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Florian Feppon

Publications and source records attributed to Florian Feppon.

6 recordsLinked to original sources
arXiv2026

Comparison of model order reduction techniques with one-shot procedure for topology optimization for thermal applications

Density-based topology optimization has become a powerful method for automatically generating optimized designs in a wide variety of applications. However, it comes with a large computational cost when solving the physical model requires large-scale simulations. Here, we investigate the use of model order reduction (MOR) techniques to accelerate the simulations in the context of thermal design applications. We project the governing and the adjoint equations onto a low-dimensional subspace by constructing two distinct reduced bases -- one for the forward state and one for the adjoint system -- using solution snapshots from previous design iterations. These snapshots are generated using either the high-fidelity solver or inaccurate fast solvers, such as the one-shot method \citep{amir2024one}. Additionally, we demonstrate that properly selecting the stopping criterion for the iterative linear solver is crucial for the effective use of reduced models. In our 3D example, the proposed framework reduces the overall total simulation time relative to the high-fidelity workflow by a factor up to $3$ when combined with high-fidelity solves and a factor up to $16$ when combined with the one-shot method. Moreover, we find that the reduced order model approach is able to achieve a speed up of $1.54$ with respect to the one-shot method.

math.NA↗
arXiv2026

The Riemannian Landing Method: From projected gradient flows to SQP

Landing methods have recently emerged in Riemannian matrix optimization as efficient schemes for handling nonlinear equality constraints without resorting to costly retractions. These methods decompose the search direction into tangent and normal components, enabling asymptotic feasibility while maintaining inexpensive updates. In this work, we provide a unifying geometric framework which reveals that, under suitable choices of Riemannian metric, the landing algorithm encompasses several classical optimization methods such as projected and null-space gradient flows, Sequential Quadratic Programming (SQP), and a certain form of the augmented Lagrangian method. In particular, we show that a quadratically convergent landing method essentially reproduces the quadratically convergent SQP method. These connections also allow us to propose a globally convergent landing method using adaptive step sizes. The backtracking line search satisfies an Armijo condition on a merit function, and does not require prior knowledge of Lipschitz constants. Our second key contribution is to analyze landing methods through a geometric parameterization of the metric in terms of fields of oblique projectors and associated metric restrictions. This viewpoint disentangles the roles of orthogonality, tangent and normal metrics, and elucidates how to design the metric to obtain explicit tangent and normal updates. For matrix optimization, this framework not only recovers recent constructions in the literature for problems with orthogonality constraints, but also provides systematic guidelines for designing new metrics that admit closed-form search directions.

math.OC↗
arXiv2023

Edge modes in subwavelength resonators in one dimension

We present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyse both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantised and the latter is not. Furthermore, we show the existence of localised edge modes arising from defects in the periodicity in both the Hermitian and non-Hermitian cases. In the non-Hermitian case, we provide a complete characterisation of the edge modes.

math.AP↗
arXiv2017

A Geometric Approach to Dynamical Model-Order Reduction

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 data.

math.DS↗
arXiv2016

One single static measurement predicts wave localization in complex structures

A recent theoretical breakthrough has brought a new tool, called \emph{localization landscape}, to predict 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 sub-regions 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 to the control and engineering of eigenmodes in any vibratory system, especially where a structural or microscopic description is not accessible.

cond-mat.dis-nn↗