Computational Mesh Research Articles

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.

Investigation of mesh independence in numerical modeling of sharp-crested weirs at different heights

Mesh independence is a term used in numerical analysis and simulation to refer to a situation where the numerical solution of a problem is insensitive to changes in the size or density of the computational mesh used to discretize the problem domain. In other words, a numerical solution is said to be mesh independent if it remains stable and accurate even when the mesh is refined. This is important because the accuracy and stability of numerical simulations depend on the quality of the mesh used to discretize the problem domain. The shape, size, and number of meshes used while performing numerical modeling in accordance with flow physics in the solution of hydraulic problems significantly affect the result. In this study, a total of 12 experiments were carried out for 4 different discharge values (Q=5 L/s, 10 L/s,15 L/s and 20 L/s) for sharp-crested weirs with three different weir heights (P=20 cm, 30 cm and 40 cm). Numerical models of each experimental setup with four different mesh sizes (m=1.25 mm, 2.50 mm, 5.00 mm and 10.00 mm) were created with the ANSYS-Fluent program. Numerical analyses were performed using the grid resolution method to obtain a mesh size-independent result where the mesh size would not affect the results after the specified error rate.

Generation of 1D channel networks for overland flow simulations on 2D complex domains

Conventionally two-dimensional (2D) models are used to simulate 2D overland flow with a non-overlapping 2D computational mesh. However, 2D models are not computationally efficient when applied to large domains. Due to their computing efficiency, one-dimensional (1D) models can be used to simulate 2D overland flows by replacing 2D computational meshes with 1D computational channel networks with topography described by closely-spaced transverse cross sections that fully cover the whole domain. Channel networks must enforce the basic physical laws of gravity-driven flows ensuring water flow paths converge and connect from high to low elevations. This manuscript presents a novel algorithm to generate channel networks following two principles, a geometric principle and a hydrologic principle. The geometric principle requires that the generated channel network covers the whole study domain without overlaps or intersections, while the hydrologic principle requires that the generated channel network follows the local steepest slope. Correspondingly, the proposed generation algorithm includes two main processes: (1) the extraction of channel network using a terrain slope-calculation-based delineation algorithm; and, (2) the generation of cross sections with varying channel widths determined by considering topographic complexity, the hydrological correctness, and computational requirements. The proposed cross section generation algorithm is described using several 2D geometrically complex domains at the laboratory scale, and is also applied to a natural watershed. An overflow simulation on a 2D domain with obliques walls using a 1D model demonstrated the effectiveness of the proposed generation algorithm for 1D channel networks of 1D model.

Domain independent post-processing with graph U-nets: applications to electrical impedance tomographic imaging⋆ ⋆SH and BH were supported by the National Institute Of Biomedical Imaging And Bioengineering of the National Institutes of Health under Award Number R21EB028064. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. AH

Objective. To extend the highly successful U-Net Convolutional Neural Network architecture, which is limited to rectangular pixel/voxel domains, to a graph-based equivalent that works flexibly on irregular meshes; and demonstrate the effectiveness on electrical impedance tomography (EIT). Approach. By interpreting the irregular mesh as a graph, we develop a graph U-Net with new cluster pooling and unpooling layers that mimic the classic neighborhood based max-pooling important for imaging applications. Main results. The proposed graph U-Net is shown to be flexible and effective for improving early iterate total variation (TV) reconstructions from EIT measurements, using as little as the first iteration. The performance is evaluated for simulated data, and on experimental data from three measurement devices with different measurement geometries and instrumentations. We successfully show that such networks can be trained with a simple two-dimensional simulated training set, and generalize to very different domains, including measurements from a three-dimensional device and subsequent 3D reconstructions. Significance. As many inverse problems are solved on irregular (e.g. finite element) meshes, the proposed graph U-Net and pooling layers provide the added flexibility to process directly on the computational mesh. Post-processing an early iterate reconstruction greatly reduces the computational cost which can become prohibitive in higher dimensions with dense meshes. As the graph structure is independent of ‘dimension’, the flexibility to extend networks trained on 2D domains to 3D domains offers a possibility to further reduce computational cost in training.

Generalisation of the Penalised Wall Function Method for the Simulation of Turbulent flows With Unfavourable Pressure Gradients

The penalized wall function method for simulation of compressible near-wall turbulent flow regions in the numerical modeling of viscous compressible flows is developed. The method is formulated as a differential condition to match the outer and the wall function solutions and is based on a generalized characteristic-based volume penalization method to transfer shear stress from the outer region of the boundary layer to the wall. The method is modified to extend its applicability to turbulent flows with adverse pressure gradient, when separation and reattachment zones are formed, as well as to use computational meshes with coarser near-wall resolution. These advantages are demonstrated for two test problems, namely, the flow over a flat plate with zero and adverse pressure gradients.

An unfitted HDG method for a distributed optimal control problem

We analyze a high order hybridizable discontinuous Galerkin (HDG) method for an optimal control problem where the computational mesh does not necessarily fit the domain. The method is based on transferring the boundary data to the computational boundary by integrating the approximation of the gradient. We prove optimal order of convergence in the L2-norm for all the variables of the state and adjoint problems, and the control variable as well. More precisely, order hk+1 if the local discrete spaces are constructed using polynomials of degree at most k on a triangulation of meshsize h. We present numerical experiments illustrating the performance of method.

Rapid spatio-temporal flood modelling via hydraulics-based graph neural networks

Abstract. Numerical modelling is a reliable tool for flood simulations, but accurate solutions are computationally expensive. In recent years, researchers have explored data-driven methodologies based on neural networks to overcome this limitation. However, most models are only used for a specific case study and disregard the dynamic evolution of the flood wave. This limits their generalizability to topographies that the model was not trained on and in time-dependent applications. In this paper, we introduce shallow water equation–graph neural network (SWE–GNN), a hydraulics-inspired surrogate model based on GNNs that can be used for rapid spatio-temporal flood modelling. The model exploits the analogy between finite-volume methods used to solve SWEs and GNNs. For a computational mesh, we create a graph by considering finite-volume cells as nodes and adjacent cells as being connected by edges. The inputs are determined by the topographical properties of the domain and the initial hydraulic conditions. The GNN then determines how fluxes are exchanged between cells via a learned local function. We overcome the time-step constraints by stacking multiple GNN layers, which expand the considered space instead of increasing the time resolution. We also propose a multi-step-ahead loss function along with a curriculum learning strategy to improve the stability and performance. We validate this approach using a dataset of two-dimensional dike breach flood simulations in randomly generated digital elevation models generated with a high-fidelity numerical solver. The SWE–GNN model predicts the spatio-temporal evolution of the flood for unseen topographies with mean average errors in time of 0.04 m for water depths and 0.004 m2 s−1 for unit discharges. Moreover, it generalizes well to unseen breach locations, bigger domains, and longer periods of time compared to those of the training set, outperforming other deep-learning models. On top of this, SWE–GNN has a computational speed-up of up to 2 orders of magnitude faster than the numerical solver. Our framework opens the doors to a new approach to replace numerical solvers in time-sensitive applications with spatially dependent uncertainties.

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.

Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.

MDSSN: An end-to-end deep network on triangle mesh parameterization

Mesh data plays a crucial role in a wide range of applications worldwide within the field of 3D computer vision. In contrast to traditional Euclidean space arrangements, meshes encompass spatial information, including edges, faces, angles, and graph structures that embed face information within the mesh data. Nevertheless, conventional deep learning frameworks such as convolutional neural networks (CNNs) have struggled to achieve significant advancements in handling meshes. This paper proposes a simple mesh computation framework called Mesh Decomposition Second-order Sobolev Network (MDSSN) to model triangle meshes and represent their shape. The construction of face-based Riemannian and edge-based Riemannian graphs is motivated by the remarkable representation capabilities of edges and faces in mesh data, achieved through a graph composition mechanism. Furthermore, we design end-to-end operators, including the Sobolev-filter block, pooling, and unpooling blocks, which draw inspiration from traditional deep learning frameworks for modeling the mesh structure. Importantly, the design of the mesh-filter block incorporates the second-order Sobolev metric. To address common challenges in mesh classification and segmentation tasks, we have designed dedicated classification and segmentation modules. We utilize a joint loss function that integrates losses from adjacent faces and categories. We evaluate the performance of the MDSSN in mesh classification and segmentation tasks, focusing on its ability to represent 3D shapes. Experimental results demonstrate that MDSSN achieves superior performance compared to other state-of-the-art approaches.

On Dihedral Angle Sums of Prisms and Hexahedra

Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.

Evolution of the Surface Computational Mesh in the Ice Accretion Process

Evolution of the Surface Computational Mesh in the Ice Accretion Process

Numerical Approach Based on Solving 3D Navier–Stokes Equations for Simulation of the Marine Propeller Flow Problems

The report presents the approach implemented in the Russian LOGOS software package for the numerical simulation of the marine propeller flow problems using unstructured computational meshes automatically generated by the mesh generator. This approach includes a computational model based on the Navier–Stokes equation system and written with respect to the physical process: the turbulent nature of flow with transient points is accounted using the Reynolds Averaged Navier–Stokes method and the k–ω SST model of turbulence by Menter along with the γ–Reθ (Gamma Re Theta) laminar-turbulent transition model; the Volume of Fluid method supplemented with the Schnerr–Sauer cavitation model is used to simulate the cavitation processes; a rotating propeller is simulated by a moving computational mesh and the GGI method to provide conformity of the solutions on adjacent boundaries of arbitrarily-shaped unstructured meshes of the two domains. The specific features of the numerical algorithms in use are described. The method validation results are given; they were obtained because of the problems of finding the performance curves of model-scale propellers in open water, namely the problems of finding the performance of propellers KP505 and IB without consideration of cavitation and the performance of propellers VP1304 and C5 under cavitation conditions. The paper demonstrates that the numerical simulation method presented allows for obtaining sufficiently accurate results to predict the main hydrodynamic characteristics for most modes of operation of the propellers.

Efficient multi-fidelity computation of blood coagulation under flow.

Clot formation is a crucial process that prevents bleeding, but can lead to severe disorders when imbalanced. This process is regulated by the coagulation cascade, a biochemical network that controls the enzyme thrombin, which converts soluble fibrinogen into the fibrin fibers that constitute clots. Coagulation cascade models are typically complex and involve dozens of partial differential equations (PDEs) representing various chemical species' transport, reaction kinetics, and diffusion. Solving these PDE systems computationally is challenging, due to their large size and multi-scale nature. We propose a multi-fidelity strategy to increase the efficiency of coagulation cascade simulations. Leveraging the slower dynamics of molecular diffusion, we transform the governing PDEs into ordinary differential equations (ODEs) representing the evolution of species concentrations versus blood residence time. We then Taylor-expand the ODE solution around the zero-diffusivity limit to obtain spatiotemporal maps of species concentrations in terms of the statistical moments of residence time, [Formula: see text], and provide the governing PDEs for [Formula: see text]. This strategy replaces a high-fidelity system of N PDEs representing the coagulation cascade of N chemical species by N ODEs and p PDEs governing the residence time statistical moments. The multi-fidelity order (p) allows balancing accuracy and computational cost providing a speedup of over N/p compared to high-fidelity models. Moreover, this cost becomes independent of the number of chemical species in the large computational meshes typical of the arterial and cardiac chamber simulations. Using a coagulation network with N = 9 and an idealized aneurysm geometry with a pulsatile flow as a benchmark, we demonstrate favorable accuracy for low-order models of p = 1 and p = 2. The thrombin concentration in these models departs from the high-fidelity solution by under 20% (p = 1) and 2% (p = 2) after 20 cardiac cycles. These multi-fidelity models could enable new coagulation analyses in complex flow scenarios and extensive reaction networks. Furthermore, it could be generalized to advance our understanding of other reacting systems affected by flow.

Numerical study of the steady core-annular flow in a focusing geometry in the presence of soluble surfactants

The steady state core annular flow of two immiscible fluids in the presence of soluble surfactants is investigated numerically in an axisymmetric tubular or flow focusing geometry. The two immiscible fluids form an interface onto which surfactants can be adsorbed/desorbed from the annular fluid where they are dissolved. The finite element method is employed to discretize the governing equations using the spine method to create the computational mesh, follow the shape of the interface and the axisymmetric wall and capture the flow domain. An unknown pressure distribution is imposed along the interface that quantifies the pressure discontinuity, while allowing the bulk pressure to remain continuous in each phase. GMRES is employed to accommodate the nearly singular nature of the jacobian matrix. The bulk surfactant concentration varies only inside a thin boundary layer of thickness δ∼Pe−1, which justifies omission of the advection-diffusion equation in the annulus bulk. Simulations reveal that the onset of Marangoni stresses tends to decelerate the velocity at the interface which is seen to translate towards the wall/axis of symmetry when the more viscous fluid is the inner/outer fluid, respectively. Finally, simulations in flow-focusing nozzles capture the development of spatially growing instabilities in the downstream region of the interface, with wavelength on the order of the nozzle radius, especially when the outer fluid is more viscous or when the surfactant concentration increases. Raising the volumetric flow rate in the annular region suppresses growth of waves due to interfacial friction. Spatial stability analysis verifies downstream growth of waves and recovers the relevant wavelength. Possible repercussions on the onset of jetting instabilities in devices with liquid focusing are discussed.

Empowering nuclear reactor analysis: OpenNode for accurate 3D core simulation through nodal expansion neutron diffusion equations solving

OpenNode is a sophisticated reactor core simulation tool, meticulously designed for educational purposes. Its main function is to solve the complex partial differential equation governing multi-group neutron diffusion in a Cartesian geometry. OpenNode relies on the second- and fourth-order Nodal Expansion Method (NEM) to provide accurate solutions to three-dimensional steady-state problems, encompassing both the direct and adjoint modes. At the heart of this method, the neutron balance equation serves as the fundamental framework for calculating the nodal neutron flux and the effective multiplication factor, enabling the complex evaluation of neutron flux distributions within individual nodes.The NEM methodology, carefully programmed in Fortran-90 in OpenNode, meticulously decomposes the reactor core into nodes of reasonable size. This approach balances computational efficiency and accuracy, achieving outstanding precision even when using a reduced number of computational meshes. To further increase the number of equations while maintaining the balance between unknowns, OpenNode judiciously incorporates two essential approximations: the Weighted Residual Method (WRM) and the Quadratic Transverse Leakage (QTL) approximation.OpenNode transcends simple computational prowess by adopting an intuitive Graphical User Interface (GUI), meticulously developed by combining Python 3, Qt Designer, and Blender software. This Python-powered graphical interface offers users a seamless, user-friendly experience, facilitating tasks such as data entry, execution of complex simulations, calculation of results, data visualization, 3D design of nuclear core geometry, and in-depth analysis of results. One of OpenNode’s key features is its innate flexibility, enabling it to solve problems in both 2D and 3D Cartesian geometries. This adaptability makes it a versatile tool suitable for a wide range of research and power reactor applications, offering users the ability to model and design reactors with unrivaled ease.To substantiate OpenNode’s reliability, comprehensive criticality calculations are carried out in various scenarios, including single-cell verification, BIBLIS, KOEBERG cores in two dimensions (2D), and IAEA cores in three dimensions (3D). The results clearly demonstrate OpenNode’s robustness and reliability in multidimensional criticality analysis. In particular, OpenNode adheres to an open-source philosophy, making it readily available for download. This commitment to open access fosters collaboration and paves the way for wider adoption within the education and research communities, nurturing a thriving ecosystem of shared knowledge and exploration.

Simple or simplistic? Sensitivity of an operating room CFD model to refinement and detailing

Indoor air quality is critical in healthcare facilities since it directly relates to infection risk, mainly in operating rooms (OR). Nowadays, air movement in such environments is often evaluated with computational fluid dynamics (CFD) simulations. However, many users typically oversimplify their OR models by including only major equipment (such as operating tables and surgical lights) that do not require refined computational meshes. This work analyzes the sensitivity of CFD simulations to mesh refinement, air circulation layout, geometry detailing, placement of furniture and equipment, as well as the presence of the surgical staff in the OR. An OR of a Brazilian university hospital was modeled and validated with environmental parameters assessed in-loco. Results showed relative differences up to 375% in air velocity depending on mesh resolution. Including small furniture and changing the OR layout resulted in nine-fold differences in air velocity. Finally, modeling the surgical staff in the OR significantly interfered with air circulation patterns, preventing inflow from the polluted towards the clean and sterile zone. The present work demonstrates the alarming fact that even a validated, sound, but overly simple CFD model may be phenomenologically insufficient to assess real contamination risks in an OR — and, by extension, in other healthcare facilities. Therefore, detailed CFD models regarding the number of elements in the simulation mesh, the positioning and the operating condition of the ventilation system, the inclusion and detailing of furniture, and the presence of the surgical staff in the OR should be employed since early design stages.

Design of the Space Camera Protection System during Launch Process

In order to improve the safety and reliability and maintain the stability of imaging quality when a space camera with a cargo spaceship is launched, a protection system for the space camera is designed. Firstly, according to the mechanical properties of the space camera, a working principle of the protection system is elaborated, and a model of the system is proposed. Secondly, a protective cover and a vibration isolation block are designed on the basis of shape, size, and requirements of the space camera. Thirdly, based on the finite element mesh, static and sinusoidal vibration simulation calculations of the space camera and its protection system are carried out. Finally, the protection system is validated after mechanical experiment. The results reveal that the three directional fundamental frequencies of space camera are 43.39 Hz, 26.74 Hz, and 22.83 Hz, respectively, and the maximum response of sinusoidal vibration acceleration is 17.02 g, which is amplified by 3.4 times. The image quality of the space camera lens is consistent before and after the test, which satisfies the requirement of the cargo ship.

Análise da influência da malha numérica na modelagem hidrodinâmica do lago Paranoá utilizando o MIKE 3

Purpose: This work aims to evaluate the effects of the different meshes constructed in MIKE 3 software on the simulation and calibration results of the model. Theoretical framework: 3D hydrodynamic models, such as MIKE 3, provide the closest representation of reality by simulating the gradients in the three spatial dimensions and solutioning the Navier-Stokes equations. In these models, meshes are used to represent complex geometries. An efficient computational mesh is required to allow convergence and stability of the solution of the equations and, furthermore, of the modelling result. Method/design/approach: Simulation of four meshes with distinct discretization, calibration, comparison, and assessment of the model performance for these four conceptual models considering: mesh’s number of elements, simulation time, mean absolute error (MAE), coefficient of determination (R2), and relative difference. Results and conclusions: For the meshes adopted for comparison, refinement only in the “throat” (region near the dam) did not show significant influences on the results that would justify its use, considering the high computational cost. Therefore, in this case, a sparse mesh and without refinement can be used in detriment of a mesh with refinement only in the “throat”. Research implication: Understand how different meshes discretization can significantly alter simulation time and highlight that optimized simulation requires an equilibrium between simulation time and mesh discretization to maintain model’s performance. Originality/value: Understanding and quantifying the influence of the discretization of the model's mesh on the simulation time and the performance of the model allows the optimization of the modeling, considering the cost-effectiveness of different discretizations leading to smaller simulation time with similar performance.

High-order implicit shock tracking boundary conditions for flows with parametrized shocks

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. In this work, we introduce a robust implicit shock tracking framework specialized for problems with parameter-dependent lead shocks (i.e., shocks separating a farfield condition from the downstream flow), which commonly arise in high-speed aerodynamics and astrophysics applications. After a shock-aligned mesh is produced at one parameter configuration, all elements upstream of the lead shock are removed and the nodes on the lead shock are positioned for new parameter configurations using the implicit shock tracking solver. The proposed framework can be used for most many-query applications involving parametrized lead shocks such as optimization, uncertainty quantification, parameter sweeps, “what-if” scenarios, or parameter-based continuation. We demonstrate the robustness and flexibility of the framework using a one-dimensional space-time Riemann problem, and two- and three-dimensional supersonic and hypersonic benchmark problems.