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In this paper, a novel isogeometric method for Biot’s consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.

An efficient method for the low-dimensional semiconductor structure is investigated. The method is applied to symmetric double rectangular quantum well as an example. A basis set of Cubic B-Splines is used as localized basis functions. The method compares well with analytical solutions and the finite difference method.

This paper introduces methods of creating, modifying and visualizing free-form geometric models, by use of the B-spline hyperpatch. Modeling tools for the hyperpatch, including local control-point interactions, lofting operations and sweeping operations, are described. These operations, together with modeling tools generated for free-form surface design, form a powerful tool hierarchy for the design of complex solids. Techniques are also described by which these hyperpatches may be integrated into conventional visible-surface algorithms. With the defined methods, and the hyperpatch model, the complexity of free-form solid object construction was found to be significantly simplified.

The problem of progressive lens design can be posed as a variational problem. The necessary condition is a fourth-order nonlinear elliptic partial differential equation. The partial differential equation can be linearized using a perturbation approach. A numerical method using a special type of splines, chosen for their smoothness properties, is devised to solve the resulting PDE. The computational method is shown to be both convergent and efficient.

We begin the mathematical study of Isogeometric Analysis based on NURBS (non-uniform rational B-splines). Isogeometric Analysis is a generalization of classical Finite Element Analysis (FEA) which possesses improved properties. For example, NURBS are capable of more precise geometric representation of complex objects and, in particular, can exactly represent many commonly engineered shapes, such as cylinders, spheres and tori. Isogeometric Analysis also simplifies mesh refinement because the geometry is fixed at the coarsest level of refinement and is unchanged throughout the refinement process. This eliminates geometrical errors and the necessity of linking the refinement procedure to a CAD representation of the geometry, as in classical FEA. In this work we study approximation and stability properties in the context of h-refinement. We develop approximation estimates based on a new Bramble–Hilbert lemma in so-called "bent" Sobolev spaces appropriate for NURBS approximations and establish inverse estimates similar to those for finite elements. We apply the theoretical results to several cases of interest including elasticity, isotropic incompressible elasticity and Stokes flow, and advection-diffusion, and perform numerical tests which corroborate the mathematical results. We also perform numerical calculations that involve hypotheses outside our theory and these suggest that there are many other interesting mathematical properties of Isogeometric Analysis yet to be proved.

We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C⁰ and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.

We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C⁰ and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

This paper proposes to use the method of principal components to reduce the dimensionality of input space and the B-splines to represent the membership functions of input variables. A model reduction strategy, which is based on Johansen’s optimality theorem, also suggested. The utility of this approach is illustrated using a fuzzy system modeling example.

We clarify the relations among different Fourier-based approaches to option pricing, and improve the B-spline probability density projection method using the sinh-acceleration technique. This allows us to efficiently separate the control of different sources of errors better than the FFT-based realization allows; in many cases, the CPU time decreases as well. We demonstrate the improvement of the B-spline projection method through several numerical experiments in option pricing, including European and barrier options, where the SINH acceleration technique proves to be robust and accurate.

We consider Gabor systems with compactly supported windows and rational sampling rate. Two modification techniques for tight frame Gabor windows, splitting for shorter support and smoothing for better smoothness, are introduced. As an illustration of such techniques, we construct multiwindow tight frame Gabor windows with smaller support and C^∞ continuity from certain tight frame Gabor windows.

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly=0$ where $L$ is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

The frame set of a function $g\in L^2(ℝ)$ is the subset of all parameters $(a,b)\in {ℝ}_{+}^2$ for which the time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(ℝ).$ In this paper, we investigate the frame set of a class of compactly supported continuous functions which includes the $B$-splines. In particular, we add some new points to the frame sets of these functions. In the process, we generalize and unify some recent results on the frame sets for this class of functions.

This paper proposes a two-level optimization approach for a multi-robot navigation problem. Our aim is to achieve tasks with optimal strategies, optimal trajectory planning and to give better and feasible solutions. Optimal alternatives are selected by combining game theory strategic decision with an optimal trajectory planning method. This is solved at the top level by applying Sequential Quadratic Programming (SQP). At the bottom, Pure Nash equilibrium is solved using the Simulated Annealing (SA) algorithm. The trajectory planning for the multi-robot system is performed by decomposition where the b-spline technique is initially used to describe the robots path in an environment with obstacles. The cubic spline technique is then introduced to set up the motion along the desired path. The kinematic and dynamic constraints inherent to the robot behavior are taken into account and collision constraints are handled by penalization. The proposed approach is applied to heterogeneous Unicycle Wheeled Mobile Robots and simulated in endowed obstacles environment. Optimal collision-free trajectories of the robot team are obtained with smooth line paths and good-quality solutions. Simulation results are given to show the effectiveness and robustness of proposed algorithm.