For safety studies of nuclear reactor cores, a finite-volume thermal-hydraulic code with a porous medium approach named THYC-coeur has been developed by EDF to simulate steady-state flows of a two-phase mixture within the core of a nuclear reactor. The steady-state solution is obtained through a fictitious transient. Despite a relatively low individual computational time, many state points are considered in the safety studies, which can represent significant CPU time. To speed up the computation, one possible objective is to reduce the number of iterations (i.e. the number of time steps) required to reach convergence. In the present work, the idea is to train a neural network (NN) to predict steady-state solutions and use this prediction to initialize the transient computation. This method allows combinating the advantages of NN prediction, in terms of rapidity, with that of the THYC-coeur model, in terms of the physical validation of the solution. To evaluate the potential of this method, a simplified one-dimensional code was designed. This code simulates a two-phase water vapor flow in a heated channel. A NN was trained to predict the solution fields from imposed boundary conditions. In this paper, we present the methodology, the database selection process, the structure of the NN, and the optimization of the network’s hyperparameters. This paper examines two different physical models: the 3-equation model and the 4-equation model. Initially, a simple loss function is used to predict the fields of these two models. The observed accelerations are approximately 70% for the 3-equation model and 48% for the 4-equation model. A detailed study of the correlation between prediction accuracy and the resulting acceleration has shown that the low-frequency components in the entropy field prediction error significantly slow down the search for the steady state. The highlight of this work is that by introducing spatial frequencies in the error for the optimization of the NN, the number of iterations is reduced between 61% (4-equation model) and 83% (3-equation model) from baseline simulations.

Three Finite Volume schemes are proposed in this note to satisfy the maximum principle for the mass fraction y, solution of an unsteady balance equation, including a relative velocity between phases and a source term. The continuous maximum principle is examined first. Then, linear implicit discrete schemes are detailed in a multi-dimensional and unstructured framework.

Finding a robust and efficient solver for (non-)symmetric systems that arise in incompressible Computational Fluid Dynamics (CFD) is of great interest to both academia and industry. We consider the Compatible Discrete Operator (CDO) discretization that has recently been devised for CFD simulations in the context of incompressible Stokes and Navier–Stokes flows. The discrete problems resulting from CDO schemes yield large saddle-point systems that require relevant numerical methods suitable to deal with large indefinite and poorly conditioned linear systems. In this paper, we focus on two segregated methods: the augmented Lagrangian Uzawa method and the generalized Golub–Kahan bidiagonalization, as well as a monolithic method based on an algebraic transformation by change of variables. We also employ algebraic multigrid (AMG) preconditioned Krylov solvers such as the Flexible Conjugate Gradient (FCG) method, and the Flexible Generalized Minimal Residual (FGMRES) method, to solve the linear systems. Using the CFD software code_saturne, we compare the numerical performance with respect to the choice of linear solvers and numerical strategies for the saddle-point problem. In the numerical experiments, the AMG preconditioned Krylov methods show robustness in test cases of Stokes and Navier–Stokes problems.

We address the numerical solution of linear systems arising from the hybrid discretizations of second-order elliptic partial differential equations. Such discretizations hinge on a hybrid set of degrees of freedom (DoFs), respectively, defined in cells and faces, which naturally gives rise to a global hybrid system of linear equations. Assuming that the cell unknowns are only locally coupled, they can be efficiently eliminated from the system, leaving only face unknowns in the resulting Schur complement, which is also called the statically condensed matrix. We propose in this work an algebraic multigrid (AMG) preconditioner specifically targeting condensed systems corresponding to lowest-order discretizations (piecewise constant). Like traditional AMG methods, we retrieve geometric information on the coupling of the DoFs from algebraic data. However, as the condensed matrix only gives information on the faces, we use the uncondensed version to reconstruct the connectivity graph between elements and faces. An aggregation-based coarsening strategy mimicking a geometric coarsening or semicoarsening can then be set up to build coarse levels. Numerical experiments are performed on diffusion problems discretized by the hybrid high-order method at the lowest order. Our approach uses a K-cycle to precondition an outer flexible Krylov method. The results demonstrate similar performances, in most cases, compared to a standard AMG method and a notable improvement on anisotropic problems with Cartesian meshes.

We consider a second-order elliptic PDE discretized by the hybrid high-order method, for which globally coupled unknowns are located at faces. To efficiently solve the resulting linear system, we propose a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level. The core of the algorithm lies in the design of the prolongation operator that passes information from coarse to fine faces through the reconstruction of an intermediary polynomial of higher degree on the cells. High orders are natively handled by the use of the same polynomial degree at every grid level. The proposed algorithm requires a hierarchy of nested meshes, such that the faces (and not only the elements) are successively coarsened. Numerical tests on homogeneous and heterogeneous diffusion problems show fast convergence, scalability in the mesh size and polynomial order, and robustness with respect to heterogeneity of the diffusion coefficient.

The use of modern discretization technologies such as hybrid high-order (HHO) methods, coupled with appropriate linear solvers, allow for the robust and fast solution of partial differential equations (PDEs). Although efficient linear solvers have recently been made available for simpler cases, complex geometries remain a challenge for large scale problems. To address this problem, we propose in this work a geometric multigrid algorithm for unstructured non-nested meshes. The non-nestedness is handled in the prolongation operator through the use of the L2-orthogonal projection from the coarse elements onto the fine ones. However, as the exact evaluation of this projection can be computationally expensive, we develop a cheaper approximate implementation that globally preserves the approximation properties of the L2-orthogonal projection. Additionally, as the multigrid method requires not only the coarsening of the elements, but also that of the faces, we leverage the geometric flexibility of polytopal elements to define an abstract non-nested coarsening strategy based on element agglomeration and face collapsing. Finally, the multigrid method is tested on homogeneous and heterogeneous diffusion problems in two and three space dimensions. The solver exhibits near-perfect asymptotic optimality for moderate degrees of approximation.

Abstract Goal-based error estimation due to spatial discretization and adaptive mesh refinement (AMR) has previously been investigated for the one dimensional, diamond difference, discrete ordinate (1-D DD-SN) method for discretizing the Neutron Transport Equation (NTE). This paper investigates the challenges of extending goal-based error estimation to multi-dimensions with supporting evidence provided on 2-D fixed (extraneous) source and Keff eigenvalue (criticality) verification test cases. It was found that extending Hennart’s weighted residual view of the lowest order 1-D DD equations to multi-dimensions gave what has previously been called the box method. This paper shows how the box method can be extended to higher orders. The paper also shows an equivalence between the higher order box methods and the higher order DD methods derived by Hébert et al. Though, less information is retained in the final solution in the latter case. These extensions allow for the definition of dual weighted residual (DWR) error estimators in multi-dimensions for the DD and box methods. However, they are not applied to drive AMR in the multi-dimensional case due to the various challenges explained in this paper.

The quantity of interest (QoI) associated with a solution of a partial differential equation (PDE) is not, in general, the solution itself, but a functional of the solution. Dual weighted residual (DWR) error estimators are one way of providing an estimate of the error in the QoI resulting from the discretisation of the PDE. This paper aims to provide an estimate of the error in the QoI due to the spatial discretisation, where the discretisation scheme being used is the diamond difference (DD) method in space and discrete ordinate (SN) method in angle. The QoI are reaction rates in detectors and the value of the eigenvalue (Keff) for 1-D fixed source and eigenvalue (Keff criticality) neutron transport problems respectively. Local values of the DWR over individual cells are used as error indicators for goal-based mesh refinement, which aims to give an optimal mesh for a given QoI.

This paper illustrates how GPU computing can be used to accelerate computational fluid dynamics (CFD) simulations. For sparse linear systems arising from finite volume discretization, we evaluate and optimize the performance of Conjugate Gradient (CG) routines designed for manycore accelerators and compare against an industrial CPU-based implementation. We also investigate how the recent advances in preconditioning, such as iterative Incomplete Cholesky (IC, as symmetric case of ILU) preconditioning, match the requirements for solving real world problems.

The paper describes practical approach for minimising the parallelisation overhead for a solution to the boundary value problems by geometric multigrid methods. It is shown that proposed multiple coarse grid correction strategy makes it possible not only to create the task of the smoother least demanding, but also to avoid load imbalance and limit the communication overhead. Estimation of maximum speedup and efficiency of parallel robust multigrid technique and parallel V-cycle are given.

The paper focuses on an aggregation-based algebraic multigrid method applied to convection/diffusion problems. We show that for an unstructured finite volume approach on arbitrary shaped cells, the separation of the two operators associated with suitable smoothers improves the aggregation-based multigrid. While the convection is treated by a piecewise constant prolongation, the off-diagonals entries of the diffusion P 0 Galerkin operator are scaled by a parameter representative of the mesh spacing ratio between the fine and coarse mesh in the vicinity of the coarse mesh cell boundaries. Some numerical examples are shown to assess the rate of convergence and the robustness of the proposed approach.

This paper describes a short and simple way of improving the performance of vector operations (e.g. X = aY +bZ +..) applied to large vectors. In a previous paper [1] we described how to take advantage of high performance vector copy operation provided by the ATLAS library [2] in the context of C++ Expression Template (ET) mechanism. Here we present a multi-threaded implementation of this approach. The proposed ET implementation that involves a parallel blocking technique, leads to significant performance increase compared to existing implementations (up to x2.7) on dual socket x86_64 targets.

This paper presents the Benchmark Template Library in C++, in short BTL++, which is a flexible framework to assess the run time of user defined computational kernels. When the same kernel is implemented in several different ways, the collected performance data can be used to automatically construct an interface library that dispatches a function call to the fastest variant available.The benchmark examples in this article are mostly functions from the dense linear algebra BLAS API. However, BTL++can be applied to any kernel that can be called by a function from a C++ main program. Within the same framework, we are able to compare different implementations of the operations to be benchmarked, from libraries such as ATLAS, over procedural solutions in Fortran and C to more recent C++ libraries with a higher level of abstraction. Results of single threaded and multi-threaded computations are included.

This paper describes a short and simple way of improving the performance of vector operations (e.g. X = aY + bZ + ..) applied to large vectors. The principle is to take advantage of high performance vector copy operation provided by the ATLAS library [1] used as a kernel for a C++ Expression Template (ET) mechanism. The proposed ET implementation that involves a simple blocking technique, leads to significant performance increase compared to existing implementations (up to 50%) and extends the ATLAS scope.

In an ideal situation, all performance optimization of computationally intensive software would take place automatically, allowing the researchers to concentrate on the development of more efficient algorithms (in terms of computational complexity) rather than having to worry about performance. However, for the time being, optimizing compilers are unable to synthesize long chains of complicated code transformations to optimize program execution. As a consequence, the need to identify and to remove the performance bottlenecks of computationally intensive codes remains. As an example of a class of computationally intensive problems, this minisymposium concentrated on the numerical solution of partial differential equations (PDEs). As with every computer program, the run times of PDE solvers depend both on the algorithms and on the data structures used in the implementations. In the context of numerical PDEs, algorithms with optimal asymptotic complexity are known for certain types of problems; e.g., multigrid methods for elliptic problems. In those cases where the optimal algorithms are applicable, only the data structures and the implementation details offer scope for improvement.

Amongst other properties, PDE solvers for large scale problems should be flexible, as they are time consuming to write, and obviously run time efficient. We report on the experiences with a regularity centred approach for grid based PDE software that aims to combine geometric flexibility with run time efficiency. An unstructured coarse grid that describes the problem geometry is repeatedly subdivided in a regular fashion to yield a hierarchy of grids on which the approximation is sought. By construction, the grid hierarchy is well suited for multilevel methods. The gain in run time performance that results from the exploitation of the patch wise regularity of the refined grids over standard implementations will be illustrated.