1175-9
language:
- iso: eng
page: 353-370
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
quality_controlled: '1'
status: public
title: Parameter identification for rubber materials with artificial spatially distributed
  data
type: journal_article
user_id: '335'
year: '2015'
...
---
_id: '45431'
author:
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
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>
  apa: Mahnken, R. (2008). A Newton-Multigrid algrithm for elasto-plastic/viscoplastic
    problems. <i>Computational Mechanics</i>, <i>15</i>(5), 408–425. <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} }'
  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>.'
  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>.'
  mla: Mahnken, Rolf. “A Newton-Multigrid Algrithm for Elasto-Plastic/Viscoplastic
    Problems.” <i>Computational Mechanics</i>, vol. 15, no. 5, Springer Science and
    Business Media LLC, 2008, pp. 408–25, doi:<a href="https://doi.org/10.1007/bf00350355">10.1007/bf00350355</a>.
  short: R. Mahnken, Computational Mechanics 15 (2008) 408–425.
date_created: 2023-05-31T12:26:42Z
date_updated: 2023-05-31T12:27:11Z
department:
- _id: '9'
- _id: '154'
doi: 10.1007/bf00350355
intvolume: '        15'
issue: '5'
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
page: 408-425
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems
type: journal_article
user_id: '335'
volume: 15
year: '2008'
...
---
_id: '45417'
author:
- first_name: C.
  full_name: Döbert, C.
  last_name: Döbert
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: E.
  full_name: Stein, E.
  last_name: Stein
citation:
  ama: Döbert C, Mahnken R, Stein E. Numerical simulation of interface debonding with
    a combined damage/friction constitutive model. <i>Computational Mechanics</i>.
    2002;25(5):456-467. doi:<a href="https://doi.org/10.1007/s004660050493">10.1007/s004660050493</a>
  apa: Döbert, C., Mahnken, R., &#38; Stein, E. (2002). 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}
    }'
  chicago: 'Döbert, C., Rolf Mahnken, and E. Stein. “Numerical Simulation of Interface
    Debonding with a Combined Damage/Friction Constitutive Model.” <i>Computational
    Mechanics</i> 25, no. 5 (2002): 456–67. <a href="https://doi.org/10.1007/s004660050493">https://doi.org/10.1007/s004660050493</a>.'
  ieee: 'C. Döbert, R. Mahnken, and E. Stein, “Numerical simulation of interface debonding
    with a combined damage/friction constitutive model,” <i>Computational Mechanics</i>,
    vol. 25, no. 5, pp. 456–467, 2002, doi: <a href="https://doi.org/10.1007/s004660050493">10.1007/s004660050493</a>.'
  mla: Döbert, C., et al. “Numerical Simulation of Interface Debonding with a Combined
    Damage/Friction Constitutive Model.” <i>Computational Mechanics</i>, vol. 25,
    no. 5, Springer Science and Business Media LLC, 2002, pp. 456–67, doi:<a href="https://doi.org/10.1007/s004660050493">10.1007/s004660050493</a>.
  short: C. Döbert, R. Mahnken, E. Stein, Computational Mechanics 25 (2002) 456–467.
date_created: 2023-05-31T12:04:03Z
date_updated: 2023-05-31T12:04:35Z
department:
- _id: '9'
- _id: '154'
doi: 10.1007/s004660050493
intvolume: '        25'
issue: '5'
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
page: 456-467
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Numerical simulation of interface debonding with a combined damage/friction
  constitutive model
type: journal_article
user_id: '335'
volume: 25
year: '2002'
...