Higher-order B-spline Research Articles

CHARMS

Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

Dynamic and Seismic Analysis of Foundations based on Free Field B-Spline Characteristic Response Histories

In this research, a direct time domain formulation for analysis of unbounded media and foundations is developed that treats dynamic excitations and ground motion in a uniform manner. The boundary element method with higher order B-Spline fundamental solutions is employed to compute the characteristic responses of the surface of the elastodynamic domain. Subsequently, time histories of the system response to general excitations are computed by a superposition scheme that accommodates in a uniform manner arbitrary time histories of external loads and/or ground motion. The characteristic responses are computed in the form of time dependent flexibility matrices of the media that are sparse due to the finite duration of the B-Spline excitation signal and characteristics of the wave propagation. The duration of the B-Spline impulse response is limited to a few time steps. Significant savings in computing time and storage requirements are achieved. The proposed approach also greatly reduces the size of the problems under consideration and still fully considers the effects of the free field. The significance of nonrelaxed boundary conditions and correct representation of the free field is established. The method is demonstrated and validated through applications pertaining to the analysis of foundations and inclusions subjected to transient loads and seismic excitations.

Higher-order B-spline strip models for laminated composite structures with integrated sensors and actuators

This paper deals with the development of families of higher- and first-order isoparametric B-spline finite strip models, applied to the static and free vibration analysis of composite laminated plate/shell structures, which can have arbitrary lay-ups, loading and boundary conditions. The models consider the possibility of existing embedded and/or surface bonded piezoelectric actuators and sensors, enabling for the passive control of parameters, such as displacements and natural frequencies of this type of structures. The structure is discretised in a finite number of strips and the geometry and the displacement components of each strip are represented through interpolating functions. These functions are products of linear or cubic B-spline functions and linear or quadratic Lagrange functions along the local y, x axes of each section within a strip, respectively. The whole structure is modelled through the assembly of a finite number of flat section strips. The accuracy and capability of the models will be discussed through two illustrative examples.

Higher-order B-spline finite strip model for laminated adaptive structures

This paper presents the development of a higher-order isoparametric B-spline finite strip model, applied to the static and free vibration analysis of composite laminated plate/shell structures, which can have arbitrary lay-ups, loading and boundary conditions. The model considers the possibility of existing embedded and/or surface bonded piezoelectric actuators and sensors, enabling passive control of parameters, such as displacements and natural frequencies. The structure is discritized in a finite number of strips and the geometry and the displacement components of each strip are represented by interpolating functions written as products of cubic B-spline functions and Lagrange and Hermite functions along the y and x strip directions, respectively. The whole structure is modeled through the assembly of a finite number of flat section strips. Several examples are presented and discussed to illustrate the accuracy and capability of the model.

Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions

A specially developed method is proposed to retrieve the particle volume distribution, the mean refractive index, and other important physical parameters, e.g., the effective radius, volume, surface area, and number concentrations of tropospheric and stratospheric aerosols, from optical data by use of multiple wavelengths. This algorithm requires neither a priori knowledge of the analytical shape of the distribution nor an initial guess of the distribution. As a result, even bimodal and multimodal distributions can be retrieved without any advance knowledge of the number of modes. The nonlinear ill-posed inversion is achieved by means of a hybrid method combining regularization by discretization, variable higher-order B-spline functions and a truncated singular-value decomposition. The method can be used to handle different lidar devices that work with various values and numbers of wavelengths. It is shown, to my knowledge for the first time, that only one extinction and three backscatter coefficients are sufficient for the solution. Moreover, measurement errors up to 20% are allowed. This result could be achieved by a judicious fusion of different properties of three suitable regularization parameters. Finally, numerical results with an additional unknown refractive index show the possibility of successfully recovering both unknowns simultaneously from the lidar data: the aerosol volume distribution and the refractive index.

Approximation of B-Spline Geometries With Lower Degree Representations

Abstract Exchange of data between geometric modeling systems of different inherent capabilities frequently requires approximate conversion of high degree, piecewise, polynomial curves and surface patches to lower degree representations. The objective of this work is to provide reliable methods for the approximation of high-order B-spline curves and surface patches by low-order B-spline representations. Our method guarantees a prespecified accuracy at all isoparametric points of the curve and patch, a critical feature for accurate exchange of data.

On Multivariate B-Splines

In this paper multivariate B-splines are constructed in terms of certain multivariate “fundamental solutions” which are analogous to the usual univariate truncated powers. Furthermore, this approach is shown to provide the recurrence relations for the construction of higher order B-splines from lower order ones which were recently found by C. A. Micchelli, A constructive approach to Kergin interpolation in $R^k $ : MultivariateB-splines and Lagrange interpolation, MRC Technical Summary Report 1978. It also allows to relate the smoothness of a B-spline to its knot configuration.

B-spline expansion bases for excited states and discretized scattering states

High-order B-splines provide effective basis functions for constructing solutions to the Schrödinger equation. For high-lying levels of the Morse potential septic splines yield an accuracy comparable to harmonic oscillators. They can also approximate hydrogenic wavefunctions; typically one-third of the N eigenvalues of an N×N matrix are more accurate than 0.01%. Matrix eigenvectors can also represent scattering states. The approximation is uniformly valid spatially when basis functions terminate sharply.