Quadratic Basis Research Articles

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.

Quadratic Basis Pursuit in Model Updating of Underconstrained Problems

Quadratic Basis Pursuit in Model Updating of Underconstrained Problems

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.

Solution of the Schrödinger equation using quadratic B-Spline collocation on non-uniform grids

This paper applies orthogonal collocation based on quadratic B-spline basis functions to solve the Schrödinger equation on uniform and non-uniform grids. The method is well suited for solving different cases of soliton solutions for which the solution behaviour is very complicated. The solutions are found to be consistent with the exact solutions for various non-uniform cases. The overall numerical scheme is proven to be unconditionally stable and utilizes minimal memory storage due to the sparse matrix systems associated with B-spline basis functions. This could prove particularly useful in machine learning applications where repetitive calculations are needed and large amounts of data need to be processed. We found that using non-uniform grids as compared to uniform grids had a profound effect in capturing the correct solution dynamics especially in the 3-soliton case.

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.

Transcendence criterion with $$(\beta ,{\mathcal {A}})$$-representations in some quadratic integer bases

Transcendence criterion with $$(\beta ,{\mathcal {A}})$$-representations in some quadratic integer bases

Consistent C-bar radial-basis Galerkin method for finite strain analysis

For finite-strain problems of quasi-incompressible materials, with hyperelastic and hyperelasto-plastic behavior, we develop a mixed polynomial-enriched radial-basis function (RBF) interpolation. It is characterized by decoupling quadrature and polynomials for the deviatoric and volumetric terms of the right Cauchy–Green tensor. With an enriched interpolation, first and second derivatives of the deformation gradient are continuous. A finite-element mesh is employed for quadrature purposes with the corresponding distribution of Gauss points. In terms of discretization, linear, quadratic and cubic polynomials are combined and function support is established from the number of pre-assigned nodes. Due to the adoption of a Lagrangian kernel, finite strain elasto-plastic constitutive developments are based on the Mandel stress. Combined with the C¯ technique, these developments are found to be convenient from an implementation perspective, as RBF formulations for finite strain plasticity have been limited in terms of generality and amplitude of deformations in the quasi-incompressibility case. Three benchmark tests are successfully solved, and it was found that a combination of reduced integration with a deviatoric cubic basis and volumetric quadratic basis provides superior results in the quasi-incompressible case. Numerical testing shows very good convergence behavior of the results. Besides absence of locking with selective interpolation, outstanding Newton convergence was observed regardless of strain levels.

Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

High order discontinuous Galerkin simulation of hypersonic shock-boundary layer interaction using subcell limiting approach

Numerical simulation of shock-boundary layer interactions in the hypersonic regime has for long been a challenge, plagued by high sensitivity of the surface predictions to grid and numerical details. In the course of developing adaptive high order methods to mitigate this issue, discontinuous Galerkin method has emerged as a promising alternative to the conventionally used finite volume method. However, till date, among the reported finite element method-based simulations of these flows, only a few have presented high order simulations using a higher than linear basis. In this work, a subcell limiting technique devised for the compressible Navier-Stokes equations is extended for use in the hypersonic regime. The proposed extension is tested by performing simulations of several shock-boundary layer interaction cases in the freestream Mach number range 6 to 14 using a quadratic basis. For a set of cases at Mach 6, grid converged simulations are presented that agree reasonably well with experimental data and other numerical solutions. More importantly, simulations of a Mach 11 case that are performed using two different flux schemes are shown to yield nearly identical predictions of the surface pressure and heat transfer, suggesting that high order solutions can reduce the sensitivity of numerical predictions to the computational details. Grid convergence studies of two cases at Mach numbers 11 and 14 led to observing large scale oscillations in the separation bubble in finer grid simulations, suggesting that explicit time integration may not always be suitable for high order computation of hypersonic separated flows at very high Mach numbers.

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.

Ridge Regression for Functional Form Identification of Continuous Predictors of Clinical Outcomes in Glomerular Disease

Introduction: Penalized regression models can be used to identify and rank risk factors for poor quality of life or other outcomes. They often assume linear covariate associations, but the true associations may be nonlinear. There is no standard, automated method for determining optimal functional forms (shapes of relationships) between predictors and the outcome in high-dimensional data settings. Methods: We propose a novel algorithm, ridge regression for functional form identification of continuous predictors (RIPR) that models each continuous covariate with linear, quadratic, quartile, and cubic spline basis components in a ridge regression model to capture potential nonlinear relationships between continuous predictors and outcomes. We used a simulation study to test the performance of RIPR compared to standard and spline ridge regression models. Then, we applied RIPR to identify top predictors of Patient-Reported Outcomes Measurement Information System (PROMIS) adult global mental and physical health scores using demographic and clinical characteristics among N = 107 glomerular disease patients enrolled in the Nephrotic Syndrome Study Network (NEPTUNE). Results: RIPR resulted in better predictive accuracy than the standard and spline ridge regression methods in 56–80% of simulation repetitions under a variety of data characteristics. When applied to PROMIS scores in NEPTUNE, RIPR resulted in the lowest error for predicting physical scores, and the second-lowest error for mental scores. Further, RIPR identified hemoglobin quartiles as an important predictor of physical health that was missed by the other models. Conclusion: The RIPR algorithm can capture nonlinear functional forms of predictors that are missed by standard ridge regression models. The top predictors of PROMIS scores vary greatly across methods. RIPR should be considered alongside other machine learning models in the prediction of patient-reported outcomes and other continuous outcomes.

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

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.

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.

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.