Pyramidal Elements Research Articles

Effects of solid particles and wall roughness on turbulent boundary layer in a two-phase horizontal channel flow

Particle image velocimetry (PIV) is used to study the horizontal channel flow over smooth and rough walls, and single-phase flow and particle-laden flow with two different particle diameters are analyzed. The phenomenon of the wall similarity hypothesis is verified in single-phase flow but affected by particles in particle-laden flows. The pyramidal rough elements cause a velocity defect and shorten the length scale of the vortex. The streamwise and wall-normal Reynolds stresses are enhanced by wall roughness. Two kinds of particles with different Stokes numbers are used in the experiment. The collision and dragging effect of particles in the inner region offset the influence of wall roughness on the mean velocity profile. The introduction of particles tends to increase the streamwise Reynolds stress, but makes the peak of the Reynolds stress move to a higher wall-normal position, which is contrary to the effect of wall roughness. Quadrant analysis indicates that the wall roughness and particles show a converse effect on quadrant events, but neither change the majority of second and forth quadrant events. The analysis of the spatial structure of turbulence shows that the small scale spatial structure is less affected by wall roughness compared with the large scale spatial structure. According to the statistics of the particle distribution, it was found that particles with a larger Stokes number show a more distinct prioritized deposition in the inner region, and the deposition of particles is reduced due to the existence of rough elements.

New enriched 3D solid finite elements: 8-node hexahedral, 6-node prismatic, and 5-node pyramidal elements

Herein, we propose new 8-node hexahedral, 6-node prismatic, and 5-node pyramidal 3D solid finite elements enriched by cover functions. To resolve the linear dependence problem, piecewise linear shape functions are adopted as the partition of unity functions. Various mesh patterns are considered for investigation of the linear dependence problem, and no rank deficiency is observed. We demonstrate convergence behavior and effectiveness of the new enriched elements, and adaptive use of cover functions through six problems: an ad hoc problem, a tool jig problem, a slender beam problem, a gear problem, and a connecting rod problem.

A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes

A virtual element method is introduced for the mixed approximation of a simple model problem for the Laplace operator on a polyhedron. The method is fully analysed when the meshes are made up of triangular right prisms, pyramids and tetrahedra. The local discrete spaces coincide with the lowest order Raviart–Thomas spaces on tetrahedral and triangular right prismatic elements, and extend them to pyramidal elements. The discrete scheme is well posed and optimal interpolation error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the discrete element space are optimal and anisotropic on anisotropic right prisms. Furthermore, a discretization of the model problem in the presence of edge and vertex singularities is analysed for the proposed method on a family of suitably designed graded meshes, and optimal estimates for the approximation error are obtained, extending in this way the results of Farhloul et al. (ESAIM Math Model Numer Anal 35:907–920, 2001) where cylindrical domains with edge singularities were considered.

An assessment of atypical mesh topologies for low-Mach large-eddy simulation

An implicit, low-dissipation, low-Mach, variable density control volume finite element formulation is used to explore foundational understanding of numerical accuracy for large-eddy simulation applications on hybrid meshes. Detailed simulation comparisons are made between low-order hexahedral, tetrahedral, pyramid, and wedge/prism topologies against a third-order, unstructured hexahedral topology. Using smooth analytical and manufactured low-Mach solutions, design-order convergence is established for the hexahedral, tetrahedral, pyramid, and wedge element topologies using a new open boundary condition based on energy-stable methodologies previously deployed within a finite-difference context. A wide range of simulations demonstrate that low-order hexahedral- and wedge-based element topologies behave nearly identically in both computed numerical errors and overall simulation timings. Moreover, low-order tetrahedral and pyramid element topologies also display nearly the same numerical characteristics. Although the superiority of the hexahedral-based topology is clearly demonstrated for trivial laminar, principally-aligned flows, e.g., a 1x2x10 channel flow with specified pressure drop, this advantage is reduced for non-aligned, turbulent flows including the Taylor–Green Vortex, turbulent plane channel flow (Reτ395), and buoyant flow past a heated cylinder. With the order of accuracy demonstrated for both homogenous and hybrid meshes, it is shown that solution verification for the selected complex flows can be established for all topology types. Although the number of elements in a mesh of like spacing comprised of tetrahedral, wedge, or pyramid elements increases as compared to the hexahedral counterpart, for wall-resolved large-eddy simulation, the increased assembly and residual evaluation computational time for non-hexahedral is offset by more efficient linear solver times. Finally, most simulation results indicate that modest polynomial promotion provides a significant increase in solution accuracy.

Mineralogical, petrographic and physical-mechanical study of Roman construction materials from the Maritime Theatre of Hadrian's Villa (Rome, Italy)

This paper presents the study of various Roman materials used in the construction of the Maritime Theatre, one of the main buildings in the Hadrian’s Villa complex, a designated UNESCO World Heritage Site located in Tivoli (Rome, Italy), dating to the first half of the II century A.D. The plaster layers (arriccio and intonachino) and overlying original Roman paintings that form the concave wall of the portico as well as some bedding mortars of the pyramidal stone elements (i.e. cubilia) of the circular masonry have been studied in particular. In addition, the acid volcanic rocks of the cubilia have been investigated, aiming to understand their state of alteration and geological origin.By mineralogical-petrographic microscopy (OM), diffractometry (XRPD), Raman spectroscopy, Point Load Tests (PLT), helium pycnometry, and particle size analysis, the composition and granulometric distribution of the aggregate, type and characteristics of the binder, and various physical-mechanical properties (density, porosity, water absorption, imbibition and saturation indices, mechanical resistance) of mortars and stones were defined. In addition, through digital image analysis of thin sections, the binder/aggregate ratio and some geometric characteristics of the aggregates (e.g. circularity) were determined.The research aims to improve the knowledge of the constructive technologies of the Maritime Theatre through the analysis of its materials.

Turbulence investigation in the roughness sub-layer of a near wall flow

The turbulence behaviour along a wall roughened by pyramidal elements was analysed in the region extending from the apex of the roughness elements up to the external limit of the roughness sub-layer. The data used for the analysis were obtained by particle image velocimetry technique. The rough wall turbulent boundary layer flow is characterized by a relatively low Reynolds number. All the results on the rough wall were compared with data referring to the canonical flow on a smooth wall turbulent boundary layer. Mean values and turbulence quantities for the two flows collapse when approaching the external limit of the roughness sublayer. The quadrant analysis of the Reynolds shear stress, in the region near the surface, shows that the contribution of the sweep motions is about equivalent for the two flows (except for wall distances lower than 40 viscous units). The contribution of the ejection motions appears to be more important over the smooth wall than over the rough wall with increasing differences approaching the wall. The probability density functions of the streamwise fluctuating velocity field for the rough wall case appear to be positively skewed in the zone very close to the pyramid apex, in contrast with the behavior observed for the smooth wall case at corresponding distances from the wall. The integral and Taylor scales for the rough wall case appear to be strongly reduced by the presence of the roughness, while the Kolmogorov microscale shows higher values.

Pyramid finite elements for discontinuous and continuous discretizations of the neutron diffusion equation with applications to reactor physics

When using unstructured mesh finite element methods for neutron diffusion problems, hexahedral elements are in most cases the most computationally efficient and accurate for a prescribed number of degrees of freedom. However, it is not always practical to create a finite element mesh consisting entirely of hexahedral elements, particularly when modelling complex geometries, making it necessary to use tetrahedral elements to mesh more geometrically complex regions. In order to avoid hanging nodes, wedge or pyramid elements can be used in order to connect hexahedral and tetrahedral elements, but it was not until 2010 that a study by Bergot established a method of developing correct higher order basis functions for pyramid elements. This paper analyses the performance of first and second-order pyramid elements created using the Bergot method within continuous and discontinuous finite element discretisations of the neutron diffusion equation. These elements are then analysed for their performance using a number of reactor physics benchmarks. The accuracy of solutions using pyramid elements both alone and in a mixed element mesh is shown to be similar to that of meshes using the more standard element types. In addition, convergence rate analysis shows that, while problems discretized with pyramids do not converge as well as those with hexahedra, the pyramids display better convergence properties than tetrahedra.

Performance Analysis of Thermoelectric Modules Consisting of Square Truncated Pyramid Elements Under Constant Heat Flux

System design of a thermoelectric (TE) power generation module is pursued in order to improve the TE performance. Square truncated pyramid shaped P–N pairs of TE elements are connected electronically in series in the open space between two flat insulator boards. The performance of the TE module consisting of 2-paired elements is numerically simulated using commercial software and original TE programs. Assuming that the heat radiating into the hot surface is regulated, i.e., the amount of heat from the hot surface to the cold one is steadily constant, as it happens for solar radiation heating, the performance is significantly improved by changing the shape and the alignment pattern of the elements. When the angle θ between the edge and the base is smaller than 72°, and when the cold surface is kept at a constant temperature, two patterns in particular, amongst the 17 studied, show the largest TE power and efficiency. In comparison to other geometries, the smarter square truncated pyramid shape can provide higher performance using a large cold bath and constant heat transfer by heat radiation.

Discrete $H^1$-Inequalities for Spaces Admitting M-Decompositions

We find new discrete $H^1$- and Poincar\'e-Friedrichs inequalities by studying the invertibility of the DG approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent HDG and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier-Stokes equations defined on unstructured meshes using, in 2D, general polygonal elements and, in 3D, general, flat-faced tetrahedral, prismatic, pyramidal and hexahedral elements.

Discontinuous Galerkin spatial discretisation of the neutron transport equation with pyramid finite elements and a discrete ordinate (SN) angular approximation

In finite element analysis it is well known that hexahedral elements are the preferred type of three dimensional element because of their accuracy and convergence properties. However, in general it is not possible to mesh complex geometry problems using purely hexahedral meshes. Indeed for highly complex geometries a mixture of hexahedra and tetrahedra is often required. However, in order to geometrically link hexahedra and tetrahedra, in a conforming finite element mesh, pyramid elements will be required. Until recently the basis functions of pyramid elements were not fully understood from a mathematical or computational perspective. Indeed only first-order pyramid basis functions were rigorously derived and used within the field of finite elements. This paper makes use of a method developed by Bergot that enables the generation of second and higher-order basis functions, applying them to finite element discretisations of the neutron transport equation in order to solve nuclear reactor physics, radiation shielding and nuclear criticality problems. The results demonstrate that the pyramid elements perform well in almost all cases in terms of both solution accuracy and convergence properties.

Five node pyramid elements for explicit time integration in nonlinear solid dynamics

Pyramid finite elements have become increasingly popular to facilitate meshing due to their distinctive shape and innate ability to transition naturally between elements with quadrilateral and triangular faces. This is especially important for explicit time integration, as the alternative of constraints can be problematic for wave propagation applications frequently modeled by such methods. Although formulations to alleviate singularity problems near the apex node have existed for decades, pyramid elements are not available in typical explicit solid dynamics software. Several 5-node pyramid approaches are evaluated herein for suitability as transition elements in lumped mass explicit methods for nonlinear solid dynamics. Several typical pyramid elements generated from both hexahedral shape functions and with Bedrosian rabbit functions are extended to explicit temporal methods by the development of mass lumping and critical time increment estimation schemes. Standard and uniform strain hexahedrons are also degenerated into a pyramid by simple nodal duplication in the connectivity. A focus of the study is on the viability of using only existing hexahedron capabilities typically available in many explicit codes. Performance is assessed in standard benchmark problems and practical applications using various elastic and elastic-plastic material models and involving large strains/deformations, severe distortion, and contact-impact. Examples first evaluate the elements on their own and then for the principal case as transitions within a hexahedral-dominant model. Row-summation mass lumping is shown to be the best method for any pyramid element approach, which may require slight coding changes for degenerated hexahedrons. The results indicate that it may also be a good systematic method for mass lumping of general degenerate hexahedral types. The single quadrature point standard and Bedrosian pyramid elements are also found to be robust and the best performers, particularly requiring significantly fewer computations than the degenerated uniform strain hexahedron. If used properly, however, all element types are demonstrated to perform satisfactorily (and identically) and thus demonstrate their viability and benefits for practical applications using hexahedral-dominant meshing.

The finite cell method: Polygonal and tetrahedral cells

AbstractPartial differential equations arising in many physical problems are most commonly solved by using the finite element method (FEM). Despite being very versatile, the FEM has one crucial drawback when heterogeneous material systems are considered, i.e. it relies on geometry‐conforming discretization. This meshing process often constitutes a bottleneck in the simulation pipeline and therefore needs to be alleviated. One idea is to exploit the benefits of the fictitious domain concept. Here, Cartesian grids can be deployed to straightforwardly discretize an extended domain. In conjunction with higher order shape functions the recently introduced method is referred to as the finite cell method (FCM).The main objective of the contribution at hand is to extend the FCM to arbitrary unstructured meshes. In two‐dimensional application polygonal finite elements based on generalized barycentric coordinates are deployed, while the three‐dimensional implementation is based on tetrahedral finite elements. A further extension to pyramidal and pentahedral (wedge) elements is, however, straightforward.There are two distinct advantages of using unstructured meshes: (i) a local mesh refinement can easily be conducted and (ii) existing computational models can be re‐used for parametric studies. The latter point is important if virtual defects need to be introduced in the ideal model to study their effect on the material behaviour. These possibilities make the unstructured FCM a powerful numerical tool for the investigation of complex highly heterogeneous materials. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Heat transfer in rotating Rayleigh–Bénard convection with rough plates

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

Numerical Study of Turbulent Channel Flow over Surfaces with Variable Spanwise Heterogeneities: Topographically-driven Secondary Flows Affect Outer-layer Similarity of Turbulent Length Scales

Wall-bounded turbulent flows over surfaces with spanwise heterogeneous surface roughness – that is, spanwise-adjacent patches of relatively high and low roughness – exhibit mean flow phenomena entirely different to what would otherwise exist in the absence of spanwise heterogeneity. In the outer layer, mean counter-rotating rolls occupy the depth of the flow, and are positioned such that “upwelling” and “downwelling” occurs above the low and high roughness, respectively. It has been comprehensively shown that these secondary flows are Prandtl’s secondary flow of the second kind (Anderson et al., J. Fluid Mech. 768, 316–347 2015). This behaviour indicates that spanwise spacing, s y , between adjacent patches of high and low roughness is, itself, a problem parameter; in this study, we have systematically assessed how s y affects turbulence structure in high Reynolds number channel flows via two-point correlations. “High roughness” is imposed with streamwise-aligned pyramid elements with height, h, selected to be ≈ 5% of the channel half height, H. For $s_{y}/H \gtrsim 1$ , we find that the aforementioned domain-scale mean circulations exist and the surface may be regarded as a topography. For s y /H ≲ 0.2, turbulence statistics show characteristics very similar to a homogeneous roughness and thus the surface may be regarded as a roughness. For 0.2 ≲ s y /H ≲ 2, the spatial extent of the counter-rotating rolls is controlled by proximity to adjacent rows, and we define such surfaces as being intermediate. We refer to such surfaces as intermediate state.

A lowest-order composite finite element exact sequence on pyramids

Composite basis functions for pyramidal elements on the spaces H1(Ω), H(curl,Ω), H(div,Ω) and L2(Ω) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart–Thomas–Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, etc.) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions.

Virtual gap element approach for the treatment of non-matching interface using three-dimensional solid elements

A method for three-dimensional non-matching interface treatment with a virtual gap element is developed. When partitioned structures contain curved interfaces and have different brick meshes, the discretized models have gaps along the interfaces. As these gaps bring unexpected errors, special treatments are required to handle the gaps. In the present work, a virtual gap element is introduced to link the frame and surface domain nodes in the frame work of the mortar method. Since the surface of the hexahedron element is quadrilateral, the gap element is pyramidal. The pyramidal gap element consists of four domain nodes and one frame node. Zero-strain condition in the gap element is utilized for the interpolation of frame nodes in terms of the domain nodes. This approach is taken to satisfy the momentum and energy conservation. The present method is applicable not only to curved interfaces with gaps, but also to flat interfaces in three dimensions. Several numerical examples are given to describe the effectiveness and accuracy of the proposed method.

A finite volume multi-moment method with boundary variation diminishing principle for Euler equation on three-dimensional hybrid unstructured grids

Abstract In this work, we have developed a novel numerical model for Euler equations on 3D hybrid unstructured grids including tetrahedral, hexahedral, prismatic and pyramidal elements. The model integrates the VPM (Volume integrated average and Point value based Multi-moment) spatial discretization scheme, the limiting projection with BVD (Boundary Variation Diminishing) manipulation and the Roe Riemann solver for unstructured grids. Distinguished from conventional finite volume method, both the volume-integrated average (VIA) and the point values (PV) at the cell vertices are memorized as prognostic variables and updated in time simultaneously. The VIA is computed by a finite volume formulation of flux form while the PV is point-wisely updated using the differential formulation, where the Roe solver is used to compute both conventional and differential Riemann problems. A special technique is introduced to the limiting projection that effectively suppresses both numerical oscillation and dissipation. The resulting numerical model provides remarkably improved numerical accuracy and robustness with a moderate increase in algorithmic complexity and computational cost, which makes it of practical significance for real-case applications. The numerical results of benchmark tests are presented to demonstrate the appealing solution quality of the present model in comparison with other existing methods.

Superconvergence byM-decompositions. Part III: Construction of three-dimensional finite elements

We apply the concept of an M -decomposition in the framework of steady-state diffusion problems to construct local spaces defining superconvergent hybridizable discontinuous Galerkin methods as well as their companion sandwiching mixed methods in ℝ 3 with tetrahedral, pyramidal, prismatic, and hexahedral elements.

An unstructured-grid numerical model for interfacial multiphase fluids based on multi-moment finite volume formulation and THINC method

We present a practical numerical framework for incompressible interfacial multiphase flows on unstructured grids with arbitrary and hybrid elements. The numerical framework is constructed by combining VPM (volume-average/point-value multi-moment) and UMTHINC (unstructured multi-dimensional tangent of hyperbola interface capturing) schemes. To facilitate accurate and reliable simulations for interfacial multiphase flows on arbitrary and hybrid unstructured grids, we have made the following major new efforts in this work. (1) UMTHINC scheme on prismatic and pyramidal elements to facilitate computations on hybrid arbitrary unstructured grids; (2) Consistent numerical formulation for mass and momentum transports to simulate multiphase flows of large density ratio; (3) Combined FVM-FEM for accurate solution to diffusion equation; (4) Pressure-projection formulation in consistent with the balanced-force model. Integrating all these numerical techniques effectively enhances the accuracy and robustness in interface capturing and numerical solution of multiphase fluid dynamics, which results in a numerical framework of great significance for practical applications. Numerical verifications have been carried out through benchmark tests ranging from surface tension dominant flows of small scale to large scale flows with violently-changing interfaces. Numerical results demonstrate that the present framework is robust with adequate accuracy for simulating multiphase flows in complex geometries.

Converting heterogeneous complex geological models to consistent finite element models: methods, development, and application to deep geothermal reservoir operation

Static geological models representing complex geological systems are the prerequisite of dynamic model simulations applied for assessing subsurface processes. The corner point grid approach has been applied to represent the complexity in geometry, hydraulic connectivity, and heterogeneity found in these static geological models. Due to the occurrence of faults, pinch-outs, and eroded geological layers, corner point grids easily degenerate, which leads to model inconsistencies. This study introduces a workflow for converting heterogeneous geological models to consistent finite element models, accounting for regular and irregular hexahedral blocks of the corner point grid by converting to a set of hexahedra, prism, pyramid, and tetrahedral elements, based on the individual degeneration situation. Heterogeneous geological data such as permeability or porosity can be transferred layer-wise or on a block-wise basis. Additionally, well trajectories can be accurately mapped to the converted finite element mesh, to place the corresponding source terms. The developed workflow is tested on dedicated test cases and applied to convert a real complex field site from the North German Basin for use in a deep geothermal reservoir operation. The field application demonstrates the robustness and applicability of the newly developed conversion workflow and the suitability of the converted mesh for dynamic finite element reservoir model simulations.