High-order Solver Research Articles

Discovering artificial viscosity models for discontinuous Galerkin approximation of conservation laws using physics-informed machine learning

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models. We refer to the proposed approach as an “hybrid” approach which stands at the edge between supervised and unsupervised learning. More precisely, the proposed “hybrid” paradigm is not supervised in the classical sense as it does not utilize labeled data in the traditional way but relies on the intrinsic properties of the reference solution. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

A reinforcement learning strategy to automate and accelerate h/p-multigrid solvers

We explore a reinforcement learning strategy to automate and accelerate h/p-multigrid methods in high-order solvers. Multigrid methods are very efficient but require fine-tuning of numerical parameters, such as the number of smoothing sweeps per level and the correction fraction (i.e., proportion of the corrected solution that is transferred from a coarser grid to a finer grid). The objective of this paper is to use a proximal policy optimization algorithm to automatically tune the multigrid parameters and, by doing so, improve stability and efficiency of the h/p-multigrid strategy.Our findings reveal that the proposed reinforcement learning h/p-multigrid approach significantly accelerates and improves the robustness of steady-state simulations for one-dimensional advection-diffusion and nonlinear Burgers' equations, when discretized using high-order h/p methods, on uniform and nonuniform grids.

Power system fault diagnosis with quantum computing and efficient gate decomposition

Power system fault diagnosis is crucial for identifying the location and causes of faults and providing decision-making support for power dispatchers. However, most classical methods suffer from significant time-consuming, memory overhead, and computational complexity issues as the scale of the power system concerned increases. With rapid development of quantum computing technology, the combinatorial optimization method based on quantum computing has shown certain advantages in computational time over existing methods. Given this background, this paper proposes a quantum computing based power system fault diagnosis method with the quantum approximate optimization algorithm. The proposed method reformulates the fault diagnosis problem as a Hamiltonian by using Ising model, which completely preserves the coupling relationship between faulty components and various operations of protective relays and circuit breakers. Additionally, to enhance problem-solving efficiency under current equipment limitations, the symmetric equivalent decomposition method of multi-z-rotation gate is utilized. Furthermore, the small probability characteristics of power system events is utilized to reduce the number of qubits. Simulation results based on the test system show that the proposed methods can achieve the same optimal results with a faster speed compared with the classical higher-order solver provided by D-Wave.

Development of a high-order solver for large eddy simulation of turbulent heat transfer at supercritical pressure based on Nek5000

Turbulent heat transfer at supercritical pressure is a complex flow phenomenon due to drastic variations in fluid properties near the pseudocritical point. Numerical simulation is an important method to reveal the underlying physics. Currently, low-order numerical methods together with Reynolds-averaged Navier–Stokes equations are the mainstream in which empirical parameters are required, preventing high-fidelity simulations. Through inventing iterative properties updating and density-weighted explicit filtering, this work develops a high-order spectral element solver based on the open-source code Nek5000. By simulating a classical problem of supercritical CO2 flowing in a heated pipe and comparing it with benchmark data, the capability of the solver in direct numerical simulation is validated. Further results suggest lowering the mesh resolution leads to inaccurate predictions of bulk parameters and turbulent statistics. Therefore, filtering-based large eddy simulation (LES) is explored with different filter weights under a coarse mesh. Results show such a method can significantly improve most of the bulk parameters, including the bulk Nusselt number. The optimal filter weight can be determined from a simple optimization problem minimizing the deviation of overall energy conservation. Being high-order and capable of LES without empirical parameter, the current solver is a powerful tool for high-fidelity simulation of turbulent heat transfer at supercritical pressure.

A polynomial-correction Navier-Stokes characteristic boundary condition

In an effort to assess the efficacy of different non-reflecting boundary conditions (NRBCs) in high-order solvers, several NRBCs have been implemented in the context of the discontinuous Galerkin (DG) discretization. The methods considered are based on buffer zone techniques and boundary-imposed conditions. The former mainly utilizes the sponge layer (SL), the perfectly matched layer (PML) and the supersonic layer (SSL), while the latter uses a Riemann extrapolation (RE) technique and a novel polynomial-correction Navier-Stokes characteristic boundary condition (PC-NSCBC). The developed PC-NSCBC method reconstructs an explicit boundary state, instead of modifying the equations on the boundary or specifying ad hoc compatibility conditions on corners and edges. This property pairs well with a Riemann solver, as is typical in a DG method. Moreover, the boundary state can be assembled either from a one-dimensional perspective (NSCBC1D) or using partial knowledge of multi-dimensional effects (NSCBC3D).Four different benchmark problems are considered: a 1D linear acoustic pulse in a quiescent flow, a 2D non-linear pressure pulse in a quiescent flow, an isentropic vortex and a shear-flow vortex shedding instability. Results imply that mainly the PML performs better than most other approaches. This is especially true for acoustically-dominated flows. Alternatively, for flow conditions predominantly less acoustic in nature and strongly non-linear, instabilities in the PML may arise. The SL and SSL both suggest that a larger layer is required to deliver comparable results to the PML. The proposed (PC-)NSCBC methods demonstrate reasonable success with no instabilities reported, which is impressive given the comparatively less computational effort required. Partial inclusion of the transverse terms in the NSCBC3D noticeably enhances the output of the NSCBC1D, albeit further investigation is required to determine an optimal, universal expression for its transverse tuning coefficient.

Accelerating high order discontinuous Galerkin solvers using neural networks: Wall bounded flows

High order solvers are accurate but computationally expensive as they require small time steps to advance the solution in time. In this work we include a corrective forcing to a low order solution to improve the accuracy while advancing in time with larger time steps, and achieve fast computations. The work uses a discontinuous Galerkin framework, where the polynomial order, inside each mesh element, can be varied to provide low or high accuracy. The corrective forcing is included for each high order Gauss nodal point in the mesh. This work is a continuation of [1, 2], where we extend the methodology to wall bounded flows. Namely, we adapt the methodology to a turbulent channel at Reτ = 182. In this case, we use three neural networks to correct different regions of the flow, which are distinguished by their y+ distance to the wall. The methodology is able to correct the low resolution simulation to attain flow statistics that are comparable to high order simulations. We include comparisons for the mean, Reynolds stresses and shear stress on the wall. We achieve good predictions using the corrected low order solution, in mean velocity and its corresponded fluctuations, as well as the shear stress on the wall.

Neko: A modern, portable, and scalable framework for high-fidelity computational fluid dynamics

Computational fluid dynamics (CFD), in particular applied to turbulent flows, is a research area with great engineering and fundamental physical interest. However, already at moderately high Reynolds numbers the computational cost becomes prohibitive as the range of active spatial and temporal scales is quickly widening. Specifically scale-resolving simulations, including large-eddy simulation (LES) and direct numerical simulations (DNS), thus need to rely on modern efficient numerical methods and corresponding software implementations. Recent trends and advancements, including more diverse and heterogeneous hardware in High-Performance Computing (HPC), are challenging software developers in their pursuit for good performance and numerical stability. The well-known maxim “software outlives hardware” may no longer necessarily hold true, and developers are today forced to re-factor their codebases to leverage these powerful new systems. In this paper, we present Neko, a new portable framework for high-order spectral element discretization, targeting turbulent flows in moderately complex geometries. Neko is fully available as open software. Unlike prior works, Neko adopts a modern object-oriented approach in Fortran 2008, allowing multi-tier abstractions of the solver stack and facilitating hardware backends ranging from general-purpose processors (CPUs) down to exotic vector processors and FPGAs. We show that Neko’s performance and accuracy are comparable to NekRS, and thus on-par with Nek5000’s successor on modern CPU machines. Furthermore, we develop a performance model, which we use to discuss challenges and opportunities for high-order solvers on emerging hardware.

Efficient implementation of complex equations of state in a high-order framework

In the last decades, the interest in non-ideal compressible fluid dynamics (NICFD) flows and high-order accurate numerical methods, such as discontinuous Galerkin (dG), has quickly grown. In fact, advanced simulation capabilities are of paramount importance to develop new sustainable technologies with higher efficiency and low environmental impact, and to decrease the use of expensive experimental facilities. Nowadays however, the coupling of accurate Computational Fluid Dynamics (CFD) solvers with sophisticated thermodynamic models has been investigated mainly in the finite volume (FV) framework. Moreover, the solution of complex equations of state (EoS) has a too high computational cost for real-life use in industry, which is often overcome with the look-up table (LuT) interpolation approach. LuTs seem to be more suitable for FV solvers rather than high-order or finite element ones, since the interpolation error introduced may deteriorate the higher accuracy provided by these numerical methods. The novelty of this work lies in the assessment of an efficient implementation for real gas simulations in a high-order solver, by resorting to structured LuTs and automatic differentiation (AD). The proposed implementation is assessed with the computation of the inviscid and turbulent flow around a NACA0012. This approach guarantees an overhead of the computational cost around 17% with respect to an ideal gas solver with manually derived jacobian matrix.

A hybrid finite volume - spectral element method for aeroacoustic problems

We propose a hybrid Finite Volume (FV) - Spectral Element Method (SEM) for modeling aeroacoustic phenomena based on the Lighthill's acoustic analogy. First the fluid solution is computed employing a FV method. Then, the sound source term is projected onto the acoustic grid and the inhomogeneous Lighthill's wave equation is solved employing the SEM. The novel projection method computes offline the intersections between the acoustic and the fluid grids in order to preserve the accuracy. The proposed intersection algorithm is shown to be robust, scalable and able to efficiently compute the geometric intersection of arbitrary polyhedral elements. We then analyze the properties of the projection error, showing that if the fluid grid is fine enough we are able to exploit the accuracy of the acoustic solver and we numerically assess the obtained theoretical estimates. Finally, we address two relevant aeroacoustic benchmarks, namely the corotating vortex pair and the noise induced by a laminar flow around a squared cylinder, to demonstrate in practice the effectiveness of the projection method when dealing with high order solvers. The flow computations are performed with OpenFOAM [50], an open-source finite volume library, while the inhomogeneous Lighthill's wave equation is solved with SPEED [34], an open-source spectral element library.

Evaluating performance portability of five shared-memory programming models using a high-order unstructured CFD solver

This paper presents implementing and optimizing a high-order unstructured computational fluid dynamics (CFD) solver using five shared-memory programming models: CUDA, OpenACC, OpenMP, Kokkos, and OP2. The study aims to evaluate the performance of these models on different hardware architectures, including NVIDIA GPUs, x86-based Intel/AMD, and Arm-based systems. The goal is to determine whether these models can provide developers with performance-portable solvers running efficiently on various architectures. The paper forms a more holistic view of a high-order solver across multiple platforms by visualizing performance portability (PP) and measuring productivity. It gives guidelines for translating existing codebases and their data structures to these models.

A reinforcement learning strategy for p-adaptation in high order solvers

Reinforcement learning (RL) has emerged as a promising approach to automating decision processes. This paper explores the application of RL techniques to optimise the polynomial order in the computational mesh when using high-order solvers. Mesh adaptation plays a crucial role in improving the efficiency of numerical simulations by increasing accuracy while reducing the cost. Here, actor-critic RL models based on Proximal Policy Optimization offer an approach for agents to learn optimal mesh modifications based on evolving conditions.The paper provides a strategy for p-adaptation in high-order solvers and includes insights into the main aspects of RL-based mesh adaptation, including the formulation of appropriate reward structures and the interaction between the RL agent and the simulation environment. The proposed strategy does not require a high-fidelity solution during the training process and the formulation is general for any computational mesh and partial differential equation (PDE), solved in a discontinuous Galerkin solver. We discuss the impact of RL-based mesh p-adaptation on computational efficiency and accuracy. We apply the RL p-adaptation strategy to a one-dimensional inviscid Burgers' equation, focusing our analysis on smooth solutions of the equation to showcase its effectiveness. The RL strategy reduces the computational cost and improves accuracy over uniform adaptation, while minimising human intervention.

Neural responses to peripheral infection in Alzheimer’s disease model mice

AbstractBackgroundClinical and animal studies have suggested that peripheral inflammation affects the central nervous system (CNS) and may be a major factor in the pathophysiology of neurodegenerative diseases like AD. Furthermore, a variety of signaling factors originating in the periphery, such as inflammatory mediators, have been associated with the regulation of brain barrier functions. Here, we hypothesized that the presence of peripheral inflammation dysregulates brain barriers’ integrity and induces neuroinflammation with consequences on amyloid pathology.MethodIn this study, we employed an AD mouse model nasally infected with Staphylococcus aureus to assess the impact of chronic or acute peripheral inflammation on brain transcriptome and amyloid pathology. We utilized both single cell and Spatial Transcriptomics to better determine transcriptional response to peripheral inflammation.ResultThe chronic exposure increased the diffuse and compact amyloid plaques and blood cytokine levels. Following a short‐term exposure, single‐cell and spatial transcriptomics uncovered cell type‐ and spatial‐specific transcriptional changes indicating a dysregulation of the brain barriers, including the blood‐brain and the blood‐cerebrospinal fluid barriers. Brain macrophages exhibited increased Apoe expression and macrophage‐specific genes were upregulated at ventricular areas of infected mice. In addition, we report an increase of disease associated microglia genes around the Ab plaques, together with disbalances in neuronal expression in response to peripheral inflammation.ConclusionWe observed that low‐grade peripheral infection triggers transcriptional responses linked to brain barriers and brain infiltrated/resident macrophages indicate a fundamental importance of them in the mechanisms linking peripheral inflammation and AD pathogenesis. Finally, we showed that bacterial infection caused molecular changes restricted to specific cell types and/or brain spatial microenvironments, which highlights the relevance of studying brain molecular responses with high resolution in order to have new insights on AD etiology.

Planetary scientific target detection via deep learning: A case study for finding shatter cones in Mars rover images

AbstractPast, present, and forthcoming planetary rover missions to Mars and other planetary bodies are equipped with a large number of scientific cameras. The very large number of images resulting from this, combined with tight time constraints for navigation, measurements, and analyses, pose a major challenge for the mission teams in terms of scientific target evaluation. Shatter cones are the only macroscopic evidence for impact‐induced shock metamorphism and therefore impact craters on Earth. The typical features of shatter cones, such as striations and horsetail structures, are particularly suitable for machine learning methods. The necessary training images do not exist for such a case; therefore, we pursued the approach of producing them artificially. Using PRo3D, a viewer developed for the interactive exploration and geologic analysis of high‐resolution planetary surface reconstructions, we virtually placed shatter cones in 3‐D background scenes processed from true Mars rover imagery. We use PRo3D‐rendered images of such scenes as training data for machine learning architectures. Terrestrial analog studies in Ethiopia supported our lab work and were used to test the resulting neural network of this feasibility study. The result showed that our approach with shatter cones in artificial Mars rover scenes is suitable to train neural networks for automatic detection of shatter cones. In addition, we have identified several aspects that can be used to improve the training of the neural network and increase the recognition rate. For example, using background data with a higher resolution in order to have equal resolution of object (shatter cone) and Martian background and increase the number of objects that can be placed in the training data set. Also using better lighting reconstructions and a better radiometric adaption between object and Martian background would further improve the results.

A three-dimensional high-order flux reconstruction lattice boltzmann flux solver for incompressible laminar and turbulent flows

A three-dimensional (3D) high-order flux reconstruction lattice Boltzmann flux solver (FR-LBFS) is developed in this paper for accurate and efficient simulations of incompressible laminar and turbulent flows. The original lattice Boltzmann flux solver (LBFS) is a second-order finite volume method (FVM) which combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. But it is not convenient to construct high-order FVM, and the compactness is always a bottleneck. The present work separates LBFS from FVM and combines LBFS with high-order FR scheme. Thus, the FR-LBFS inherits the superiority of LBFS and FR and shares the following attractive features: (i) a fully-explicit arbitrary order compact method, (ii) weakly compressible (WC) model for unsteady incompressible flows and (iii) evaluating inviscid and viscous fluxes simultaneously and no particular technique requirement, such as local discontinuous Galerkin method (LDG), for viscous term discretization. To validate the current method, various 3D incompressible laminar isothermal and thermal numerical experiments are presented first. Good agreements are achieved between the current results and the previous experimental and numerical data whilst the present high-order solver adopt coarse meshes. Furthermore, the ability of the proposed method to perform accurate and stable computations of incompressible decaying homogeneous isotropic turbulent flows and turbulent channel flows with different mesh resolutions and accuracy orders are investigated. The results show that the present 3D FR-LBFS is a promising tool for under-resolved or implicit large eddy simulation (ILES) of incompressible turbulent flows.

The Knowns and Unknowns in Protein-Metabolite Interactions.

Increasing attention has been focused on the study of protein-metabolite interactions (PMI), which play a key role in regulating protein functions and directing an orchestra of cellular processes. The investigation of PMIs is complicated by the fact that many such interactions are extremely short-lived, which requires very high resolution in order to detect them. As in the case of protein-protein interactions, protein-metabolite interactions are still not clearly defined. Existing assays for detecting protein-metabolite interactions have an additional limitation in the form of a limited capacity to identify interacting metabolites. Thus, although recent advances in mass spectrometry allow the routine identification and quantification of thousands of proteins and metabolites today, they still need to be improved to provide a complete inventory of biological molecules, as well as all interactions between them. Multiomic studies aimed at deciphering the implementation of genetic information often end with the analysis of changes in metabolic pathways, as they constitute one of the most informative phenotypic layers. In this approach, the quantity and quality of knowledge about PMIs become vital to establishing the full scope of crosstalk between the proteome and the metabolome in a biological object of interest. In this review, we analyze the current state of investigation into the detection and annotation of protein-metabolite interactions, describe the recent progress in developing associated research methods, and attempt to deconstruct the very term "interaction" to advance the field of interactomics further.

A matrix–free high–order solver for the numerical solution of cardiac electrophysiology

We propose a matrix–free solver for the numerical solution of the cardiac electrophysiology model consisting of the monodomain nonlinear reaction–diffusion equation coupled with a system of ordinary differential equations for the ionic species. Our numerical approximation is based on the high–order Spectral Element Method (SEM) to achieve accurate numerical discretization while employing a much smaller number of Degrees of Freedom than first–order Finite Elements. We combine vectorization with sum–factorization, thus allowing for a very efficient use of high–order polynomials in a high performance computing framework. We validate the effectiveness of our matrix–free solver in a variety of applications and perform different electrophysiological simulations ranging from a simple slab of cardiac tissue to a realistic four–chamber heart geometry. We compare SEM to SEM with Numerical Integration (SEM–NI), showing that they provide comparable results in terms of accuracy and efficiency. In both cases, increasing the local polynomial degree p leads to better numerical results and smaller computational times than reducing the mesh size h. We also implement a matrix–free Geometric Multigrid preconditioner that results in a comparable number of linear solver iterations with respect to a state–of–the–art matrix–based Algebraic Multigrid preconditioner. As a matter of fact, the matrix–free solver proposed here yields up to 45× speed–up with respect to a conventional matrix–based solver.

A combined volume penalization / selective frequency damping approach for immersed boundary methods applied to high-order schemes

There has been an increasing interest in developing efficient immersed boundary method (IBM) based on Cartesian grids, recently in the context of high-order methods. IBM based on volume penalization is a robust and easy to implement method to avoid body-fitted meshes and has been recently adapted to high order discretizations [1]. This work proposes an improvement over the classic penalty formulation for high-order solvers based on flux reconstruction. We include a selective frequency damping (SFD) approach [2] acting only inside solid body defined through the immersed boundary masking, to damp spurious oscillations. An encapsulated formulation for the SFD method is implemented, which can be used as a wrapper around an existing time-stepping code. The numerical properties have been studied through eigensolution analysis based on the advection equation. These studies not only show the advantages of using the SFD method as an alternative of the traditional volume penalization, but also show the favorable properties of combining both approaches. This new approach is then applied to the Navier-Stokes equation to simulate steady flow past an airfoil and unsteady flow past a circular cylinder. The advantages of the SFD method in providing improved accuracy are reported.

3D wave problems evaluation and forecasting through an innovative technique

The NovelApproach is a straightforward methodology that may be used to solve both nonlinearity and initial value issues. One of the most significant challenges in numerical methods is the solution of numerical methods. Several iterative approaches for solving complex equations have received a lot of attention. Using the decomposition methodology, scientists devised and investigated numerous techniques for iterative methods having higher-order resolution. The well-known approach to computing the Initial Value Problem and the Hyperbolic Limit Problem was offered in the supplied work (IBVP). The presented method differs significantly from the numerical method. The outcomes of the analysis are compared to the currently used exact or analytical approach. The proposed novel technique is basic and straightforward to use, improving the result's accuracy and convergence. As a result, the current process is known as the creative method. The creative method converges very quickly for three-dimensional thick plate results and is more precise than computerized methods. The solutions obtained from this approach are more convergent and quicker than any other way, which encourages people to solve issues using this method rather than others.

On the use of high order central difference schemes for differential equation based wall distance computations

A computationally efficient high-order solver is developed to compute the wall distances by solving the relevant partial differential equations, namely: Eikonal, Hamilton–Jacobi (HJ) and Poisson equations. In contrast to the upwind schemes widely used in the literature, we explore the suitability of high-order central difference schemes (explicit/compact) for the wall-distance computation. While solving the Hamilton–Jacobi equation, the high-order central difference schemes performed approximately 1.4–2.8 times faster than the upwind schemes with a marginal improvement in the solution accuracy. A new pseudo HJ formulation based on the localized artificial diffusivity (LAD) approach has been proposed. It is demonstrated to predict results with an accuracy comparable to that of the Eikonal equation and the simulations are ≈ 1.5 times faster than the baseline HJ solver using upwind schemes. A curvature correction is also incorporated in the HJ equation to correct for the near-wall errors due to concave/convex wall curvatures. We demonstrate the efficacy of the proposed methods on both the steady and unsteady test cases and exploit the unsteady wall-distance solver to estimate the instantaneous shape and burning surface area of a dendrite propellant grain in a solid propellant rocket motor.

Parallel high-order resolution of the Shallow-water equations on real large-scale meshes with complex bathymetries

The resolution of the Shallow-water equations is of practical interest in the study of inundations and often requires very large and dense meshes to accurately simulate river flows. Those large meshes are often decomposed into multiple sub-domains to allow for parallel processing. When such a decomposition process is used in the context of distributed parallel computing, each sub-domain requires an exchange of one or more layers of ghost cells at each time step of the simulation due to the spatial dependency of numerical methods. In the first part of this paper, we show how the domain decomposition and ghost-layer generation process can be performed in a parallel manner for large meshes, and show a new way of storing the resulting sub-domains with all their send/receive information within a single CGNS mesh file. The performance of the ghost-layer generation process is studied both in terms of time and memory on 2D and 3D meshes containing up to 70 million cells. In the second part of the paper, the program developed in the first part is used to generate the domain decomposition of large meshes of practical interest in the study of natural free surface flows. We use our in-house multi-CPU multi-GPU (MPI+CUDA) solver to show the impact of multiple layers of ghost cells on the execution times of the first-order HLLC, and second-order WAF and MUSCL methods. Finally, the parallel solver is used to perform the second-order resolution on real large-scale meshes of rivers near Montréal, using up to 32 GPUs on meshes of 13 million cells.