---
_id: '63676'
abstract:
- lang: eng
text: "\r\n Purpose\r\n The
purpose of this paper is to develop new methods of error representation to improve
the accuracy and numerical efficiency of a posteriori and goal-oriented adaptive
framework of elastoplasticity with Prandtl–Reuss type material laws.\r\n
\ \r\n \r\n Design/methodology/approach\r\n
\ To obtain new methods of error representation for
a posteriori and goal-oriented error estimators, weak forms of primal and dual
problems are investigated starting with the initial boundary value problem (IBVP).
Then, we approximate both problems using temporal discretization. Additionally,
we introduce a secant form considering the nonlinearity of elasto-plastic constitutive
equations, which is approximated by a tangent form. Finally, we obtain numerical
primal and dual solutions and their corresponding error approximations of discretized
primal and dual problems, allowing to build several goal-oriented a posteriori
error estimators on temporal and spatial adaptive refinement by inserting primal
solutions, dual solutions and their error approximations as arguments in residuals
of both weak forms as well as in the secant form of the bilinear residual.\r\n
\ \r\n \r\n Findings\r\n
\ An elasto-plastic material is investigated in a framework
of goal-oriented error estimator by using separately several methods of error
representation to deal with either temporal or spatial adaptive refinement, as
well as with both refinements leading to an effective reduction of computational
effort. Specifically, new error representations based on goal-oriented error estimators
are presented and obtained from primal and dual residuals, which use only primal
solutions or only dual solutions or a combination of primal and dual solutions
as arguments. Error representations obtained from primal residuals and evaluated
using only primal arguments do not require the formulation of a dual problem.\r\n
\ \r\n \r\n Research
limitations/implications\r\n The effectiveness
of the different proposed methods is illustrated by an example of a perforated
sheet for adaptive spatial refinement where new mesh adaptation methods of error
representation are compared against existing mesh adaptation methods such as uniform
mesh refinement, mesh refinement based on gradient indicators and adjoint-based
methods in literature. The framework generates a balanced mesh consisting of fine,
medium and coarse elements for accurate results, avoiding a numerically costly
simulation with only fine elements.\r\n \r\n
\ \r\n Originality/value\r\n
\ All new proposed methods of error representation successfully
estimate actual errors during mesh adaptivity. Furthermore, the proposed methods
of error representation allow us to obtain significant reduction and equidistribution
of spatial error at the end of the mesh adaptivity process. Their application
to a framework of goal-oriented error estimation due to time and mesh adaptivity
remains an open issue.\r\n "
author:
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
- first_name: Ismail
full_name: Caylak, Ismail
id: '75'
last_name: Caylak
- first_name: Richard
full_name: Ostwald, Richard
id: '106876'
last_name: Ostwald
orcid: 0000-0003-2147-8444
citation:
ama: Tchomgue Simeu A, Caylak I, Ostwald R. Error representations for goal-oriented
a posteriori error estimation in elasto-plasticity
with applications to mesh adaptivity. Engineering Computations. Published
online 2026:1-40. doi:10.1108/ec-12-2023-0975
apa: Tchomgue Simeu, A., Caylak, I., & Ostwald, R. (2026). Error representations
for goal-oriented a posteriori error
estimation in elasto-plasticity with applications to mesh adaptivity. Engineering
Computations, 1–40. https://doi.org/10.1108/ec-12-2023-0975
bibtex: '@article{Tchomgue Simeu_Caylak_Ostwald_2026, title={Error representations
for goal-oriented a posteriori error
estimation in elasto-plasticity with applications to mesh adaptivity}, DOI={10.1108/ec-12-2023-0975}, journal={Engineering
Computations}, publisher={Emerald}, author={Tchomgue Simeu, Arnold and Caylak,
Ismail and Ostwald, Richard}, year={2026}, pages={1–40} }'
chicago: Tchomgue Simeu, Arnold, Ismail Caylak, and Richard Ostwald. “Error Representations
for Goal-Oriented a Posteriori Error
Estimation in Elasto-Plasticity with Applications to Mesh Adaptivity.” Engineering
Computations, 2026, 1–40. https://doi.org/10.1108/ec-12-2023-0975.
ieee: 'A. Tchomgue Simeu, I. Caylak, and R. Ostwald, “Error representations for
goal-oriented a posteriori error
estimation in elasto-plasticity with applications to mesh adaptivity,” Engineering
Computations, pp. 1–40, 2026, doi: 10.1108/ec-12-2023-0975.'
mla: Tchomgue Simeu, Arnold, et al. “Error Representations for Goal-Oriented
a Posteriori Error Estimation in Elasto-Plasticity
with Applications to Mesh Adaptivity.” Engineering Computations, Emerald,
2026, pp. 1–40, doi:10.1108/ec-12-2023-0975.
short: A. Tchomgue Simeu, I. Caylak, R. Ostwald, Engineering Computations (2026)
1–40.
date_created: 2026-01-21T06:49:18Z
date_updated: 2026-01-21T06:50:58Z
department:
- _id: '9'
- _id: '952'
- _id: '321'
doi: 10.1108/ec-12-2023-0975
language:
- iso: eng
page: 1-40
publication: Engineering Computations
publication_identifier:
issn:
- 0264-4401
- 1758-7077
publication_status: published
publisher: Emerald
quality_controlled: '1'
status: public
title: Error representations for goal-oriented a posteriori error
estimation in elasto-plasticity with applications to mesh adaptivity
type: journal_article
user_id: '85414'
year: '2026'
...
---
_id: '56212'
abstract:
- lang: eng
text: AbstractTo increase the quality of computational
results for heterogeneous materials like fiber‐reinforced composites with Prandtl–Reuss‐type
material laws, goal‐oriented measures of the adaptive finite element method coupled
to model adaptivity is established. The former is an adaptive mesh refinement
on the macroscale, which allows to control the spatial discretization errors.
The latter is an efficient combination of a numerically low cost nonuniform transformation
field analysis (NTFA) and numerically high cost full‐field elasto‐plastic homogenization
methods on the microscale. The present contribution deals with the application
of the concept of downwind and upwind approximations to a goal‐oriented a posteriori
error estimator based on duality techniques by means of reduced order homogenization
schemes like NTFA, and with accuracy and numerical efficiency of the proposed
goal‐oriented adaptive framework. NTFA consists of an offline phase and an online
phase. During the offline phase, some relevant information of the micro system
under consideration is precomputed allowing a reduced set of equations to be solved
in the online phase. Thus, NTFA leads to a quite efficient homogenization method
but less accurate compared to the full‐field homogenization method which is characterized
with a high computational demand for accounting nonlinear microstructural mechanisms.
Due to nonlinearities and time‐dependency of plasticity, the estimation of error
transport and error generation are obtained with a backward‐in‐time dual method
despite a high demand on memory capacity. In this contribution, the dual problem
is solved with a forward‐in‐time dual method that allows estimating the full error
during the resolution of the primal problem without the need for extra memory
capacity. Several numerical examples illustrate the effectiveness of the proposed
adaptive approach based on downwind and upwind approximations.
author:
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
- first_name: Rolf
full_name: Mahnken, Rolf
last_name: Mahnken
citation:
ama: Tchomgue Simeu A, Mahnken R. Mesh‐ and model adaptivity for NTFA and full‐field
elasto‐plastic homogenization based on downwind and upwind approximations. PAMM.
Published online 2024. doi:10.1002/pamm.202400074
apa: Tchomgue Simeu, A., & Mahnken, R. (2024). Mesh‐ and model adaptivity for
NTFA and full‐field elasto‐plastic homogenization based on downwind and upwind
approximations. PAMM. https://doi.org/10.1002/pamm.202400074
bibtex: '@article{Tchomgue Simeu_Mahnken_2024, title={Mesh‐ and model adaptivity
for NTFA and full‐field elasto‐plastic homogenization based on downwind and upwind
approximations}, DOI={10.1002/pamm.202400074},
journal={PAMM}, publisher={Wiley}, author={Tchomgue Simeu, Arnold and Mahnken,
Rolf}, year={2024} }'
chicago: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Mesh‐ and Model Adaptivity for
NTFA and Full‐field Elasto‐plastic Homogenization Based on Downwind and Upwind
Approximations.” PAMM, 2024. https://doi.org/10.1002/pamm.202400074.
ieee: 'A. Tchomgue Simeu and R. Mahnken, “Mesh‐ and model adaptivity for NTFA and
full‐field elasto‐plastic homogenization based on downwind and upwind approximations,”
PAMM, 2024, doi: 10.1002/pamm.202400074.'
mla: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Mesh‐ and Model Adaptivity for NTFA
and Full‐field Elasto‐plastic Homogenization Based on Downwind and Upwind Approximations.”
PAMM, Wiley, 2024, doi:10.1002/pamm.202400074.
short: A. Tchomgue Simeu, R. Mahnken, PAMM (2024).
date_created: 2024-09-23T11:23:21Z
date_updated: 2024-09-23T11:26:52Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202400074
language:
- iso: eng
publication: PAMM
publication_identifier:
issn:
- 1617-7061
- 1617-7061
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Mesh‐ and model adaptivity for NTFA and full‐field elasto‐plastic homogenization
based on downwind and upwind approximations
type: journal_article
user_id: '85414'
year: '2024'
...
---
_id: '56721'
article_number: '117277'
author:
- first_name: Rolf
full_name: Mahnken, Rolf
last_name: Mahnken
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
citation:
ama: Mahnken R, Tchomgue Simeu A. Downwind and upwind approximations for primal
and dual problems of elasto-plasticity with Prandtl–Reuss type material laws.
Computer Methods in Applied Mechanics and Engineering. 2024;432. doi:10.1016/j.cma.2024.117277
apa: Mahnken, R., & Tchomgue Simeu, A. (2024). Downwind and upwind approximations
for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material
laws. Computer Methods in Applied Mechanics and Engineering, 432,
Article 117277. https://doi.org/10.1016/j.cma.2024.117277
bibtex: '@article{Mahnken_Tchomgue Simeu_2024, title={Downwind and upwind approximations
for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material
laws}, volume={432}, DOI={10.1016/j.cma.2024.117277},
number={117277}, journal={Computer Methods in Applied Mechanics and Engineering},
publisher={Elsevier BV}, author={Mahnken, Rolf and Tchomgue Simeu, Arnold}, year={2024}
}'
chicago: Mahnken, Rolf, and Arnold Tchomgue Simeu. “Downwind and Upwind Approximations
for Primal and Dual Problems of Elasto-Plasticity with Prandtl–Reuss Type Material
Laws.” Computer Methods in Applied Mechanics and Engineering 432 (2024).
https://doi.org/10.1016/j.cma.2024.117277.
ieee: 'R. Mahnken and A. Tchomgue Simeu, “Downwind and upwind approximations for
primal and dual problems of elasto-plasticity with Prandtl–Reuss type material
laws,” Computer Methods in Applied Mechanics and Engineering, vol. 432,
Art. no. 117277, 2024, doi: 10.1016/j.cma.2024.117277.'
mla: Mahnken, Rolf, and Arnold Tchomgue Simeu. “Downwind and Upwind Approximations
for Primal and Dual Problems of Elasto-Plasticity with Prandtl–Reuss Type Material
Laws.” Computer Methods in Applied Mechanics and Engineering, vol. 432,
117277, Elsevier BV, 2024, doi:10.1016/j.cma.2024.117277.
short: R. Mahnken, A. Tchomgue Simeu, Computer Methods in Applied Mechanics and
Engineering 432 (2024).
date_created: 2024-10-22T10:44:02Z
date_updated: 2024-11-08T08:54:41Z
department:
- _id: '321'
- _id: '154'
- _id: '321'
doi: 10.1016/j.cma.2024.117277
intvolume: ' 432'
language:
- iso: eng
project:
- _id: '1218'
grant_number: 'Geschäftszeichen: MA 1979/30-2'
name: Hier20 - Zielorientierte Adaptivität für nichtlineare Homogenisierungen mittels
hierarchischer Modelle
publication: Computer Methods in Applied Mechanics and Engineering
publication_identifier:
issn:
- 0045-7825
publication_status: published
publisher: Elsevier BV
quality_controlled: '1'
status: public
title: Downwind and upwind approximations for primal and dual problems of elasto-plasticity
with Prandtl–Reuss type material laws
type: journal_article
user_id: '85414'
volume: 432
year: '2024'
...
---
_id: '49866'
abstract:
- lang: eng
text: AbstractThe use of heterogeneous materials,
such as composites with Prandtl‐Reuss‐type material laws, has increased in industrial
praxis, making finite element modeling with homogenization techniques a well‐accepted
tool. These methods are particularly advantageous to account for microstructural
mechanisms which can be related to nonlinearities and time‐dependency due to elasto‐plasticity
behavior. However, their advantages are diminished by increasing computational
demand. The present contribution deals with the balance of accuracy and numerical
efficiency of nonlinear homogenization associated with a framework of goal‐oriented
adaptivity, which takes into account error accumulation over time. To this end,
model adaptivity of homogenization methods is coupled to mesh adaptivity on the
macro scale. Our new proposed adaptive procedure is driven by a goal‐oriented
a posteriori error estimator based on duality techniques using downwind and upwind
approximations. Due to nonlinearities and time‐dependency of the plasticity, the
estimation of error transport and error generation is obtained with a backward‐in‐time
dual method despite a high demand on memory capacity. In this contribution, the
dual problem is solved with a forward‐in‐time dual method that allows estimating
the full error during the resolution of the primal problem without the need for
extra memory capacity. Finally, a numerical example illustrates the effectiveness
of the proposed adaptive approach.
author:
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
- first_name: Rolf
full_name: Mahnken, Rolf
id: '335'
last_name: Mahnken
citation:
ama: Tchomgue Simeu A, Mahnken R. Downwind and upwind approximations for mesh and
model adaptivity of elasto‐plastic composites. PAMM. Published online 2023.
doi:10.1002/pamm.202300136
apa: Tchomgue Simeu, A., & Mahnken, R. (2023). Downwind and upwind approximations
for mesh and model adaptivity of elasto‐plastic composites. PAMM. https://doi.org/10.1002/pamm.202300136
bibtex: '@article{Tchomgue Simeu_Mahnken_2023, title={Downwind and upwind approximations
for mesh and model adaptivity of elasto‐plastic composites}, DOI={10.1002/pamm.202300136},
journal={PAMM}, publisher={Wiley}, author={Tchomgue Simeu, Arnold and Mahnken,
Rolf}, year={2023} }'
chicago: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Downwind and Upwind Approximations
for Mesh and Model Adaptivity of Elasto‐plastic Composites.” PAMM, 2023.
https://doi.org/10.1002/pamm.202300136.
ieee: 'A. Tchomgue Simeu and R. Mahnken, “Downwind and upwind approximations for
mesh and model adaptivity of elasto‐plastic composites,” PAMM, 2023, doi:
10.1002/pamm.202300136.'
mla: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Downwind and Upwind Approximations
for Mesh and Model Adaptivity of Elasto‐plastic Composites.” PAMM, Wiley,
2023, doi:10.1002/pamm.202300136.
short: A. Tchomgue Simeu, R. Mahnken, PAMM (2023).
date_created: 2023-12-19T12:20:05Z
date_updated: 2023-12-19T12:20:51Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202300136
keyword:
- Electrical and Electronic Engineering
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
publication: PAMM
publication_identifier:
issn:
- 1617-7061
- 1617-7061
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Downwind and upwind approximations for mesh and model adaptivity of elasto‐plastic
composites
type: journal_article
user_id: '335'
year: '2023'
...
---
_id: '44890'
author:
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
- first_name: Rolf
full_name: Mahnken, Rolf
id: '335'
last_name: Mahnken
citation:
ama: Tchomgue Simeu A, Mahnken R. Goal‐oriented adaptivity based on a model hierarchy
of mean‐field and full‐field homogenization methods in elasto‐plasticity. PAMM.
2023;22(1). doi:10.1002/pamm.202200053
apa: Tchomgue Simeu, A., & Mahnken, R. (2023). Goal‐oriented adaptivity based
on a model hierarchy of mean‐field and full‐field homogenization methods in elasto‐plasticity.
PAMM, 22(1). https://doi.org/10.1002/pamm.202200053
bibtex: '@article{Tchomgue Simeu_Mahnken_2023, title={Goal‐oriented adaptivity based
on a model hierarchy of mean‐field and full‐field homogenization methods in elasto‐plasticity},
volume={22}, DOI={10.1002/pamm.202200053},
number={1}, journal={PAMM}, publisher={Wiley}, author={Tchomgue Simeu, Arnold
and Mahnken, Rolf}, year={2023} }'
chicago: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Goal‐oriented Adaptivity Based
on a Model Hierarchy of Mean‐field and Full‐field Homogenization Methods in Elasto‐plasticity.”
PAMM 22, no. 1 (2023). https://doi.org/10.1002/pamm.202200053.
ieee: 'A. Tchomgue Simeu and R. Mahnken, “Goal‐oriented adaptivity based on a model
hierarchy of mean‐field and full‐field homogenization methods in elasto‐plasticity,”
PAMM, vol. 22, no. 1, 2023, doi: 10.1002/pamm.202200053.'
mla: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Goal‐oriented Adaptivity Based on
a Model Hierarchy of Mean‐field and Full‐field Homogenization Methods in Elasto‐plasticity.”
PAMM, vol. 22, no. 1, Wiley, 2023, doi:10.1002/pamm.202200053.
short: A. Tchomgue Simeu, R. Mahnken, PAMM 22 (2023).
date_created: 2023-05-16T12:18:15Z
date_updated: 2023-05-25T10:02:34Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202200053
intvolume: ' 22'
issue: '1'
keyword:
- Electrical and Electronic Engineering
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
publication: PAMM
publication_identifier:
issn:
- 1617-7061
- 1617-7061
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Goal‐oriented adaptivity based on a model hierarchy of mean‐field and full‐field
homogenization methods in elasto‐plasticity
type: journal_article
user_id: '335'
volume: 22
year: '2023'
...
---
_id: '46762'
author:
- first_name: Arnold
full_name: Tchomgue Simeu, Arnold
id: '83075'
last_name: Tchomgue Simeu
- first_name: Rolf
full_name: Mahnken, Rolf
id: '335'
last_name: Mahnken
citation:
ama: 'Tchomgue Simeu A, Mahnken R. Mesh- and model adaptivity for elasto-plastic
mean-field and full-field homogenization based on downwind and upwind approximations.
In: XI International Conference on Adaptive Modeling and Simulation. CIMNE;
2023. doi:10.23967/admos.2023.054'
apa: Tchomgue Simeu, A., & Mahnken, R. (2023). Mesh- and model adaptivity for
elasto-plastic mean-field and full-field homogenization based on downwind and
upwind approximations. XI International Conference on Adaptive Modeling and
Simulation. https://doi.org/10.23967/admos.2023.054
bibtex: '@inproceedings{Tchomgue Simeu_Mahnken_2023, title={Mesh- and model adaptivity
for elasto-plastic mean-field and full-field homogenization based on downwind
and upwind approximations}, DOI={10.23967/admos.2023.054},
booktitle={XI International Conference on Adaptive Modeling and Simulation}, publisher={CIMNE},
author={Tchomgue Simeu, Arnold and Mahnken, Rolf}, year={2023} }'
chicago: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Mesh- and Model Adaptivity for
Elasto-Plastic Mean-Field and Full-Field Homogenization Based on Downwind and
Upwind Approximations.” In XI International Conference on Adaptive Modeling
and Simulation. CIMNE, 2023. https://doi.org/10.23967/admos.2023.054.
ieee: 'A. Tchomgue Simeu and R. Mahnken, “Mesh- and model adaptivity for elasto-plastic
mean-field and full-field homogenization based on downwind and upwind approximations,”
2023, doi: 10.23967/admos.2023.054.'
mla: Tchomgue Simeu, Arnold, and Rolf Mahnken. “Mesh- and Model Adaptivity for Elasto-Plastic
Mean-Field and Full-Field Homogenization Based on Downwind and Upwind Approximations.”
XI International Conference on Adaptive Modeling and Simulation, CIMNE,
2023, doi:10.23967/admos.2023.054.
short: 'A. Tchomgue Simeu, R. Mahnken, in: XI International Conference on Adaptive
Modeling and Simulation, CIMNE, 2023.'
date_created: 2023-09-01T07:52:20Z
date_updated: 2023-09-01T07:54:31Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.23967/admos.2023.054
language:
- iso: eng
publication: XI International Conference on Adaptive Modeling and Simulation
publication_status: published
publisher: CIMNE
quality_controlled: '1'
status: public
title: Mesh- and model adaptivity for elasto-plastic mean-field and full-field homogenization
based on downwind and upwind approximations
type: conference
user_id: '335'
year: '2023'
...