Quadratic B-spline Finite Element Method Research Articles

Quadratic B-spline finite element method for a rotating nonuniform Euler–Bernoulli beam

In this article, we solve the free vibration problem of a rotating non-uniform Euler-Bernoulli beam using the quadratic B-spline finite element method. The Galerkin method is used to obtain the weak form of the problem. The quadratic B-spline approximation provides the required continuity. This approximation yields the mass and stiffness matrices, which are half the size of the matrices obtained by the conventional finite element approximation. The resulting polynomials are quadratic while the Hermite polynomials are cubic. The mass and the stiffness matrix are derived. Results are matched with the published literature and compared with the conventional finite element results. A non-uniform approximation is used as well to observe an impact of the centrifugal force in numerical solution. Results are found to be more accurate with the non-uniform B-spline approximation for the first three natural frequencies of a rotating beam. Since the centrifugal force makes a difference to the first few modes, it is a desirable outcome.

Numerical solutions of the improved boussinesq equation by the galerkin quadratic B-spline finite element method

In this paper, we are going to obtain numerical solutions of the improved Boussinesq equation with the aid of Galerkin quadratic B-spline finite element method. To test the accuracy and efficiency of the current method, four test problems have been used. These are solitary wave movement, interaction of two solitary waves, wave break-up and blow-up of solutions. Their results have been compared with those available in the literature for different values of space and time steps. Also, the error norms L2 and L1 have been computed and presented in comparison.

Quadratic B-spline finite element method for a rotating non-uniform Rayleigh beam

The quadratic B-spline finite element method yields mass and stiffness matrices which are half the size of matrices obtained by the conventional finite element method. We solve the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Rayleigh beam theory includes the rotary inertia effects in addition to the Euler-Bernoulli theory assumptions and presents a good mathematical model for rotating beams. Galerkin's approach is used to obtain the weak form which yields a system of symmetric matrices. Results obtained for the natural frequencies at different rotating speeds show an accurate match with the published results. A comparison with Euler-Bernoulli beam is done to decipher the variations in higher modes of the Rayleigh beam due to the slenderness ratio. The results are obtained for different values of non-uniform parameter (<TEX>$\bar{n}$</TEX>).

INTERNAL FEEDBACK CONTROL OF THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION

A numerical scheme is proposed to control the BBMB (Benjamin-Bona-Mahony-Burgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.

The quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations

In this paper, a quadratic B-spline finite-element method is proposed for solving the coupled Schrödinger–Boussinesq equations numerically. A semi-discrete finite-element scheme is constructed for this system. The existence and uniqueness of the numerical solutions and the convergence of the discrete scheme are discussed. Numerical results indicate that the proposed method is accurate and efficient.

Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrödinger–KdV equations

We propose a novel split-step quadratic B-spline finite element method for solving the initial-boundary value problem of the coupled Schrödinger–KdV equations. A full-discrete finite element scheme is constructed. The conserved properties of the full-discrete scheme are proved. Detailed numerical results show the efficiency of our scheme.

Quadratic B-spline finite element method for numerical solution of the Burgers’ equation

In this paper, the one-dimensional nonlinear Burgers’ equation was converted to a set of nonlinear ordinary differential equations by the method of discretization in time and then each of them was solved by applying the quadratic B-spline finite element method. In order to show the efficiency of the presented method two problems were considered. The numerical solutions obtained for various values of viscosity were compared with the exact solutions. It is seen that the results obtained by this way are quite satisfactory.

An application for a modified KdV equation by the decomposition method and finite element method

A numerical solution to a Modified Korteweg–de Vries (MKdV) equation is obtained using a “lumped” Galerkin method with quadratic B-spline finite element method (FEM) and using the Adomian’s Decomposition Method (ADM). Test problems concerning the motion and interaction of soliton solutions are used to compare the FEM with the ADM. Classical problems concerning the development, motion and interaction of solitons are used to validate the methods. The present methods extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation

A numerical solution to a generalized Korteweg–de Vries (KdV) equation is obtained using the Galerkin method with quadratic B-spline finite element method (FEM) over which the nonlinear term is locally linearized and using the Adomian’s Decomposition Method (ADM). Test problems concerning the motion and interaction of soliton solutions are used to compare the FEM with the ADM. The present methods extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

Numerical solution of Burgers’ equation by quadratic B-spline finite elements

In this paper, a Galerkin quadratic B-spline finite element method is used to solve numerically the one-dimensional Burgers’ equation reduced by the Hopf–Cole transformation. The performance of the method is tested on two model problems involving moderately Reynolds numbers with known exact solutions. The obtained numerical results show that the method is efficient, robust and reliable for solving Burgers’ equation accurately even involving high Reynolds numbers for which the exact solution fails. Computed results are compared with other numerical results in the literature. A stability analysis of the method is also investigated.

Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method

In this study, a least-squares quadratic B-spline finite element method for calculating the numerical solutions of the one-dimensional Burgers-like equations with appropriate boundary and initial conditions is presented. Three test problems have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each problem and are found to be in good agreement with each other. A Fourier stability analysis of the method is also investigated.

Least-squares quadratic B-spline finite element method for the regularised long wave equation

An approximate solution consisting of a combination of the quadratic B-spline functions is incorporated into the least-squares method. An application of this method is presented for computing the solution of the Regularised Long Wave (RLW) equation. Quadratic B-spline solution of the RLW equation leads to tridiagonal matrix system which is solved easily by using the Thomas algorithm. Performance of this scheme is tested by obtaining a solitary wave solution of the equation and by studying the development of the undular bore. Numerical results of the proposed algorithm is shown to have higher accuracy in terms of L 2-error norm compared studying the migration of the solitary wave. A Fourier stability analysis of the method is investigated.

A quadratic B-spline finite element method for solving nonlinear Schrödinger equation

A numerical solution of the nonlinear Schrödinger (NLS) equation is obtained by using the quadratic B-spline finite element method. Test problems concerning motion, birth and bound states of solitons of the NLG equation are studied to evaluate the method. Results are compared with various numerical methods.