Blending Functions Research Articles

A generic framework for design of interface capturing schemes for multi-fluid flows

A generic methodology for the design of interface capturing schemes for immiscible multi-fluid flows using the VOF (Volume-Of-Fluid) approach is outlined. Interface capturing schemes are devised as a blend of high-resolution and compressive schemes and their efficiency and accuracy are dependent on the choice of the constituent schemes and the blending function. On the basis of a set of design principles proposed in this work, we show that interface capturing schemes proposed in literature may be encompassed into a single class of GPL (Generalised Piecewise Linear)-κ schemes allowing for a unified approach for development of such schemes. This methodology is used to de-construct four existing schemes and to propose two new schemes for interface capture in an unstructured finite volume framework. The schemes are tested on challenging advection problems using both structured and unstructured grids to evaluate their performance. The new schemes perform as well as existing schemes and even outperform them on unstructured grids, while also exhibiting near Courant number independence for all test problems and grid topologies. The CUIBS (Cubic Upwind Interpolation based Blending Scheme) proposed in the study shows the best performance among all interface capturing schemes discussed herein and is applied to the study of immiscible binary fluid flows. Numerical simulations of viscous sloshing in a tank and Rayleigh–Taylor instability demonstrate the ability of the CUIBS scheme to resolve fluid–fluid interfaces with minimal diffusion.

G2 Surface Interpolation Over General Topology Curve Networks

AbstractThe basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature continuous connections (G2) may also be required. Examples include special curve network configurations with supplemented internal edges, “master‐slave” curvature constraints, and general topology surface approximations over meshes.The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G2 continuous. We extend Gregory's multi‐sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper.

Compressive sensing image-fusion algorithm in wireless sensor networks based on blended basis functions

Compressive sensing (CS) has given us a new idea at data acquisition and signal processing. It has proposed some novel solutions in many practical applications. Focusing on the pixel-level multi-source image-fusion problem in wireless sensor networks, the paper proposes an algorithm of CS image fusion based on multi-resolution analysis. We present the method to decompose the images by nonsubsampled contourlet transform (NSCT) basis function and wavelet basis function successively and fuse the images in compressive domain. It means that the images can be sparsely represented by more than one basis function. We named this process as blended basis functions representation. Since the NSCT and wavelet basis functions have complementary advantages in multi-resolution image analysis, and the signals are sparser after being decomposed by two kinds of basis functions, the proposed algorithm has perceived advantages in comparison with CS image fusion in wavelet domain which is widely reported by literatures. The simulations show that our method provides promising results.

A dynamic model of ball bearing for simulating localized defects on outer race using cubic hermite spline

In this paper a dynamic model is presented for predicting the vibration behavior of a ball bearing under the influence of localized defects on the outer race. The calculation of contact force is based on Hertzian contact deformation theory. The pulse generated by the ball striking the defect on outer race is modeled by using the blending functions of the cubic hermite spline. The effect of change in the angular position of the defect, size of the defect on outer race, multiple defects on outer race and the variation of load on the vibration amplitude is predicted by this model. A computer program in MATLAB is developed and the governing equation of motion is solved by Euler’s method. The numerical results are presented as a function of variation of the geometry of the outer race due to the impact at the defect and normal race contact w.r.t. time and the conclusion about the health of the bearing is determined by the spectral analysis. To validate the results, experimentation has also been performed.

Spline-based reservoir’s geometry reconstruction and mesh generation for coupled flow and mechanics simulation

In this paper, the geometry of oil reservoirs is reconstructed by using B-splines surfaces. The technique exploits the reservoir’s static model’s simplicity to build a robust piecewise continuous geometrical representation by means of Bezier bicubic patches. Interpolation surfaces can manage the reservoir’s topology while translational surfaces allow extrapolating it towards its sideburdens. After that, transfinite interpolation (TFI) can be applied to generate decent hexahedral meshes. In order to test the procedure, several open-to-the-public oil reservoir datasets are reconstructed and hexahedral meshes around them are generated. This reconstruction workflow also allows having different meshes for flow and mechanics by computing a projection operator in order to map pressures from the original flow mesh to the generated reference mechanics mesh. As an update respect to a previous version of this research, we already incorporate blending functions to the TFI procedure in order to attract the mesh towards the reservoir, which allows grading the hexahedral meshes in the appropriate manner. Finally, field scale reservoir compaction and subsidence computations are carried out by using continuous Galerkin FEM for both flow and mechanics in order to demonstrate the applicability of the proposed algorithm.

Algebraic distance estimations for enriched isogeometric analysis

In problems with evolving boundaries, interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the propagating boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. In this paper a purely algebraic, and computationally efficient technique is described for constructing distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The constructed distance measures are level sets of the implicitized constituent Bezier patches of the NURBS surfaces that are obtained purely algebraically. Since, in general, the implicitized functions extend beyond the parametric range of the generating Bezier patch, algorithmic procedures are developed to trim these global implicit functions to the boundaries of the Bezier patch. Boolean compositions are then carried out between adjoining Bezier patches to construct a composite distance field over the domain. The compositions rely on R-functions that are also algebraic in nature. The developed technique is demonstrated by constructing algebraic distance field for complex geometries and by solving a variety of examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks.

Ribbon-based transfinite surfaces

One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves.The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch (Várady et al., 2011); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results.

A DDES model with a Smagorinsky-type eddy viscosity formulation and log-layer mismatch correction

The current work develops a variant of delayed detached eddy simulation (DDES) that could be characterized as limiting the production term. Previous formulations have been based on limiting the dissipation rate (Spalart et al., 2006). A clipped length scale is applied directly to the eddy viscosity, yielding a Smagorinsky-like formulation when the model is on the eddy simulation branch. That clipped eddy viscosity limits the production rate. The length scale is modified in order to account for the log-layer mismatch (a well-known issue with DDES), without using additional blending functions. Another view of our approach is that the subgrid eddy-viscosity is represented by a mixing length formula l2ω; in the eddy field ω acts like a filtered rate of strain. Our model is validated for channel flow as well as separated flows (backward-facing step, 2D periodic hills) and illustrated via an air-blast atomizer.

Mach 6 Wake Flow Simulations Using a Large-Eddy Simulation/Reynolds-Averaged Navier–Stokes Model

A hybrid large-eddy simulation/Reynolds-averaged Navier–Stokes turbulence model is used to simulate the Mach 6 flow around a scaled model similar to NASA’s Orion multipurpose crew vehicle. The results for surface pressure and heat transfer are compared with experimental data from previous base flow experiments conducted at the Calspan—University at Buffalo Research Center. Using the highest Reynolds number test case ( based on capsule diameter), different numerical aspects of the hybrid approach are addressed, such as use of a low-dissipation scheme, a modification to the eddy-viscosity blending function, time-averaging results, filtering computational results, and sensitivity to grid resolution. In addition, results are compared with Reynolds-Averaged Navier–Stokes using Menter’s two-equation baseline model and to detached-eddy simulation predictions. By introducing a new modification to the blending from Reynolds-averaged Navier–Stokes to large-eddy simulation within boundary layers, very good agreement with the experiment is obtained in regions where the boundary-layer grid spacing is too coarse for large-eddy simulation. The findings show that the high-fidelity schemes produce results that agree much better with the experimental data than Reynolds-averaged Navier–Stokes methods, which tend to underpredict base pressures and overpredict heat fluxes. The overall accuracy of each scheme is evaluated using a normalized root mean square error in different regions of the flow, and the analysis shows that, in separated regions, the integrated error is over 120% using a Reynolds-averaged Navier–Stokes model, and 30–50% using the higher fidelity schemes. Additionally, the hybrid methodology presented is further validated by considering a lower Reynolds number (), where the flow is nominally transitional, and very accurate heat transfer predictions are also obtained.

Laminar thermally developing flow in rectangular channels and parallel plates: uniform heat flux

Numerical simulations were conducted for thermally developing laminar flow in rectangular channels with aspect ratios ranging from 1 to 100, and for parallel plates. The simulations were for laminar, thermally developing flow with H1 boundary conditions: uniform heat flux along the length of the channel and constant temperature around the perimeter. In the limit as the non-dimensional length, x* = x/(D h RePr), goes to zero, the Nusselt number is dependent on x* to the negative exponent m. As the non-dimensional length goes to infinity the Nusselt number approaches fully developed values that are independent of x*. General correlations for the local and mean heat transfer coefficients are presented that use an asymptotic blending function to transition between these limiting cases. The discrepancy between the correlation and the numerical results is less than 2.5 % for all aspect ratios. The correlations presented are applicable to all aspect ratios and all non-dimensional lengths, and decrease the discrepancy relative to existing correlations.

Assessment of RANS and improved near-wall modeling for forced convection at low Prandtl numbers based on LES up to [formula omitted

Most promising Generation IV nuclear reactor concepts are based on a liquid metal coolant. However, at low Prandtl (Pr) numbers such as those of liquid metal, classical approaches derived for unity, or close to unity, Pr fail to accurately predict the heat transfer. This paper assesses the RANS modeling of forced turbulent convection at low Pr and in channel flow. Reference results at high Reynolds (Re) number are required to ensure that the Peclet number is sufficiently high. Therefore, new reference results were obtained by performing a wall-resolved Large-Eddy Simulation of turbulent channel flows at a friction Reynolds number Reτ=2000 and at Pr=0.01 and 0.025 (this also corresponds to the highest Re Direct Numerical Simulation (DNS) available in the literature for the flow, but without heat transfer). The LES velocity statistics are in very good agreement with those of the DNS and, as validated by the authors in previous publications, the LES approach used here accurately predicts the temperature statistics at low Pr. The LES results are used to assess RANS heat transfer modeling based on the effective turbulent Prandtl number (Prt) concept. Among existing Prt correlations, the correlation by Kays (1994) [10] is shown to yield the best results. Since it is also shown that the near-wall temperature profile does not follow a log-law, a new “law of the wall for temperature” is here proposed, which does not use any blending function. Its use as a wall-function is also validated in actual RANS simulations.

Modeling of the time-averaged gas–solid drag force in a fluidized bed based on results from transient 2D Eulerian–Eulerian simulations

In the present paper, the possibilities to cover all fluidization states by a single drag law in steady-state CFD multiphase flow modeling are evaluated. The time-averaged drag force is expressed as the product of the drag force calculated from the traditional drag laws for homogeneous conditions, and a correction function. Closure correlations for the correction function are developed by nonlinear regression modeling based on data collected from 69 transient 2D simulations of bubbling, turbulent and circulating fluidized beds. The correlations are given as functions of eight variables: the solid volume fraction, the distance from the nearest wall, the height above the air distributor, the slip velocity between the phases, the gas velocity, the particle size, the solid density and the gas viscosity. The results indicate that covering all fluidized bed conditions in a single drag correlation is feasible, although fully satisfactory results were not obtained for the surface and freeboard regions of a bubbling fluidized bed (BFB) with correlations that were acceptable in circulating fluidized bed (CFB) conditions. A correlation that covers the whole range of fluidized states is complicated and thus the modeling task could be divided into development of separate correlations for different regions that could be combined into a single correlation by means of blending functions. The validity of this approach was demonstrated by developing a separate correlation for the dilute conditions above a height of 1.5m in CFB risers. Results show that the accuracy of the predictions significantly improved in dilute CFB conditions where a much simpler correlation with six input variables could be used. The modeling approach is a good starting point for the development of a general drag law for CFD simulations of fluidized beds.

Positive blending Hermite rational cubic spline fractal interpolation surfaces

Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS.

Discontinuous Galerkin discretization of the Reynolds-averaged Navier–Stokes equations with the shear-stress transport model

In this article we consider the development of Discontinuous Galerkin (DG) methods for the numerical approximation of the Reynolds-averaged Navier–Stokes (RANS) equations with the shear-stress transport (SST) model by Menter. This turbulence model is based on a blending of the Wilcox k–ω model used near the wall and the k–ϵ model used in the rest of the domain where the blending functions depend on the distance to the nearest wall. For the computation of the distance of each quadrature point in the domain to the nearest of the curved, piecewise polynomial wall boundaries, we propose a stabilized continuous finite element (FE) discretization of the eikonal equation. Furthermore, we propose a new wall boundary condition for the dissipation rate ω based on the projection of the analytic near-wall behavior of ω onto the discrete ansatz space of the DG discretization. Finally, we introduce an artificial viscosity to the discretization of the turbulence kinetic energy (k-)equation to suppress oscillations of k near the underresolved boundary layer edge.The wall distance computation based on the continuous FE discretization of the eikonal equation is demonstrated for an internal and three external/aerodynamic flow geometries including a three-element high-lift configuration. The DG discretization of the RANS equations with the SST model is demonstrated for turbulent flows past a flat plate and the RAE2822 airfoil (Cases 9 and 10). The results are compared to the underlying k–ω model and experimental data.

Simulation of the flow past a circular cylinder in the supercritical regime by blending RANS and variational-multiscale LES models

A strategy which blends a variational multiscale large eddy simulation (VMS-LES) model and a RANS model in a hybrid approach is investigated. A smooth blending function, which is based on the value of a blending parameter, is used for switching from VMS-LES to RANS. Different definitions of the blending parameter are investigated. The capabilities of the novel hybrid approach are appraised in the simulation of the flow around a circular cylinder at a Reynolds number 1.4×105, based on the freestream velocity and on the cylinder diameter, in the presence of turbulent boundary-layer due to turbulent inflow conditions. A second study at Reynolds numbers from Re=6.7×105 to 1.25×106 is also presented. The effect of using the VMS-LES approach in the hybrid model is evaluated. Results are compared to those of other RANS, LES and hybrid simulations in the literature and with experimental data

Convex data modelling using rational bi-cubic spline function

A piecewise rational cubic function in the form of a cubic/quadratic has been extended to a rational bicubic blended function to visualize 3D convex data. It involves twelve shape parameters in each rectangular mesh for its representation. Simple data-dependent constraints are imposed on four shape parameters to conserve convexity of convex data. The remaining shape parameters are left up to the user for modification of convex surfaces. Several numerical examples are presented to show the visually pleasing convexity preserving surfaces as compared to existing interpolants.

Construction of Polynomial Spline Spaces Over Quadtree and Octree T-meshes

We present a new strategy for constructing tensor product spline spaces over quadtree and octree T-meshes. The proposed technique includes some simple rules for inferring local knot vectors to define spline blending functions. These rules allow to obtain for a given T-mesh a set of cubic spline functions that span a space with nice properties: it can reproduce cubic polynomials, the functions are C2-continuous, linearly independent, and spaces spanned by nested T-meshes are also nested. In order to span spaces with these properties applying the proposed rules, the T-mesh should fulfill the only requirement of being a 0-balanced quadtree or octree. The straightforward implementation of the proposed strategy and the simplicity of tree structures can make it attractive for its use in geometric design and isogeometric analysis. In this paper we give a detailed description of our technique and illustrate some examples of its application in isogeometric analysis performing adaptive refinement for 2D and 3D problems.

Monotonicity-preserving rational bi-cubic spline surface interpolation

We discuss the problem of monotonicity preservation of surfaces through 3D monotone data. This can be done using a rational bi-cubic blended function that is an extension of a rational cubic function in the form of a cubic numerator and quadratic denominator. The function involves twelve shape parameters in each rectangular patch. Data- dependent constraints are derived on four of these shape parameters to conserve the shape of the data while the other eight are left free to modify the monotone surface as desired. Several numerical examples are presented to show the effectiveness and capability of the scheme. The present scheme isC 1 , flexible, simple, local, and economical.

Behavioral Current-Voltage Model with Intermediate States for Unipolar Resistive Memories

In this paper, a behavioral current-voltage model with intermediate states is proposed for analog applications of unipolar resistive memories, where intermediate resistance values between SET and RESET state are used to store analog data. In this model, SET and RESET behaviors are unified into one equation by the blending function and the percentage volume fraction of each region is modeled by the Johnson-Mehl-Avrami (JMA) equation that can describe the time-dependent phase transformation of unipolar memory. The proposed model is verified by the measured results of TiO₂ unipolar memory and tested by the SPECTRE circuit simulation with CMOS read and write circuits for unipolar resistive memories. With the proposed model, we also show that the behavioral model that combines the blending equation and JMA kinetics can universally describe not only unipolar memories but also bipolar ones. This universal behavioral model can be useful in practical applications, where various kinds of both unipolar and bipolar memories are being intensively studied, regardless of polarity of resistive memories.

Linear independence of the blending functions of T-splines without multiple knots

The purpose of the present paper is to answer the following open question given by Sederberg’s research team: are there T-splines with linearly dependent blending functions that do not have multiple knots? First, the mathematical properties of the blending functions of T-splines without multiple knots are analyzed. Then, the linear independence of them is proved, and meanwhile a necessary condition is deduced for the linear dependence of T-spline blending functions. Finally, the answer to the open question is obtained: there are no T-splines with linearly dependent blending functions that do not have multiple knots.