Abstract: We propose a mathematical theory of acoustic wave scattering in one-dimensional finite high-contrast media. The system considered is constituted of a finite alternance of high-contrast segments of arbitrary lengths and interdistances, called the ``resonators'', and a background medium. We prove the existence of subwavelength resonances, which are the counterparts of the well-known Minnaert resonances in 3D systems. Our main contribution is to show that the resonant frequencies, as well as the transmission and reflection properties of the system can be accurately predicted by a ``capacitance'' eigenvalue problem, analogously to the 3D setting. Numerical results considering different situations with $N=1$ to $N=6$ resonators are provided to support our mathematical analysis, and to illustrate the various possibilities offered by high-contrast resonators to manipulate waves at subwavelength scales.

Abstract: This article provides a rigorous mathematical analysis of acoustic wave scattering induced by a high-contrast subwavelength resonator whose material density is periodically modulated in time, and with a modulation frequency that is much larger than the one of the incident wave. We find that in general, the effect of the fast modulation is averaged over time and that the system behaves as an unmodulated resonator with an apparent effective density. However, under a suitable tuning of the modulation, which achieves a matching between temporal Sturm-Liouville and spatial Neumann eigenvalues, the low frequency incident wave becomes suddenly able to excite high frequency modes in the resonator. This phenomenon leads to the generation of scattered waves carrying high frequency components in the far field, and to the existence of exponentially growing outgoing modes. From these findings, it is expected that such time-modulated system could serve as a spontaneously radiating device, or as a high harmonic generator.

Abstract: A systematic two-step procedure is proposed for the derivation of full asymptotic expansions of the solution of elliptic partial differential equations set on a domain perforated with a small hole on which a Dirichlet boundary condition is applied. First, an integral representation of the solution is sought, which enables to exploit the explicit dependence with respect to the small parameter to predict the correct form of a two-scale ansatz. Second, the terms of the ansatz are characterized by the method of matched asymptotic expansions as the solutions of a cascade of successive interior and exterior domains. This allows to interpret them as high-order correctors, for which error bounds can be proved using variational estimates. The methodology is illustrated on two different problems: we start by revisiting the perforated Poisson problem with Dirichlet boundary conditions on both the hole and the outer boundary, where we highlight how the method enables to obtain very naturally the correct ansatz in the most delicate two-dimensional setting. Then, we provide original and complete asymptotic expansions for a perforated cell-problem featuring periodic conditions.

Abstract: We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number $N$ of small heterogeneities of characteristic size $s$, randomly and independently distributed in a bounded domain. We first consider a ``sound-absorbing'' material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the ``sub-critical'' regime $sN=O(1)$, we obtain that the effective medium is governed by a dissipative Lippmann-Schwinger equation which approximates the total field with a relative mean-square error of order $O(\max((sN)^{2}N^{-\frac{1}{3}}, N^{-\frac{1}{2}}))$. We retrieve the critical size $s\sim 1/N$ of the literature at which the effects of the obstacles can be modelled by a ``strange term'' added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the $N$ heterogeneities are packets of $K$ inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, $\delta\rightarrow 0$, the effective medium admits $K$ resonant characteristic sizes $(s_i(\delta))_{1\leqslant i\leqslant K}$ and is governed by a Lippmann-Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies $\omega$ respectively slightly larger or slightly smaller than the corresponding $K$ resonant frequencies $(\omega_i(\delta))_{1\leqslant i\leqslant K}$. These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the ``subcritical regime'' where the contrast parameter is small enough, i.e. $\delta=o(N^{-2})$, while the considered frequency is ``not too close'' to the resonance, i.e. $N\delta^{\frac{1}{2}}=O(|1-s/s_i(\delta)|)$. Our mathematical analysis and the current literature allow us to conjecture that ``solidification'' phenomena are expected to occur in the ``super-critical'' regime $N\delta^{\frac{1}{2}}|1-s/s_i(\delta)|^{-1}\rightarrow +\infty$.

Abstract: This paper provides several contributions to the mathematical analysis of subwavelength resonances in a high-contrast medium containing N acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of a N dimensional linear system, and characterizes resonant frequencies as the solutions to a N-dimensional nonlinear eigenvalue problem. Under a simplicity assumptions on the eigenvalues of the capacitance matrix, we prove the analyticity of the scattering resonances with respect to the square root of the contrast parameter, and we provide a deterministic algorithm allowing to compute all terms of the corresponding Puiseux series. We then establish a nonlinear modal decomposition formula for the scattered field as well as point scatterer approximations for the far field pattern of the sound wave scattered by N bodies. As a prerequisite to our analysis, a first part of the work establishes various novel results about the capacitance matrix, since qualitative properties of the resonances, such as the leading order of the scattering frequencies or of the corresponding far field pattern are closely related to its spectral decomposition.

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.

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.

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.

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.

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 $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$.

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 $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Abstract: This work is devoted to shape and topology optimization of multiphysics systems motivated by aeronautic industrial applications. Shape derivatives of arbitrary objective functionals are computed for a weakly coupled thermal fluid-structure model. A novel gradient flow type algorithm is then developed for solving generic constrained shape optimization problems without the need for tuning non-physical metaparameters. Motivated by the need for enforcing non-mixing constraints in the design of liquid-liquid heat exchangers, a variational method is developed in order to simplify the numerical evaluation of geometric constraints: it allows to compute line integrals on a mesh by solving a variational problem without requiring the explicit knowledge of these lines on the spatial discretization. All these ingredients allowed us to implement a variety of 2-d and 3-d multiphysics shape optimization test cases: from single, double or three physics problems in 2-d, to moderately large-scale 3-d test cases for structural design, thermal conduction, aerodynamic design and a fluid-structure interacting system. A final opening chapter derives high order homogenized equations for the Stokes system in a porous medium. These high order equations encompass the three classical homogenized regimes---namely Stokes, Brinkman and Darcy---associated with different obstacle's size scalings. They could allow, in future works, to develop new topology optimization methods for the design of fluid systems characterized by multi-scale patterns such as industrial heat exchangers.

Abstract: This work focuses on developing theory and methodologies for the analysis of material transport in stochastic fluid flows. In a first part, two dominant classes of techniques for extracting Lagrangian Coherent Structures are reviewed and compared and some improvements are suggested for their pragmatic applications on realistic high-dimensional deterministic ocean velocity fields. In the stochastic case, estimating the uncertain Lagrangian motion can require to evaluate an ensemble of realizations of the flow map associated with a random velocity flow field, or equivalently realizations of the solution of a related transport partial differential equation. The Dynamically Orthogonal (DO) approximation is applied as an efficient model order reduction technique to solve this stochastic advection equation. With the goal of developing new rigorous reduced-order advection schemes, the second part of this work investigates the mathematical foundations of the method. Riemannian geometry providing an appropriate setting, a framework free of tensor notations is used to analyze the embedded geometry of three popular matrix manifolds, namely the fixed rank manifold, the Stiefel manifold and the isospectral manifold. Their extrinsic curvatures are characterized and computed through the study of the Weingarten map. As a spectacular by-product, explicit formulas are found for the differential of the truncated Singular Value Decomposition, of the Polar Decomposition, and of the eigenspaces of a time dependent symmetric matrix. Convergent gradient flows that achieve related algebraic operations are provided. A generalization of this framework to the non-Euclidean case is provided, allowing to derive analogous formulas and dynamical systems for tracking the eigenspaces of non-symmetric matrices. In the geometric setting, the DO approximation is a particular case of projected dynamical systems, that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. It is obtained that the error committed by the DO approximation is controlled under the minimal geometric condition that the original solution stays close to the low-rank manifold. The last part of the work focuses on the practical implementation of the DO methodology for the stochastic advection equation. Fully linear, explicit central schemes are selected to ensure stability, accuracy and efficiency of the method. Riemannian matrix optimization is applied for the dynamic evaluation of the dominant SVD of a given matrix and is integrated to the DO time-stepping. Finally the technique is illustrated numerically on the uncertainty quantification of the Lagrangian motion of two bi-dimensional benchmark flows..

Abstract: The purpose of these notes is to offer a comprehensive introduction to topology optimization for automated generation of complex heat exchanger designs, based on the methodof Hadamard whereby the design variable is the shape of the fluid-solid interfaces and is updated iteratively until convergence to a nearly optimal design. The material is intended to be an introductory exposure to our recent work (Feppon et al.(2021)) and PhD thesis (Feppon, 2019).