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.

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 [4]. It is demonstrated how the implementation enables to treat a variety of shape optimization problems.

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.

An efficient framework is described for the shape and topology optimization of realistic threedimensional, 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, highquality 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.

A recent theoretical breakthrough has brought a new tool, called 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.

The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimiza- tion applications. We rely on a variant of the ODE approach proposed by Yamashita for equality constrained problems: the search direction is a combina- tion of a null space step and a range space step, aiming to decrease 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. Here, we solve their local combinatorial character by computing the projection of the gradient of the ob- jective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem. The solution to this problem allows to identify the inequality constraints to which the optimization tra- jectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of|in nite-dimensional|Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard's boundary variation method.

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 η ≡ aε/ε between the radius aε of the holes and the size ε 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 (uε, pε) as a linear combination of the derivatives of their formal average (u * ε , p * ε ) weighted by suitable corrector tensors. The formal average (u * ε , p * ε ) 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 [53,27], we derive, for any K ∈ N, a well-posed model of order 2K + 2 which yields approximations of the original solutions with an error of order O(ε K+3 ) in the L 2 norm. Furthermore, the error improves up to the order O(ε 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 η → 0 and in dimension d ≥ 3. This allows us to obtain that our effective equations converge coefficient-wise to either of the Brinkman or Darcy regimes which arise when η is respectively equivalent, or greater than the critical scaling η crit ∼ ε 2/(d−2) .

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.