B-spline Basis Functions Research Articles

Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function

In this paper, we solve a class of fractional variational problems (FVPs) by using operational matrix of fractional integration which derived from second order spline (B-spline) basis function. The fractional derivative is defined in the Caputo and Riemann-Liouville fractional integral operator. The B-spline function with unknown coefficients and B-spline operational matrix of integration are used to replace the fractional derivative which is in the performance index. Next, we applied the method of constrained extremum which involved a set of Lagrange multipliers. As a result, we get a system of algebraic equations which can be solve easily. Hence, the value for unknown coefficients of B-spline function is obtained as well as the solution for the FVPs. Finally, the illustrative examples shown the validity and applicability of this method to solve FVPs.

APPLICATION OF B-SPLINE METHOD IN SURFACE FITTING PROBLEM

Abstract. Fitting a smooth surface on irregular data is a problem in many applications of data analysis. Spline polynomials in different orders have been used for interpolation and approximation in one or two-dimensional space in many researches. These polynomials can be made by different degrees and they have continuous derivative at the boundaries. The advantage of using B-spline basis functions for obtaining spline polynomials is that they impose the continuity constraints in an implicit form and, more importantly, their calculation is much simpler. In this study, we explain the theory of the least squares B-spline method in surface approximation. Furthermore, we present numerical examples to show the efficiency of the method in linear, quadratic and cubic forms and it’s capability in modeling changes in numerical values. This capability can be used in different applications to represent any natural phenomenon which can’t be experienced by humans directly. Lastly, the method’s accuracy and reliability in different orders will be discussed.

Simulations of nonlinear advection-diffusion models through various finite element techniques

In this study, the Burgers equation is analyzed in both numerically and mathematically by considering various finite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary differential equation system is approximately solved by α-family of time approximation. All these methods are theoretically explained using cubic B-spline basis and weight functions for a strong form of the model equation. Von Neumann matrix stability analysis is performed for each of these methods and stability criteria are determined in terms of the problem parameters. Some challenging examples of the Burgers equation are numerically solved and compared with the literature and exact solutions. Also, the proposed techniques have been compared with each other in terms of their advantageous and disadvantageous depending on the problem types. The more advantageous method of the three, comparison to other two, has been found out for the special cases of the present problem in detail.

Spatiotemporal Thermal Field Modeling Using Partial Differential Equations With Time-Varying Parameters

Accurate modeling of a thermal field is one of the fundamental requirements in engineering thermal management in numerous industries. Existing studies have shown that using differential equations to model a thermal field delivers good performance when the parameters are predetermined through physical or experimental analysis. However, due to variations of the inner medium affected by certain latent factors, the parameters in differential equation models may not be treated as constants while the thermal field is estimated, and this fact poses a new challenge to field estimation by directly solving the differential equation models. In this study, a novel approach to thermal field modeling is developed by considering the parameters as functional variables that vary temporally in partial differential equations (PDEs). This approach provides a new perspective to model the dynamic thermal field by fully using the collected sensor data from the thermal system. Specifically, time-varying parameters can be constructed through a combination of basis functions whose coefficients can be efficiently estimated through the sensor data. A two-level iterative parameter estimation algorithm is also tailored to obtain the parameters in the PDE model. Both simulation and real case studies show that our proposed approach provides satisfactory estimation performance compared with the benchmark method that uses the constant parameter estimation. Note to Practitioners —The proposed method aims to model a thermal field using PDEs with time-varying parameters. To better implement this method in practice, three things are noteworthy: first, the proposed method models a thermal field by fully considering physics-specific engineering knowledge using PDEs and the collected sensor data from thermal systems. Second, because time-varying parameters in PDEs cannot be estimated directly, the proposed model represents the time-varying parameters by a combination of B-spline basis functions in terms of time. Estimating time-varying parameters is converted into estimating the constant coefficients of the basis functions. Because the derivatives of a thermal field might not have an analytical expression, the proposed model represents the thermal field by a combination of B-spline basis functions. Taking the derivatives of the thermal field is converted into taking the derivatives of the corresponding basis functions. Third, the proposed method can not only model a thermal field but can also be applied in other physics-specific engineering cases.

An artificial neural network-differential evolution approach for optimization of bidirectional functionally graded beams

A novel and effective artificial neural network (ANN)-differential evolution (DE) approach as an integration of ANN into DE is first introduced to the material distribution optimization of bidirectional functionally graded (BFG) beams under free vibration. In this methodology, ANN is utilized as an analyzer to predict responses of BFG beams instead of directly solving eigenvalue problems via time-consuming finite element analyses (FEAs). Meanwhile, DE is employed as an optimizer for optimization problems without complex sensitivity analyses. Accordingly, the ANN-DE significantly reduces the computational cost, yet still achieving a high-quality global solution. The material volume fraction at control points defined based on the isogeometric analysis (IGA) concept is taken as continuous design variables. Optimal material profiles are represented by two-dimensional Non-Uniform Rational B-spline (NURBS) basis functions. Obtained results are compared with those of existing literature to demonstrate the accuracy and reliability of the proposed paradigm. Additionally, the ANN-DE is also applied to several other examples to further prove its effectiveness and robustness in dramatically saving the computational efforts, while still yielding optimal outcomes with high accuracy.

Predicting Longitudinal Traits Derived from High-Throughput Phenomics in Contrasting Environments Using Genomic Legendre Polynomials and B-Splines

Recent advancements in phenomics coupled with increased output from sequencing technologies can create the platform needed to rapidly increase abiotic stress tolerance of crops, which increasingly face productivity challenges due to climate change. In particular, high-throughput phenotyping (HTP) enables researchers to generate large-scale data with temporal resolution. Recently, a random regression model (RRM) was used to model a longitudinal rice projected shoot area (PSA) dataset in an optimal growth environment. However, the utility of RRM is still unknown for phenotypic trajectories obtained from stress environments. Here, we sought to apply RRM to forecast the rice PSA in control and water-limited conditions under various longitudinal cross-validation scenarios. To this end, genomic Legendre polynomials and B-spline basis functions were used to capture PSA trajectories. Prediction accuracy declined slightly for the water-limited plants compared to control plants. Overall, RRM delivered reasonable prediction performance and yielded better prediction than the baseline multi-trait model. The difference between the results obtained using Legendre polynomials and that using B-splines was small; however, the former yielded a higher prediction accuracy. Prediction accuracy for forecasting the last five time points was highest when the entire trajectory from earlier growth stages was used to train the basis functions. Our results suggested that it was possible to decrease phenotyping frequency by only phenotyping every other day in order to reduce costs while minimizing the loss of prediction accuracy. This is the first study showing that RRM could be used to model changes in growth over time under abiotic stress conditions.

A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate ([formula omitted]) angular discretisation

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation. The IGA spatial discretisation is based upon non-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. In addition a source iteration compatible maximum principle is used to derive the IGA spatially discretised SAAF equation. It is demonstrated that this maximum principle is mathematically equivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions (MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions, and for the same number of degrees of freedom, the FE based spatial discretisation methods are numerically less accurate than IGA methods. The difference in numerical accuracy between the IGA and FE methods is shown to be because of the higher-order continuity of NURBS basis functions within a NURBS patch as well as the preservation of both the volume and surface area throughout the solution domain within the IGA spatial discretisation. Finally, the numerical results of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensional C5G7 quarter core nuclear reactor physics verification benchmark test case are presented. The results, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transport equation for the same order of discrete ordinate (SN) angular approximation.

Numerical simulation of arbitrary holes in orthotropic media by an efficient computational method based on adaptive XIGA

We present in this paper an efficient computational approach based on an adaptive extended isogeometric analysis (XIGA) for simulation of arbitrary holes in orthotropic materials. The interfaces of holes are described by multiple level set functions so that the developed XIGA can capture the hole geometry without considering its interfaces. The approach is further enhanced by using the locally refined (LR) B-spline basis functions, which dominate over B-spline or non-uniform rational B-spline functions (NURBS) due to the local refinement. This study only deals with the structured mesh strategy for local refinement. The implementation of the local refinement is guided by posteriori error estimator based on stress recovery. Accuracy study is performed for isotropic media due to the availability of analytical solutions. For numerical experiments, different types of holes in orthotropic media are studied, and the computed numerical results are compared with reference solutions derived from ABAQUS (FEM). We also compare the convergence rate obtained by our adaptive local refinement with that derived from uniform global refinement to indicate the greater advantage of the developed adaptive XIGA.

Variable selection for semiparametric varying-coefficient spatial autoregressive models with a diverging number of parameters

In this article, we consider variable selection in semiparametric varying-coefficient spatial autoregressive models with a diverging number of parameters. With the nonparametric functions approximated by B-spline basis functions and combining 2SLS method with the SCAD penalty, we propose a variable selection procedure. Under mild conditions, we establish the consistency and oracle property of the resulting estimators for parameter components and consistency of the regularized estimator for nonparametric component. Some simulation studies are conducted to assess the finite sample performance of the proposed variable selection procedure, and the developed methodology is illustrated by an analysis of the Boston housing price data.

Stock market trend prediction using a functional time series approach

Thanks to advanced technologies, ultra-high-frequency limit order book (LOB) data are now available to data analysts. An LOB contains comprehensive information on all transactions in a market. We use LOB data to investigate the high-frequency dynamics of market supply and demand (S–D) and inspect their impacts on intra-daily market trends. The intra-daily S–D curves are fitted with B-spline basis functions. Technique of multi-resolution is introduced to capture inhomogeneous curvature of the S–D curves and a lasso-type criterion is employed to select a common basis set. Based on empirical evidence, we model the time varying coefficients in the B-spline interpolation by vector autoregressive models of order . The Xgboost algorithm is employed to extract information from the areas under the S–D curves to predict the intra-daily market trends. In the empirical study, we analyze the LOB data from LOBSTER (https://lobsterdata.com/). The results show that the proposed approach is able to recover the S–D curves and has satisfactory performance on both curve and market trend predictions.

Isogeometric FE analysis of CNT-reinforced composite plates: free vibration

Free vibration behavior of carbon nanotube-reinforced composite plates is studied here, using isogeometric analysis (IGA) based on a higher-order shear deformation theory (HSDT). Carbon nanotubes are considered dispersed into matrix in four different types of volume fraction distributions varying across the thickness. The formulation developed is approximated according to the HSDT model and is incorporated into a computer program developed in-house. IGA uses the same basis function for exact geometric modeling and for analysis, i.e., non-uniform rational B-spline basis function which can easily model the complex geometries exactly, and it can also achieve any desired degree of continuity through the choice of interpolation order so that this method can easily fulfill the C1 continuity for the HSDT model. The effect of variation of width-to-thickness ratio, plate aspect ratio and other parameters, on natural frequencies, are evaluated from parametric studies after due validation with the given literature which is done in FORTRAN.

Analysis of laminated composite plates based on THB-RKPM method using the higher order shear deformation plate theory

In the present investigation, static, free vibration and buckling response of laminated composite plates based on the coupling of truncated hierarchical B-splines (THB-splines) and reproducing kernel particle method (RKPM) within higher order shear deformation plate theory are presented. The coupled THB-RKPM method blends the advantages of the isogeometric analysis and meshfree methods. Since under certain conditions, the isogeometric B-spline and NURBS basis functions are exactly represented by reproducing kernel meshfree shape functions, recursive process of producing isogeometric bases can be omitted. More importantly, a seamless link between meshfree methods and isogeometric analysis can be easily defined which provide an authentic meshfree approach to refine the model locally in isogeometric analysis. This procedure can be accomplished using truncated hierarchical B-splines to construct new bases and adaptively refine them. It is shown that THB-RKPM method is ideally appropriate for local refinement of laminated composite plates in the framework of isogeometric analysis. The flexibility of the proposed method for refining basis functions leads to decrease the computational cost without losing the accuracy of the solution. Numerical examples considering different boundary conditions, various aspect ratios, stiffness ratios and fiber orientations demonstrate validity and versatility of the proposed method.

Isogeometric boundary integral formulation for Reissner’s plate problems

PurposeThis paper aims to develop a new isogeometric boundary element formulation based on non-uniform rational basis splines (NURBS) curves for solving Reissner’s shear-deformable plates.Design/methodology/approachThe generalized displacements and tractions along the problem boundary are approximated as NURBS curves having the same rational B-spline basis functions used to describe the geometrical boundary of the problem. The source points positions are determined over the problem boundary by the well-known Greville abscissae definition. The singular integrals are accurately evaluated using the singularity subtraction technique.FindingsNumerical examples are solved to demonstrate the validity and the accuracy of the developed formulation.Originality/valueThis formulation is considered to preserve the exact geometry of the problem and to reduce or cancel mesh generation time by using NURBS curves employed in computer aided designs as a tool for isogeometric analysis. The present formulation extends such curves to be implemented as a stress analysis tool.

Maximum energy dissipation for elasto-plastic plates via isogeometric shape optimization

In this research, optimum shape of plate structures is sought to maximize the energy dissipation via structural shape optimization. To achieve this, isogeometric analysis (IGA) is utilized for structural analysis of plates considering elasto-plastic behavior of materials. The von Mises material model is employed for this purpose. Non-uniform rational B-splines basis functions are used for both geometry definition and approximating the unknown deformation field. The optimization problem is to maximize the structural dissipated energy until a prescribed displacement is reached and a fixed amount of material is considered in the design domain. A direct shape sensitivity analysis is performed and a mathematical based approach is employed for the optimization process. To demonstrate the efficiency of the proposed algorithm three examples are illustrated. Using the IGA prevents adjusting analysis model during the optimization process, which is time-consuming especially when iterative nonlinear analysis is performed. The results also show that large geometry modifications can be properly managed by the proposed algorithm.

Controllable optimal design of auxetic structures for extremal Poisson’s ratio of −2

We develop a lattice structure of a desired Poisson’s ratio that is maintained when subjected to finite deformation. Using a gradient-based design optimization in an isogeometric computational framework, the representative unit cell is sophisticatedly tailored to attain the controlled Poisson’s ratio of up to −2. The ligaments of the lattice structure are modeled using geometrically exact beams whose configuration and cross-sectional area are regarded as design variables, which are parameterized by the higher order B-spline basis functions. The obtained optimal design is fabricated using a PolyJet 3-D printing machine and validated by physical experiments. Excellent quantitative agreement is observed between the numerical and the experimental results measured from an optical deformation measurement system.

Optimal design of lattice structures for controllable extremal band gaps

This paper presents very large complete band gaps at low audible frequency ranges tailored by gradient-based design optimizations of periodic two- and three-dimensional lattices. From the given various lattice topologies, we proceed to create and enlarge band gap properties through controlling neutral axis configuration and cross-section thickness of beam structures, while retaining the periodicity and size of the unit cell. Beam neutral axis configuration and cross-section thickness are parameterized by higher order B-spline basis functions within the isogeometric analysis framework, and controlled by an optimization algorithm using adjoint sensitivity. Our optimal curved designs show much more enhanced wave attenuation properties at audible low frequency region than previously reported straight or simple undulated geometries. Results of harmonic response analyses of beam structures consisting of a number of unit cells demonstrate the validity of the optimal designs. A plane wave propagation in infinite periodic lattice is analyzed within a unit cell using the Bloch periodic boundary condition.

Isogeometric Residual Minimization Method (iGRM) with direction splitting for non-stationary advection–diffusion problems

In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each space-direction, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection–diffusion computational examples, including model “membrane” problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.

Visualization of Intuitionistic Fuzzy B-Spline Space Curve and Its Properties

In this paper, an intuitionistic fuzzy B-spline space curve is defined and some of its properties is introduced. Firstly, intuitionistic fuzzy control point is defined based on intuitionistic fuzzy and geometrical modeling concepts. Each of the control point relation that consists of three function is find and shown. Later, the control point is blended with B-spline basis function and intuitionistic fuzzy B-spline curve is visualized. Some of the control point and space curve properties in the Euclidean space is also discussed throughout this paper.

Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

The non-linear Schrödinger (NLS) equation has often been used as a model equation in the study of quantum states of physical systems. Numerical solution of NLS equation is obtained using cubic B-spline Galerkin method. We have applied the Crank–Nicolson scheme for time discretization and the cubic B-spline basis function for space discretization. Three numerical problems, including single soliton, interaction of two solitons and birth of standing soliton, are demonstrated to evaluate to the performance and accuracy of the method. The error norms and conservation laws are determined and found to be in good agreement with the published results. The obtained results show that the approach is feasible and accurate. The proposed method has almost second order convergence. The linear stability of the method is performed using the Von Neumann method.

Solid–liquid coupled material point method for simulation of ground collapse with fluidization

An improved version of the solid–liquid coupled material point method (MPM) is proposed to simulate ground collapses with fluidization involving transition processes from soil structures to flowing mixture. A water-saturated soil is assumed based on porous media theory, and the physical quantities of the soil and water phases are assigned to two separate sets of particles (material points). The main contribution of this study is the introduction of the fractional step projection method for the time discretization of the momentum equation of the water phase on the assumption of incompressibility. Thanks to this, the proposed solid–liquid coupled MPM is capable of suppressing the pressure oscillations caused by the weak incompressibility of water, which is commonly assumed in the previous studies, and of representing the wide range of behavior of the soil–water mixture at relatively low computational cost. Also, B-spline basis functions are utilized for the spatial discretization to suppress the cell-crossing errors caused by particles crossing element (cell) boundaries. Several numerical tests are conducted to examine the performance of the proposed method that inherits the beneficial features of MPM and demonstrate the capability of reproducing a model experiment of wave collision to sandpile that exhibits the water flow-induced fluidization process of soil involving scouring, transportation and sedimentation.