From Lagrange to Bernstein: Generalized Transfinite Elements with Arbitrary Nodes

Polynomial basis conversion made stable by truncated singular value decomposition

Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions

A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

This manuscript presents a framework for trajectory generation for soft continuum robots using principles from optimal control. The problem is constrained over the partial differential kinematic equations of the Cosserat rod model, capturing all modes of deformation which soft continuum systems can achieve. The derived optimal control problem is transformed to a nonlinear programming problem which can be solved using the Bernstein polynomial basis. Non-unit quaternions are used to discretely apply rotational transformations to values approximated over Bernstein polynomials, allowing individual components of strain to be constrained separately. Included within this manuscript are numerical results as well as validation through experimental results.

In this paper, we introduce a new interpolation method that is easy to use and its interpolating function can be explicitly expressed. This interpolation method can be used for a wide spectral type of functions, since it has the ability to change the form of the interpolating function. This interpolation inspired by the Shepard type interpolation. We examine the ability of this interpolation to piecewise functions and rational functions and compare it with Lagrange interpolation and linear interpolation by using Bernstein polynomials basis. We applied this interpolation for an integral equation, and we presented a linear system of equations to approximate the solution of a Fredholm integral equation by collocation method. In the collocation method, the Reverse Interpolation expresses the approximate solution in the form of a linear combination of some basic functions. Several examples are presented to illustrate the effectiveness of the proposed method and the numerical results confirm the desired accuracy.

Chapter 3 - Hermite Interpolation and the Extended Neville Algorithm

In this brief, numerical techniques for an efficient implementation of the spline transcription (ST) method by using the pseudospectral framework are proposed. It intends to develop a new approach to the ST method, analyses with which can be carried out with nearly the same level of computational burden without resorting to a specific type of collocation points as in the pseudospectral method. Therefore, the piecewise continuous polynomial interpolation is converted into a global spline interpolation, which has the same forms as the Lagrange interpolation and is used to derive the formulas for the partial integration and the differentiation. Using these formulas, the nonlinear optimal control problem can be transcribed in a similar manner as in the pseudospectral method and analyzed with numerical techniques and mathematical theories established well for the pseudospectral method. Also, this brief shows the Lagrange and tension spline interpolation can be used for the proposed methods. Numerical accuracy and efficiency of the proposed method are accessed through its applications to the solution of two nonlinear optimal control problems on uniform nodes. The predicted results of states, controls, and costates are well correlated with the optimal solutions. Finally, this brief recommends the use of advanced grid topologies to enhance both accuracy and convergence by using the flexibility in node selections with the proposed methods.

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

In this paper, we propose a framework for online quadrotor motion planning for autonomous navigation in unknown environments. Based on the onboard state estimation and environment perception, we adopt a fast marching-based path searching method to find a path on a velocity field induced by the Euclidean signed distance field (ESDF) of the map, to achieve better time allocation. We generate a flight corridor for the quadrotor to travel through by inflating the path against the environment. We represent the trajectory as piecewise Bezier curves by using Bernstein polynomial basis and formulate the trajectory generation problem as typical convex programs. By using Bezier curves, we are able to bound positions and higher order dynamics of the trajectory entirely within safe regions. The proposed motion planning method is integrated into a customized light-weight quadrotor platform and is validated by presenting fully autonomous navigation in unknown cluttered indoor and outdoor environments. We also release our code for trajectory generation as an open-source package.

Multi-node higher order expansions of a function

An identity for multivariate Bernstein polynomials

Adaptive reconstruction of free-form surfaces using Bernstein basis function networks

Conversion from NURBS to Bézier representation

Saddlepoint Approximations to Tail Expectations Under Non-Gaussian Base Distributions