43 research outputs found
An error indicator-based adaptive reduced order model for nonlinear structural mechanics -- application to high-pressure turbine blades
The industrial application motivating this work is the fatigue computation of
aircraft engines' high-pressure turbine blades. The material model involves
nonlinear elastoviscoplastic behavior laws, for which the parameters depend on
the temperature. For this application, the temperature loading is not
accurately known and can reach values relatively close to the creep
temperature: important nonlinear effects occur and the solution strongly
depends on the used thermal loading. We consider a nonlinear reduced order
model able to compute, in the exploitation phase, the behavior of the blade for
a new temperature field loading. The sensitivity of the solution to the
temperature makes {the classical unenriched proper orthogonal decomposition
method} fail. In this work, we propose a new error indicator, quantifying the
error made by the reduced order model in computational complexity independent
of the size of the high-fidelity reference model. In our framework, when the
{error indicator} becomes larger than a given tolerance, the reduced order
model is updated using one time step solution of the high-fidelity reference
model. The approach is illustrated on a series of academic test cases and
applied on a setting of industrial complexity involving 5 million degrees of
freedom, where the whole procedure is computed in parallel with distributed
memory
A nonintrusive Reduced Basis Method applied to aeroacoustic simulations
The Reduced Basis Method can be exploited in an efficient way only if the
so-called affine dependence assumption on the operator and right-hand side of
the considered problem with respect to the parameters is satisfied. When it is
not, the Empirical Interpolation Method is usually used to recover this
assumption approximately. In both cases, the Reduced Basis Method requires to
access and modify the assembly routines of the corresponding computational
code, leading to an intrusive procedure. In this work, we derive variants of
the EIM algorithm and explain how they can be used to turn the Reduced Basis
Method into a nonintrusive procedure. We present examples of aeroacoustic
problems solved by integral equations and show how our algorithms can benefit
from the linear algebra tools available in the considered code.Comment: 28 pages, 7 figure
Variants of the Empirical Interpolation Method: symmetric formulation, choice of norms and rectangular extension
The Empirical Interpolation Method (EIM) is a greedy procedure that
constructs approximate representations of two-variable functions in separated
form. In its classical presentation, the two variables play a non-symmetric
role. In this work, we give an equivalent definition of the EIM approximation,
in which the two variables play symmetric roles. Then, we give a proof for the
existence of this approximation, and extend it up to the convergence of the
EIM, and for any norm chosen to compute the error in the greedy step. Finally,
we introduce a way to compute a separated representation in the case where the
number of selected values is different for each variable. In the case of a
physical field measured by sensors, this is useful to discard a broken sensor
while keeping the information provided by the associated selected field.Comment: 7 page
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
The reduced basis method is a model reduction technique yielding substantial
savings of computational time when a solution to a parametrized equation has to
be computed for many values of the parameter. Certification of the
approximation is possible by means of an a posteriori error bound. Under
appropriate assumptions, this error bound is computed with an algorithm of
complexity independent of the size of the full problem. In practice, the
evaluation of the error bound can become very sensitive to round-off errors. We
propose herein an explanation of this fact. A first remedy has been proposed in
[F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced
basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012)
539--542.]. Herein, we improve this remedy by proposing a new approximation of
the error bound using the Empirical Interpolation Method (EIM). This method
achieves higher levels of accuracy and requires potentially less
precomputations than the usual formula. A version of the EIM stabilized with
respect to round-off errors is also derived. The method is illustrated on a
simple one-dimensional diffusion problem and a three-dimensional acoustic
scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical
Analysis, 201
A nonintrusive method to approximate linear systems with nonlinear parameter dependence
We consider a family of linear systems with system matrix
depending on a parameter and for simplicity parameter-independent
right-hand side . These linear systems typically result from the
finite-dimensional approximation of a parameter-dependent boundary-value
problem. We derive a procedure based on the Empirical Interpolation Method to
obtain a separated representation of the system matrix in the form
for some selected values of the
parameter. Such a separated representation is in particular useful in the
Reduced Basis Method. The procedure is called nonintrusive since it only
requires to access the matrices . As such, it offers a crucial
advantage over existing approaches that instead derive separated
representations requiring to enter the code at the level of assembly. Numerical
examples illustrate the performance of our new procedure on a simple
one-dimensional boundary-value problem and on three-dimensional acoustic
scattering problems solved by a boundary element method.Comment: 17 pages, 9 figure
Nonintrusive approximation of parametrized limits of matrix power algorithms -- application to matrix inverses and log-determinants
We consider in this work quantities that can be obtained as limits of powers
of parametrized matrices, for instance the inverse matrix or the logarithm of
the determinant. Under the assumption of affine dependence in the parameters,
we use the Empirical Interpolation Method (EIM) to derive an approximation for
powers of these matrices, from which we derive a nonintrusive approximation for
the aforementioned limits. We derive upper bounds of the error made by the
obtained formula. Finally, numerical comparisons with classical intrusive and
nonintrusive approximation techniques are provided: in the considered
test-cases, our algorithm performs well compared to the nonintrusive ones
MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability
When learning simulations for modeling physical phenomena in industrial
designs, geometrical variabilities are of prime interest. While classical
regression techniques prove effective for parameterized geometries, practical
scenarios often involve the absence of shape parametrization during the
inference stage, leaving us with only mesh discretizations as available data.
Learning simulations from such mesh-based representations poses significant
challenges, with recent advances relying heavily on deep graph neural networks
to overcome the limitations of conventional machine learning approaches.
Despite their promising results, graph neural networks exhibit certain
drawbacks, including their dependency on extensive datasets and limitations in
providing built-in predictive uncertainties or handling large meshes. In this
work, we propose a machine learning method that do not rely on graph neural
networks. Complex geometrical shapes and variations with fixed topology are
dealt with using well-known mesh morphing onto a common support, combined with
classical dimensionality reduction techniques and Gaussian processes. The
proposed methodology can easily deal with large meshes without the need for
explicit shape parameterization and provides crucial predictive uncertainties,
which are essential for informed decision-making. In the considered numerical
experiments, the proposed method is competitive with respect to existing graph
neural networks, regarding training efficiency and accuracy of the predictions
A multiscale problem in thermal science
International audienceWe consider a multiscale heat problem in civil aviation: determine the temperature field in a plane in flying conditions, with air conditioning. Ventilated electronic components in the bay bring a heat source, introducing a second scale in the problem. First, we present three levels of modelling for the physical phenomena, which are applied to the two sub-problems: the plane and the electronic component. Then, having reduced the complexity of the problem to a linear non-symmetric coercive PDE, we will use the reduced basis method for the electronic component problem
Boundary Element and Finite Element Coupling for Aeroacoustics Simulations
We consider the scattering of acoustic perturbations in a presence of a flow.
We suppose that the space can be split into a zone where the flow is uniform
and a zone where the flow is potential. In the first zone, we apply a
Prandtl-Glauert transformation to recover the Helmholtz equation. The
well-known setting of boundary element method for the Helmholtz equation is
available. In the second zone, the flow quantities are space dependent, we have
to consider a local resolution, namely the finite element method. Herein, we
carry out the coupling of these two methods and present various applications
and validation test cases. The source term is given through the decomposition
of an incident acoustic field on a section of the computational domain's
boundary.Comment: 25 page