This article is devoted to the mathematical and numerical treatments of a shape optimization problem emanating from the desire to reconcile quantum theories of chemistry and classical heuristic models: we aim to identify Maximum Probability Domains (MPDs), that is, domains $$\Omega $$ of the 3d space where the probability $${\mathbb {P}}_\nu (\Omega )$$ to find exactly $$\nu $$ among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional $${\mathbb {P}}_\nu (\Omega )$$ arises as the integral over $$\nu $$ copies of $$\Omega $$ and $$(n-\nu )$$ copies of the complement $${\mathbb {R}}^3 \setminus \Omega $$ of an analytic function defined over the space $${\mathbb {R}}^{3n}$$ of all the spatial configurations of the n electron system. Our first task is to explore the mathematical well-posedness of the shape optimization problem: under mild hypotheses, we prove that global maximizers of the probability functions $${\mathbb {P}}_\nu (\Omega )$$ do exist as open subsets of $${\mathbb {R}}^3$$ ; meanwhile, we identify the associated necessary first-order optimality condition. We then turn to the numerical calculation of MPDs, for which we resort to a level set based mesh evolution strategy: the latter allows for the robust tracking of complex evolutions of shapes, while leaving the room for accurate chemical computations, carried out on high-resolution meshes of the optimized shapes. The efficiency of this procedure is enhanced thanks to the addition of a fixed-point strategy inspired from the first-order optimality conditions resulting from our theoretical considerations. Several three-dimensional examples are presented and discussed to appraise the efficiency of our algorithms.

Gradient-based optimization algorithms, where gradient information is extracted using adjoint equations, are efficient but can quickly slow down when applied to unsteady and nonlinear flow problems. This is mainly due to the sequential nature of the algorithm, where the primal problem is first integrated forward in time, providing the initial condition for the adjoint problem, which is then integrated backward. In order to address the sequential nature of this optimization procedure parallel-in-time algorithms can be employed. However, the characteristics of the governing equations of interest in this work, and in particular, the divergence-free constraint (incompressibility effect) as well as the nonlinearity and the unsteadiness of the flow, make direct application of existing parallel-in-time algorithms less than straightforward. In this work, we introduce a parallel-in-time procedure, applied to the integration of the adjoint problem, which addresses all the existing constraints and allows quick access to local gradients. The performance of the proposed algorithm is assessed for both steady and unsteady actuation; in both cases it readily outperforms the sequential algorithm.

The purpose of this article is to discuss several modern aspects of remeshing, which is the task of modifying an ill-shaped tetrahedral mesh with bad size elements so that it features an appropriate density of high-quality elements. After a brief sketch of classical stakes about meshes and local mesh operations, we notably expose (i) how the local size of the elements of a mesh can be adapted to a user-defined prescription (guided, e.g., by an error estimate attached to a numerical simulation), (ii) how a mesh can be deformed to efficiently track the motion of the underlying domain, (iii) how to construct a mesh of an implicitlydefined domain, and (iv) how remeshing procedures can be conducted in a parallel fashion when large-scale applications are targeted. These ideas are illustrated with several applications involving high-performance computing. In particular, we show how mesh adaptation and parallel remeshing strategies make it possible to achieve a high accuracy in large-scale simulations of complex flows, and how the aforementioned methods for meshing implicitly defined surfaces allow to represent faithfully intricate geophysical interfaces, and to account for the dramatic evolutions of shapes featured by shape optimization processes.

In this paper, an hp -adaptation strategy designed for discontinuous Galerkin methods is extended and applied to hybrid RANS/LES simulations. The 3D hp- adaptive strategy is suited for tetrahedral and hybrid prismatic/tetrahedral meshes, and relies on a metric-based remeshing approach. The metric field and the polynomial map of the adapted meshes are built from an a posteriori error estimator which couples the measure of the energy associated with the highest-order modes and the inter-element jumps of the solution, combined with a smoothness sensor which guides the choice between h- and p- adaptation. The turbulence modeling relies on a Zonal Detached Eddy Simulation approach. The developed hp -adaptation algorithm is assessed in the context of hybrid RANS/LES simulations of the PPRIME nozzle configuration at diameter-based Reynolds number equal to 10^6 , starting the adaptive process from previously RANS-adapted meshes. Far-field acoustic analysis are performed using a Ffowcs Williams−Hawkings method.

In this paper, we present h- and hp-adaptive strategies suited for the discontinuous Galerkin formulation of the compressible laminar and Reynolds-averaged Navier–Stokes equations on unstructured grids, relying on a metric-based simplicial remeshing approach. An a posteriori error estimator, combining the measure of the energy associated with the highest-order modes and the inter-element jumps, is used to build both the metric field and the polynomial degree distribution map. The choice of refining either in h or p is driven by a smoothness indicator based on the decay of the modal coefficients in each element. The performance of the developed adaptation algorithms is assessed for the 2D laminar viscous flow past a NACA0012 airfoil, and for the 3D laminar viscous flows past a sphere and past a delta wing. The adaptive hp-strategy is applied to a 3D turbulent jet issued from a nozzle. Finally, the gain in accuracy provided by the adaptive algorithms with respect to uniformly refined simulations, for a given number of degrees of freedom, with polynomial degrees p=1,2,3, is demonstrated.

In this paper, we present a mesh h-adaptation strategy suited for the discontinuous Galerkin formulation of the compressible Navier-Stokes equations on unstructured grids, based on the simplicial remeshing library MMG. A novel a posteriori error estimator, combining the measure of the energy associated with the highest-order modes and the inter-element jumps, is used to build the metric field. The performance of the developed mesh adaptation algorithm is assessed for steady laminar viscous flows past a square cylinder and past a NACA0012 airfoil, and for the unsteady laminar viscous flow past a circular cylinder. The gain in accuracy for a given number of degrees of freedom (DoFs) is demonstrated for p=1, p=2 and p=3 polynomial degrees, with respect to uniformly refined simulations.

Haptic feedback has become crucial to enhance the user experiences in Virtual Reality (VR). This justifies the sudden burst of novel haptic solutions proposed these past years in the HCI community. This article is a survey of Virtual Reality interactions, relying on haptic devices. We propose two dimensions to describe and compare the current haptic solutions: their degree of physicality, as well as their degree of actuation. We depict a compromise between the user and the designer, highlighting how the range of required or proposed stimulation in VR is opposed to the haptic interfaces flexibility and their deployment in real-life use-cases. This paper (1) outlines the variety of haptic solutions and provides a novel perspective for analysing their associated interactions, (2) highlights the limits of the current evaluation criteria regarding these interactions, and finally (3) reflects the interaction, operation and conception potentials of "encountered-type of haptic devices".

Haptic feedback has become crucial to enhance the user experiences in Virtual Reality (VR). This justifies the sudden burst of novel haptic solutions proposed these past years in the HCI community. This article is a survey of Virtual Reality interactions, relying on haptic devices. We propose two dimensions to describe and compare the current haptic solutions: their degree of physicality, as well as their degree of actuation. We depict a compromise between the user and the designer, highlighting how the range of required or proposed stimulation in VR is opposed to the haptic interfaces flexibility and their deployment in real-life use-cases. This paper (1) outlines the variety of haptic solutions and provides a novel perspective for analysing their associated interactions, (2) highlights the limits of the current evaluation criteria regarding these interactions, and finally (3) reflects the interaction, operation and conception potentials of ”encountered-type of haptic devices”.

We present CoVR, a novel robotic interface providing strong kinesthetic feedback (100 N) in a room-scale VR arena. It consists of a physical column mounted on a 2D Cartesian ceiling robot (XY displacements) with the capacity of (1) resisting to body-scaled users' actions such as pushing or leaning; (2) acting on the users by pulling or transporting them as well as (3) carrying multiple potentially heavy objects (up to 80kg) that users can freely manipulate or make interact with each other. We describe its implementation and define a trajectory generation algorithm based on a novel user intention model to support non-deterministic scenarios, where the users are free to interact with any virtual object of interest with no regards to the scenarios' progress. A technical evaluation and a user study demonstrate the feasibility and usability of CoVR, as well as the relevance of whole-body interactions involving strong forces, such as being pulled through or transported.

In this paper a shape optimization approach based on the Hadamard geometric optimization method is developed. Four case studies representing typical fluid flow patterns in fluid dynamics, i.e. flow around an obstacle, flow in a 90. or 180. elbow pipe and flow in a dyadic tree, are considered. Low velocities are imposed at the inlet of each case study in order to operate in laminar flow regime. The objective is to determine the shape that minimizes the energy dissipated by viscous friction subjected to the Navier-Stokes equations and to iso-volumic constraint. The required gradients of the performance index and constraint with respect to the shape are computed by means of the adjoint system method. The momentum equations are implemented and solved using the OpenFOAM CFD software, and the solver ''adjointShapeOptimizationFoam'' is modified in order to compute the solution of the resulting optimization problems. The optimal shapes obtained in the four case studies are in very good agreement with the available literature works. Moreover, they allow a significant reduction of the dissipated energy ranging from 10.8 to 53.3 %.

In this paper, a simple and robust numerical shape optimization approach is presented. This approach is based on the Hadamard geometric optimization method and tested on 3 two-dimensional case studies representing standard flows in fluid dynamics, namely the flow around an obstacle, in a 90° elbow pipe and in a dyadic tree. This viscous flows are driven by the stationary Navier-Stokes equations without turbulence model. Low velocities are imposed at the inlet of each case study in order to operate in laminar flow regime. The objective is to determine the shape of the 3 aforementioned case studies that minimizes the energy dissipation in the fluid due to the work of viscous forces under a volume constraint. The required gradients of the performance index and constraint with respect to the shape are computed by means of the adjoint system method. The Navier-Stokes equations and the adjoint system are implemented and solved by using the finite volume method within OpenFOAM CFD software. The solver "adjointShapeOptimizationFoam" is modified in order to implement the optimization algorithm and determine the best shape in each of the three considered case studies. The optimal shapes obtained in the three case studies are in very good agreement with the available literature works. Moreover, they allow a significant reduction of the dissipated energy ranging from 10.8 to 53.3 %. Therefore, a decrease of the pressure losses in each case is also achieved in the same proportion.

In this paper, we present a numerical scheme for solving 2-phase or free-surface flows. Here, the interface/free surface is modeled using the level-set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free-surface location. In addition, it enables us to solve the time-discretized fluid equation only in the fluid domain in the case of free-surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier-Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first-order Lagrange-Galerkin method. The level-set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level-set function (redistancing) as well as the construction of a convenient flow for the level-set advection equation. Numerical results are presented for both bifluid and free-surface problems.

In this article, we present simple and robust numerical methods for two-dimensional geometrical shape optimization problems, in the context of viscous flows driven by the stationary Navier-Stokes equations at low Reynolds number. The salient features of our algorithm are exposed with an educational purpose; in particular, the numerical resolution of the nonlinear stationary Navier-Stokes system, the Hadamard boundary variation method for calculating the sensitivity of the minimized function of the domain, and the mesh update strategy are carefully described. Several pedagogical examples are discussed. The corresponding program is written in the FreeFem++ environment, and it is freely available. Its chief features—and notably the implementation details of the main steps of our algorithm—are carefully presented, so that it can easily be handled and elaborated upon to deal with different, or more complex physical situations.

This note addresses the following shape matching problem: given a 'template' shape, numerically described by means of a computational mesh, and a 'target' shape, known only via a signed distance function to its boundary, we aim at deforming iteratively the mesh of the template shape into a computational mesh of the target shape. To achieve this goal, we rely on techniques from shape optimization. Under the sole assumption that both shapes share the same topology, the desired transformation is realized as a sequence of elastic displacements, which are obtained by minimizing an energy functional based on the distance between the two shapes. The proposed method has been implemented in a finite elements setting and numerical examples in two and three dimensions are presented to illustrate its efficiency. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS.

For many years, providing an algorithm to generate hexahedral meshes that fulfill minimal geometric criteria (boundary-alignment, minimum of singularity vertices) and that is not limited to a category of geometries has been an open issue. In the past couple of years, techniques using 3D frame fields have emerged to design such meshes (Huang et al., 2011; Li et al., 2012). Those methods are based on a two-step process where a 3D frame field is built by assigning a frame to each cell of a tetrahedral mesh, then a parametrization algorithm is applied to generate a hexahedral mesh. In this work, we propose a novel algorithm to generate block-structured hexahedral meshes for any CAD domain Q. This work differs from previous ones in many points: (1) the proposed approach does not need to start from a pre-meshed quad boundary; (2) The frame field initialization does not put singularity lines around the medial object of Q; (3) Conceptually, frames are assigned to the vertices and not to the cells of the tetrahedral mesh; (4) The parametrization process is replaced by a constructive algorithm that generates a block structure, which partitions Q in meshable regions. (C) 2015 Elsevier Ltd. All rights reserved.

The formal language of Clifford’s algebras is attracting an increasingly large community of mathematicians, physicists and software developers seduced by the conciseness and the efficiency of this compelling system of mathematics. This contribution will suggest how these concepts can be used to serve the purpose of scientific visualization and more specifically to reveal the general structure of complex vector fields. We will emphasize the elegance and the ubiquitous nature of the geometric algebra approach, as well as point out the computational issues at stake.

The purpose of this exercise session is to give an overview of several aspects of scientific computing. First, we focus on the solving of an elliptic boundary value problem with the Finite Element Method. In a first part, we focus on the implementation of a very basic PDE model, hoping this may get the reader to be familiar with the software FreeFem++, which makes it very easy to handle finite element spaces and mesh modifications, within a very few lines of black-box code. Then, we turn to more realistic PDE models from fluid dynamics.

In the past couple of years, techniques using 3D frame fields have emerged to design hexahedral meshes[1,2]. Those methods are based on a two-step process where a 3D frame field is built by assigning a frame to each cell of a tetrahedral mesh, then a parametrization algorithm is applied to generate a hexahedral mesh. In this paper, we propose a novel algorithm to generate block-structured hexahedral meshes for any CAD domain Ω. This work differs from previous ones in several points: (1) the proposed approach does not start from a pre-meshed boundary; (2) The frame field initialization does not put singularity lines around the medial object of Ω; (3) Frames are assigned to the vertices and not to the cells of the tetrahedral mesh; (4) We do not perform a parametrization process but we generate a block structure that partition Ω in meshable regions.