Quadratic Basis Functions Research Articles

One Approach to the Numerical Solution of Mass-Transfer Problems with Large Péclet Numbers

We consider a mathematical model of propagation of drugs in the wall of a vessel. The model is described by the advection-diffusion boundary-value problem with a system of two differential equations. In view of the specific features of the input parameters of the problem, when the advection coefficient significantly exceeds the diffusion coefficient, the application of the classical finite-element method with linear and quadratic basis functions leads to the loss of stability of the computational process. We propose a new approach to the solution of the advection-diffusion problems with large Peclet numbers based on the replacement of the unknown function by an exponential function in the formulation of the problem and the inverse replacement prior to the application of the finite-element method. We also perform the numerical analysis of the results obtained by the application of the proposed method for the approximate solution of the problem of propagation of drugs in the wall of a vessel.

Generalized Petrov-Galerkin time finite element weighted residual methodology for designing high-order unconditionally stable algorithms with controllable numerical dissipation

In this paper, a novel stabilized time-weighted residual methodology under the umbrella of Petrov-Galerkin time finite element formulation is developed to design a generalized computational framework, which permits unconditionally stable, high-order time accuracy, and features with controllable numerical dissipation for solving transient first-order systems. Various unconditionally stable (A/L-stable) algorithms can be readily obtained in the proposed framework having not only high-order accuracy but also controllable numerical dissipation in the high frequency. Quadratic and cubic basis functions are utilized to illustrate the specific design process of the proposed method, which consequently ends up with numerous third-/fifth-order time accurate algorithms, QUAD3(γ,ρ∞) and CUBE5(γ,ρ∞), with controllable numerical dissipation. Comparing with the well-known implicit Runge-Kutta (RK) family of algorithms, these newly developed QUAD3(γ,ρ∞) and CUBE5(γ,ρ∞) can (a) recover the Radau IIA3/RK3 and Radau IIA5/RK5 schemes by certain selection of algorithmic parameters (γ,ρ∞); (b) obtain numerous new algorithms with improved solution accuracy that is superior to the RK family of algorithms, and (c) has similar computational efficiency as that of the implicit RK family of algorithms. Several single/multi-degree of freedom (SDOF and MDOF) problems are investigated to validate the proposed developments. In addition, it is worth noting that the proposed methodology can also use high-order (not limited to the quadratic and cubic) basis functions to design more advanced schemes and can be integrated with high-order spatial discretization methods in the solution of space-time PDEs, such as isogeometric methods, SEM, Discontinuous Galerkin (DGM), p-versions FEM, etc.

Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method

The fractional Langevin equation is of great scientific significance and engineering application value. Based on the classical block-by-block method, the numerical solution of a class of fractional Langevin equations with Caputo derivatives was obtained. Through introduction of the quadratic Lagrange basis function interpolation, the block-by-block convergent nonlinear equations were constructed, and the numerical solution of the Langevin equation was obtained by coupling in each block. Under the condition of 0<α<1, the stochastic Taylor expansion was used to prove that the block-by-block method is (3+α)-order convergent. Numerical experiments show that, the block-by-block method is stable and convergent under different values of α and time step <i>h</i>,and overcomes the existing methods’ disadvantages of slow speed and poor accuracy for solving fractional Langevin equations.

A numerical study on inverse quadratic optimal shape parameter in interpolation problems

The main purpose of this work is to shed more light into the inverse quadratic radial basis function (RBF) in the application of interpolation. This RBF contains a parameter that plays a crucial role in determining the final result quality. Five strategies of variable shape parameters are numerically investigated. Both types of node distribution; normally and scattered, are considered. It is discovered that good results can be obtained when using some strategies particularly in 1D problem. Challenges become appearing when dealing with 2D and deserves further study.

Modelling and optimal state-delay control in microbial batch process

In this paper, we consider optimal control problem involving a time-varying state-delay system arising in 1,3-propanediol microbial batch process. The dynamic system in this problem includes unknown time-varying delay function and unknown kinetic parameters. To optimally determine the unknown delay function and unknown kinetic parameters in the system, the weighted least-squares error between the computed values and experimental data is minimized subject to path constraints. By parameterizing the delay function with piecewise quadratic basis functions, the optimal state-delay control problem is approximated by a sequence of parameter optimization problems. Furthermore, an exact penalty method is utilized to transform these parameter optimization problems into the ones only with box constraints. On this basis, a modified differential evolution algorithm is developed to solve the resulting optimization problems. Finally, numerical results are presented to verify the effectiveness of the developed solution approach.

Algorithms for Numerical Solution to the Problem of Diffusion in Nanoporous Media

Introduction. Mathematical modeling of mass transfer in heterogeneous media of microporous structure and construction of solutions to the corresponding problems of mass transfer was considered by many authors [1–9, etc.]. In [6, 7] authors proposed a methodology for modeling mass transfer systems and parameter identification in nanoporous particle media (diffusion, adsorption, competitive diffusion of gases, filtration consolidation), which are described by non-classical boundary and initial-boundary value problems taking into account the mutual influence of micro- and macro-transfer flows, heteroporosity, the structure of microporous particles, multicomponent and other factors. In [8, 9] for a mathematical model of nonstationary diffusion of a single substance in a nanoporous medium described in [2] in the form of a multi-scale differential mathematical problem, the classical problems in the weak formulation were obtained. In this paper, algorithms for solving the above mathematical problems are constructed by using the finite element method. The results of the numerical solution of the test problem are presented. The results confirm the efficiency of the developed algorithms. The purpose is to solve a problem of nonstationary diffusion of single substance in nanoporous medium by constructing discretization algorithms using FEM quadratic basis functions. Results. Algorithms for the numerical solution of the problem of nonstationary diffusion of single substance in a nanoporous medium are proposed. Peculiarities of discretization of the region and construction of the matrix of masses, stiffness, and vector of right-hand sides when solving the problem by using FEM are described. The efficiency of the developed algorithms is confirmed by the results of solving a model example. Keywords: mathematical modeling, numerical methods, nonstationary diffusion, nanoporous medium, finite element method.

Optical diagnosis of hepatitis B virus infection in blood plasma using Raman spectroscopy and chemometric techniques

AbstractThe potential of Raman spectroscopy has been utilized for the diagnosis of hepatitis B virus (HBV) infection in blood plasma. Raman spectra of 24 diseased and 10 healthy samples were used to develop distinct types of support vector machine (SVM) models, including linear, quadratic, and radial basis function (RBF) using multivariate method of principal component analysis (PCA) to reduce the dimensions of the obtained datasets. To assess the diagnostic power of these algorithms, developed models were tested on independent dataset. RBF‐based PCA‐SVM model achieved the best performance and yielded accuracy of 98.82%, sensitivity of 98.89%, and specificity of 98.80%. The performance of the SVM models was compared with rerated chemometric method of partial least square regression (PLSR), which has been developed by using the same dataset. The PLSR model attained the diagnostic accuracy of 88%, sensitivity of 93%, and specificity of 78% for same dataset. Our developed model has established promising results compared with state‐of‐the‐art approaches. The results reveal the improved performance of the developed chemometric techniques and clinical prediction potential of HBV by PCA‐SVMs in conjunction with Raman spectroscopy.

An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

A quadratic finite volume (FV) method for steady‐state Riesz space‐fractional diffusion equations (sFDEs) with variable diffusivity coefficient is developed using piecewise quadratic basis functions, and a resulting linear algebra system with two‐by‐two block‐type Toeplitz‐like coefficient matrix is formulated. It is proved that the method requires a minimum memory of order , where is the number of spatial partitions. Moreover, as two of the produced Toeplitz‐like submatrices are not square, a new fast nonsquare Toeplitz‐like matrix‐vector product is specially designed, which requires an almost linear computational complexity of order . Then, a fast version of biconjugate gradient stabilized (BiCGSTAB) solution algorithm, named FBiCGSTAB, is proposed for the FV scheme. The FV method combined with Crank‐Nicolson (CN) time discretization is applied to solve time‐dependent sFDEs, and an efficient CN‐FV scheme is developed and analyzed. Finally, numerical results are presented to show the utility of the fast FV and fast CN‐FV methods.

Educational Data Classification using Data Mining and Kernel Ensemble Classifier

The success of students gives the good name for institution and it become popular. Due to the large number of student’s database it is difficult to identify the performance and activities of each student. The educational data mining is used to identify the performance and status of the students individually. In this study, the Educational Data Classification (EDC) using data mining technique and kernel ensemble classification using Support Vector Machine (SVM) based kernels like linear, polynomial, quadratic and Radial Basis Function (RBF) is discussed. Initially the data preprocessing is made to remove the raw data into understandable format. The SVM kernels like linear, polynomial, quadratic and radial basis function based ensemble classifier is used for classification of student’s data. The data mining is used for making final decision of student’s performance in class like activities and interaction with electronic learning system. The performance of the system is evaluated by kalboard 360 database. The performance of the system is made by classification accuracy of 72.52% using SVM kernel ensemble classification

FINMHD: An Adaptive Finite-element Code for Magnetic Reconnection and Formation of Plasmoid Chains in Magnetohydrodynamics

Abstract Solving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small-scale magnetic islands called plasmoids. A new code is thus presented for the solution of two-dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end. A current–vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite-element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic-Galerkin scheme. The adaptivity of the code is illustrated on simplified test equations and finally for magnetic reconnection associated with the nonlinear development of the tilt instability between two repelling current channels. Varying the Lundquist number S has allowed us to study the transition between the steady-state Sweet–Parker reconnection regime (for S ≲ 104) and the plasmoid-dominated reconnection regime (for S ≳ 105). The implications for the understanding of the mechanism explaining the fast conversion of free magnetic energy in astrophysical environments such as the solar corona are briefly discussed.

Pulse shape discrimination based on Fractional Discrete Cosine Transform

One of the most serious errors in PET systems is the parallax error, which occurs due to spatial resolution limitation in PET. This error can be solved by using multilayer scintillation detector (phoswish detector) where each layer has a different decay constant. Each phoswish detector is optically coupled to a single PMT. Hence, by applying pulse shape discrimination (PSD) methods to identify the scintillated layer, the parallax error can be reduced. This paper proposes a PSD method relied on a Fractional Discrete Cosine Transform (FrDCT) and a Support Vector Machine (SVM) classifier. The proposed method can be used to differentiate between two arbitrary types of scintillation pulses where each type has a different decay constant. For demonstration of its performance, the proposed FrDCT-based method is applied to a data set of scintillation pulses of LSO and LuYAP crystals. Furthermore, FrDCT with different factors as well as different kernels of SVM; such as Linear, Quadratic, and Radial Basis Function (RBF) kernels, are used to study their effects on the discrimination efficiency. The highest achieved efficiency is equal to 95.9% for the 0.9 factor of FrDCT using a quadratic kernel SVM. On the other hand, the linear kernel consumes the lowest execution time at the expense of slightly reducing the discrimination efficiency.

An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture

A reproducing kernel gradient smoothing meshfree formulation is proposed for the fourth-order phase field modeling of brittle fracture. In order to circumvent the complexity and lower efficiency of meshfree gradient computation, a reproducing kernel gradient smoothing formulation is presented with particular reference to quadratic basis functions. Both first- and second-order meshfree smoothed gradients are discussed, in which the first-order smoothed gradients are expressed as a reproducing kernel form and meet the quadratic consistency conditions of Galerkin weak form, and the second-order smoothed gradients are then computed through performing a direct differentiation on the first-order smoothed gradients. It turns out that the resulting second-order smoothed gradients satisfy the standard gradient reproducing conditions of meshfree approximations. Subsequently, the smoothed gradients of meshfree shape functions are employed to discretize the Galerkin weak forms of equilibrium and crack phase field equations, where the numerical integration and smoothed gradient construction are specially emphasized. The tensile–compressive strain decomposition is adopted to prevent compressive crack evolution. Numerical results demonstrate the effectiveness of the proposed gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture.

Numerical prediction of wave added resistance using a Rankine Panel method

Based on a time domain Rankine panel method, a computer code was developed to predict wave added resistance. The unknown variables of the discretized flow field boundaries were represented using the quadratic B-spline basis function; the temporal derivative of the free surface conditions were approximated using an Euler scheme. The added resistance was computed via a near-field method. Three numerical approaches were introduced, namely, choosing the boundary end condition, neglecting the second order derivative of the disturbed potential, and computing the water line integral. Numerical computations were conducted in short and long waves for a Wigley hull, the S-175 container ship, and the KVLCC2 tanker. The double-body linearization and the Neumann–Kelvin linearization were tested, and numerical results were compared to published experimental measurements and numerical data. Fairly good agreement was found between the results of double-body linearization and the published data. Effects of wave radiation and diffraction on added resistance were analyzed, demonstrating that the interaction between radiation and diffraction should not be omitted in long waves.

Superconvergent isogeometric analysis of natural frequencies for elastic continua with quadratic splines

A superconvergent isogeometric formulation is presented to accurately analyze the natural frequencies for elastic continua. This formulation is realized by a set of superconvergent quadrature rules which are designed for the numerical integration of isogeometric mass and stiffness matrices. In order to obtain these quadrature rules, a natural frequency error measure for elastic continua is systematically deduced using the quadratic basis functions, where the mass and stiffness matrices are formulated by the assumed quadrature rules. In contrast to the quadrature-based superconvergent isogeometric formulation for the scalar-valued wave equations, it is shown that herein different quadrature rules are required for the mass and stiffness matrices to achieve the superconvergence of natural frequency computation for the vector-valued elastic continuum problems. Consequently, the superconvergent quadrature rules are established through optimizing the natural frequency accuracy. It turns out that with these quadrature rules, the accuracy of natural frequencies for elastic continua is improved by two orders compared with the standard isogeometric formulation employing the consistent mass matrix. Meanwhile, it is found that the superconvergent quadrature rules involve the wave propagation angle and consequently simplified quadrature rules without the angle dependence are further proposed for straightforward practical applications. Numerical results reveal the superconvergence of the proposed method regarding the natural frequencies for elastic continua.

Compressible-flow geometric-porosity modeling and spacecraft parachute computation with isogeometric discretization

One of the challenges in computational fluid–structure interaction (FSI) analysis of spacecraft parachutes is the “geometric porosity,” a design feature created by the hundreds of gaps and slits that the flow goes through. Because FSI analysis with resolved geometric porosity would be exceedingly time-consuming, accurate geometric-porosity modeling becomes essential. The geometric-porosity model introduced earlier in conjunction with the space–time FSI method enabled successful computational analysis and design studies of the Orion spacecraft parachutes in the incompressible-flow regime. Recently, porosity models and ST computational methods were introduced, in the context of finite element discretization, for compressible-flow aerodynamics of parachutes with geometric porosity. The key new component of the ST computational framework was the compressible-flow ST slip interface method, introduced in conjunction with the compressible-flow ST SUPG method. Here, we integrate these porosity models and ST computational methods with isogeometric discretization. We use quadratic NURBS basis functions in the computations reported. This gives us a parachute shape that is smoother than what we get from a typical finite element discretization. In the flow analysis, the combination of the ST framework, NURBS basis functions, and the SUPG stabilization assures superior computational accuracy. The computations we present for a drogue parachute show the effectiveness of the porosity models, ST computational methods, and the integration with isogeometric discretization.

Optimization of black-box problems using Smolyak grids and polynomial approximations

A surrogate-based optimization method is presented, which aims to locate the global optimum of box-constrained problems using input–output data. The method starts with a global search of the n-dimensional space, using a Smolyak (Sparse) grid which is constructed using Chebyshev extrema in the one-dimensional space. The collected samples are used to fit polynomial interpolants, which are used as surrogates towards the search for the global optimum. The proposed algorithm adaptively refines the grid by collecting new points in promising regions, and iteratively refines the search space around the incumbent sample until the search domain reaches a minimum hyper-volume and convergence has been attained. The algorithm is tested on a large set of benchmark problems with up to thirty dimensions and its performance is compared to a recent algorithm for global optimization of grey-box problems using quadratic, kriging and radial basis functions. It is shown that the proposed algorithm has a consistently reliable performance for the vast majority of test problems, and this is attributed to the use of Chebyshev-based Sparse Grids and polynomial interpolants, which have not gained significant attention in surrogate-based optimization thus far.

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

Angular discretization impacts nearly every aspect of a deterministic solution to the linear Boltzmann transport equation, especially in the presence of magnetic fields, as modeled by a streaming operator in angle. In this work a novel stabilization treatment of the magnetic field term is developed for an angular finite element discretization on the unit sphere, specifically involving piecewise partitioning of path integrals along curved element edges into uninterrupted segments of incoming and outgoing flux, with outgoing components updated iteratively. Correct order-of-accuracy for this angular framework is verified using the method of manufactured solutions for linear, quadratic, and cubic basis functions in angle. Higher order basis functions were found to reduce the error especially in strong magnetic fields and low density media. We combine an angular finite element mesh respecting octant boundaries on the unit sphere to spatial Cartesian voxel elements to guarantee an unambiguous transport sweep ordering in space. Accuracy for a dosimetrically challenging scenario involving bone and air in the presence of a 1.5 T parallel magnetic field is validated against the Monte Carlo package GEANT4. Accuracy and relative computational efficiency were investigated for various angular discretization parameters. 32 angular elements with quadratic basis functions yielded a reasonable compromise, with gamma passing rates of 99.96% (96.22%) for a 2%/2 mm (1%/1 mm) criterion. A rotational transformation of the spatial calculation geometry is performed to orient an arbitrary magnetic field vector to be along the z-axis, a requirement for a constant azimuthal angular sweep ordering. Working on the unit sphere, we apply the same rotational transformation to the angular domain to align its octants with the rotated Cartesian mesh. Simulating an oblique 1.5 T magnetic field against GEANT4 yielded gamma passing rates of 99.42% (95.45%) for a 2%/2 mm (1%/1 mm) criterion.

A novel approach for forecasting global horizontal irradiance based on sparse quadratic RBF neural network

Integrating solar energy into the electricity grid is an important but challenging task. Forecasting errors can not only break the supply–demand balance but also cause additional costs. Therefore, accurately and effectively forecast the global horizontal irradiance is the key feature to the photovoltaic installation. In this paper, sparse quadratic radial basis function neural network (QRBF) is established. Through mining the association rules, Eclat algorithm is applied to determine relevant meteorological variables to forecast the global horizontal irradiance. QRBF is reformulated as a linear-in-the-parameters problem and a novel approach called square root progressive quantile variable selection procedure (SRPQVSP) is proposed to reduce the complexity of model structure. Furthermore, cuckoo search (CS) algorithm is utilized to optimize the parameters in the model so as to boost forecasting accuracy. Finally, the developed model is verified at four sites of Qinghai province in China with different features of terrain, latitude and other meteorological sources. The experimental results reveal that the developed models composing of selected variables deliver superior performances over other existing approaches.

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

CLASSIFICATION OF NORMAL AND KNEE JOINT DISORDER VIBROARTHROGRAPHIC SIGNALS USING MULTIFRACTALS AND SUPPORT VECTOR MACHINES

The development of reliable Computer Aided Diagnosis (CAD) systems would help in the early detection of Knee Joint Disorder (KJD). In this work, normal and KJD vibroarthrographic (VAG) signals are classified using multifractals and Support Vector Machines (SVM). Multifractal dimension [Formula: see text] is calculated from the VAG signals for various [Formula: see text]-values ([Formula: see text]). Geometrical features are calculated from the multifractal spectrum. The dimension of the feature set is reduced using Principal Component Analysis (PCA). The significant features obtained from the multifractal spectrum are fed as the input to the SVM classifier. The accuracy of the classifier is analyzed using kernels such as linear, quadratic, polynomial and Radial Basis Functions (RBF). The results suggest that VAG signals exhibits the multifractal property. The fluctuations in the normal and abnormal signals are well predicted in small scales of segments of time series. The features such as [Formula: see text] and Mean[Formula: see text] are high in abnormal VAG signals. These features give statistically significant values in differentiating the normal and abnormal subjects ([Formula: see text]). The area under the Receiver Operating Characteristic (ROC) curve is high for polynomial function (0.98). The SVM classifier with polynomial function gives 92.13% of accuracy in differentiating the normal and abnormal subjects. The calculation of multifractal spectrum and geometrical features from VAG signals requires optimization of few parameters, easy to compute, computationally inexpensive, and less time consuming. Hence, the CAD system seems to be clinically significant for the classification of normal and KJD subjects.