I am a Hermann-Weyl postodoctoral instructor at ETH Zürich. I am mentored by Habib Ammari in the Seminar for Applied Mathematics.

My research focuses on topology optimization and inverse problems of wave and fluid systems with the level-set method or the homogenization method.

Before, I completed my PhD thesis at Centre de Mathématiques Appliquées (École polytechnique, Palaiseau) under the supervision of Grégoire Allaire and that of Charles Dapogny (Laboratoire Jean Kuntzmann, Grenoble) with a CIFRE funding provided by Safran.

And even before, I worked with Pierre Lermusiaux at Massachusetts Institute of Technology where I wrote a Master’s thesis on geometric methods for dynamical Model Order Reduction of Lagrangian transport.

**Email:** florian.feppon [AT] sam.math.ethz.ch

**Address:**

Department of Mathematics

ETH Zürich

Office HG G 56.1

Rämistrasse 101

8092 Zürich

Switzerland

- Homogenization
- Optimal Design
- Partial Differential Equations
- Numerical optimization
- Computational Science and Engineering
- Shape and Topology optimization
- Geometric Model Order Reduction

[18] Feppon, F. and Ammari, H. Analysis of a Monte-Carlo Nystrom method (2021). *Submitted*. HAL preprint hal-03281401.
(abstract)
(bibtex)

**Abstract**:
This paper considers a Monte-Carlo Nystrom method for solving integral equations of
the second kind, whereby the values $(z(y_i))_{1\leqslant i\leqslant N}$ of the solution $z$ at a
set of $N$ random and
independent points $(y_i)_{1\leqslant i\leqslant N}$ are approximated by the solution
$(z_{N,i})_{1\leqslant i\leqslant N}$ of a discrete, $N$-dimensional linear system obtained by replacing the integral with
the empirical average over the samples $(y_i)_{1\leqslant i\leqslant N}$. Under the unique
assumption that the integral equation admits a unique solution $z(y)$, we prove the
invertibility of the linear system for sufficiently large $N$ with probability
one, and the convergence of the solution $(z_{N,i})_{1\leqslant i\leqslant N}$ towards
the point values $(z(y_i))_{1\leqslant i\leqslant N}$ in a mean-square sense at a rate
$O(N^{-\frac{1}{2}})$.
For particular choices of kernels, the discrete linear system arises as the
Foldy-Lax approximation for the scattered field generated by a
system of $N$ sources emitting waves at the points $(y_i)_{1\leqslant i\leqslant N}$. In this
context, our result can equivalently be considered as
a proof of the well-posedness of the Foldy-Lax approximation for
systems of $N$ point scatterers, and of its convergence as $N\rightarrow
+\infty$ in a mean-square sense to the
solution of a Lippmann-Schwinger equation characterizing the effective medium.
The convergence of Monte-Carlo solutions at the rate $O(N^{-1/2})$ is numerically
illustrated on 1D examples and for solving a 2D Lippmann-Schwinger equation.

@unpublished{feppon:hal-03281401,
TITLE = {{Analysis of a Monte-Carlo Nystrom method}},
AUTHOR = {Feppon, Florian and Ammari, Habib},
URL = {https://hal.archives-ouvertes.fr/hal-03281401},
NOTE = {working paper or preprint},
YEAR = {2021},
MONTH = Jul,
KEYWORDS = {Monte-Carlo method ; Nystrom method ; Foldy-Lax approximation ; point scatterers ; effective medium},
PDF = {https://hal.archives-ouvertes.fr/hal-03281401/file/ex_article.pdf},
HAL_ID = {hal-03281401},
HAL_VERSION = {v1},
}

[17] Feppon, F. and Lermusiaux, P. F. J. Rigid Sets and Coherent Sets in Realistic Ocean Flows (2021). *Submitted*. HAL preprint hal-03176348.
(abstract)
(bibtex)

**Abstract**:
This paper focuses on the extractions of Lagrangian Coherent Sets from realistic
velocity fields obtained from ocean data and simulations, each of which can be
highly resolved and non volume-preserving. Two classes of methods have emerged
for such purpose: those relying on the flow map diffeomorphism associated with
the velocity field, and those based on spectral decompositions of the Koopman or
Perron-Frobenius operators. The two classes of methods are reviewed, synthesized,
augmented, and compared numerically on three velocity fields. First, we propose a
new "diffeomorphism-based" criterion to extract "rigid sets", defined as sets
over which the flow map acts approximately as a rigid transformation. Second, we
develop a matrix-free methodology that provides a simple and efficient framework
to compute "coherent sets" with operator methods. Both new methods and their
resulting rigid sets and coherent sets are illustrated and compared using three
numerically simulated flow examples, including a realistic, submesocale to
large-scale dynamic ocean current field in the Palau Island region of the western
Pacific Ocean.

@unpublished{feppon:hal-03176348,
TITLE = {{Rigid Sets and Coherent Sets in Realistic Ocean Flows}},
AUTHOR = {Feppon, F and Lermusiaux, P F J},
URL = {https://hal.archives-ouvertes.fr/hal-03176348},
NOTE = {working paper or preprint},
YEAR = {2021},
MONTH = Mar,
KEYWORDS = {LCS ; Rigid sets ; Koopman operator ; Arnoldi Iterations ; Ocean Modeling ; Lagrangian transport ; Realistic data},
PDF = {https://hal.archives-ouvertes.fr/hal-03176348/file/main.pdf},
HAL_ID = {hal-03176348},
HAL_VERSION = {v1},
}

[16] Feppon, F. and Jing, W. High order homogenized Stokes models capture all three regimes (2021). *Submitted*. HAL preprint hal-03098222.
(abstract)
(bibtex)

**Abstract**:
This article is a sequel to our previous work \cite{feppon2020Homog} concerned
with the derivation of high-order homogenized models for the Stokes equation in a
periodic porous medium. We provide an improved asymptotic analysis of the
coefficients of the higher order models in the low-volume fraction regime whereby
the periodic obstacles are rescaled by a factor $\eta$ which converges to zero.
By introducing a new family of order $k$ corrector tensors with a
controlled growth as $\eta\rightarrow 0$ uniform in $k\in\mathbb{N}$, we are
able to show that both the infinite order and
the finite order models converge in a coefficient-wise sense to the three
classical asymptotic regimes. Namely, we retrieve the Darcy model, the
Brinkman equation or the Stokes equation in the homogeneous cubic domain
depending on whether $\eta$ is respectively larger, proportional to, or
smaller than the critical size $\eta_{\rm crit}\sim \epsilon^{2/(d-2)}$. For
completeness, the paper first establishes the analogous results for the
perforated Poisson equation,
considered as a simplified scalar version of the Stokes system.

@unpublished{feppon:hal-03098222,
TITLE = {{High order homogenized Stokes models captureall three regimes}},
AUTHOR = {Feppon, Florian and Jing, Wenjia},
URL = {https://hal.archives-ouvertes.fr/hal-03098222},
NOTE = {working paper or preprint},
YEAR = {2021},
MONTH = Jan,
KEYWORDS = {Homogenization ; higher order models ; perforated Poisson problem ; Stokes system ; low volume fraction asymptotics ; strange term},
PDF = {https://hal.archives-ouvertes.fr/hal-03098222/file/ex_article.pdf},
HAL_ID = {hal-03098222},
HAL_VERSION = {v1},
}

[15] Feppon, F., Allaire, G., Dapogny D. and Jolivet, P. Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers (2021). *Computer Methods in Applied Mechanics and Engineering, 376, 113638*. HAL preprint hal-02924308.
(abstract)
(bibtex)

**Abstract**:
We present a topology optimization approach for the design of
fluid-to-fluid heat exchangers which rests on an explicit meshed
discretization of the phases at stake, at every iteration of the optimization
process. The considered physical situations involve a weak coupling between the
Navier--Stokes equations for the velocity and the pressure in the fluid, and the
convection--diffusion equation for the temperature field. The proposed framework
combines several recent techniques from the field of shape and topology
optimization, and notably a level-set based mesh evolution algorithm for tracking
shapes and their deformations, an efficient optimization algorithm for
constrained shape optimization problems, and a numerical method to handle a wide
variety of geometric constraints such as thickness constraints and non-penetration
constraints. Our strategy is applied to the optimization of various types
of heat exchangers. At first, we consider a simplified 2D cross-flow model where
the optimized boundary is the section of the hot fluid phase flowing in the
transverse direction, which is naturally composed of multiple holes. A minimum
thickness constraint is imposed on the cross-section so as to account for
manufacturing and maximum pressure drop constraints. In a second part, we
optimize the design of 2D and 3D heat exchangers composed of two types of fluid
channels (hot and cold), which are separated by a solid body. A non-mixing
constraint between the fluid components containing the hot and cold phases is
enforced by prescribing a minimum distance between them. Numerical results are
presented on a variety of test cases, demonstrating the efficiency of our
approach in generating new, realistic, and unconventional heat exchanger designs.

@article{FEPPON2021113638,
title = "Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers",
journal = "Computer Methods in Applied Mechanics and Engineering",
volume = "376",
pages = "113638",
year = "2021",
issn = "0045-7825",
doi = "https://doi.org/10.1016/j.cma.2020.113638",
url = "http://www.sciencedirect.com/science/article/pii/S0045782520308239",
author = "F. Feppon and G. Allaire and C. Dapogny and P. Jolivet",
}

[14] Feppon, F. High order homogenization of the Stokes system in a periodic porous medium (2021). *SIAM J. Math. Anal., 53(3), 2890–2924*. HAL preprint hal-02880030.
(abstract)
(bibtex)

**Abstract**:
We derive high order homogenized models for the incompressible
Stokes system in a cubic domain filled with periodic obstacles. These models have
the potential to unify the three classical limit problems (namely the
``unchanged' Stokes system, the Brinkman model, and the Darcy's law)
corresponding to various asymptotic regimes of the ratio $\eta\equiv
a_{\epsilon}/\epsilon$ between the radius $a_{\epsilon}$ of the holes and the
size $\epsilon$ of the periodic cell. What is more, a novel, rather surprising
feature of our higher order effective equations is the occurrence of odd order
differential operators when the obstacles are not symmetric. Our derivation
relies on the method of two-scale power series expansions and on the existence of
a ``criminal' ansatz, which allows to reconstruct the oscillating velocity and
pressure $(\mathbf{u}_{\epsilon},p_{\epsilon})$ as a linear combination of the
derivatives of their formal average $(\mathbf{u}_{\epsilon}^{*},p_{\epsilon}^{*})$
weighted by suitable corrector tensors. The formal average
$(\mathbf{u}_\epsilon^{*},p_{\epsilon}^{*})$ is itself the solution to a formal, infinite
order homogenized equation, whose truncation at any finite order is in general
ill-posed. Inspired by the variational truncation method of
\cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any
$K\in\mathbb{N}$, a well-posed model of order $2K+2$ which yields approximations of the
original solutions with an error of order $O(\epsilon^{K+3})$ in the $L^{2}$
norm. Furthermore, the error improves up to the order $O(\epsilon^{2K+4})$ if a
slight modification of this model remains well-posed. Finally, we find
asymptotics of all homogenized tensors in the low volume fraction limit
$\eta\rightarrow 0$ and in dimension $d\geqslant 3$. This allows us to obtain that our
effective equations converge coefficient-wise to either of the Brinkman or Darcy
regimes which arise when $\eta$ is respectively equivalent, or greater than
the critical scaling $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$.

@article{feppon2021highorderStokes,
author = {Feppon, Florian},
title = {High Order Homogenization of the Stokes System in a Periodic Porous Medium},
journal = {SIAM Journal on Mathematical Analysis},
volume = {53},
number = {3},
pages = {2890-2924},
year = {2021},
doi = {10.1137/20M1348078},
URL = {https://doi.org/10.1137/20M1348078},
eprint = {https://doi.org/10.1137/20M1348078}
}

[13] Feppon, F. High order homogenization of the Poisson equation in a perforated periodic domain. *Submitted*. HAL preprint hal-02518528.
(abstract)
(bibtex)

**Abstract**:
We derive high order homogenized models for the Poisson problem in
a cubic domain periodically perforated with holes where Dirichlet boundary
conditions are applied. These models have the potential to unify the three
possible kinds of limit problems derived by the literature for various
asymptotic regimes (namely the ``unchanged'' Poisson equation, the Poisson
problem with a strange reaction term, and the zeroth order limit problem) of the
ratio $\eta\equiv a_{\epsilon}/\epsilon$ between the size $a_{\epsilon}$ of
the holes and the size $\epsilon$ of the periodic cell. The derivation relies on
algebraic manipulations on formal two-scale power series in terms of $\epsilon$
and more particularly on the existence of a ``criminal'' ansatz, which allows to
reconstruct the oscillating solution $u_{\epsilon}$ as a linear combination of
the derivatives of its formal average $u_{\epsilon}^{*}$ weighted by suitable
corrector tensors. The formal average is itself the solution of a formal,
infinite order homogenized equation. Classically, truncating the infinite order
homogenized equation yields in general an ill-posed model. Inspired by a
variational method introduced in
\cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any
$K\in\mathbb{N}$, well-posed corrected homogenized equations of order $2K+2$ which
yields approximations of the original solutions with an error of order
$O(\epsilon^{2K+4})$ in the $L^{2}$ norm. Finally, we find asymptotics of all
homogenized tensors in the low volume fraction regime $\eta\rightarrow 0$ and in
dimension $d\>3$. This allows us to show that our higher order effective
equations converge coefficient-wise to either of the classical homogenized
regimes of the literature which arise when $\eta$ is respectively equivalent, or
greater than the critical scaling $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$.

@unpublished{feppon:hal-02518528,
TITLE = {{High order homogenization of the Poisson equation in a perforated periodic domain}},
AUTHOR = {Feppon, Florian},
URL = {https://hal.archives-ouvertes.fr/hal-02518528},
NOTE = {working paper or preprint},
YEAR = {2020},
MONTH = Mar,
KEYWORDS = {Homogenization ; higher order models ; perforated Poisson problem ; homogeneous Dirichlet boundary conditions ; strange term},
PDF = {https://hal.archives-ouvertes.fr/hal-02518528/file/homogenization_poisson.pdf},
HAL_ID = {hal-02518528},
HAL_VERSION = {v1},
}

[12] Feppon, F., Allaire, G., Dapogny D. and Jolivet, P. Topology optimization of thermal fluid-structure systems using body-fitted meshes and parallel computing (2020). *Journal of Computational Physics, 109574*. HAL preprint hal-02518207.
(abstract)
(bibtex)

**Abstract**:
An efficient framework is described
for the shape and topology optimization of realistic three-dimensional,
weakly-coupled fluid-thermal-mechanical systems. At the theoretical
level, the proposed methodology relies on the boundary variation of
Hadamard for describing the sensitivity of functions with respect to the
domain. From the numerical point of view, three key ingredients are
used:
(i) a level set based mesh evolution method allowing to describe large
deformations of the shape while maintaining an adapted, high-quality mesh of
the latter at every stage of the optimization process;
(ii) an efficient constrained optimization algorithm which is very well
adapted to the infinite-dimensional shape optimization context;
(iii) efficient preconditioning techniques for the solution of large finite
element systems in a reasonable computational time.
The performance of our strategy is illustrated with two examples of coupled
physics: respectively fluid--structure interaction and convective heat
transfer. Before that, we perform three other test cases, involving a
single physics (structural, thermal and aerodynamic design), for comparison
purposes and for assessing our various tools: in particular, they prove the
ability of the mesh evolution technique to capture very thin bodies or
shells in 3D.

@article{FEPPON2020109574,
title = "Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing",
journal = "Journal of Computational Physics",
volume = "417",
pages = "109574",
year = "2020",
issn = "0021-9991",
doi = "https://doi.org/10.1016/j.jcp.2020.109574",
url = "http://www.sciencedirect.com/science/article/pii/S002199912030348X",
author = "F. Feppon and G. Allaire and C. Dapogny and P. Jolivet",
keywords = "Shape and topology optimization, Fluid–structure interaction, Convective heat transfer, Aerodynamic design, Mesh adaptation, Distributed computing",
}

[11] Feppon, F., Allaire, G. and Dapogny, C. Null space gradient flows for constrained optimization with applications to shape optimization (2020). *ESAIM: COCV, 26 90 (Open Access)*. HAL preprint hal-01972915.
(abstract)
(bibtex)

**Abstract**:
The purpose of this article is to introduce a gradient-flow
algorithm for solving generic equality or inequality constrained
optimization problems, which is suited for shape optimization applications.
We rely on a variant of the Ordinary Differential Equation (ODE) approach
proposed by Yamashita for equality constrained problems: the search
direction is a combination of a null space step and a range space step,
which are aimed to reduce the value of the minimized objective function and
the violation of the constraints, respectively. Our first contribution is
to propose an extension of this ODE approach to optimization problems
featuring both equality and inequality constraints. In the literature, a
common practice consists in reducing inequality constraints to equality
constraints by the introduction of additional slack variables. Here, we
rather solve their local combinatorial character by computing the
projection of the gradient of the objective function onto the cone of
feasible directions. This is achieved by solving a dual quadratic
programming subproblem whose size equals the number of active or violated
constraints, and which allows to identify the inequality constraints which
should remain tangent to the optimization trajectory. Our second
contribution is a formulation of our gradient flow in the context
of-infinite-dimensional-Hilbert space settings. This allows to extend the
method to quite general optimization sets equipped with a suitable manifold
structure, and notably to sets of shapes as it occurs in shape optimization
with the framework of Hadamard's boundary variation method. The cornerstone
of this latter setting is the classical operation of extension and
regularization of shape derivatives. Some numerical comparisons on simple
academic examples are performed to illustrate the behavior of our
algorithm. Its numerical efficiency and ease of implementation are finally
demonstrated on more realistic shape optimization problems.

@article{feppon2020optim,
author = {{Feppon, F.} and {Allaire, G.} and {Dapogny, C.}},
doi = {10.1051/cocv/2020015},
journal = {ESAIM: COCV},
pages = {90},
title = {Null space gradient flows for constrained optimization with applications to shape optimization},
url = {https://doi.org/10.1051/cocv/2020015},
volume = 26,
year = 2020
}

[10] Feppon, F., Allaire, G. and Dapogny, C. A variational formulation for computing shape derivatives of geometric constraints along rays (2020). *ESAIM: M2AN, 54 1 181-228 (Open Access)*. HAL preprint hal-01879571.
(abstract)
(bibtex)

**Abstract**:
In the formulation of shape optimization problems, multiple geometric constraint
functionals involve the signed distance function to the optimized shape $\Omega$.
The numerical evaluation of their shape derivatives requires to integrate some
quantities along the normal rays to $\Omega$, a challenging operation to implement, which is usually achieved thanks
to the method of characteristics. The goal of the present paper is to propose an
alternative, variational approach for this purpose. Our method amounts, in full
generality, to compute integral quantities along the characteristic curves of a given
velocity field without requiring the explicit knowledge of these curves on the
spatial discretization; it rather relies on a variational problem which can be
solved conveniently by the finite element method. The well-posedness of this problem
is established thanks to a detailed analysis of weighted graph spaces of the
advection operator $\beta\cdot\nabla$ associated to a $\mathcal{C}^1$ velocity field
$\beta$. One novelty of our approach is the ability to handle velocity fields with
possibly unbounded divergence: we do not assume $\text{div}(\beta)\in L^\infty$. Our
working assumptions are fulfilled in the context of shape optimization of
$\mathcal{C}^2$ domains $\Omega$, where the velocity field $\beta=\nabla d_\Omega$ is
an extension of the unit outward normal vector to the optimized shape. The
efficiency of our variational method with respect to the direct integration of
numerical quantities along rays is evaluated on several numerical examples.
Classical albeit important implementation issues such as the calculation of a shape's
curvature and the detection of its skeleton are discussed. Finally, we demonstrate
the convenience and potential of our method when it comes to enforcing maximum and
minimum thickness constraints in structural shape optimization.

@article{feppon2020variational,
author = {{Feppon, Florian} and {Allaire, Gr\'egoire} and {Dapogny, Charles}},
title = {A variational formulation for computing shape derivatives of geometric constraints along rays},
DOI= "10.1051/m2an/2019056",
url= "https://doi.org/10.1051/m2an/2019056",
journal = {ESAIM: M2AN},
year = 2020,
volume = 54,
number = 1,
pages = "181-228",
}

[9] Feppon, F., Allaire, G., Bordeu, F., Cortial, J. and Dapogny, C. Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework (2019). *SeMA, 76: 413*. HAL preprint hal-01686770.
(abstract)
(bibtex)

**Abstract**:
Hadamard’s method of shape differentiation is applied to
topology optimization of a weakly coupled three physics problem. The
coupling is weak because the equations involved are solved consecutively,
namely the steady state Navier–Stokes equations for the fluid domain,
first, the convection diffusion equation for the whole domain, second, and
the linear thermo-elasticity system in the solid domain, third. Shape
sensitivities are derived in a fully Lagrangian setting which allows us to
obtain shape derivatives of general objective functions. An emphasis is
given on the derivation of the adjoint interface condition dual to the one
of equality of the normal stresses at the fluid solid interface. The
arguments allowing to obtain this surprising condition are specifically
detailed on a simplified scalar problem. Numerical test cases are presented
using the level set mesh evolution framework of Allaire et al. (Appl Mech
Eng 282:22–53, 2014). It is demonstrated how the implementation enables
to treat a variety of shape optimization problems.

@article{Feppon2019Sep,
author = {Feppon, F. and Allaire, G. and Bordeu, F. and Cortial, J. and Dapogny, C.},
title = {{Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework}},
journal = {SeMA},
volume = {76},
number = {3},
pages = {413--458},
year = {2019},
month = {Sep},
issn = {2254-3902},
publisher = {Springer International Publishing},
doi = {10.1007/s40324-018-00185-4}
}

[8] Grejtak T., Jia X., Feppon F., Joynson S.G., Cunniffe A.R., Shi Y., Kauffman D.P., Vermaak N. and Krick B.A. Topology Optimization of Composite Materials for Wear: A Route to Multifunctional Materials for Sliding Interfaces (2019). *Advanced Engineering Materials*.
(abstract)
(bibtex)

**Abstract**:
Predicting and optimizing the wear performance of tribological
systems is of great interest in many mechanical applications. Wear modeling based
on elastic foundation models can be used to predict the wear behavior of
composite materials. Topology optimization has previously been used to improve
the wear performance of a bi‐material composite surface without direct
experimental validation. In this paper, three multi‐material composite wear
surfaces are presented and fabricated that are the product of topology
optimization. The wear surfaces are designed for optimal wear performance
including minimized run‐in wear volume lost. In this work, the designs are
evaluated with high‐accuracy simulations prior to fabrication. Extensive testing
is conducted including for wear volume, wear rate, surface height distribution,
and profile measurements throughout the wear process. The effects of boundary
conditions and the importance of taking wear sliding directionality into account
in the modeling process are discussed.

@article{doi:10.1002/adem.201900366,
author = {Grejtak, Tomas and Jia, Xiu and Feppon, Florian and Joynson,
Sam G. and Cunniffe, Annaliese R. and Shi, Yupin and Kauffman,
David P. and Vermaak, Natasha and Krick, Brandon A.},
title = {Topology Optimization of Composite Materials for Wear: A Route to
Multifunctional Materials for Sliding Interfaces},
journal = {Advanced Engineering Materials},
volume = {0},
number = {0},
pages = {1900366},
keywords = {composite design, level-set method, mechanical design, topology optimization, tribology, wear},
doi = {10.1002/adem.201900366},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/adem.201900366},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/adem.201900366}
}

[7] Feppon, F. and Lermusiaux, P. F. J. The Extrinsic Geometry of Dynamical Systems tracking nonlinear matrix projections (2019). *SIAM Journal on Matrix Analysis and Applications, 40(2), 814-844*. HAL preprint hal-02096001.
(abstract)
(bibtex)

**Abstract**:
A generalization of the concepts of extrinsic curvature and
Weingarten endomorphism is introduced to study a class of nonlinear maps
over embedded matrix manifolds. These (nonlinear) oblique projections,
generalize (nonlinear) orthogonal projections, i.e. applications mapping a
point to its closest neighbor on a matrix manifold. Examples of such maps
include the truncated SVD, the polar decomposition, and functions mapping
symmetric and non-symmetric matrices to their linear eigenprojectors. This
paper specifically investigates how oblique projections provide their image
manifolds with a canonical extrinsic differential structure, over which a
generalization of the Weingarten identity is available. By diagonalization
of the corresponding Weingarten endomorphism, the manifold principal
curvatures are explicitly characterized, which then enables us to (i)
derive explicit formulas for the differential of oblique projections and
(ii) study the global stability ofgeneric Ordinary Differential Equation
(ODE) computing their values. This methodology, exploited for the truncated
SVD in [22], is generalized to non-Euclidean settings, and applied to the
four other maps mentioned above and their image manifolds: respectively,
the Stiefel, the isospectral, the Grassmann manifolds, and the manifold of
fixed rank (non-orthogonal) linear projectors. In all cases studied, the
oblique projection of a target matrix is surprisingly the unique stable
equilibrium point of the above gradient flow. Three numerical applications
concerned with ODEs tracking dominant eigenspaces involving possibly
multiple eigenvalues finally showcase the results.

@article{doi:10.1137/18M1192780,
author = {Feppon, F. and Lermusiaux, P.},
title = {The Extrinsic Geometry of Dynamical Systems Tracking Nonlinear Matrix Projections},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {40},
number = {2},
pages = {814-844},
year = {2019},
doi = {10.1137/18M1192780},
}

[6] Feppon, F. and Lermusiaux, P. F. J. Dynamically orthogonal numerical schemes for efficient stochastic advection and Lagrangian transport (2018). *SIAM Review, 60(3), 595-625*. HAL preprint hal-01881442.
(abstract)
(bibtex)

**Abstract**:
Quantifying the uncertainty of Lagrangian motion can be performed by solving a
large number of ordinary differential equations with random velocities or,
equivalently, a stochastic transport partial differential equation (PDE) for
the ensemble of flow-maps. The dynamically orthogonal (DO) decomposition is
applied as an efficient dynamical model order reduction to solve for such
stochastic advection and Lagrangian transport. Its interpretation as the method
that applies the truncated SVD instantaneously on the matrix discretization of
the original stochastic PDE is used to obtain new numerical schemes. Fully
linear, explicit central advection schemes stabilized with numerical filters
are selected to ensure efficiency, accuracy, stability, and direct consistency
between the original deterministic and stochastic DO advections and flow-maps.
Various strategies are presented for selecting a time-stepping that accounts
for the curvature of the fixed-rank manifold and the error related to closely
singular coefficient matrices. Efficient schemes are developed to dynamically
evolve the rank of the reduced solution and to ensure the orthogonality of the
basis matrix while preserving its smooth evolution over time. Finally, the new
schemes are applied to quantify the uncertain Lagrangian motions of a 2D
double-gyre flow with random frequency and of a stochastic flow past a
cylinder.

@article{doi:10.1137/16M1109394,
author = {Feppon, F. and Lermusiaux, P.},
title = {Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport},
journal = {SIAM Review},
volume = {60},
number = {3},
pages = {595-625},
year = {2018},
doi = {10.1137/16M1109394},
}

[5] Feppon, F. and Lermusiaux, P. F. J. A geometric approach to dynamical model order reduction (2018). *SIAM Journal on Matrix Analysis and Applications, 39(1), 510-538*. Arxiv preprint 1705.08521.
(abstract)
(bibtex)

**Abstract**:
Any model order reduced dynamical system that evolves a modal decomposition to
approximate the discretized solution of a stochastic PDE can be related to a
vector field tangent to the manifold of fixed rank matrices. The dynamically
orthogonal (DO) approximation is the canonical reduced-order model for which
the corresponding vector field is the orthogonal projection of the original
system dynamics onto the tangent spaces of this manifold. The embedded geometry
of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the
manifold is characterized and related to the smallest singular value through
the study of the Weingarten map. Differentiability results for the orthogonal
projection onto embedded manifolds are reviewed and used to derive an explicit
dynamical system for tracking the truncated singular value decomposition (SVD)
of a time-dependent matrix. It is demonstrated that the error made by the DO
approximation remains controlled under the minimal condition that the original
solution stays close to the low rank manifold, which translates into an
explicit dependence of this error on the gap between singular values. The DO
approximation is also justified as the dynamical system that applies
instantaneously the SVD truncation to optimally constrain the rank of the
reduced solution. Riemannian matrix optimization is investigated in this
extrinsic framework to provide algorithms that adaptively update the best low
rank approximation of a smoothly varying matrix. The related gradient flow
provides a dynamical system that converges to the truncated SVD of an input
matrix for almost every initial datum.

@article{doi:10.1137/16M1095202,
author = {Feppon, F. and Lermusiaux, P.},
title = {A Geometric Approach to Dynamical Model Order Reduction},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {39},
number = {1},
pages = {510-538},
year = {2018},
doi = {10.1137/16M1095202},
}

[4] Feppon, F., Michailidis, G., Sidebottom, M. A., Allaire, G., Krick, B. A. and Vermaak, N. Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints (2017). *Structural and Multidisciplinary Optimization, 55(2), 547-568*. HAL preprint hal-01336301.
(abstract)
(bibtex)

**Abstract**:
The wear of materials continues to be a limiting factor in the
lifetime and performance of mechanical systems with sliding surfaces. As
the demand for low wear materials grows so does the need for models and
methods to systematically optimize tribological systems. Elastic foundation
models offer a simplified framework to study the wear of multimaterial
composites subject to abrasive sliding. Previously, the evolving wear
profile has been shown to converge to a steady-state that is characterized
by a time-independent elliptic equation. In this article, the steady-state
formulation is generalized and integrated with shape optimization to
improve the wear performance of bi-material composites. Both macroscopic
structures and periodic material microstructures are considered. Several
common tribological objectives for systems undergoing wear are identified
and mathematically formalized with shape derivatives. These include (i)
achieving a planar wear surface from multimaterial composites and (ii)
minimizing the run-in volume of material lost before steady-state wear is
achieved. A level-set based topology optimization algorithm that
incorporates a novel constraint on the level-set function is presented. In
particular, a new scheme is developed to update material interfaces; the
scheme (i) conveniently enforces volume constraints at each iteration, (ii)
controls the complexity of design features using perimeter penalization,
and (iii) nucleates holes or inclusions with the topological gradient. The
broad applicability of the proposed formulation for problems beyond wear is
discussed, especially for problems where convenient control of the
complexity of geometric features is desired.

@article{feppon2017introducing,
title={Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints},
author={Feppon, Florian and Michailidis, G and Sidebottom, MA and Allaire, Gr{\'e}goire and Krick, BA and Vermaak, N},
journal={Structural and Multidisciplinary Optimization},
volume={55},
number={2},
pages={547--568},
year={2017},
publisher={Springer}
}

[3] Feppon, F., Sidebottom, M. A., Michailidis, G., Krick, B. A. and Vermaak, N. Efficient Steady-State Computation for Wear of Multimaterial Composites (2016). *Journal of Tribology, 138(3), 031602*. Preprint.
(abstract)
(bibtex)

**Abstract**:
'Traditionally, iterative schemes have been used to predict
evolving material profiles under abrasive wear. In this work, more
efficient continuous formulations are presented for predicting the wear of
tribological systems. Following previous work, the formulation is based on
a two parameter elastic Pasternak foundation model. It is considered as a
simplified framework to analyze the wear of multimaterial surfaces. It is
shown that the evolving wear profile is also the solution of a parabolic
partial differential equation (PDE). The wearing profile is proven to
converge to a steady-state that propagates with constant wear rate. A
relationship between this velocity and the inverse rule of mixtures or
harmonic mean for composites is derived. For cases where only the final
steady-state profile is of interest, it is shown that the steady-state
profile can be accurately and directly determined by solving a simple
elliptic differential system—thus avoiding iterative schemes altogether.
Stability analysis is performed to identify conditions under which an
iterative scheme can provide accurate predictions and several comparisons
between iterative and the proposed formulation are made. Prospects of the
new continuous wear formulation and steady-state characterization are
discussed for advanced optimization, design, manufacturing, and control
applications.

@article{feppon2016efficient,
title={Efficient steady-state computation for wear of multimaterial composites},
author={Feppon, Florian and Sidebottom, Mark A and Michailidis, Georgios and Krick, Brandon A and Vermaak, Natasha},
journal={Journal of Tribology},
volume={138},
number={3},
pages={031602},
year={2016},
publisher={American Society of Mechanical Engineers}
}

[2] Sidebottom, M. A., Feppon, F., Vermaak, N. and Krick, B. A. Modeling Wear of Multimaterial Composite Surfaces (2016). *Journal of Tribology, 138(4), 041605*. Preprint.
(abstract)
(bibtex)

**Abstract**:
Iterative numerical wear models provide valuable insight into
evolving material surfaces under abrasive wear. In this paper, a holistic
numerical scheme for predicting the wear of rubbing elements in
tribological systems is presented. In order to capture the wear behavior of
a multimaterial surface, a finite difference model is developed. The model
determines pressure and height loss along a composite surface as it slides
against an abrasive compliant countersurface. Using Archard's wear law, the
corresponding nodal height loss is found using the appropriate material
wear rate, applied pressure, and the incremental sliding distance. This
process is iterated until the surface profile reaches a steady-state
profile. The steady-state is characterized by the incremental height loss
at each node being nearly equivalent to the previous loss in height.
Several composite topologies are investigated in order to identify key
trends in geometry and material properties on wear performance.

@article{sidebottom2016modeling,
title={Modeling wear of multimaterial composite surfaces},
author={Sidebottom, Mark A and Feppon, Florian and Vermaak, Natasha and Krick, Brandon A},
journal={Journal of Tribology},
volume={138},
number={4},
pages={041605},
year={2016},
publisher={American Society of Mechanical Engineers}
}

[1] Lefebvre, G., Gondel, A., Dubois, M., Atlan, M., Feppon, F., Labbé, A., Gillot C., Garelli A., Ernoult M. and Filoche, M. One single static measurement predicts wave localization in complex structures (2016). *Physical review letters, 117(7), 074301*. Arxiv preprint 1604.03090.
(abstract)
(bibtex)

**Abstract**:
A recent theoretical breakthrough has brought a new tool,
called the localization landscape, for predicting the localization regions
of vibration modes in complex or disordered systems. Here, we report on the
first experiment which measures the localization landscape and demonstrates
its predictive power. Holographic measurement of the static deformation
under uniform load of a thin plate with complex geometry provides direct
access to the landscape function. When put in vibration, this system shows
modes precisely confined within the subregions delineated by the landscape
function. Also the maxima of this function match the measured
eigenfrequencies, while the minima of the valley network gives the
frequencies at which modes become extended. This approach fully
characterizes the low frequency spectrum of a complex structure from a
single static measurement. It paves the way for controlling and engineering
eigenmodes in any vibratory system, especially where a structural or
microscopic description is not accessible.

@article{PhysRevLett.117.074301,
title = {One Single Static Measurement Predicts Wave Localization in Complex Structures},
author = {Lefebvre, Gautier and Gondel, Alexane and Dubois, Marc and Atlan, Michael and Feppon, Florian and Labb'e,
Aim'e and Gillot, Camille and Garelli, Alix and Ernoult, Maxence and Mayboroda, Svitlana and Filoche,
Marcel and Sebbah, Patrick},
journal = {Phys. Rev. Lett.},
volume = {117},
issue = {7},
pages = {074301},
numpages = {5},
year = {2016},
month = {Aug},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.117.074301},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.117.074301}
}