Higher-order B-spline Research Articles

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

Multi-sensor data fusion for autonomous flight of unmanned aerial vehicles in complex flight environments

The flight environment of unmanned aerial vehicles faces various challenges. To effectively navigate and perform tasks, they need to effectively integrate multiple sensors. This study applies the adaptive weighted average method, combined with data from global positioning system, inertial measurement unit, three-dimensional optical detection and ranging, and uses linear Kalman filtering to smooth the merged velocity data. High-order B-spline curves for route planning and applying flight constraint formulas to better adapt are used to the dynamics of unmanned aerial vehicles. The research results indicated that the improved adaptive weighting algorithm had high comprehensive performance for multi-sensor data fusion, with the highest accuracy, robustness, real-time performance, and consistency of 94.2%, 93.7%, 100%, and 95.6%, respectively. The flight path lengths planned by the A* algorithm and higher-order B-spline curve were 15.7 and 16.3 m, respectively, and the flight time was 8.2 and 7.1 s, respectively. The flight path planned by higher-order B-spline curve was further away from obstacles. The use of adaptive weighted fusion and linear Kalman filtering facilitates the fusion of multi-sensor data, and autonomous flight routes planned using high-order B-spline curves can also meet the needs of unmanned aerial vehicle flight in complex flight environments.

XFEM level set-based topology optimization for turbulent conjugate heat transfer problems

Solving conjugate heat transfer design problems is relevant for various engineering applications requiring efficient thermal management. Heat exchange between fluid and solid can be enhanced by optimizing the system layout and the shape of the flow channels. As heat is transferred at fluid/solid interfaces, it is crucial to accurately resolve the geometry and the physics responses across these interfaces. To address this challenge, this work investigates for the first time the use of an eXtended Finite Element Method (XFEM) approach to predict the physical responses of conjugate heat transfer problems considering turbulent flow. This analysis approach is integrated into a level set-based optimization framework. The design domain is immersed into a background mesh and the geometry of fluid/solid interfaces is defined implicitly by one or multiple level set functions. The level set functions are discretized by higher-order B-splines. The flow is predicted by the Reynolds Averaged Navier–Stokes equations. Turbulence is described by the Spalart–Allmaras model and the thermal energy transport by an advection–diffusion model. Finite element approximations are augmented by a generalized Heaviside enrichment strategy with the state fields being approximated by linear basis functions. Boundary and interface conditions are enforced weakly with Nitsche’s method, and the face-oriented ghost stabilization is used to mitigate numerical instabilities associated with the emergence of small integration subdomains. The proposed XFEM approach for turbulent conjugate heat transfer is validated against benchmark problems. Optimization problems are solved by gradient-based algorithms and the required sensitivity analysis is performed by the adjoint method. The proposed framework is illustrated with the design of turbulent heat exchangers in two dimensions. The optimization results show that, by tuning the shape of the fluid/solid interface to generate turbulence within the heat exchanger, the transfer of thermal energy can be increased.

XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

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

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

Magnification-driven B-spline interpolation for cone-beam projection and backprojection.

Discretizing tomographic forward and backward operations is a crucial step in the design of model-based reconstruction algorithms. Standard projectors rely on linear interpolation, whose adjoint introduces discretization errors during backprojection. More advanced techniques are obtained through geometric footprint models that may present a high computational cost and an inner logic that is not suitable for implementation on massively parallel computing architectures. In this work, we take a fresh look at the discretization of resampling transforms and focus on the issue of magnification-induced local sampling variations by introducing a new magnification-driven interpolation approach for tomography. Starting from the existing literature on spline interpolation for magnification purposes, we provide a mathematical formulation for discretizing a one-dimensional homography. We then extend our approach to two-dimensional representations in order to account for the geometry of cone-beam computed tomography with a flat panel detector. Our new method relies on the decomposition of signals onto a space generated by nonuniform B-splines so as to capture the spatially varying magnification that locally affects sampling. We propose various degrees of approximations for a rapid implementation of the proposed approach. Our framework allows us to define a novel family of projector/backprojector pairs parameterized by the order of the employed B-splines. The state-of-the-art distance-driven interpolation appears to fit into this family thus providing new insight and computational layout for this scheme. The question of data resampling at the detector level is handled and integrated with reconstruction in a singleframework. Results on both synthetic data and real data using a quality assurance phantom, were performed to validate our approach. We show experimentally that our approximate implementations are associated with reduced complexity while achieving a near-optimal performance. In contrast with linear interpolation, B-splines guarantee full usage of all data samples, and thus the X-ray dose, leading to more uniform noise properties. In addition, higher-order B-splines allow analytical and iterative reconstruction to reach higher resolution. These benefits appear more significant when downsampling frames acquired by X-ray flat-panel detectors with small pixels. Magnification-driven B-spline interpolation is shown to provide high-accuracy projection operators with good-quality adjoints for iterative reconstruction. It equally applies to backprojection for analytical reconstruction and detector datadownsampling.

Computational design of weingarten surfaces

In this paper we study Weingarten surfaces and explore their potential for fabrication-aware design in freeform architecture. Weingarten surfaces are characterized by a functional relation between their principal curvatures that implicitly defines approximate local congruences on the surface. These symmetries can be exploited to simplify surface paneling of double-curved architectural skins through mold re-use. We present an optimization approach to find a Weingarten surface that is close to a given input design. Leveraging insights from differential geometry, our method aligns curvature isolines of the surface in order to contract the curvature diagram from a 2D region into a 1D curve. The unknown functional curvature relation then emerges as the result of the optimization. We show how a robust and efficient numerical shape approximation method can be implemented using a guided projection approach on a high-order B-spline representation. This algorithm is applied in several design studies to illustrate how Weingarten surfaces define a versatile shape space for fabrication-aware exploration in freeform architecture. Our optimization algorithm provides the first practical tool to compute general Weingarten surfaces with arbitrary curvature relation, thus enabling new investigations into a rich, but as of yet largely unexplored class of surfaces.

Impact force analysis using the B-spline material point method

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

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

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

Thermally Induced Vibration Analysis of Flexible Beams Based on Isogeometric Analysis

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

Algorithm for Probability Calculation of B- Spline Curve

This paper presents a probability calculation method for B-spline curve, which is made up of coordinate transformation unit, normalization unit, binarization unit, probability calculation unit, data decoding unit. This thesis offers the general structure and algorithm principle of generating B-spline curve, and draws the B-spline curve on the simulation hardware of VC++6.0 software. Experimental results show that compared with the traditional method of directly calculating B-spline, Paper’s method has lower algorithm complexity and faster curve generation, which is more beneficial to the realization of high-order B-spline curve.

The implementation of B-splines to Hashin and Shtrikman variational principle based FFT method for the homogenization of composite

Fast Fourier transform (FFT) has been successfully used to estimate the effective properties of composites with meso periodic structure for more than two decades. Numerous improvements have been done to make it adequate for infinite contrast, bound elastic energy and composite voxels. The Hashin and Shtrikman variational principle based FFT method handles these three problems simultaneously and is very attractive. In this paper, B-splines were adopted to discretize the polarized field. The approximate polarized field was substituted into the Hashin and Shtrikman variational principle to obtain the discretized system. The constitutive matrix of a composite voxel was calculated by both the energetically consistent way and the laminate mixing rule. Numerical example of a square matrix with a circular inclusion shows that the results are more accurate than the remaining cases when the first and second order B-splines, as well as the laminate mixing constitutive matrix, are used. For higher order B-spline, the summation to compute the discrete Green operator for periodic boundary conditions converges faster while the discretized polarized field needs more iterations to converge. Only when the polarized field is sufficiently smooth, high order B-spline improves the numerical accuracy and efficiency.

Reversibility Error of Image Interpolation Methods: Definition and Improvements

There is no universal procedure in image processing for evaluating the quality and performance of an interpolation method. In this work, we introduce a new quantity: the reversibility error. For a given image, it measures the error after applying successively a homography close to the identity, a crop (removing boundary artifacts) and the inverse homography. An average over random homographies is made to remove the dependency on the homography. A more precise measurement discarding very high-frequency artifacts is obtained by clipping the spectrum of the difference. We also propose new fine-tuned interpolation methods that are based on the DFT zoom-in and pre-existing (or base) interpolation methods. The zoomed version of an interpolation method is obtained by applying it to the DFT zoom-in of the image. In the periodic plus smooth version of interpolation methods, the non-periodicity is handled by applying the zoomed version to the periodic component and a base interpolation method to the smooth component. In an experimental part, we show that the proposed fine-tuned methods have smaller reversibility errors than their base interpolation methods and that the error is mainly localized in a small high-frequency band. We recommend to use the periodic plus smooth versions of high order B-spline. It is more efficient and provides better results than trigonometric polynomial interpolation.

An immersed boundary hierarchical B-spline method for flexoelectricity

This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche’s formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin’s strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

A High-Order B-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions: $$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0<x\le 1, \\ y'(0)= & {} 0,\quad ay(1)+by'(1)=c, \end{aligned}$$ where $$p(x)=x^{\alpha }g(x),\alpha \ge 0$$ is a general class of non-negative function. The error analysis for the quartic B-spline interpolation is discussed. To demonstrate the applicability and efficiency of our method we consider eight numerical examples, seven of which arise in various branches of applied science and engineering: (1) equilibrium of isothermal gas sphere; (2) thermal explosion; (3) thermal distribution in the human head; (4) oxygen diffusion in a spherical cell; (5) stress distribution on shallow membrane cap; (6) reaction diffusion process in a spherical permeable catalyst; (7) heat and mass transfer in a spherical catalyst. It is shown that our method has fourth-order convergence and is more accurate than finite difference methods (Chawla et al., in BIT 28:88–97, 1988; Pandey et al. in J Comput Appl Math 224:734–742, 2009) and B-spline collocation methods (Abukhaled et al. in Int J Numer Anal Model 8:353–363, 2011; Khuri and Sayfy in Int J Comput Methods 11(1):1350052, 2014).

A PIA optimization algorithm for non-uniform B-spline curves and surfaces

This paper aims to address an important issue in B-spline interpolation. In order to reduce shape distortion due to parameterization in the iterative interpolation of B-spline curves and surfaces, we consider both the initial iteration direction of the control vertices and the parameter values corresponding to the nearest points on the initial iterative curves or surfaces. Genetic algorithm with a carefully designed objective function is employed in finding the optimal parameter solutions for high order B-spline curves and surfaces. Experiments show that the shape distortion due to parameterization has been significantly reduced.

Numerical study for cohesive zone model in delamination analysis based on higher-order B-spline functions

A numerical aspect of the implementation cohesive zone model for delamination analysis is presented in this work by employing interface elements and higher-order B-spline functions. The stress oscillation is addressed, and Newton–Cotes integration scheme is considered as a good candidate to overcome this phenomenon. In order to track the nonlinear equilibrium path, a general arc-length constraint named dissipation-based arc-length method is applied. A numerical example is presented to perform the ability of the formulation in predicting the delamination behavior of studied structures.

Randomly Perturbed B-Splines for Nonrigid Image Registration.

B-splines are commonly utilized to construct the transformation model in free-form deformation (FFD) based registration. B-splines become smoother with increasing spline order. However, a higher-order B-spline requires a larger support region involving more control points, which means higher computational cost. In general, the third-order B-spline is considered as a good compromise between spline smoothness and computational cost. A lower-order function is seldom used to construct the transformation model for registration since it is less smooth. In this research, we investigated whether lower-order B-spline functions can be utilized for more efficient registration, while preserving smoothness of the deformation by using a novel random perturbation technique. With the proposed perturbation technique, the expected value of the cost function given probability density function (PDF) of the perturbation is minimized by a stochastic gradient descent optimization. Extensive experiments on 2D synthetically deformed brain images, and real 3D lung and brain scans demonstrated that the novel randomly perturbed free-form deformation (RPFFD) approach improves the registration accuracy and transformation smoothness. Meanwhile, lower-order RPFFD methods reduce the computational cost substantially.

Accurate Solution of Magnetic Field Integral Equation with Higher-Order Basis Functions and Non-Uniform Rational B-Splines Modeling

This article presents an accurate higher-order moment method to solve the magnetic field integral equation by higher-order basis functions and the non-uniform rational B-splines geometric modeling. Instead of the existing techniques for planar triangulations, it applies two simple integral transformations to extract the singularities and near singularities in the impedances. The performance of the proposed method is illustrated on two examples: a sphere and a missile scatterer. The resulting radar cross-section of this method is often more accurate than that of the magnetic field integral equation with a low-order moment method. Additionally, when the matrix equation is solved by iterative algorithms, the proposed method does not always converge faster than its electric field integral equation counterpart.

An extended wavelet Galerkin method with a high-order B-spline for 2D crack problems

Two-dimensional (2D) crack problems are solved employing a novel technique based on a combination of wavelet Galerkin method and X-FEM with a high-order interpolant. Multiresolution analysis of the wavelet basis functions (scaling/wavelet functions) plays an important role in the numerical simulation. High-order B-spline scaling/wavelet functions are chosen as the basis functions. Severe stress concentration near a crack tip is represented by superposing the multiresolution wavelet functions. In addition, the crack modeling is easy to treat by introducing enrichment functions of the X-FEM. In the proposed approach, the governing equation is discretized based on fixed grid, and fracture mechanics problems with complicated shaped geometries can be analyzed effectively, reducing the model generation tasks. 2D linear fracture mechanics problems are solved, and the accuracy is studied for numerical examples.