A new auxiliary volume-based gradient algorithm for triangular and tetrahedral meshes

Athkuri, Sai Saketha Chandra; Eswaran, V.

doi:10.1016/j.jcp.2020.109780

Cited by 10 publications

(15 citation statements)

References 22 publications

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“…An alternate way to obtain first-order accurate gradients using the divergence theorem is to apply it on a different volume (not the grid cell) so that the interpolation of the face midpoint solution is estimated to second-order accuracy. This approach has been followed by Athkuri and Eswaran, 16 in constructing an AV around a cell-center with the help of a circle. Linear interpolation is then used to calculate the solution values on the periphery of the circle using neighboring cell solutions, as…”

Section: Divergence Theorem-based Gradient Methodsmentioning

confidence: 99%

The mid‐point Green‐Gauss gradient method and its efficient implementation in a 3D unstructured finite volume solver

Athkuri

Nived

Eswaran

2021

Numerical Methods in Fluids

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To achieve consistent second-order spatial accuracy in cell-centered finite volume methods using the divergence theorem, the gradient computation at the cell-centers should at least be first-order accurate, which is not assured by the Traditional Green-Gauss (TGG) method on irregular meshes. The Circle Green-Gauss (CGG) method, however, achieves this by constructing a new auxiliary volume (AV) around the cell of interest using a circle and then employs linear interpolation to obtain estimates of the solution at the face centers of this AV, which are then used to compute the cell-center gradients. We propose a modified method, which we call the mid-point Green-Gauss method (mpGG), that alleviates the complexity in the implementation while maintaining the same accuracy as the CGG method. We offer detailed construction of the three-dimensional AV for the most commonly used multi-hydral elements in unstructured meshes, that is, hexahedral, pyramidal, prism, and tetrahedral cells, using an in-house finite volume solver that reads and writes data based on the computational fluid dynamics (CFD) General Notation System (CGNS).Accuracy analysis of the proposed method is carried out in comparison with the TGG, CGG, and weighted least squares methods. The proposed method is further validated using the case of laminar flow over a sphere at a Reynolds number of 100. An inviscid vortex dissipation study is performed to compare the level of numerical dissipation produced by these methods. Further, we study the performance of these methods in a problem that involves turbulent shock-wave boundary layer interaction.

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Section: Divergence Theorem-based Gradient Methodsmentioning

confidence: 99%

The mid‐point Green‐Gauss gradient method and its efficient implementation in a 3D unstructured finite volume solver

Athkuri

Nived

Eswaran

2021

Numerical Methods in Fluids

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“…Linear interpolation is used for 2nd-order accuracy, but the gradients at C 0 and C f are likely only 1st-order accurate anyway. Furthermore, a first-order accurate approximation of the gradient at c f is all that is needed in (24) because the O(h) error of the gradient, multiplied by (c f − c f ), contributes only a O(h 2 ) error to (24). The most convenient 1st-order approximation to ∇φ(c f ) is just ∇φ(C 0 ), because this avoids any coupling between the gradients of neighbouring cells 2 .…”

Section: Self-corrected Green-gauss Gradientsmentioning

confidence: 99%

“…The most convenient 1st-order approximation to ∇φ(c f ) is just ∇φ(C 0 ), because this avoids any coupling between the gradients of neighbouring cells 2 . So, replacing ∇φ(C 0 ) for ∇φ(c f ) in (24) and substituting the latter expression in (23) gives:…”

Section: Self-corrected Green-gauss Gradientsmentioning

confidence: 99%

“…In a previous publication [19] we analysed the properties of both of these gradient families, and showed in theory and verified in practice that the GG gradients require special grid smoothness in order to be consistent, whereas LS gradients are always at least first-order accurate, which is a prerequisite for a second-order accurate FVM on grids of general geometry. Despite their shortcomings, GG gradients remain popular and a number of recent publications have addressed the problem of their inconsistency [20,21,22,23,24]. One way to remove the inconsistency is through an iterative procedure that couples the gradients at neighbouring grid cells [25,13,19,21,22] (equivalently, a large linear system that includes the gradients at all grid cells can be solved [26]).…”

Section: Introductionmentioning

confidence: 99%

“…This procedure is expensive, but if the gradient iterations are interwoven with the solution iterations of an implicit FVM then the gradient computation cost becomes almost as small as for the uniterated GG gradient [19,21]. Another option for obtaining consistent GG gradient schemes, without the need for iteration, is to apply the Gauss theorem to suitably selected auxiliary cells rather than to the actual grid cells [19,24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A unification of least-squares and Green-Gauss gradients under a common projection-based gradient reconstruction framework

Syrakos¹,

Oxtoby²,

Varchanis³

et al. 2021

Preprint

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We propose a family of gradient reconstruction schemes based on the solution of over-determined systems by orthogonal or oblique projections. In the case of orthogonal projections, we retrieve familiar weighted least-squares gradients, but we also propose new direction-weighted variants. On the other hand, using oblique projections that employ cell face normal vectors we derive variations of consistent Green-Gauss gradients, which we call Taylor-Gauss gradients. The gradients are tested and compared on a variety of grids such as structured, locally refined, randomly perturbed, unstructured, and with high aspect ratio. The tests include quadrilateral and triangular grids, and employ both compact and extended stencils, and observations are made about the best choice of gradient and weighting scheme for each case. On high aspect ratio grids, it is found that most gradients can exhibit a kind of numerical instability that may be so severe as to make the gradient unusable. A theoretical analysis of the instability reveals that it is triggered by roundoff errors in the calculation of the cell centroids, but ultimately is due to truncation errors of the gradient reconstruction scheme, rather than roundoff errors. Based on this analysis, we provide guidelines on the range of weights that can be used safely with least squares methods to avoid this instability.

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On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

Nived

Athkuri

Eswaran

2022

Computers & Mathematics with Applications

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A new auxiliary volume-based gradient algorithm for triangular and tetrahedral meshes

Cited by 10 publications

References 22 publications

The mid‐point Green‐Gauss gradient method and its efficient implementation in a 3D unstructured finite volume solver

The mid‐point Green‐Gauss gradient method and its efficient implementation in a 3D unstructured finite volume solver

A unification of least-squares and Green-Gauss gradients under a common projection-based gradient reconstruction framework

On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

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