Anisotropic Grids Research Articles

Computational fluid dynamics for naval hydrodynamics

This article describes key issues which have to be addressed to apply Computational Fluid Dynamics to Naval Hydrodynamics. The specific aspects of Naval Hydrodynamics are discussed and illustrated by recent simulations and comparisons with available experiments. Free-surface flows with or without waves and even violent phenomena such as ventilation or cavitation can be modelled with mixture-fluid surface capturing. Turbulence modelling of thick boundary layers and vortical flows requires anisotropic RANS models or hybrid RANS/LES in case of strongly separated flows. Moreover, fluid–structure interaction in the form of rigid or flexible body motion and multi-body systems is crucial to represent ship manoeuvring and propulsion. Finally, the paper underlines the central role played by anisotropic adaptive grid refinement in the accurate simulation of marine flows.

Anisotropic media hybrid grid tomographic inversion method based on angle domain imaging gathers

The grid tomography velocity inversion method prevails as the most efficacious technique for acquiring background velocity requisite for migration. Originating from imaging gathers, this method not only inverses velocity but also subsequently ascertains the anisotropic parameters field. Traditional approaches utilize a parameter field discretized in a rectangular grid for both ray tracing and parameter inversion. However, the rectangular grids may not provide optimal accuracy for structures with pronounced lateral parameter variations. In contrast, triangular grid discretization allows for grid division along steep slopes, offering sparsity and flexibility when juxtaposed with conventional rectangular grid methods. This paper introduces an optimized triangular grid travel time parameter tomography inversion method, formulated on the parameter sensitivity to travel time and employing angle gathers to derive travel time residuals. Analytical modeling and real data processing substantiate that the triangular grid ray tracing method ameliorates the ill-conditioned nature of the tomography equation and enhances the accuracy of the inversion parameter field effectively.

Assessment of a Flow-dependent Subgrid Characteristic Length for Large-Eddy Simulation on Anisotropic Grids

This paper presents the latest results of a long track development activity in the context of low-dissipative finite volume method for compressible flows. Specifically, here we focus our attention on the Large-Eddy Simulation (LES) approach which can be considered a good candidate for turbulent flow simulations over the next decades. One of the key ingredients of LES models is the subgrid length scale which is typically evaluated based on the local mesh size. This standard approach suffers from loss of accuracy on anisotropic grids that are commonly employed to obtain sufficient wall-normal resolution, whilst keeping the total cell count to a minimum. In order to avoid this issue, we assess the effectiveness of a velocity-gradient-based length scale, referred to as least square length (LSQ) [1]. In this paper, we present for the first time results obtained with the LSQ length scale in the context of compressible LES. The superiority of the LSQ approach over the standard cubic-root length scale is demonstrated in terms of accuracy and overall time to solution.

Wall-Modeled Large-Eddy Simulation of Turbulent Boundary Layer with Spatially Varying Pressure Gradients

Wall-modeled large-eddy simulations of turbulent boundary layers subjected to spatially varying streamwise pressure gradients are conducted to assess the predictive performance of three wall models: ordinary differential equation equilibrium, integral nonequilibrium, and partial differential equation (PDE) nonequilibrium models. The test case is based on experiments conducted by Volino (Journal of Fluid Mechanics, Vol. 897, Aug. 2020, p. A2), where the flow is subjected successively to zero pressure gradient (ZPG), favorable pressure gradient (FPG), recovery ZPG, and adverse pressure gradient (APG) regions. Skin friction is overpredicted in FPG by all the wall models. For equilibrium and integral models, this overprediction is attributed to the strong deviation of mean velocity profiles within FPG from the log law, used explicitly in the equilibrium and implicitly in the integral model. The overprediction is more pronounced for the PDE model, which is attributed to dynamic correction of wall-model eddy viscosity for resolved stresses and a lack of correction for pressure gradient. Potential remedies to mitigate this problem are proposed. Grid refinement improves wall-stress predictions in both FPG and APG but only affects the outer profiles in APG, revealing that accurate wall-flux modeling is more critical for APG. Anisotropic grid analysis shows streamwise grid refinement to be more crucial than spanwise for the convergence of skin-friction, outer velocity, and Reynolds stress profiles.

Advanced FFF of PEEK: Infill Strategies and Material Characteristics for Rapid Tooling.

Traditional vulcanization mold manufacturing is complex, costly, and under pressure due to shorter product lifecycles and diverse variations. Additive manufacturing using Fused Filament Fabrication and high-performance polymers like PEEK offer a promising future in this industry. This study assesses the compressive strength of various infill structures (honeycomb, grid, triangle, cubic, and gyroid) when considering two distinct build directions (Z, XY) to enhance PEEK's economic and resource efficiency in rapid tooling. A comparison with PETG samples shows the behavior of the infill strategies. Additionally, a proof of concept illustrates the application of a PEEK mold in vulcanization. A peak compressive strength of 135.6 MPa was attained in specimens that were 100% solid and subjected to thermal post-treatment. This corresponds to a 20% strength improvement in the Z direction. In terms of time and mechanical properties, the anisotropic grid and isotropic cubic infill have emerged for use in rapid tooling. Furthermore, the study highlights that reducing the layer thickness from 0.15 mm to 0.1 mm can result in a 15% strength increase. The study unveils the successful utilization of a room-temperature FFF-printed PEEK mold in vulcanization injection molding. The parameters and infill strategies identified in this research enable the resource-efficient FFF printing of PEEK without compromising its strength properties. Using PEEK in rapid tooling allows a cost reduction of up to 70% in tool production.

Sparse grid time-discontinuous Galerkin method with streamline diffusion for transport equations

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

Local sensitivity informed anisotropic sparse grid method for uncertainty quantification of chemical kinetics

The determination of rate coefficient parameters in detailed chemical kinetic mechanisms through experiments often suffers from avoidable aleatory uncertainty, while the use of reduced mechanisms, based on various reduction methods, introduces epistemic uncertainty. Both sources of uncertainty pose significant challenges for modeling and numerical simulation, highlighting the need for calibrating these uncertain parameters to achieve robust chemical mechanisms in the next generation. However, the high-dimensional parameter space of chemical kinetic mechanisms remains a significant obstacle in uncertainty computing. The Anisotropic Sparse Grid (ASG) technique has been successful in dealing with high-dimensional uncertainty problems, but it lacks prior knowledge of anisotropy. To address this issue, we propose the Local Sensitivity-Informed Anisotropic Sparse Grid (LSIASG) method, which utilizes local sensitivity as prior information for the ASG method, thereby accelerating the entire uncertainty quantification process. We test the LSIASG method on a theoretical model, a detailed hydrogen kinetic mechanism, and a methane mechanism and demonstrate that it can capture high-dimensional uncertainty characteristics, including expectation and deviation.

An Improved Cellular Automata Solidification Model Considering Kinetic Undercooling

A cellular automata (CA) model has been developed for solidification simulation considering the kinetic undercooling at the interface. The state-of-the-art model incorporates a decentered growth algorithm to suppress the grid anisotropy and a generalized height function method to calculate the curvature accurately. To develop a CA model which is independent of the mesh size, a new diffusion term is proposed to handle the diffusion between the interface cells and liquid cells. The developed CA model is employed to simulate the single-dendritic solidification of an Al–3Cu (wt pct) alloy. The simulated tip velocities agree with the prediction of the Kurz–Giovanola–Trivedi (KGT) model. Further studies show that the developed CA model converges to an equilibrium model with increasing kinetic mobility values. Moreover, it is found that the virtual liquid cell assumption which is commonly used in existing CA models may lead to a deviation in the mass balance. The mass balance error has been resolved by redistributing solutes from neighboring liquid cells in each time step. The developed CA model could be potentially used in solidification simulations with a high undercooling, which is common in welding and additive manufacturing.

Anisotropic Cartesian Grid Generation Strategy for Arbitrarily Complex Geometry Based on a Fully Threaded Tree

In this work, we present a powerful method for anisotropic Cartesian grid generation for arbitrarily complex 3D geometry. The proposed method addresses a number of challenging topics to achieve robustness and efficiency in mesh generation. In addition to displaying the basic information, including the generation of the initial uniform grid, the cell size, the cell number, and the cell centre coordinates, a modified fully threaded tree (FTT) data structure for use in the anisotropic Cartesian grid framework is also developed. Moreover, three criteria are proposed to ensure reasonable neighbourhood relations. A five-step adaptive mesh refinement strategy is developed to portray complex geometric details with high fidelity and maintain a reasonable number of cells. Three three-dimensional meshing experiments are implemented to illustrate and evaluate the present mesh generation performance, such as the number of cells, neighbour query runtime, total runtime, and memory consumption. Our results show that the proposed method overcomes the problems of a large number of cells and high memory consumption compared to the conventional isotropic Cartesian grid method. Additionally, mesh generation based on the modified FTT data structure outperforms the conventional 2 n tree data structure in terms of the neighbour query runtime, total runtime, and memory consumption. At last, a solution example is given to verify the proposed strategy. • A powerful strategy for anisotropic Cartesian grid generation is proposed. • A fully threaded tree (FTT) data structure for use in the anisotropic Cartesian grid framework is developed. • A five-step adaptive mesh refinement strategy is described. • The efficiency of anisotropic Cartesian grid generation for complex 3D geometry is assessed.

Grid anisotropy reduction method for cellular automata based solidification models

The reliability of a cellular automata (CA) simulation for a free dendritic growth problem relies heavily on its ability to reduce the artificial grid anisotropy. Hence, a computationally efficient, accurate and elegant cell capturing methodology is essential to achieve reliable results. Therefore, a novel cell capturing method termed limited circular neighbourhood (LCN) is proposed in the present study for solidification models. The LCN method is applied to the canonical test cases with an isotropic growth rate and is compared with other grid anisotropy reducing methods. It is observed that the LCN method is able to capture the growth orientation accurately. Moreover, the mass loss and shape error in the proposed method is significantly reduced as compared with the other methods. In addition, its performance is also evaluated for a free dendrite growth problem in a pure material in which the growth captured by the LCN method is found to be accurate. Finally, its efficacy is also demonstrated in the results presented for a constrained dendritic growth problem in a binary alloy with multiple growth sites.

Globally supported surrogate model based on support vector regression for nonlinear structural engineering applications

This work presents a global surrogate modelling of mechanical systems with elasto-plastic material behaviour based on support vector regression (SVR). In general, the main challenge in surrogate modelling is to construct an approximation model with the ability to capture the non-smooth behaviour of the system under interest. This paper investigates the ability of the SVR to deal with discontinuous and high non-smooth outputs. Two different kernel functions, namely the Gaussian and Matèrn 5/2 kernel functions, are examined and compared through one-dimensional, purely phenomenological elasto-plastic case. Thereafter, an essential part of this paper is addressed towards the application of the SVR for the two-dimensional elasto-plastic case preceded by a finite element method. In this study, the SVR computational cost is reduced by using anisotropic training grid where the number of points are only increased in the direction of the most important input parameters. Finally, the SVR accuracy is improved by smoothing the response surface based on the linear regression. The SVR is constructed using an in-house MATLAB code, while Abaqus is used as a finite element solver.

Dependency of Mesoscale Organization on Grid Anisotropy in Large‐Eddy Simulations of Convective Boundary Layers at Gray Zone Resolutions

AbstractA new generation of operational atmospheric models operating at horizontal resolutions in the range 200 m ∼ 2 km is becoming increasingly popular for operational use in numerical weather prediction and climate applications. Such grid spacings are becoming sufficiently fine to resolve a fraction of the turbulent transports. Here we analyze Large‐eddy simulation results of a convective boundary layer obtained by coarsening horizontal grid spacings up to 800 m. The aim is to explore the dependency of the mean state and turbulent fluxes on the grid resolution. Both isotropic and anisotropic eddy diffusion approaches are evaluated, where in the latter case the horizontal and vertical eddy diffusivities differ in accord with their horizontal and vertical grid spacings. For coarsening horizontal grid sizes entrainment at the top of the boundary layer tends to get slightly enhanced for isotropic diffusion, whereas for the anisotropic diffusion approach the vertically well‐mixed boundary‐layer structure becomes severely degraded. An analysis of the energy spectrum shows that anisotropic diffusion causes relatively more dissipation of variance at smaller length scales. This leads, in turn, to a shift of spectral energy toward larger length scales that also becomes apparent from a rather different kind of spatial organization of convection. The present study therefore suggests that details with regards to the representation of processes at small scales might impact the organization at length scales much larger than the smallest scales that can be resolved by the model.

Immersed Boundary Methods: Historical Perspective and Future Outlook

Immersed boundary methods (IBMs) are versatile and efficient computational techniques to solve flow problems in complex geometric configurations that retain the simplicity and efficiency of Cartesian structured meshes. Although these methods became known in the 1970s and gained credibility only in the new millennium, they had already been conceived and implemented at the beginning of the 1960s, even if the early computers of those times did not allow researchers to exploit their potential. Nowadays IBMs are established numerical schemes employed for the solution of many complex problems in which fluid mechanics may account for only part of the multiphysics dynamics. Despite the indisputable advantages, these methods also have drawbacks, and each problem should be carefully analyzed before deciding which particular IBM implementation is most suitable and whether additional modeling is necessary. High–Reynolds number flows constitute one of the main limitations of IBMs owing to the resolution of thin wall shear layers, which cannot benefit from anisotropic grid refinement at the boundaries. To alleviate this weakness, researchers have developed IBM-compliant wall models and local grid refinement strategies, although in these cases possible pitfalls must also be considered.

Sharp phase-field modeling of isotropic solidification with a super efficient spatial resolution

The phase-field method provides a powerful framework for microstructure evolution modeling in complex systems, as often required within the framework of integrated computational materials engineering. However, spurious grid friction, pinning and grid anisotropy seriously limit the resolution efficiency and accuracy of these models. The energetic resolution limit is determined by the maximum dimensionless driving force at which reasonable model operation is still ensured. This limit turns out to be on the order of 1 for conventional phase-field models. In 1D, grid friction and pinning can be eliminated by a global restoration of Translational Invariance (TI) in the discretized phase-field equation. This is called the sharp phase-field method, which allows to choose substantially coarser numerical resolutions of the diffuse interface without the appearance of pinning. In 3D, global TI restricts the beneficial properties to a few specific interface orientations. We propose an accurate scheme to restore TI locally in the local interface normal direction. The new sharp phase-field model overcomes grid friction and pinning in three-dimensional simulations, and can accurately operate at dimensionless driving forces up to the order of 10^{4}. At one-grid-point interface resolutions, exceptional degrees of isotropy can be achieved, if further the largely inhomogeneous latent heat release at the advancing solid-liquid interface is mitigated. Imposing a newly proposed source term regularization, the new model captures the formation of isotropic seaweed structures without spurious dendritic selection by grid anisotropy, even at one-grid-point interface resolutions.

A suite of Richardson preconditioners for semi-implicit all-scale atmospheric models

The paper documents a suite of preconditioners for Krylov-subspace solvers of elliptic boundary-value problems (BVPs) that underlie semi-implicit integrations of the nonhydrostatic equations governing the dynamics of all-scale atmospheric flows. Effective preconditioning of the linear operators inherent in the semi-implicit models lies at the heart of the state-of-the-art multiscale-flow simulation. This is especially evident in simulations of global weather and climate—posed on a thin spherical shell—where some form of direct tridiagonal inversion of the operator in the vertical is crucial to relax the often enormous stiffness of the problem. The documented preconditioners stem from the Richardson's (1910) idea of augmenting an elliptic BVP with a transient diffusion equation. Exploiting this idea for mixed explicit-implicit pseudo-time-stepping schemes leads to a broad suite of stationary-iteration solvers, including the many classical algorithms. Here, the high-performance all-scale EULAG model (Smolarkiewicz et al. (2014) [58]), with a flexible three-dimensional decomposition of MPI tasks, is furnished with the preconditioners akin to the classical alternating-direction-implicit (ADI) algorithms, generalized to optional permutations of parallel tridiagonal inversions. The utility of various options is found to be problem dependent, in terms of computational accuracy as well as efficiency. The main thrust of the work is on the long-range forecasts using large anisotropic grids. The relative efficiency and/or accuracy gains attainable with the developed preconditioners are illustrated for idealized scenarios representative of atmospheric flows from planetary to a single-cloud and laboratory scales. The key insight that best encapsulates the significance and novelty of the present work is that there is no single “super-preconditioner” that will perform best in all cases, yet the suite as a whole offers substantial gains in the model performance.

Impact of Anisotropic Mesh Adaptation on the Aerothermodynamics of Atmospheric Reentry

The presence of complex geometries and/or multiple bodies during atmospheric reentry may lead to complex and directional flow features, such as shock waves and shear layers, that need to be correctly predicted to ensure accurate calculation of the aerothermal loads during reentry. Central in ensuring reliable numerical prediction of loads is the adoption of meshes that ensure grid independence and minimize the misalignment between the directional flow features and the grid cell interfaces, a situation that is known to give rise to nonphysical behaviors and spurious oscillations. The use of anisotropic unstructured grid adaptation is here presented as a means to ensure appropriate grid resolution and alignment with directional flow features for cases where the use of structured grids is not always possible or practical. Results highlight the effectiveness and reliability of anisotropic mesh adaptation in successfully predicting the location of shock discontinuities as well as surface aerothermodynamic quantities, while providing results comparable with established approaches relying on structured meshes. Results are presented for single- and multiple-body cases through comparison with experimental data and reference numerical solutions.

An automatic isotropic/anisotropic hybrid grid generation technique for viscous flow simulations based on an artificial neural network

Based on the author's previous research, a novel hybrid grid generation technique is developed by introducing an Artificial Neural Network (ANN) approach for realistic viscous flow simulations. An initial hybrid grid over a typical geometry with anisotropic quadrilaterals in the boundary layer and isotropic triangles in the off-body region is generated by the classical mesh generation method to train two ANNs on how to predict the advancing direction of the new point and to control the grid size. After inputting the initial discretized fronts, the ANN-based Advancing Layer Method (ALM) is adopted to generate the anisotropic quadrilaterals in boundary layers. When the high aspect ratio of the anisotropic grid reaches a specified value, the ANN-based Advancing Front Method (AFM) is adopted to generate isotropic triangles in the off-body computational domain. The initial isotropic triangles are smoothed to further improve the grid quality. Three typical cases are tested and compared with experimental data to validate the effectiveness of grids generated by the ANN-based hybrid grid generation method. The experimental results show that the two ANNs can predict the advancing direction and the grid size very well, and improve the adaptability of the isotropic/anisotropic hybrid grid generation for viscous flow simulations.

UNSTRUCTURED MESH SIZE CONTROL METHOD BASED ON ARTIFICIAL NEURAL NETWORK

Automatic mesh generation and adaptation are bottleneck problems restricting computational fluid dynamics (CFD). Grid quality, efficiency, flexibility, automation level, and robustness are several key issues in grid generation. Mesh size control is significant in unstructured mesh generation which directly impacts the mesh quality, efficiency, and solution accuracy. Controlling mesh size by the background grid method requires mesh size defined on a background mesh by solving differential equations and interpolating from background mesh to specific location, which is very tedious and time-consuming in traditional unstructured grid generation. In this paper, two novel mesh size control methods are proposed in terms of efficiency and automation level. Firstly, radial basis function (RBF) interpolation was developed to control mesh size. In order to improve the efficiency of RBF interpolation, the greedy algorithm was applied to reduce the list of reference nodes. Meanwhile, an artificial neural network (ANN) is used to control the mesh size, relative wall distance, and relative mesh size are introduced as input and output parameters for the ANN. Training models are established and samples (2D cylinder and airfoil grids) are generated by commercial software. The relationship is established between wall distance and mesh size by machine learning. Several meshes are generated with the aforementioned three methods, the results demonstrate that the RBF method and the ANN method are 5-10 times more efficient than the background mesh method, which contributes to efficiency improvement of the grid generation process. Finally, the ANN method is extended to mesh size control of anisotropic hybrid grids, which also obtained meshes of good quality.

A Plane-Dependent Model of 3D Grid Cells for Representing Both 2D and 3D Spaces Under Various Navigation Modes.

Grid cells are crucial in path integration and representation of the external world. The spikes of grid cells spatially form clusters called grid fields, which encode important information about allocentric positions. To decode the information, studying the spatial structures of grid fields is a key task for both experimenters and theorists. Experiments reveal that grid fields form hexagonal lattice during planar navigation, and are anisotropic beyond planar navigation. During volumetric navigation, they lose global order but possess local order. How grid cells form different field structures behind these different navigation modes remains an open theoretical question. However, to date, few models connect to the latest discoveries and explain the formation of various grid field structures. To fill in this gap, we propose an interpretive plane-dependent model of three-dimensional (3D) grid cells for representing both two-dimensional (2D) and 3D space. The model first evaluates motion with respect to planes, such as the planes animals stand on and the tangent planes of the motion manifold. Projection of the motion onto the planes leads to anisotropy, and error in the perception of planes degrades grid field regularity. A training-free recurrent neural network (RNN) then maps the processed motion information to grid fields. We verify that our model can generate regular and anisotropic grid fields, as well as grid fields with merely local order; our model is also compatible with mode switching. Furthermore, simulations predict that the degradation of grid field regularity is inversely proportional to the interval between two consecutive perceptions of planes. In conclusion, our model is one of the few pioneers that address grid field structures in a general case. Compared to the other pioneer models, our theory argues that the anisotropy and loss of global order result from the uncertain perception of planes rather than insufficient training.

Three-Dimensional Central Moment Lattice Boltzmann Method on a Cuboid Lattice for Anisotropic and Inhomogeneous Flows

Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.