We study numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs). This problem describes the heat propagation in a liquid-solid phase change system. It implies a heat equation and a discontinuity at the interface where the phase change occurs. In the context of PINNs, this model leads to difficulties in the learning process, especially near the interface of phase change. We present different strategies that can be used in this context. We illustrate our results and compare with classical solvers for PDEs (finite differences).

This package performs parametric and non-parametric estimation and simulation for multi-state discrete-time semi-Markov processes (Barbu et al., submitted).For the parametric estimation, several discrete distributions are considered for the sojourn times: Uniform, Geometric, Poisson, Discrete Weibull of type 1 and Negative Binomial.The non-parametric estimation concerns the sojourn time distributions, where no assumptions are done on the shape of distributions.Moreover, the estimation can be done on the basis of one or several sample paths, with or without censoring at the beginning or/and at the end of the sample paths.Estimation and simulation of discrete-time k-th order Markov chains are also considered.

In this article, we investigate the movement and vibrations of a blade due to the presence of the mast. When the blade passes in front of the mast, a sudden pressure spike induces vibrations in the blade. To study the influence of stiffness, two different structures were studied. We present our numerical schemes concerning the resolution of the flow, the behavior of the structure and the coupling of the two systems. Then, we validate two methods against an experiment (Bahaj et al., 2007). In a third section, we present cases of fluid-structure interaction. Several structures are setup by modifying the stiffness of the material. Their steady open-water (without a mast) behaviors are compared. And finally, two dynamic fluid-structure computations are performed to compare the behavior of an elastic blade passing next to a mast. For all the cases, we use K–FSI developed by K-Epsilon to solve the fluid-structure interaction (FSI).

Efficient global optimization (EGO) has become a standard approach for the global optimization of complex systems with high computational costs. EGO uses a training set of objective function values computed at selected input points to construct a statistical surrogate model, with low evaluation cost, on which the optimization procedure is applied. The training set is sequentially enriched, selecting new points, according to a prescribed infilling strategy, in order to converge to the optimum of the original costly model. Multifidelity approaches combining evaluations of the quantity of interest at different fidelity levels have been recently introduced to reduce the computational cost of building a global surrogate model. However, the use of multifidelity approaches in the context of EGO is still a research topic. In this work, we propose a new effective infilling strategy for multifidelity EGO. Our infilling strategy has the particularity of relying on non-nested training sets, a characteristic that comes with several computational benefits. For the enrichment of the multifidelity training set, the strategy selects the next input point together with the fidelity level of the objective function evaluation. This characteristic is in contrast with previous nested approaches, which require estimation of all lower fidelity levels and are more demanding to update the surrogate. The resulting EGO procedure achieves a significantly reduced computational cost, avoiding computations at useless fidelity levels whenever possible, but it is also more robust to low correlations between levels and noisy estimations. Analytical problems are used to test and illustrate the efficiency of the method. It is finally applied to the optimization of a fully nonlinear fluid-structure interaction system to demonstrate its feasibility on real large-scale problems, with fidelity levels mixing physical approximations in the constitutive models and discretization refinements.

We present high-performance and high-accuracy numerical simulations of quantum turbulence modelled by the Gross–Pitaevskii equation for the time-evolution of the macroscopic wave function of the system. The hydrodynamic analogue of this model is a flow in which the viscosity is absent and all rotational flow is carried by quantized vortices with identical topological line-structure and circulation. Numerical simulations start from an initial state containing a large number of quantized vortices and follow the chaotic vortex interactions leading to a vortex-tangle turbulent state. The Gross–Pitaevskii equation is solved using a parallel (MPI-OpenMP) code based on a pseudo-spectral spatial discretization and second order splitting for the time integration. We define four quantum-turbulence simulation cases based on different methods used to generate initial states: the first two are based on the hydrodynamic analogy with classical Taylor–Green and Arnold–Beltrami–Childress vortex flows, while the other two methods use a direct manipulation of the wave function by generating a smoothed random phase field, or seeding random vortex-ring pairs. The dynamics of the turbulent field corresponding to each case is analysed in detail by presenting statistical properties (spectra and structure functions) of main quantities of interest (energy, helicity, etc.). Some general features of quantum turbulence are identified, despite the variety of initial states. Numerical and physical parameters of each case are presented in detail by defining corresponding benchmarks that could be used to validate or calibrate new Gross–Pitaevskii codes. The efficiency of the parallel computation for a reference case is also reported.

We present and distribute a FreeFem++ Toolbox for the parallel computing of two- or three-dimensional liquid-solid phase-change systems involving natural convection. FreeFem++ (www. freefem.org) is a free finite-element software available for all existing operating systems. We use the recent library ffddm that makes available in FreeFem++ state-of-the-art scalable Schwarz domain decomposition methods (DDM). The single domain approach used in our previous contribution (Rakotondrandisa et al., 2020) is adapted for the use of the DDM method. As a result, the computational time is considerably reduced for 2D configurations and furthermore 3D problems become affordable. The numerical method is based on an enthalpy-porosity model. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. A Carman-Kozeny-type penalty term is added to the momentum equations to bring progressively the velocity to zero into the solid. Model equations are discretized using Galerkin triangular or tetrahedral finite elements. The coupled system of equations is integrated in time using a second-order Gear implicit scheme. The resulting discrete equations are solved using a Newton algorithm. The DDM approach is based on an overlapping Schwarz method. The mesh is first split in subdomains using Scotch or Metis libraries. The final linear system is then solved in parallel using a GMRES Krylov method, with a Restricted Additive Schwarz (RAS) preconditioner. The mesh is adapted during the computation using metrics control. The 3D-mesh adaptivity uses the mmg (www.mmgtools.org) open source library. Parallel 2D and 3D computations of benchmark cases of increasing difficulty are presented: natural convection of air, natural convection of water, melting or solidification of a phase-change material, and, finally, a water freezing case. For each case, careful validations are provided and the performance of the code is assessed. The robustness of the Toolbox in 3D is also demonstrated by adapting the number of processors to the number of tetrahedra, which can considerably vary after the mesh adaptation. Program summary Program Title: PCM_Toolbox_DDM_2D and PCM_Toolbox_DDM_3D CPC Library link to program files: http://dx.doi.org/10.17632/dk49rfrz9y.1 Licensing provisions: Apache License, 2.0 Programming language: FreeFem++(www.freefem.org), mmg (www.mmgtools.org) Nature of problem: The software is scoped to parallel computations of 2D or 3D configurations of liquid-solid phase-change problems with convection in the liquid phase. Natural convection, melting and solidification processes are illustrated in the paper. The software can be easily modified to take into account different related physical models. Solution method: We use a single domain approach, solving the incompressible Navier-Stokes equations with Boussinesq approximation in both liquid and solid phases. A Carman-Kozeny-type penalty term is added to the momentum equations to bring the velocity to zero into the solid phase. An enthalpy model is used in the energy equation to take into account the phase change. Discontinuous variables (latent heat, material properties) are regularized through an intermediate (mushy) region. Space discretization is based on Galerkin triangular/tetrahedral finite elements. A second order Gear implicit scheme is used for the time integration of the coupled system of equations. The resulting discrete equations are solved using a Newton algorithm. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretization are possible. The mesh is first split in subdomains using Scotch or Met is libraries, which are interfaced with FreeFem++. Then, a Schwarz domain decomposition method is used through the FreeFem++library ffddm. The final linear system is solved in parallel using a GMRES Krylov method, with a Restricted Additive Schwarz (RAS) preconditioner. Mesh adaptivity using metrics control makes possible the optimization of the distribution of mesh elements. For 3D case, the mmg open source library is used to adapt the mesh. (C) 2020 Elsevier B.V. All rights reserved.

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach. (C) 2016 Elsevier Inc. All rights reserved.

In this report, the mean power and thrust are compared between CFD computations and experimental data. The parameters are the flow speed, the blade pitch angle and the rotation speed. Then a surrogate model with a Latin Hypecube Sampling is used to find the optimal pitch angle and rotation speed. The experiment used is the one made by Bahaj et al. 1 and was the first to be performed on tidal turbines. It contains many results for different pitch and is very useful to compare against. Though the experiment has a high blockage correction (up to ' 18%), it is well documented and provide much insight. Many people used this experiment to validate codes, Blade Element Momentum Theory (BEMT) for example in 2.O Other experiments exist today but will not be used in this paper (Ifremer 3, Liverpool 4, 5, Manchester6). BEMT is a good approach to assess the performance of one turbine, but it fails to perform for multiple turbines. To avoid this problem, other approach has been developed such as the Vortex Lattice Method (VLM) in 7. Their main focus is the wake of the turbine to study the interaction between two or more turbines 8. Their results are good until stall which is expected since their method force the flow to be attached until the trailing edge. Later 9 included turbulence. Attempts to use Computational Fluid Dynamics (CFD) on wind or tidal has been performed in the past. To avoid too much computational efforts, many authors modeled the behavior of the turbine instead of resolve the full geometry. For instance, 10 has used Large Eddy Simulation (LES) with the turbine replaced by an approximated model of a concentrated drag force to study the wake development. Also using an approximated model for the turbine, 11 performed a LES computation using an actuator disk. Fully resolved blade geometry CFD computations are computationaly expensive, but can give many more insight about the flow behavior and force distribution along the blade. 12 compared k−ω SST, Launder-ReeceRodi turbulence model (LLR) and LES on the 20° pitch angle case of 1 as an unsteady simulation, including the mast and a simplified geometry of the cavitation tunnel. The results are very interesting but lack other pitch angles to further validate the models which is what we are adressing in this paper.

We present a new algorithm based on a quasi-monolithic approach to solve strongly coupled fluid-structure interaction problems. This approach is an implicit coupling adapted to a partitioned solver while conserving the property of convergence and stability of the monolithic approach. The coupling is done between a finite element program ARA developed by K-Epsilon and the Reynolds-averaged Navier-Stokes code, ISIS-CFD, part of the commercial software FINE/Marine. The fluid mesh is deformed using a fast, robust and parallelized method which propagates the deformation state. The mesh deformation is taken into account through the ALE method. Validation of the coupling was performed against the experimental results of a flapping membrane. Application of the coupling is made to compare the unsteady flying stability of two downwind sails using an automatic trimming algorithm. Future application to a respiring tube is discussed.

Gennakers are lightweight and flexible sails, used for downwind sailing configurations. Qualities sought for this kind of sail are propulsive force and dynamic stability. To simulate accurately the flow surrounding a sail, several problems need to be solved. Firstly, the structural code has to take into account cloth behavior, orientation and reinforcements. Moreover, wrinkles need to be taken into account through modeling or fine enough discretization. Secondly, the fluid solver needs to reproduce the atmospheric boundary layer as an input boundary condition, and be able to simulate separation. Thirdly, the fluid-structure interaction (FSI) is strongly coupled due to the lightness and the flexibility of the structure. The added mass is three orders of magnitude greater than the mass of the sail, and large structural displacement occur, which makes the coupling between the two solvers difficult to achieve. Finally, the problem is unsteady, and dynamic trimming is important to the simulation of gennakers (Graf and Renzsch, 2006). As the FSI procedure is detailed in Durand (2012), the present work is rather focused on its application to downwind sail stability.The main objective of this paper is to use numerical simulations to model gennakers, in order to predict both propulsive force and sail dynamic stability. Recent developments from Durand (2012) are used to solve these problems mentioned earlier, using a finite element structural analysis program dedicated to sails and rig simulations coupled with an unsteady Reynolds averaged Navier-Stokes equations (URANSE) solver. The FSI coupling is done through a partitioned approach with quasi-monolithic properties. An arbitrary Lagrangian Eulerian (ALE) formulation is used, hence the fluid mesh follows the structural deformation while keeping the same topology. The fluid mesh deformation is carried out with a fast, robust and parallelized method based on the propagation of the deformation state of the sail boundary fluid faces (Durand et al., 2010).Tests were realized on a complete production chain: a sail designer from Incidences-Sails has designed two different shapes of an IMOCA60 gennaker with the SailPack software. An automatic procedure was developed to transfer data from Sailpack to a structure input file taking into account the orientation of sailcloth and reinforcements. The same automatic procedure is used for both gennakers, in order to compare dynamic stability and propulsion forces. A new method is then developed to quantify the practical stability of a downwind sail. (C) 2014 Elsevier Ltd. All rights reserved.