scholarly journals An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators

2014 ◽  
Vol 14 (4) ◽  
pp. 461-472 ◽  
Author(s):  
Daniele A. Di Pietro ◽  
Alexandre Ern ◽  
Simon Lemaire
Keyword(s):  
Degrees Of Freedom ◽  
Arbitrary Order ◽  
Penalty Term ◽  
Polyhedral Meshes ◽  
Gradient Reconstruction ◽  
Reconstruction Operator ◽  
Primal Method ◽  
Volume Method ◽  
Compact Stencil ◽  
General Meshes

AbstractWe develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.

scholarly journals Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

2018 ◽  
Vol 52 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Daniele Boffi ◽  
Daniele A. Di Pietro
Keyword(s):  
Mixed Method ◽  
Degrees Of Freedom ◽  
Recent Literature ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Discretization Methods ◽  
Polyhedral Meshes ◽  
Starting Point ◽  
Primal Method ◽  
Polytopal Meshes

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L2-norms.

Geometrical Evaluation of a Pair of Elliptical Tubes Subjected to a Flow with Forced Convection Heat Transfer

2019 ◽  
Vol 396 ◽  
pp. 155-163
Author(s):  
Ana Paula Del Aghenese ◽  
Eliander Manke Heinemann ◽  
Gabriel de Avila Barreto ◽  
Filipe Branco Teixeira ◽  
Liércio André Isoldi ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Forced Convection ◽  
Degrees Of Freedom ◽  
Fluid Dynamic ◽  
Convection Heat Transfer ◽  
Numerical Approach ◽  
Angle Of Incidence ◽  
Prandtl Numbers ◽  
Forced Convective Flow ◽  
Volume Method

In the present work it is performed a study on the geometric evaluation of a pair of elliptical tubes subjected to external flow with forced convection by means of numerical approach. The objectives are the maximization of Nusselt number (NuD) and the minimization of drag coefficient (CD). The degrees of freedom for the pair of tubes arrangement are: the ratio between the transverse pitch and characteristic length of tubes (ST/D), where D = (A)1/2, the ratio of the main and secondary axes of the elliptical tube (a/b) and the angle of incidence of the flow on the pair of tubes (α). The simulations were carried out considering two-dimensional forced convective flows, in the laminar regime and incompressible conditions. For all configurations, Reynolds and Prandtl numbers are constant, ReD = 100 and Pr = 0.71. The Finite Volume Method (FVM) is used to solve conservation equations of mass, momentum and energy. The software Gmsh is used for creation of the geometries and generation of the meshes. Results showed that the degrees of freedom affected the fluid dynamic and thermal performance of the forced convective flow. According to the objectives outlined in this study, the best performance for the maximization of heat transfer was obtained when α = 0o, a/b = 1⁄2 and ST/D = 3.5. In the case of the fluid dynamics study, the optimal result for CD minimization occurred when α = 0o, a/b = 2.0 and ST/D = 4.0. Thus, the optimal geometry will depend on the indicator performance where the problem is evaluated.

301 Moving-Grid Finite-Volume Method using Polyhedral Meshes

2013 ◽  
Vol 2013.26 (0) ◽  
pp. _301-1_-_301-2_
Author(s):  
Daichi TANIO ◽  
Masashi YAMAKAWA ◽  
Kenichi MATSUNO
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Moving Grid ◽  
Polyhedral Meshes ◽  
Volume Method

scholarly journals Recovered finite element methods on polygonal and polyhedral meshes

2020 ◽  
Vol 54 (4) ◽  
pp. 1309-1337
Author(s):  
Zhaonan Dong ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Diffusion Reaction ◽  
Galerkin Methods ◽  
Polyhedral Meshes ◽  
Order Polynomial ◽  
Spatial Dimensions ◽  
Practical Performance

Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.

scholarly journals GEOMETRICAL EVALUATION OF RECTANGULAR FIN MOUNTED IN LATERAL SURFACE OF LID-DRIVEN CAVITY FORCED CONVECTIVE FLOWS

2019 ◽  
Vol 18 (2) ◽  
pp. 98
Author(s):  
E. D. dos Santos ◽  
P. M. Rodrigues ◽  
L. A. Isoldi ◽  
J. F. Prolo Filho ◽  
L. A. O. Rocha ◽  
...  
Keyword(s):  
Aspect Ratio ◽  
Thermal Performance ◽  
Degrees Of Freedom ◽  
Reynolds Numbers ◽  
Cavity Surface ◽  
Square Cavity ◽  
Driven Cavity ◽  
Convective Flows ◽  
Constructal Design ◽  
Volume Method

In this work, it is investigated the geometric effect of rectangular fin inserted in a lid-driven square cavity over thermal performance of laminar, incompressible, steady and forced convective flows. This study is performed by applying Constructal Design to maximize the heat transfer between the fin and the cavity flow. For that, the problem is subjected to two constraints: area of the cavity and area of rectangular fin, and two degrees of freedom: height/length ratio of rectangular fin (H1/L1) and its position in upstream surface of the cavity (S/A1/2). It is considered here some fixed parameters, as the ratio between the fin and cavity areas (ϕ = 0.05), the aspect ratio of the cavity dimensions (H/L = 1.0) and Prandtl number (Pr = 0.71). The fin aspect ratio (H1/L1) was varied for three different placements of the fin at the upstream cavity surface (S/A1/2 = 0.1, 0.5 and 0.9) which represents a lower, intermediate and upper positions of the fin. The effects of the fin geometry over the spatial-averaged Nusselt number ( ) is investigated for three different Reynolds numbers (ReH = 10, 102 and 103). The conservation equations of mass, momentum and energy were numerically solved with the Finite Volume Method. Results showed that both degrees of freedom (H1/L1 and S/A1/2) had a strong influence over , mainly for higher magnitudes of Reynolds number. Moreover, the best thermal performance is reached when the fin is placed near the upper surface of the cavity for an intermediate ratio between height and length of rectangular fin, more precisely when (S/A1/2)o = 0.9 and (H1/L1)oo = 2.0.

Geometry Evaluation of Heat Transfer by Mixed Convention in Driven Cavities with Two Inserted Fins

2019 ◽  
Vol 396 ◽  
pp. 164-173 ◽  
Author(s):  
Priscila M. Rodrigues ◽  
Cicero C. de Escobar ◽  
Luiz Alberto Oliveira Rocha ◽  
Liércio André Isoldi ◽  
Elizaldo Domingues dos Santos
Keyword(s):  
Heat Transfer ◽  
Degrees Of Freedom ◽  
Numerical Study ◽  
Lower Surface ◽  
Stable Stratification ◽  
Geometric Constraints ◽  
Driven Cavity ◽  
Geometric Configurations ◽  
Two Dimensional Flow ◽  
Volume Method

In this work, a numerical study of a flow with heat transfer by mixed convection are carried out. The objective is the geometric evaluation through the application of the Construtal Design and the exhaustive search method. The behavior of a lid-driven cavity with stable stratification subjected to an incompressible, laminar and two-dimensional flow is investigated. The cavity has two rectangular fins inserted in the lower surface. The problem is subject to three constrains: three geometric constraints: the area of the cavity, two fin areas. The investigated geometry has three degrees of freedom: the ratio between height and cavity length (H/L) and the ratio between height and length of each fin (H1/L1 and H2/L2). The effect of the fin geometry over spatial-averaged Nusselt (NuH) is investigated for Reynolds number (ReH) = 400 and Richardson (Ri) = 0.1. The conservation equations of mass, momentum and energy are tackled with Finite Volume Method (FVM) through the use of commercial software FLUENT. The results showed that the lower H2/L2 ratios resulted in higher NuH values. An increase in NuH value of approximately 49% between the worst and the best geometrical configuration was found, thus highlighting the importance of geometric evaluation on this kind of problem. It is concluded that for the problem addressed the best behavior is obtained when the fins have a small insertion into the cavity, thus avoiding the restriction of the main vortex flow. The results found highlight the importance of the geometric evaluation for the purpose of theoretical recommendation on the geometric configurations that lead to the best thermal performance.

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