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@PHDTHESIS{Eggersmann:837746,
author = {Eggersmann, Robert},
othercontributors = {Reese, Stefanie and Ortiz, Michael},
title = {{C}onstitutive-model-free data-driven computational
mechanics},
school = {Rheinisch-Westfälische Technische Hochschule Aachen},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2022-00043},
pages = {1 Online-Ressource : Illustrationen},
year = {2021},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University 2022; Dissertation, Rheinisch-Westfälische
Technische Hochschule Aachen, 2021},
abstract = {One of the most powerful tools for the design and
engineering of technical innovations is numerical
simulation. Based on simulations, engineers have to make
decisions that influence everyone’s daily life. This can
affect how we deal with resources in any sense, personal
safety, or simply our well-being. Among all engineering
disciplines, the field of solid mechanics is essential and
also well-established. For many years, researchers have been
developing great improvements of the finite element method
to design structures and compute, e.g., critical loads.
Here, a central challenge is to formulate material models.
Over the years, these models became more and more accurate,
but also more complex and complicated. To circumvent this
complexity, a paradigm shift has taken place in recent
years. Next to classical material modeling, the idea of
data-driven computing has gained importance. The present
cumulative dissertation targets to make a helpful
contribution in this regard. It represents a merger of three
published works of the author and his coauthors
concentrating on the data-driven computing paradigm in
mechanics initially introduced by Kirchdoerfer and Oritz in
2016. The overall goal is to develop methods for finite
element simulations, which come along without the
formulation of a constitutive model. Here, the ansatz is to
treat the fundamental laws in mechanics, i.e., the
equilibrium of forces and compatibility, as boundary
conditions of a minimization problem. The material data is
used directly in the computation without replacing it by any
model simplification. This procedure makes it unnecessary to
formulate complicated constitutive equations or to fit model
parameters. On the one hand, uncertainties that come along
with the material modeling step are bypassed. On the other
hand, this method standardizes material modeling in order to
save time and resources. The current thesis begins with an
introduction, including a literature overview and a detailed
description of research-relevant questions. The first
article follows the introduction and extends the data-driven
formulation to inelasticity. This fundamental extension
enables computations with history-dependent or
path-dependent materials and, therefore, represents a
generalization to the data-driven paradigm. To derive the
underlying theory, we investigate three material
representations: (1) materials with memory, (2) differential
materials, and (3) materials de- scribed by history
variables. We use the equivalence between these three
formulations to derive possible representations of data
sets, describing, among others viscoelastic, and
elastoplastic material behavior. The second article deals
with an extension to the data-driven computing paradigm for
sparse data sets. These data sets appear, e.g. for
history-dependent materials. The article states the possible
incorporation of locally-linear tangent spaces into the
solver. Here, the key idea is that the data’s underlying
structures can be used and approximated by linear
representations. Those linear representations are computed
by the tensor voting method introduced by Mordohai and
Medioni. The tensor voting method can be seen as an
unsupervised machine learning technique based on manifold
learning. In contrast to global approximations, the method
is instance-based and, therefore, analyzes the data
structure pointwise. Numerical examples are investigated to
illustrate the higher-order convergence behavior of the
extension w.r.t. the data set size.The final article
addresses the efficiency of the data-driven solver. This
iterative solver mainly consists of two steps or projections
in each iteration. Starting from a state in the material
data set, the constraint set’s closest state is computed,
where equilibrium and kinematics are fulfilled. Afterwards,
we find the closest state in the data set. The article
focuses on treating the latter step, which is the most time
consuming for large data sets since a nearest neighbor
problem is solved at each integration point. Therefore, we
analyzed and adopted different data structures. We
discovered that approximate nearest neighbor algorithms
accelerate the search in these problems by many orders of
magnitude compared to exact algorithms. The treated
numerical examples cover computations with up to a billion
data points analyzing a 3D elastic solid.},
cin = {311510},
ddc = {624},
cid = {$I:(DE-82)311510_20140620$},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2022-00043},
url = {https://publications.rwth-aachen.de/record/837746},
}
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