Quadratic Basis Functions Research Articles

An approximate solution of multi-term fractional telegraph equation with quadratic B-spline basis functions

This paper introduces the Galerkin Method for the approximate solution ofMulti-term Fractional Telegraph Equations (MFTE). The Galerkin Method (GM)is one of the most popular techniques for the solution of Partial Differential Equations (PDEs), and it uses the idea of mapping a solution onto a set of basis functions and then seeking the residual error through minimization. In GM, the weight and basis functions are the same, and as such, the basis functions are selected appropriately to satisfy the given conditions imposed. In this paper, the quadratic B-spline functions are adopted as shape and test functions for resolving the approximate solution of MFTE. Here, the Caputo fractional derivative takes care of the fractional part, and the Gauss-Mamadu-Njoseh quadrature scheme handles numerical integration. Numerical illustrations are examined for single-term and two-term MFTE, respectively, with numerical evidence measured usingL2 and L∞ error norms. Consequently, the resulting numerical evidence, as presented in Tables and Figures, shows the accuracy and reliability of the method. Also, the study examined and presented relevant theorems of convergence and error analyses of method. MAPLE 18 was used for all computational frameworks in this research.

Numerical analysis for nonlinear wave equations with boundary conditions: Dirichlet, Acoustics and Impenetrability

In this article, we present an error estimation in the L2 norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

2D temperature field reconstruction using optimized Gaussian radial basis function networks

Monitoring temperature fields by sensors and mapping the reconstructed temperature field holds great importance in numerous contexts. Concurrently, as sensing technologies continue to advance, there is a parallel development of temperature reconstruction algorithms. In this study, we present an enhanced Quadratic Optimization Gaussian Radial Basis Function approximation (QO-GRBF). We demonstrate the effectiveness of this algorithm by reconstructing a 10 cm x 10 cm temperature field using distributed optical fiber sensing system. The absolute error across four different temperature stages from 35 °C to 65 °C ranged from 0.21 °C to 0.45 °C, indicating a strong reconstruction ability. Furthermore, we compared the performance of QO-GRBF with GRBF in different point densities. The results reveal an over 30 % improvement in scenarios with denser data points, highlighting the efficacy of proposed algorithm. The data size chosen was quantified as well. There was a 10.22 % improvement observed when transitioning from selecting 4 points to selecting 6 points.

Solving multi-dimensional European option pricing problems by integrals of the inverse quadratic radial basis function on non-uniform meshes

This paper explores multi-asset options as a means to diversify portfolios, mitigating risk across various assets. We present a numerical method using radial basis function-generated finite difference solvers via integrals of the inverse quadratic kernel. Our method introduces new weights for the task we are dealing with. We derive and compute analytical solutions to approximate function derivatives on three-node stencils with non-uniform and uniform distances. Our findings highlight the convergence order of the proposed analytical weights. Numerical examples illustrate the theory.

A Three-Grid High-Order Immersed Finite Element Method for the Analysis of CAD Models

The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains described by complex CAD models and discretised with tens of millions of degrees of freedom. The solution field is discretised using linear and quadratic Lagrange basis functions.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Stabilized mixed material point method for incompressible fluid flow analysis

This paper proposes novel and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated-Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.

Epileptic seizure detection on a compressed EEG signal using energy measurement

Epilepsy is a common neurological disorder affecting both children and adults. It can trigger seizures without any stimuli. An accurate diagnosis of epilepsy is essential to the treatment and management of the patient’s condition. Electroencephalography EEG is a non-invasive technique for the diagnosis of epilepsy. Data analysis and processing can be challenging due to the high dimensionality of EEG signals. In this study, the dimensionality reduction technique using compressive sensing is applied to EEG signals to optimize the process and improve the accuracy of the classification in the epileptic EEG signals. The CS method is applied to minimize processed data by reducing EEG signal dimensions, and energy is calculated on the compressed signal as a feature. The performance of the proposed method is evaluated using SVM with Linear, Quadratic, Cubic, and Radian Basis Function kernels through 5- and 10-fold cross-validation. The results showed 100% accuracy in the classification of ictal and inter-ictal EEG signals with compression ratios of 1/2, 1/4, and 1/8. The results demonstrate that CS is effective in analyzing EEG and diagnosing epilepsy. In addition, this study has significant implications for further development of dimensionality reduction techniques in various EEG-based applications, such as epilepsy classification, sleep stage analysis, and brain–computer interface.

A Simple Non-Conforming Isogeometric Formulation with Superior Accuracy for Free Vibration Analysis of Thin Beams and Plates

A set of non-conforming quadratic basis functions is introduced to formulate the mass and stiffness matrices that enable a superior frequency accuracy for isogeometric free vibration analysis of thin beams and plates. The non-conforming basis functions are expressed as a simple combination of the original basis functions and their second-order derivatives with an adjustable parameter. By construction, these quadratic non-conforming basis functions only affect the mass matrices and do not alter the stiffness matrices. The adjustable parameter arising from the non-conforming basis functions are determined through optimizing the frequency accuracy. In the case of thin beams, the proposed non-conforming isogeometric formulation leads to an increase of frequency accuracy order or superconvergence. For thin plates, the frequency error of the proposed method is guaranteed to be no larger than that of the standard isogeometric approach. Numerical results for thin beams and plates consistently verify that the proposed formulation with non-conforming basis functions is quite robust and produces very favorable frequency accuracy for both uniform and non-uniform meshes.

Сравнение способов аппроксимации функции множителей Лагранжа при решении контактных задач методом с независимой границей контакта

The paper considers the contact problem of the elasticity theory in a static spatial two-dimensional formulation without considering friction. For discretization of the elasticity theory equations, the finite element method was introduced using a triangular unstructured grid and linear and quadratic basis functions. To account for the contact boundary conditions, a modified method of Lagrange multipliers with independent contact boundary is proposed. This method implies the ability to construct a contact boundary with the smoothness degree required for the solution precision and to execute approximation of the Lagrange multiplier function independent of the grids inside the contacting bodies. Various types of the Lagrange multiplier function approximations were studied, including piecewise constant, continuous piecewise linear functions and piecewise linear functions with discontinuities at the difference cells boundaries. Examples of test calculations are provided both for problems with rectilinear and curvilinear contact boundaries. In both cases, the use of discontinuous approximations of the Lagrange multiplier function makes it possible to obtain a numerical solution with fewer artificial oscillations and higher rate of convergence at the grid refinement. It is shown that the numerical solution precision could be improved by more detailed discretization of the contact boundary without changing the grids inside the contacting bodies

Flow computation with the Space–Time Isogeometric Analysis and higher-order basis functions in time

We present method-evaluation incompressible-flow computations with Space–Time Isogeometric Analysis (ST-IGA) and higher-order basis functions in time. The computational-methods platform is made of the ST Variational Multiscale (ST-VMS) method, the ST-IGA with IGA basis functions in space and time, and local length scales and stabilization parameters targeting isogeometric discretization. The computations are for 2D flow past a circular cylinder at Reynolds number 100, which has an easily discernible vortex shedding frequency and widely published lift and drag coefficients and Strouhal number. We compute with quadratic basis functions in space and polynomial orders of 1, 2, 3, and 4 in time, for four different time-step sizes, and with six different sets of expressions for the stabilization parameters. The computations yield a comprehensive set of method-evaluation data that can serve as reference. They also show computational-cost efficiency in using higher-order functions in time.

Boundary layer mesh resolution in flow computation with the Space–Time Variational Multiscale method and isogeometric discretization

We present an extensive study on boundary layer mesh resolution in flow computation with the Space–Time Variational Multiscale (ST-VMS) method and isogeometric discretization. The study is in the context of 2D flow past a circular cylinder, at Reynolds number ranging from [Formula: see text] to [Formula: see text]. It was motivated by the need to have in tire aerodynamics a better understanding of the mesh resolution requirements near the tire surface. The focus in the study is on the normal-direction element length for the first layer of elements near the cylinder, with that length varying by a refinement factor ranging from 2 to 40. The evaluation is based mostly on the velocity profile near the cylinder. As the element length for the first layer is varied, the element lengths for the other layers of the disk-shaped inner mesh are adjusted, with no increase in the number of elements for the refinement factors 2, 3, and 4, and with modest increases only in the radial direction for refinement factors beyond that. The computations are performed with quadratic NURBS basis functions in space and linear basis functions in time. The expressions for the stabilization parameters used in the ST-VMS and for the related local lengths scales are those targeting isogeometric discretization, introduced in recent years. The mesh resolution study is based mostly on the strong enforcement of the Dirichlet boundary conditions on the cylinder, but also includes some computations with the weakly-enforced conditions. We expect that the data generated and observations made will be helpful in setting proper near-surface mesh resolution in VMS-based computations with isogeometric discretization, not only for cylindrical shapes but also for comparable geometries. We furthermore expect that although the data generated and observations made are based on computations with nonmoving meshes, they will also be applicable to computations with moving meshes where the mesh around the solid surface rotates with the surface in the framework of the ST Slip Interface method.

Arabic Font Desigin Using Quadratic Bezier-Like Curve

This paper has constructed a linear quadratic Bézier-like curve with two shape parameters. The proposed work is to design the Arabic fonts by using Bezier-like curves. The focus of this study is to study the new functions of the Bezier-like curve and look at their application in designing Arabic fonts. Our new procedures involve a combination of quadratic and linear polynomial basis functions. The new functions of rational and non-rational Bezier-like curve are used to design Arabic fonts. The shape of the curve is modified as desired by simply altering the values of the shape parameters without changing the control polygon. The present study also looks at the extent of the approximation of the Arabic script design to its digitized image using the Bezier-like curves. The results of this study confirm that all generated functions can give the very visually pleasing shapes of the Arabic fonts similar to the original.

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Boundary Element Method (BEM) is a numerical tool that is applied to many different types of engineering problems. BEM possesses several advantages over other numerical methods such as boundary-only discretization and its semi-analytical nature. Application of Isogeometric Analysis (IGA) to BEM is a recently proposed computational method where the main purpose is to use geometrical basis functions as the shape functions of field variables. Key features of combining these two techniques are the exact representation of the geometry, elimination or suppression of the meshing procedure, reduction of the problem dimension and obtaining highly accurate results especially for the flux values on the boundaries compared to conventional BEM. In this study, steady-state heat conduction problems with nonlinear boundary condition and surface heat source are analyzed where geometries are formed by NURBS, and field variables are described with Lagrangian (constant and quadratic) basis functions and NURBS basis functions. Convergence rates and CPU time of different types of element representations are discussed which eventually reveals the performance and capabilities of IGABEM.

Full-Vector Mode Solver for Nonlinear Optical Waveguides Using Meshfree Finite Cloud Method

A full-vector mode solver based on the meshfree (MF) finite cloud method (FCM) is developed for analysis of nonlinear optical waveguide. Self-consistent solutions are approximated through a simple iterative scheme after the nonlinear eigenvalue matrix equation derived using the quadratic basis function as well as the MF shape function. The proposed mode solver can significantly reduce the computational requirements since major parts of formulations are tackled analytically. Results for two numerical examples are presented and compared with those from the well-known finite element method (FEM) and FEM-based imaginary-distance beam propagation method (FEM-IDBPM), respectively, showing the validity and utility of the proposed method.

Comparison of meshfree displacement and stress error recovery of finite element solutions using moving least squares interpolation

The moving least square (MLS) interpolation based the recovery procedures have been successfully applied to recover the finite element solution errors in the analysis of elastic plates and pipes problems and can be advantageously applied for large deformation and fracture problems. The study presents the displacement and stress error recovery characteristics in the error estimation analysis employing Moving Least Squares interpolation approach. The study considers quartic spline, cubic spline, and exponential weight function with three different order of basis function in Moving Least Squares interpolation based error recovery analysis. The displacement/stress errors in finite element solution are quantified in energy norm. The cylinder and plate benchmark examples using triangular and quadrilateral elements are analyzed to compare the convergence, effectivity and adaptively improved meshes obtained using the various displacement/stress recovery procedures. The study shows that cubic spline weight function and quadratic basis function found to perform better in MLS based meshfree recovery technique for stress as well as displacement errors recovery of finite element solution. It is observed from the study that increasing the order of basis function will enhance the error estimation quality that is, rate of convergence become faster with improved effectivity of the results. The increase in convergence rate with the increase of the order of basis function is more in displacement recovery technique as compared to stress recovery technique. It is observed in the analysis of benchmark example with linear triangular meshing that the error reduction using meshfree MLS interpolation based displacement and stress recovery is about 10% and 150% respectively for the displacement and stress recovery over the mesh dependent least square based displacement and stress recovery. The study concludes that the effectiveness and efficiency of meshfree displacement/stress error recovery technique strongly depends on the weight and basis functions of MLS method to recover the errors.

A uniformly convergent quadratic [formula omitted]-spline based scheme for singularly perturbed degenerate parabolic problems

In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient b(x,t)=b0(x,t)xp,p≥1 of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme.

LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions

This paper develops new numerical schemes for solution to nonlinear fractional convection-diffusion equations of order β∈[1,2]. We propose the local discontinuous Galerkin methods by adopting linear, quadratic, and cubic B-spline basis functions and prove stability and optimal order of convergence O(hk+1) for the fractional diffusion problem. This method transforms the equation into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The B-Spline functions significantly improve the accuracy and stability of the method. The performed numerical results demonstrate the efficiency and accuracy of the proposed scheme in different conditions and confirm the optimal order of convergence.

A Superconvergent Isogeometric Method with Refined Quadrature for Buckling Analysis of Thin Beams and Plates

A superconvergent isogeometric method is developed for the buckling analysis of thin beams and plates, in which the quadratic basis functions are particularly considered. This method is formulated through refining the quadrature rules used for the numerical integration of geometric and material stiffness matrices. The criterion for the quadrature refinement is the optimization of the buckling load accuracy under the assumption of harmonic buckling modes for thin beams and plates. The method development starts with the thin beam buckling analysis, where the material stiffness matrix with quadratic basis functions does not involve numerical integration and thus the refined quadrature rule for geometric stiffness matrix can be obtained in a relatively easy way. Subsequently, this refined quadrature rule for thin beam geometric stiffness matrix is conveniently generalized to the thin plate geometric stiffness matrix via the tensor product operation. Meanwhile, the refined quadrature rule for the thin plate material stiffness matrix is derived by minimizing the buckling load error. It turns out that the refined quadrature rule for the thin plate material stiffness matrix generally depends on the wave numbers of buckling modes. A theoretical error analysis for the buckling loads evinces that the isogeometric method with refined quadrature rules offers a fourth-order accurate superconvergent algorithm for buckling load computation, which is two orders higher than the standard isogeometric analysis approach. Numerical results well demonstrate the superconvergence of the proposed method for the buckling loads corresponding to harmonic buckling modes, and for those related with non-harmonic modes, the buckling loads given by the proposed method are also much more accurate than their counterparts produced by the conventional isogeometric analysis.

Predicting Resilient Modulus of Cementitiously Stabilized Subgrade Soils Using Neural Network, Support Vector Machine, and Gaussian Process Regression

Artificial neural network (ANN), support vector machine (SVM), and Gaussian process regression (GPR) were developed in this study for predicting resilient modulus (Mr) values of cementitously stabilized subgrade soils. A database was developed using laboratory test results on 160 specimens prepared by using four soils stabilized with three cementitious additives, namely, lime (3%, 6%, 9%), CFA (5%, 10%, 15%), and CKD (5%, 10%, 15%). Of these, 120 specimens (development dataset) prepared using three types of soils were used in development, and the remaining 40 specimens (validation dataset) were used in the validation of the developed models. A commercial software, MATLAB, was leveraged to develop three ANN models (radial basis function network/RBFN; a multilayer perceptrons network/MLPN with one hidden layer; and MLPN with two hidden layers); four SVM models (linear, quadratic, cubic, and radial basis function kernels); and three GPR models (rational quadratic, Matérn, and exponential kernels) by using codes written in MATLAB language. The strengths and weaknesses of the developed models were examined by comparing the predicted Mr values with the observed/experimental values with respect to the mean squared error (MSE) values and determination coefficient (R2) values. Through comprehensive comparison among these three types of models, an MLPN model with one hidden layer was determined as the best performing model developed in this study. It can be used to predict Mr of cementitiously stabilized subgrade soils for Level 2 pavement design applications. This model as well as the other models could be refined using an enriched database.