Non-uniform Spline Research Articles

A non-uniform compound strip method for effective buckling analysis of stiffened structures

In this paper, a versatile compound strip method (CSM) with non-uniform spline functions is developed for buckling analysis of stiffened structures. The proposed method further extends the existing CSM by permitting the flexible modification of local knots to place the stiffeners along the strip arbitrarily. In addition, a reasonable knot arrangement can achieve an accurate solving procedure with improved convergence. The displacements of stiffeners are expressed compatibly by the fundamental parameters of skin according to the beam-plate model. Consequently, the reinforcement of stiffeners is naturally incorporated by adding the beam stiffness matrix to the corresponding strips. Based on the first-order shear deformation plate theory and non-uniform spline functions, the semi-analytical formulations of stiffened structures are established. The present method provides the possibility to achieve the local mesh refinement in the finite strip methods. The convergence and validation study is demonstrated through the comparisons with the results of the existing literature solution and finite element method. In conclusion, a more efficient and applicable CSM is proposed to analyze the buckling behaviors of stiffened structures.

Non-uniform splines for semi-Lagrangian kinetic simulations of the plasma sheath

Numerical methods based on non-uniform splines of varying degrees are used to run kinetic semi-Lagrangian simulations of the plasma sheath. The sheath describes a region of plasma in contact with a wall, acting as a heat, momentum, and particle bath. The latter regulates the current loss and consequently the electric field. At micro scale it then influences the development of plasma turbulence and at a larger scale plasma rotation. However this region is particularly difficult to simulate, due to its kinetic nature and the presence of steep gradients requiring fine numerical resolution. In this work, a new well-conditioned method for determining the coefficients for Schoenberg's “best” quadrature is presented, providing improved precision compared to a naive approach based on b-spline derivatives (4 extra significant figures of precision are obtained for coefficients of the fifth order scheme). The construction of a simulation grid from Greville abcissae of judiciously chosen non-uniform knots is shown to reduce the memory requirements by 89% for the simulation presented. Cubic splines are found to provide a good compromise between fast convergence and controlled oscillations preserving positivity. Despite the reduction in performance associated with using non-uniform spline schemes, effective parallelisation through the use of GPUs leads to non-uniform simulations running 5.5 times faster than uniform simulations providing equivalent results.

Dynamic modelling of gear–spline system with nonuniform spline clearance and wear evolution of spline

The gear–spline system consisting of the internal spline of gears and the external spline of shafts is widely used in mechanical transmission systems. However, considering the nonuniform spline clearance caused by the assembly and manufacturing errors, the vibration characteristics of the system and the spline wear mechanism are still unclear. This article proposed a dynamic model of the gear–spline system with nonuniform spline clearance and wear evolution of spline. Compared with the existing models, the new contribution of this article is to reveal the coupling vibration mechanism of the gear–spline system under the condition of nonuniform spline clearance through the dynamic model. On this basis, the support stiffness characteristics and wear evolution law of the spline pair under the dynamic gear meshing force are studied. The results show that under the condition of nonuniform clearance, the displacement response of the system is modulated by the rotation frequency, and the motion trajectory tends to be chaotic. The load distribution of spline teeth and the equivalent stiffness of the spline depends on the number of loaded spline teeth, which in turn depends on the input torque and clearance distribution. In addition, the spline teeth with smaller clearance bear more loads and wear faster, and the wear causes the nonuniform clearance to gradually increase and become uniform. This study can provide theoretical guidance for revealing the vibration mechanism of the gear–spline system and calculating the support stiffness of the spline shaft.

The “Skin Effect” Assessment of Chloride Ingress into Concrete Based on the Identification of Effective and Apparent Diffusivity

The method of assessment of the “skin effect” for chloride ingress into concrete has been proposed, based on the inverse problem for the identification of at-surface variability of chloride diffusivity under fully saturated conditions. For this purpose, experimental results of 180-day diffusion tests of five types of concrete were used, which allowed the calculation of their chloride apparent diffusivity (taking into account the chloride binding by the cement matrix) and effective diffusivity (relating to the transport of free chloride ions in the pore liquid). The tested concrete samples with a water to cement ratio of 0.5 differed only in the type of cement (high early strength Portland, low-alkali normal early strength Portland, ash Portland, blast furnace, and pozzolanic). In order to effectively describe the chloride binding isotherm, a first-degree non-uniform spline function was used. Finally, the “skin effect” depth at the untreated outer surface of the concrete samples was estimated up to about 5 mm when analyzing space variability of apparent chloride diffusivity for four types of concrete with low-alkali normal early strength Portland, ash Portland, blast furnace, and pozzolanic cement. In this respect, the “skin effect” on the concrete with high early strength Portland cement was not detected.

Non-Uniform Spline Quasi-Interpolation to Extract the Series Resistance in Resistive Switching Memristors for Compact Modeling Purposes

An advanced new methodology is presented to improve parameter extraction in resistive memories. The series resistance and some other parameters in resistive memories are obtained, making use of a two-stage algorithm, where the second one is based on quasi-interpolation on non-uniform partitions. The use of this latter advanced mathematical technique provides a numerically robust procedure, and in this manuscript, we focus on it. The series resistance, an essential parameter to characterize the circuit operation of resistive memories, is extracted from experimental curves measured in devices based on hafnium oxide as their dielectric layer. The experimental curves are highly non-linear, due to the underlying physics controlling the device operation, so that a stable numerical procedure is needed. The results also allow promising expectations in the massive extraction of new parameters that can help in the characterization of the electrical device behavior.

МОДЕЛИРОВАНИЕ АРХИТЕКТУРНОЙ ФОРМЫ ПОВЕРХНОСТЬЮ ЗАВИСИМЫХ СЕЧЕНИЙ

For the design of surfaces in architecture, as a rule, universal techniques developed for other technical industries are used. First of all, these are general kinematic surfaces and interpolation cubic splines for modeling complex piecewise smooth surfaces. The authors propose to use the fourth degree inerodifferential spline developed by them for problems of geometric modeling of architectural forms. For calculations and constructions on a computer, the proposed spline is not much more complicated than traditional cubic splines, since it has one additional parameter - a coefficient. However, this allows you to locally control the shape of a curve or surface during design, that is, to change the shape in individual areas without affecting other areas. The article proposes a method for constructing a geometric model of the kinematic surface of dependent sections with a fourth degree parabola as a generator. When using cubic splines as a guide, the surface is a 3 × 4 non-uniform (heterogeneous) spline. The article shows that the surface on the basis of the proposed mathematical apparatus can be composite piecewise-smooth. A particular case of surface design is considered on the example of creating a model of the surface of the facade of a residential building according to the existing concept. The algorithm can be easily programmed and added as a tool to existing CAD systems.

A feature point scheme for improving estimation of the temporally varying bottom friction coefficient in tidal models using adjoint method

Refined estimation of bottom friction coefficient (BFC) is significantly important for coastal engineers to determine precisely the hydrodynamic conditions and sediment transport rates, which can be realized by assimilating observations of sea surface elevation using the adjoint method. In this study, a new method, named as feature point scheme (FPS), is developed to further improve the estimation of temporally varying BFC. In FPS, BFCs at some feature points are assumed to be independent and BFCs at other temporal points are obtained by interpolating BFCs at feature points.By assimilating artificial observations in twin experiments, it is demonstrated that FPS can improve the estimated temporally varying BFC, especially with the nonuniformly distributed feature points and nonuniform spline interpolation (nonuniform FPS). The M2 tide and four principal tidal constituents in the Bohai Sea are then simulated by assimilating real satellite-retrieved observations with FPS. In all the experiments, the performance of the tidal model with nonuniform FPS outperforms those using other schemes. The experimental results indicate that the temporally varying BFC is related to the tidal constituents. Moreover, the temporal variation of estimated BFC has a significantly negative correlation with the current speed, which is possibly related with the sediment transport.

A cylindrical profile measurement method for cylindricity and coaxiality of stepped shaft

The cylindricity and coaxiality of the stepped shaft are important parameters reflecting the vibration characteristics of the rotating machines system. In order to evaluate the cylindricity and coaxiality of the stepped shaft accurately, the paper proposes a cylindrical profile measurement model, which takes into account five kinds of systematic errors and analyzes the combined effects of five systematic errors on cylindrical profile measurement results. Based on the proposed measurement model, the distribution of sample angle deviation caused by five systematic errors is studied. A non-uniform spline filter is designed, and the effect of uniform and non-uniform spline filter methods on the cylindricity and coaxiality of stepped shaft are analyzed. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors and the spline filter method proposed in the paper, a rotary measuring instrument with high precision is built. The experimental results show that compared with the traditional measurement method, the measurement accuracy of cylindricity and coaxiality can be improved by 11.43 μm and 8.99 μm for stepped shaft with large radius on the proposed method, respectively. The proposed method is suitable for small probe radius and large eccentricity error, probe offset error, workpiece tilt error, and guide rail tilt error, especially for stepped shaft with large radius, such as aerospace engine, gas turbine, and machine tools.

Construction of Scaling Partitions of Unity

Partitions of unity formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames.

Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This leads us to the definition of the generalized Beppo-Levi space M, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV) over M, subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that non-uniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or TV-like regularization constraints. Remarkably, the spline-type is fully determined by the choice of L and does not depend on the actual nature of the measurements.

Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis

We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.

Cubic Trigonometric Nonuniform Spline Curves and Surfaces

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomialB-spline curves and enjoy some other advantageous properties for engineering design. They haveC2continuity at single knots. For equidistant knots, they haveC3continuity andC5continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.

Non-uniform spline recovery from small degree polynomial approximation

We investigate the sparse spikes deconvolution problem onto spaces of algebraic polynomials. Our framework encompasses the measure reconstruction problem from a combination of noiseless and noisy moment measurements. We study a TV-norm regularization procedure to localize the support and estimate the weights of a target discrete measure in this frame. Furthermore, we derive quantitative bounds on the support recovery and the amplitude errors under a Chebyshev-type minimal separation condition on its support. Incidentally, we study the localization of the knots of non-uniform splines when a Gaussian perturbation of their inner-products with a known polynomial basis is observed (i.e. a small degree polynomial approximation is known) and the boundary conditions are known. We prove that the knots can be recovered in a grid-free manner using semidefinite programming.

Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials

In this work we consider the problem of recovering non-uniform splines from their projection onto spaces of algebraic polynomials. We show that under a certain Chebyshev-type separation condition on its knots, a spline whose inner-products with a polynomial basis and boundary conditions are known, can be recovered using Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of ‘dual’ interpolating polynomials and is based on Candès and Fernandez-Granda (2014), where the theory was developed for trigonometric polynomials. We also show results for the multivariate case.

Non-uniform interpolatory subdivision via splines

We present a framework for deriving non-uniform interpolatory subdivision algorithms closely related to non-uniform spline interpolants. Families of symmetric non-uniform interpolatory 2n-point schemes of smoothness Cn−1 are presented for n=2,3,4 and even higher order, as well as a variety of non-uniform 6-point schemes with C3 continuity.

Fast Jacobian Mie Library for Terrestrial Hydrometeors

This paper presents an approach for the fast accurate computation of several useful Mie-based parameters for homogeneous, spherical, liquid water, and ice hydrometeor distributions over a wide range of frequencies, mean hydrometeor diameters, and physical temperatures as occur in the terrestrial atmosphere. The absorption coefficient, scattering coefficient, backscattering coefficient, and phase asymmetry parameters are cast into functions of three independent variables: frequency, temperature, and mean diameter. An exponential drop size distribution with a constant fractional volume of is used to model polydispersed hydrometeors. The ranges used for frequency, temperature, and mean diameter are [1, 1000] GHz, , and [0.002, 20] mm, respectively. The functions are then sampled on a logarithmic grid. Trivariate cubic spline interpolation using nonuniform basis splines (B-splines) is then used to efficiently represent these 3-D functions in a compact library. By using this method, four important criteria are achieved: 1) fast random computability of any of these parameters given the values of frequency, temperature, and mean diameter; 2) minimal memory usage by storage of only B-spline coefficients; 3) representation of parameters using well-behaved functional forms amenable to analytical differentiation for the evaluation of Jacobians, or alternatively, for higher accuracy, B-spline coefficients calculated using true Jacobian values can be used; and 4) negligibly small and bounded error over the entire domain of the library. These procedures result in considerable acceleration of microwave radiative transfer simulations across a broad frequency spectrum, as demonstrated in calculations for both scattering and nonscattering atmospheres. The methods discussed can also be applied to other geophysical problems requiring rapid calculation of series-based functions of several independent variables, where the function evaluation is a time-consuming process, and maximum error bounds are critical.

Acquisition and Analysis on 3D Deformation of Windshield Glass in Vehicle-Pedestrian Accident

A method based on the 3D laser scanning is proposed for obtaining and analyzing the windshield's deformation in vehicle-pedestrian accident. With an actual traffic accident of vehicle-pedestrian collision taken for example, point cloud data of windshield’s deformation is scanned by EXAScan 3D laser scanner and the deformed 3D NURBS surface is obtained after fitting process. Based on the discussion about characteristics of curved surface of the vehicle’s front windshield and the principle of continuous curvature, this paper conducts a surface fitting with cubic non-uniform spline according to geometric information of grid surface around the deformation zone, and obtains the pre-deformed surface of windshield by sampling with spline curve. Compared with the pre-deformed surface of vehicle’s windshield, the maximum deformation depth is 48.509mm and its area is 0.3816m2on the right side, while the maximum deformation depth is 36.341mm and its area is 0.2529m2on the left side. The application results in actual traffic accident show that this method can be used to obtain 3D deformation information of overall impact area, which provide a new method for accurately inspecting the windshield’s deformation in vehicle-pedestrian accident. This method also provides a new research idea for quantitative saving the evidence in permanent, and it also has a great value in accident analysis and reconstruction.

Reconstruction of aperiodic FRI signals and estimation of the rate of innovation based on the state space method

In the absence of observation noise, it is known that it is possible to develop exact sampling schemes for a large class of parametric non-bandlimited signals, namely, certain signals of finite rate of innovation (FRI signals), either periodic or aperiodic, such as streams of Diracs, nonuniform splines or piecewise polynomials. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and they can be reconstructed from a finite number of uniform samples of the filtered signal. Unfortunately, the accuracy of such reconstruction substantially degrades when the samples are distorted by noise. For the case of periodic FRI signals, good algorithms based on the state space method have been proposed which are robust against noise. However, in the case of aperiodic signals, these algorithms may still fail to accurately reconstruct the signals due to ill-conditioning problems that often arise in the matrices involved. This paper proposes a new reconstruction method for aperiodic FRI signals that is also based on the state space method but has a considerably better numerical conditioning than previous reconstruction algorithms. This advantage is achieved by using a frequency domain formulation.

Sampling and reconstruction of signals in a reproducing kernel subspace of [formula omitted

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of L p ( R d ) , 1 ⩽ p ⩽ ∞ , associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of p-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of L p ( R d ) . We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation–projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap.

Wavelets-based NURBS simplification and fairing

This paper describes a novel simplification and fairing method for NURBS (Non-Uniform Rational B-Spline) curves and surfaces using orthogonal non-uniform spline wavelets. The compression nature of wavelets is utilized for NURBS simplification. Sufficient conditions for C 0 - and G 1 -continuity at the shared boundary of multiple NURBS patches are proved and utilized to preserve the boundary continuity. Our fairing method filters out unwanted imperfections and noise on the NURBS curves and surfaces while preserving geometric details via wavelet decomposition and selective reconstruction. The continuity between neighboring patches is preserved automatically during fairing. Our algorithm is accurate, automatic, robust, efficient, and can be extensively applied to any NURBS models in various research fields such as Computer Aided Design (CAD), Reverse Engineering (RE), finite element analysis and isogeometric analysis. Several examples are presented in this paper, including a propeller CAD model, a Navy boat model and a G 1 -continuous NURBS model of the thoracic aorta constructed from medical CT (Computed Tomography) data.