Order B-spline Research Articles

A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points.

Echo state network implementation for chaotic time series prediction

The implementation of an Echo State Neural Network (ESNN) for chaotic time series prediction is introduced. First, the ESNN is simulated using floating-point arithmetic and afterwards fixed-point arithmetic. The synthesis of the ESNN is done in a field-programmable gate array (FPGA), in which the activation function of the neurons’ outputs is a hyperbolic tangent one, and is approximated with a new design of quadratic order b-splines and four integer multipliers. The FPGA implementation of the ESNN is applied to predict four chaotic time series associated to the Lorenz, Chua, Lü, and Rossler chaotic oscillators. The experimental results show that with 50 hidden neurons, the fixed-point arithmetic is good enough when using 15 or 16 bits in the fractional part: using more bits does not reduce the mean-squared error prediction. The neurons are limited to four inputs in the hidden layer to achieve a more efficient hardware implementation, guaranteeing a prediction of more than 10 steps ahead.

Edge-preserving smoothing filter using fast M-estimation method with an automatic determination algorithm for basic width

Digital filtering is essential for digital imaging, image recognition, and super-resolution technology. For example, the presence of noise in images captured by digital cameras causes deterioration of the image quality and image recognition rate. In order to improve the image recognition rate, noise reduction and edge preservation must be performed during preprocessing. Noise is generally reduced using low-pass filters, such as the Gaussian filter. Although they reduce noise, such filters also have the properties of blurring edge. A strong edge blur reduces the accuracy of the feature detection in image recognition. Therefore, in our previous study, a fast M-estimation Gaussian filter for images (FMGFI) was proposed as an image filter that simultaneously achieves denoising and edge preservation. In the FMGFI, the setting of the optimal basic width of the 2nd order B-spline basis functions is important for achieving simultaneous denoising and edge preservation. In this method, the optimal basic width of the FMGFI was determined not only by manually setting the basic width but also by human judgment of the filtered images. Consequently, the inability to automatically determine the optimal basic width hindered efficient denoising during image processing Therefore, in this research, we develop and propose a method that can automatically determine the optimal basic width of the FMGFI. The previously proposed method calculates using the same basic width for all the pixels over the entire image; in contrast, the proposed method calculates using the basic width automatically determined for each pixel. The experiments confirmed that the method proposed in this study achieves higher denoising and edge preservation performance than the ones used in previous research. The results also showed that it has the highest denoising performance against salt-and-pepper noise as compared to other filters: non-local mean filter, Gaussian filter, median filter, bilateral filter, adaptive bilateral filter, and FMGFI. The experimental results for the Gaussian noise sowed that the proposed method has the same denoising and edge preservation performance as the other filters in visual evaluation. From the above, the proposed method is expected to contribute to efficient denoising and improvement of image quality by using it as a preprocessing.

An algebraic characterization of B-splines

B-splines of order k can be viewed as a mapping N taking a (k+1)-tuple of increasing real numbers a0<⋯<ak and giving as a result a certain piecewise polynomial function. Looking at this mapping N as a whole, basic properties of B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) has local support contained in the interval [a0,ak]; (2) N(a0,…,ak) allows refinement, i.e. for every a∈∪j=0k−1(aj,aj+1) we have that if (α0,…,αk+1) is the increasing rearrangement of the points {a0,…,ak,a}, the ‘old’ function N(a0,…,ak) is a linear combination of the ‘new’ functions N(α0,…,αk) and N(α1,…,αk+1); (3) N is translation and dilation invariant. It is easy to see that derivatives of N(a0,…,ak) satisfy properties (1)–(3) as well.Let F be a mapping taking (k+1)-tuples of increasing real numbers to some generalized function. In this paper we show that under some additional mild condition on the size of the supports of F(a0,…,ak) relative to the interval [a0,ak], properties (1)–(3) are already sufficient to deduce that F(a0,…,ak) is a non-zero multiple of (some derivative of) a B-spline function. However, and somewhat surprisingly, we explicitly give examples of choices of F satisfying (1)–(3) but are not of this form.

Improving the wear resistance of 3D printed spur gears through a free-form tooth flank optimization process

Involute gears have traditionally been the preferred choice for gear transmission systems due to their simplicity and interchangeability. However, there are applications where they do not provide the most durable and efficient solution. While the cost of implementing optimized non-involute gears in most applications often outweighs their comparative advantages, the advent of additive manufacturing has opened up possibilities for designers to explore alternative gear tooth profiles. This is particularly relevant in the realm of plastic gears, where optimized non-involute gears produced through 3D printing can address their primary drawbacks, such as surface durability and wear resistance. In this study, a comprehensive free-form optimization process was conducted to determine the optimal tooth profile that minimizes wear on 3D printed spur gears during operation. The tooth flank geometry was represented using a 4th order B-spline curve, and a genetic algorithm was employed to determine the optimum positions of the control points aiming to minimize wear depth across the tooth flanks. The spur gears were manufactured using Fused Deposition Modeling (FDM) with PLA material. The parameters of the additive manufacturing process were experimentally fine-tuned to achieve the best possible accuracy. To evaluate the performance of the optimized free-form gears, two case studies were implemented, demonstrating that the optimized gears achieved a remarkable reduction of average wear depth by more than 50% and a reduction of maximum wear depth by more than 69% compared to standard involute gears. To further validate the effectiveness of the optimization method, experiments were carried out using an FZG test rig. The profiles of the tooth flanks were measured on a Coordinate Measuring Machine (CMM) before and after the experiments to compare the wear depth against the standard involute gears. The results revealed a significant improvement in the wear resistance of the tooth flanks, with a reduction of wear depth of 44.1%.

English

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline finite elements method, Exponential cubic B-spline method, Septic B-spline technique, Quintic B-spline Galerkin method, and B-spline Galerkin method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Quantifying ventilator unloading in CPAP ventilation

BackgroundThe intrinsic (muscular) patient effort driving inspiration in non-invasive ventilation modes, such as continuous positive airway pressure (CPAP) therapy, has not been identified from non-invasive data. Current CPAP settings are based on clinical judgment and assessment of symptoms of respiratory distress. Non-optimal settings, including too much positive end expiratory pressure (PEEP) can cause unintended lung injury and ventilator unloading, where patient effort drops and the CPAP device enables too much work being imposed on the injured lung. Currently, there is no non-invasive means of quantifying or identifying these effects. MethodsA novel model-based method of ascertaining intrinsic patient work of breathing (WOB) in CPAP is developed based on linear single compartment and 2nd order b-spline models previously used in invasive ventilation modes. Results are compared to current clinical indications, such as total Imposed WOB from the CPAP device and beak length, the latter of which is the clinical metric used to indicate alveolar overdistension. Intrinsic and Imposed WOB are compared.The hypothesis is that ventilator unloading can be assessed as a decrease in Intrinsic WOB relative to Imposed WOB, as PEEP and associated ventilator unloading rise. This hypothesis is tested using 14 subjects from a CPAP trial of several breathing rates at two PEEP levels. ResultsThe ratio of Intrinsic to Imposed WOB, normalised per unit tidal volume, decreased with increasing PEEP (4–7 cm H2O), capturing the expected trend of ventilator unloading. Ventilator unloading was observed across all breathing rates. Beak length measurements showed no conclusive evidence of capturing overdistension at higher PEEP or ventilator unloading. ConclusionsPatient Intrinsic WOB in CPAP was non-invasively quantified using model-based methods, based on pressure and flow measurements. The ratio of Intrinsic to Imposed WOB per unit tidal volume clearly and consistently showed ventilator unloading across all patients and breathing rates, with Intrinsic WOB decreasing with increasing PEEP. This trend was not observed in the current clinical metric of beak length. Non-invasively quantifying Intrinsic WOB and ventilator unloading is the critical first step to objectively optimising clinical CPAP settings, patient care, and outcomes.

Nonic B-spline algorithms for numerical solution of the Kawahara equation

In this paper, the nonic (9th order) B-spline functions which have not been used before for the numerical solutions of the partial differential equations by finite element methods are used to solve numerically the Kawahara equation. These approaches involve the collocation and Galerkin finite element methods based on the nonic B-spline functions in space discretization and second order scheme (Crank-Nicolson method) in time discretization. To see the accuracy of the proposed methods three test problems are demonstrated and the obtained numerical results for both of the methods are compared with the exact solution of the Kawahara equation.

Prediction of variable amplitude fatigue crack growth life based on modified grey model

Grey model has been several applications on fatigue crack, with prediction on fatigue crack growth life curve often serving as milestones. A typical difficulty in predicting the fatigue crack growth life curve is the retardation of crack growth caused by the overload effect. Overload retardation, resulting in the lack of experiment data points in the retardation interval and the deflection of crack growth curve, is a long-term challenge for predicting fatigue crack growth. We improve the ability of grey model to predict fatigue crack growth life by second order B-spline interpolation, an interpolation for local data points. It combines the trend of local data point to deal with the retardation interval, and interpolates multiple data points in the retardation interval when there is a lack of local data, and uses a metabolic grey model to construct the prediction formula of crack growth. In the study involving variable amplitude load and multiple overload fatigue crack growth experiments, the proposed method is proved to improve, with statistical significance, the predictive ability on the whole range of experiment data. The method is simple and accurate. Consequently, it is conducive to solve the problem of fatigue crack growth life of engineering structure.

On semi-analytical Stochastic Boundary Element Method and its application to eigenproblem of thin elastic plate immersed into a fluid

The main objective in this work is to study an application of the B-spline approximations in the Least Squares Method recovery of the structural response for boundary value problems in some linear elastic systems. These responses were approximated before using polynomial bases in the Stochastic perturbation-based Boundary Element Method and now are replaced with second or third order B-spline functions. A majority of such an approach is that the resulting expected values and standard deviations of structural response are obtained using analytical calculus of probability integrals, so that neither statistical nor expansion methods are necessary. Numerical illustration delivered in this paper includes fundamental eigenfrequencies of elastic thin homogeneous and isotropic plate immersed into Newtonian fluid subjected to various boundary conditions at its external edges.

A unified framework of high order structure-preserving B-splines Galerkin methods for coupled nonlinear Schrödinger systems

Using a general computational framework, we derive an optimal error estimate in the L2 norm for a semi discrete method based on high order B-splines Galerkin spatial discretizations, applied to a coupled nonlinear Schrödinger system with cubic nonlinearity. A fully discrete method based on a conservative nonlinear splitting Crank-Nicolson time step is then proposed; and conservation of the mass and the energy is theoretically proven. To validate its accuracy in space and time, and its conservation properties, several numerical experiments are carried out with B-splines up to order 7.

Data reconstruction of homogeneous turbulence using Lagrangian Particle Tracking with Shake-The-Box and machine learning

This paper proposes a data reconstruction of homogeneous turbulent flow combined machine learning (ML) approach using experimental Lagrangian Particle Tracking (LPT) data with Shake-The-Box (STB). The LPT with STB was adopted to measure a von Kármán flow with a homogeneous turbulent region in the center [1]. The STB results have been stored and a temporal filter using 3rd order B-splines has been applied with optimal weighting coefficients to be used as input for FlowFit data assimilation method [2]. FlowFit data was used as ground truth to train ML algorithm. The low-resolved data of the velocity and acceleration field was reconstructed using an Adaptive Neuro-Fuzzy Inference System (ANFIS) with the downsampled LPT data as an input to predict homogeneous turbulent flow [3]. The training process can be mathematically regarded as an optimization problem to determine the weighting factor.

Design of Pelton Turbine and Bucket Surface using Non-Uniform Rational Basis Spline and its Analysis with Computational Fluid Dynamics

Hydraulic turbines are used to convert the energy in flowing water to rotational mechanical energy. The design of high head Pelton turbine is difficult due to complex flow pattern on different parts. The basic dimensions can be obtained from interpolation techniques and design trends but the main challenge is to model the hydrodynamic surface. The surface must be designed such that it would harvest energy in an efficient manner and the manufacturing of the surface is also economic and simple. For the design of Pelton turbine, the data from Kulekhani-I hydropower is taken. Pelton turbine is designed for the given head (550 m), flow rate (6.05 m3/sec) and speed (600 rpm). For basic dimensions, interpolation techniques and design trends are used. Number of buckets is calculated for maximum efficiency condition and found to be twenty. Nozzle and spear are selected based on hydraulic efficiency. Spear angle and nozzle angle of 70° and 100° respectively are selected. The bucket surface is designed with the help of second order B-Spline. Circular and Second order B-spline surfaces with different depths are analyzed using commercial CFD code. The second order B-Spline with depth 186.22 mm is found to be more effective in terms of force exerted by the jet among the given surfaces. Finally, Bucket surface is generated using non-uniform rational B‑spline (NURBS) modeling after all the basic dimensions are determined. For verification, the design is compared with the design of commercial design software and the turbine in Kulekhani-I hydropower plant.

A new technique for numerical solution of 1D and 2D non-linear coupled Burgers’ equations by using cubic Uniform Algebraic Trigonometric (UAT) tension B-spline based differential quadrature method

In present paper, a new technique is developed named Uniform Algebraic Trigonometric (UAT) tension B-spline based DQM, for the better numerical approximation of coupled 1D and coupled 2D Burgers’ equation. The formula of UAT tension B-spline of order 4 is developed in this paper, by using above-mentioned basis function. Strong stability preserving SSP-RK 43 method is implemented to solve obtained ordinary differential equations. To check the effectiveness and accuracy of the proposed scheme, L 2 and L ∞ error norms are used. Four numerical experiments are discussed to put light upon applicability of scheme. By means of tables and figures, results obtained are given. Obtained results are in a good agreement with existing ones. Stability of proposed scheme is also checked by implementing the Matrix stability analysis method. This novel scheme will surely open some new dimensions in the field of numerical approximation, for other researchers in their future work.

Impact force analysis using the B-spline material point method

In the MPM algorithm, all the particles are formulated in a single-valued velocity field hence the non-slip contact can be satisfied without any contact treatment. However, in some impact and penetration problems, the non-slip contact condition is not appropriate and may even yield unreasonable results, so it is important to overcome this drawback by using a contact algorithm in the MPM. In this paper, the variation of contact force with respect to time caused by the impact is investigated. The MPM using the Lagrange basis function, so causing the cell-crossing phenomenon when a particle moves from one cell to another. The essence of this phenomenon is due to the discontinuity of the gradient of the linear basis function. The accuracy of the results is therefore also affected. The high order B-spline MPM is used in this study to overcome the cell-crossing error. The BSMPM uses higher-order B-spline functions to make sure the derivatives of the shape functions are continuous, so that alleviate the error. The algorithm of MPM and BSMPM has some differences in defining the computational grid. Hence, the original contact algorithm in MPM needs to be modified to be suitable in order to use in the BSMPM. The purpose of this study is to construct a suitable contact algorithm for BSMPM and then use it to investigate the contact force caused by impact. Some numerical examples are presented in this paper, the impact of two circular elastic disks and the impact of a soft circular disk into a stiffer rectangular block. All the results of contact force obtained from this study are compared with finite element results and perform a good agreement, the energy conservation is also considered.

Galerkin approximation with quintic B-spline as basis and weight functions for solving second order coupled nonlinear Schrödinger equations

In this article, the Galerkin method, based on quintic B-spline function as the shape and weight functions is described for the numerical solution of the second order coupled nonlinear Schrödinger equations. Finite difference and Crank–Nicolson schemes are used to discretize the time derivative and nodal parameters respectively. Three numerical problems are presented to assess the accuracy and capability of the proposed method. The maximum errors, norms and conserved quantities are calculated. The obtained numerical results show that the present scheme with higher order B-spline as basis and weight functions performs well and accurately. The numerical results are compared with analytical and published results.

Numerical Solution of Fifth Order Boundary Value Problems by Collocation Method with Sixth Order B-Splines

Collocation method with sixth degree B-splines as basis functions has been developed to solve a fifth order special case boundary value problem. To get an accurate solution by the collocation method with sixth degree B-splines, the original sixth degree B-splines are redefined into a new set of basis functions which in number match with the number of collocation points. The method is tested for solving both linear and nonlinear boundary value problems. The proposed method is giving better results when compared with the methods available in literature.

Fractional B-spline collection method for solving fractal-differential equations

This study used the fractional B-spline collocation technique to obtain the numerical solution of fractal-fractional differential equations. The technique was considered to solve the fractal-fractional differential equations (FFDEs) with ($0 <gamma_i < 1,, i = 1, 2, cdots, N$). In this suggested technique, the B-spline of fractional order was utilised in the collocation technique. The scheme was easily attained, efficient, and relatively precise with reduced computational work numerical findings. Via the proposed technique, FFDEs can be reduced for solving a system of linear algebraic equations using an appropriate numerical approach. The verified numerical illustrative experiments were presented will show the effectiveness of the technique proposed in this study in solving FFDEs in three cases of nonlocal integral and differential operators namely power law kernel, when the kernels are exponential and the generalization of Mittag-Leffler kernel. The approximate solution is very good and accurate to the exact solution.

Thermally Induced Vibration Analysis of Flexible Beams Based on Isogeometric Analysis

Spacecraft flexible appendages may experience thermally induced vibrations (TIV) under sudden heating loads, which in consequence will be unable to complete their intended missions. Isogeometric analysis (IGA) utilizes, in an isoparametric concept, the same high order and high continuity non-uniform rational B-splines (NURBS) to represent both the geometry and the physical field of the structure. Compared to the traditional Lagrange polynomial based finite element method where only C0-continuity across elements can be achieved, IGA is geometrically exact and naturally fulfills the C1-continuity requirement of Euler–Bernoulli (EB) beam elements, therefore, does not need extra rotational degrees-of-freedom. In this paper, we present a thermally induced vibration analysis framework based on the isogeometric method where thermal and structural behaviors are coupled. We fully exploited the higher order, higher continuous and geometric exactness of the NURBS basis with both benchmarks and sophisticated problems. In particular, we studied the thermally induced vibrations of the Hubble Space Telescope (HST) solar panel where main factors influencing thermal flutters are studied, and where possible improvements of the analytical reference methods are discussed. Additionally, thermally induced vibrations of the thin-walled lenticular tubes are studied and two new configurations of the tube are proposed to effectively suppress the thermally induced vibrations. Numerical examples of both benchmarks and sophisticated problems confirm the accuracy and efficiency of the isogeometric analysis framework for thermally induced vibration analysis of space structures.

Pilot study of model-based estimation of inspiratory driving pressure in CPAP ventilation

Abstract Models of lung elastance, airway resistance, and patient work of breathing have been successfully applied to invasive mechanical ventilation data. Non-invasive mechanical ventilation data, including continuous positive airway pressure (CPAP), has presented challenges in predicting inspiratory driving pressure due to the combination of patient and device work. The model applied in this paper utilizes second order b-splines to describe inspiratory driving pressure. The model provided an accurate fit to the data, with an average root-mean-squared (RMS) error in model fit of 0.6 [0.425 – 0.675] cmH2O (median [lower quartile (LQ), upper quartile (UQ)]). Subject fit expiratory elastances were between 3.1 and 10.2 cmH2O/L and showed no correlation to set positive end-expiratory pressure (PEEP). Inspiratory driving pressure profiles approximated literature and work of breathing was shown to remain consistent between PEEP levels. Outlying data is hypothesized to be caused by subjects’ expiratory effort which was assumed negligible in the model. Further application of this model alongside validation data would provide more information on this and provide more evidence of model accuracy.