Blending Functions Research Articles

A Review on Attempts towards CAD/CAE Integration Using Macroelements

In this paper we review several aspects of older and contemporary attempts to integrate computer-aided design (CAD: geometric model) and computer-aided engineering (CAE: finite elements, boundary elements, etc.). After a short review on formulas for the description of CAD surfaces, a systematic mechanism for creating several types of corresponding isoparametric macroelements is presented. Gordon-Coons is initially applied in conjunction with piecewise linear, Lagrange polynomials and natural B-splines. Then, it is extended to more basis and blending functions. In addition to the well-known 'Lagrangian'-type elements, equivalent 'Bezierian'-type elements are introduced. Tensor product B-splines and aspects of NURBS isogeometric formulation are given. In addition to quadrilaterals, triangular macroelements based on Barnhill's interpolation are presented for the first time. The review covers applications of CAD-based macroelements in conjunction with the Galerkin-Ritz formulation, the Boundary Element Method, as well as recent Global Collocation procedures. A numerical example on a vibrating membrane elucidates the performance of the CAD-based global interpolation and depicts its superiority over the conventional finite element method.

Stress Analysis of Bonded Patched Plates with Circular Cutout Using Full-Discrete Layer Model

To predict the static response of patched plates with circular cutout, a full-discrete layer model is proposed. The shape functions are based on 2D and 1D integrals of Legendre polynomials, allowing accurate simulation of 3D behavior. Exact mapping of curved boundary is undertaken using blending functions. The easy to use modeling scheme is applied to single-patch repaired plates with circular cutout and compared with published results. The results of parametric studies on patched plates are presented by varying the relative thicknesses of patch and adhesive, patch shape, and patch size.

Trigonometric generalized T-splines

The paper’s main aim consists of extending the T-spline approach to trigonometric generalized B-splines, a particularly relevant case of non-polynomial splines. Such goal can be achieved by a careful revision of some results concerning the basic properties of the trigonometric generalized B-splines and by a formalization of the concept of T-splines in the trigonometric setting. Moreover, fundamental for the use of this new tool is the study of the noteworthy case with constant frequencies and of the linear independence of the corresponding blending functions, which can be proved to be strongly linked to the linear independence of the polynomial blending functions associated to the same T-mesh.

Evaluation of RANS Models in Predicting Low Reynolds, Free, Turbulent Round Jet

In order to study the flow behavior of multiple jets, numerical prediction of the three-dimensional domain of round jets from the nozzle edge up to the turbulent region is essential. The previous numerical studies on the round jet are limited to either two-dimensional investigation with Reynolds-averaged Navier–Stokes (RANS) models or three-dimensional prediction with higher turbulence models such as large eddy simulation (LES) or direct numerical simulation (DNS). The present study tries to evaluate different RANS turbulence models in the three-dimensional simulation of the whole domain of an isothermal, low Re (Re = 2125, 3461, and 4555), free, turbulent round jet. For this evaluation the simulation results from two two-equation (low Re k-ɛ and low Re shear stress transport (SST) k-ω), a transition three-equation (k-kl-ω), and a transition four-equation (SST) eddy-viscosity turbulence models are compared with hot-wire anemometry measurements. Due to the importance of providing correct inlet boundary conditions, the inlet velocity profile, the turbulent kinetic energy (k), and its specific dissipation rate (ω) at the nozzle exit have been employed from an earlier verified numerical simulation. Two-equation RANS models with low Reynolds correction can predict the whole domain (initial, transition, and fully developed regions) of the round jet with prescribed inlet boundary conditions. The transition models could only reach to a good agreement with the measured mean axial velocities and its rms in the initial region. It worth mentioning that the round jet anomaly is still present in the turbulent region of the round jet predicted by the low Re k-ɛ. By comparing the k and the ω predicted by different turbulence models, the blending functions in the cross-diffusion term is found one of the reasons behind the more consistent prediction by the low Re SST k-ω.

A multi-scale simulation method for high Reynolds number wall-bounded turbulent flows

We present an assessment and enhancement of the hybrid two-level large-eddy simulation method (A.G. Gungor and S. Menon, A new two-scale model for large eddy simulation of wall-bounded flows, Prog. Aerosp. Sci. 46 (2010), pp. 28–45), a multi-scale formulation for simulation of high Reynolds number wall-bounded turbulent flows. The assessment of the method is performed by examining role of static and dynamic blending functions used to perform hybridisation of two-level simulation (K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows, J. Comput. Phys. 220 (2006), pp. 290–311; K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows. Part 2: Non-homogeneous flows, J. Comput. Phys. 222 (2007), pp. 673–701) and large-eddy simulation methods. The sensitivity of first- and second-order turbulence statistics to the type of blending functions is investigated by simulating a fully developed turbulent flow in a channel at a friction Reynolds number Reτ = 395 and comparing the results with those obtained using a direct numerical simulation. The first-order statistics do not show any significant differences for different blending functions, but the second-order statistics show some minor differences. The dynamic evaluation of the hybrid region and the blending function is necessary for non-equilibrium and complex flows where use of a static blending function can lead to inaccurate results. We propose two criteria for the dynamic evaluation; first evaluates extent of the hybrid region based on the subgrid turbulent kinetic energy and the second estimates the blending function based on a characteristic length scale. The computational efficiency of the method is enhanced by incorporating a hybrid programming paradigm where a standard domain decomposition by the message-passing-interface library is combined with the open multi-processing based parallelisation. A further enhancement of the method is achieved by incorporating a closure model for the unclosed hybrid terms in the governing equations, which appear due to hybridisation of two-level- and large-eddy-simulation methods. The model is based on an order of magnitude approximation and a preliminary assessment of the model shows improvement of turbulence statistics when used to simulate turbulent flow in a periodic channel. The assessment and improvements to the multi-scale method make it more suitable for simulation of practical wall-bounded turbulent flows at higher Reynolds number than a conventional large-eddy simulation. This is demonstrated by simulating two representative cases; turbulent flow at high Reynolds number in a periodic channel and flow over a bump placed on the lower surface of a channel, where a relatively coarser computational grid is found to be sufficient for reasonably accurate results.

Analysis of Incision Effects on Upper Lip Height and Thickness After Maxillary Surgically Assisted Expansion: A Randomized Clinical Trial

To evaluate changes from maxillary circumvestibular incision performed with a #15 scalpel blade versus electrocautery on the height and thickness of the upper lip (UL) after surgically assisted maxillary expansion. Twenty-three patients who underwent surgically assisted maxillary expansion for transverse maxillary deficiency from April 2011 through April 2012 were included in the study. In group 1 (n = 11), the incision was performed with a Bard-Parker #15 scalpel blade. In group 2 (n = 12), the incision was performed with monopolar electrocautery (blend function, 1; power, 20 Watts; frequency, 480 kHz). Clinical measurements for UL height and radiographic measurements for UL height and thickness were performed preoperatively and postoperatively (mean, 6 months). The collected data were subjected to statistical analysis to test the hypotheses of the study. After excluding 2 patients, 21 patients were included in the sample. For UL height, clinical and radiographic measurements showed shortening of the UL, and the differences between pre- and postoperative values were statistically significant (P < .05) in the 2 groups. However, there was no statistically significant difference between groups 1 and 2 for UL height (P > .05) by clinical or radiographic measurements. For UL thickness, the 2 groups showed slight UL thickening in the lower portion, with no statistical difference between pre- and postoperative values (P > .05). Moreover, these results were not statistically different between the 2 groups (P = .754). In the uppermost portion of the UL, there was significant thinning in group 2 (P = .001), but not in group 1 (P = .076), and no statistical difference between groups (P = .535). Maxillary circumvestibular incision causes significant shortening of the UL and thinning of the uppermost portion when using a #15 scalpel blade or electrocautery. However, there is no difference in UL dimensional changes when using either technique for maxillary circumvestibular incision.

ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

Topology Optimization Using Nonlinear Finite Elements and Control-point-based Parametrization

Abstract This paper presents an approach to shape/topology optimization of continuous structures. The proposed approach combines the design element technique and the level set function in order to obtain an efficient topology parameterization of the domain under consideration. The shape and the level set function are both parameterized by the control points and corresponding blending functions of the design elements. For the sake of generality, nonlinear finite elements are employed, which have to be adapted adequately in order to be able to describe full material, void, and any intermediate state. In this way the design element technique has not yet been used for topology optimization, partially because it requires that the domain geometry and finite element mesh have to be defined by utilizing control-point-based design elements. In spite of this drawback, the proposed approach offers several attractive benefits. Namely, in contrast to other level set methods, the proposed approach does not make any use of the Hamilton-Jacobi differential equation. Consequently, the boundary evolution stage of the process need not to be treated separately, but is integrated with the strain/stress analysis stage into a rather conventional optimization scheme. Furthermore, the proposed approach allows for any type of finite elements (linear/nonlinear) to be implemented into the procedure if adjusted adequately. The formulation of the optimization problem is also completely arbitrary. The properties of the proposed approach are illustrated by several numerical examples.

High‐order detached eddy simulation, zonal LES and URANS of cavity and labyrinth seal flows

SUMMARYHybrid numerical large eddy simulation (NLES), detached eddy simulation (DES) and URANS methods are assessed on a cavity and a labyrinth seal geometry. A high sixth‐order discretization scheme is used and is validated using the test case of a two‐dimensional vortex. The hybrid approach adopts a new blending function. For the URANS simulations, the flow within the cavity remains steady, and the results show significant variation between models. Surprisingly, low levels of resolved turbulence are observed in the cavity for the DES simulation, and the cavity shear layer remains two dimensional. The hybrid RANS–NLES approach does not suffer from this trait.For the labyrinth seal, both the URANS and DES approaches give low levels of resolved turbulence. The zonal Hamilton–Jacobi approach on the other had given significantly more resolved content. Both DES and hybrid RANS–NLES give good agreement with the experimentally measured velocity profiles. Again, there is significant variation between the URANS models, and swirl velocities are overpredicted. Copyright © 2013 John Wiley &amp; Sons, Ltd.

An Alternative Methodology to Represent B-spline Surface for Applications in Virtual Reality Environment

AbstractB-spline representation is one of the main methods for free-form surface modeling and has become the standard for CAD systems. However, in Virtual Reality (VR) environment, when a B-spline surface deforms, the blending functions need to be continuously computed. The high computational cost of continuously calculating the blending functions for merging, collision detection and physics-based deformation system, while the model is deforming, restricts the use of B-spline representation in a VR environment. This paper presents an alternative methodology to represent B-spline surface patches for an interactive VR environment. A uniformly discretized B-spline surface patch can be represented by a set of control points and two precalculated B-spline blending matrices. The proposed technique exploits the fact that these B-spline blending matrices are independent of the position of control points and therefore can be pre-calculated. The blending matrices enable the algorithm to merge B-spline surface patch...

Analysis of circular free edge effect in composite laminates by p-convergent global–local model

In this paper the p-convergent global–local model based on layerwise theory is presented to predict the complicated patterns of stress fields around a circular hole of composite laminates under tension. A distinction of this model is to combine two-dimensional elements with three-dimensional elements in a designed mesh. In the local region with high stress gradient, three-dimensional displacement fields can be defined by layer-by-layer representation, while equivalent single-layer elements are adopted in the global region with smooth stress gradient. Also, the p-refinement in local as well as global regions is simultaneously implemented using Lobatto shape functions. Higher-order shape functions for three-dimensional elements are derived by the combination of one- and two-dimensional shape functions in a layerwise sense. In this study the orders of the shape functions are kept to be fixed as p-level (in-plane direction)=8 and q-level (thickness direction)=5 from the convergence test of in-plane and transverse stresses. The proposed model achieves compatibility displacements and stress equilibrium at the junction or interface between the different element types. Also, exact mapping of curved boundary is undertaken using blending functions. Numerical examples of curved free-edge problems have been taken into account to illustrate the performance of the present approach. Numerical results show that the proposed model is capable of predicting in-plane stresses around a circular hole as well as interlaminar stresses at the interface between layers.

Mixed-aspect fractal surfaces

In order to provide accurate tools to model original surfaces in a Computer Aided Geometric Design context, we develop a formalism based on iterated function systems. This model enables us to represent both smooth and fractal free-form curves and surfaces. But, because of the self-similarity property underlying the iterated function systems, curves and surfaces can only have homogeneous roughness. The aim of our work was to elaborate a method to build parametric shapes (curves, surfaces, …) with a non-uniform local aspect: every point is assigned a “geometric texture” that evolves continuously from a smooth to a rough aspect. The principle is to blend shapes with uniform aspects to define a shape with a variable aspect. A blending function controls the influence of each initial shape. An illustrated application is then built, joining surfaces characterized by different kinds of roughness.

Multiple Lyapunov Functions with Blending for Induced L2‐norm Control of Switched LPV Systems and its Application to an F‐16 Aircraft Model

AbstractThis paper studies the induced L2‐norm problem for switched linear parameter varying (LPV) systems using a blending method. For a switched LPV system where the parameters are grouped into slow‐varying and fast‐varying parameters, the blending method is used to construct blended Lyapunov functions based on the multiple Lyapunov functions conditions in terms of linear matrix inequalities (LMIs). The proposed method is applied to an F‐16 aircraft longitudinal model and the simulation results demonstrate the effectiveness of the approach.

Numerical Simulations of Counter Current Flow Experiments Using a Morphology Detection Algorithm

In order to improve the understanding of counter-current two-phase flows and to validate new physical models, CFD simulations of 1/3rd scale model of the hot leg of a German Konvoi PWR with rectangular cross section was performed. Selected counter-current flow limitation (CCFL) experiments at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) were calculated with ANSYS CFX 12.1 using the multi-fluid Euler-Euler modeling approach. The transient calculations were carried out using a gas/liquid inhomogeneous multiphase flow model coupled with a SST turbulence model for each phase. In the simulation, the surface drag was approached by a new correlation inside the Algebraic Interfacial Area Density (AIAD) model. The AIAD model allows the detection of the morphological form of the two phase flow and the corresponding switching via a blending function of each correlation from one object pair to another. As a result this model can distinguish between bubbles, droplets and the free surface using the local liquid phase volume fraction value. A comparison with the high-speed video observations shows a good qualitative agreement. The results indicated that quantitative agreement of the CCFL characteristics between calculation and experimental data was obtained. To validate the model and to study scaling effects CFD simulations of the CCFL phenomenon in a full scale PWR hot leg of the UPTF test facility were performed. Also these results indicated a good agreement between the calculation and experimental data. The final goal is to provide an easy usable AIAD framework for all ANSYS CFX users, with the possibility of the implementation of their own correlations.

Interpolatory blending net subdivision schemes of Dubuc–Deslauriers type

Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc–Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc–Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc–Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc–Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples.

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigono- metric interpolation introduced in Dryanov and Petrov (Interpolation and L1-approximation by trigonometric polynomials and blending func- tions, J. Approx. Theory 164:1049-1064, 2012) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite- Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with pre- scribed order.

A force-based coupling scheme for peridynamics and classical elasticity

In this work, we present a novel methodology to derive blending schemes to concurrently couple local and nonlocal models obtained from a single reference framework based upon the peridynamic theory of solid mechanics. A consistent force-based blended model that couples peridynamics and classical elasticity is presented using nonlocal weights composed of integrals of blending functions. The proposed blended model possesses desired properties of multiscale material models such as satisfying Newton’s third law and passing the patch test. This approach finds useful applications in material failure for which the peridynamics theory can be used to describe regions where fracture is expected, whereas classical elasticity could be efficiently used elsewhere. Numerical experiments demonstrating the accuracy and efficiency of the blended model are presented as well as qualitative studies of the error sensitivity on different model and problem parameters. We also generalize this approach to the coupling of peridynamics and higher-order gradient models of any order.

Construction and characterization of non-uniform local interpolating polynomial splines

This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are rational, our approach provides spline interpolants having low, and sometimes minimum degree.Thanks to its generality, the proposed framework also allows us to recover uniform local interpolating splines previously proposed in the literature, to generalize them to the non-uniform case, and to complete families of arbitrary support width. Furthermore it provides new local interpolating polynomial splines with prescribed smoothness and polynomial reproduction properties.

Development of a compressive surface capturing formulation for modelling free‐surface flow by using the volume‐of‐fluid approach

SUMMARYWith the aim of accurately modelling free‐surface flow of two immiscible fluids, this study presents the development of a new volume‐of‐fluid free‐surface capturing formulation. By building on existing volume‐of‐fluid approaches, the new formulation combines a blended higher resolution scheme with the addition of an artificial compressive term to the volume‐of‐fluid equation. This reduces the numerical smearing of the interface associated with explicit higher resolution schemes while limiting the contribution of the artificial compressive term to ensure the integrity of the interface shape is maintained. Furthermore, the computational efficiency of the the higher resolution scheme is improved through the reformulation of the normalised variable approach and the implementation of a new higher resolution blending function. The volume‐of‐fluid equation is discretised via an unstructured vertex‐centred finite volume method and solved via a Jacobian‐type dual time‐stepping approach. Copyright © 2012 John Wiley &amp; Sons, Ltd.

On the generation of the mean velocity profile for turbulent boundary layers with pressure gradient under equilibrium conditions

AbstractThe generation of a fully turbulent boundary layer profile is investigated using analytical and numerical methods over the Reynolds number range 422 ≤ Reθ ≤ 31,000. The numerical method uses a new mixing length blending function. The predictions are validated against reference wind tunnel measurements under zero streamwise pressure gradient. The methods are then tested for low and moderate adverse pressure gradients. Comparison against experiment and DNS data show a good predictive ability under zero pressure gradient and moderate adverse pressure gradient, with both methods providing a complete velocity profile through the viscous sub-layer down to the wall. These methods are useful computational fluid dynamic tools for generating an equilibrium thick turbulent boundary layer at the computational domain inflow.