@article{63676,
abstract = {{
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Research limitations/implications
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.
Originality/value
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.
}},
author = {{Tchomgue Simeu, Arnold and Caylak, Ismail and Ostwald, Richard}},
issn = {{0264-4401}},
journal = {{Engineering Computations}},
pages = {{1--40}},
publisher = {{Emerald}},
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}},
year = {{2026}},
}
@article{56212,
abstract = {{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 = {{Tchomgue Simeu, Arnold and Mahnken, Rolf}},
issn = {{1617-7061}},
journal = {{PAMM}},
publisher = {{Wiley}},
title = {{{Mesh‐ and model adaptivity for NTFA and full‐field elasto‐plastic homogenization based on downwind and upwind approximations}}},
doi = {{10.1002/pamm.202400074}},
year = {{2024}},
}
@article{56721,
author = {{Mahnken, Rolf and Tchomgue Simeu, Arnold}},
issn = {{0045-7825}},
journal = {{Computer Methods in Applied Mechanics and Engineering}},
publisher = {{Elsevier BV}},
title = {{{Downwind and upwind approximations for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material laws}}},
doi = {{10.1016/j.cma.2024.117277}},
volume = {{432}},
year = {{2024}},
}
@article{49866,
abstract = {{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 = {{Tchomgue Simeu, Arnold and Mahnken, Rolf}},
issn = {{1617-7061}},
journal = {{PAMM}},
keywords = {{Electrical and Electronic Engineering, Atomic and Molecular Physics, and Optics}},
publisher = {{Wiley}},
title = {{{Downwind and upwind approximations for mesh and model adaptivity of elasto‐plastic composites}}},
doi = {{10.1002/pamm.202300136}},
year = {{2023}},
}
@article{44890,
author = {{Tchomgue Simeu, Arnold and Mahnken, Rolf}},
issn = {{1617-7061}},
journal = {{PAMM}},
keywords = {{Electrical and Electronic Engineering, Atomic and Molecular Physics, and Optics}},
number = {{1}},
publisher = {{Wiley}},
title = {{{Goal‐oriented adaptivity based on a model hierarchy of mean‐field and full‐field homogenization methods in elasto‐plasticity}}},
doi = {{10.1002/pamm.202200053}},
volume = {{22}},
year = {{2023}},
}
@inproceedings{46762,
author = {{Tchomgue Simeu, Arnold and Mahnken, Rolf}},
booktitle = {{XI International Conference on Adaptive Modeling and Simulation}},
publisher = {{CIMNE}},
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}},
year = {{2023}},
}