Quadratic B-spline Finite Element Research Articles

English

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline finite elements method, Exponential cubic B-spline method, Septic B-spline technique, Quintic B-spline Galerkin method, and B-spline Galerkin method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

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.

Finite element method for starved hydrodynamic lubrication with film separation and free surface effects

This paper proposes a numerical method for determining the evolution of lubricant film thickness and pressure in partially and fully flooded regions of a hydrodynamic contact between two non-conformal rigid surfaces. The proposed method accounts for the classical Reynolds equation within the fully flooded region and for film separation with surface tension driven flow in the partially flooded region, while at the same time it resolves the a priori unknown boundary between the two regions. Additionally it deals with transitions between wetted, partially flooded regions to dry regions, where the film thickness is zero. Both pressure and film thickness fields are considered as unknowns to solve for in each time step and they are approximated through quadratic B-spline finite elements. The geometry of the gap between the rigid surfaces delimiting the lubricant is accounted for in the form of a unilateral contact condition. Appropriate complementarity conditions with respect to separation or no penetration and no slip between the lubricant and the rigid surfaces are enforced by means of a weighted residual formulation.

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.

Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements

Abstract In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms L2 and L∞ are presented in tables.

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>).

An approach to time fractional gas dynamics equation: Quadratic B-spline Galerkin method

In the present article, a quadratic B-spline finite element Galerkin method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the L1 discretization formula is applied to the equation in time. A numerical example is given and the obtained results show the accuracy and efficiency of the method. Therefore, the present method can be used as an efficient alternative one to find out the numerical solutions of other both linear and nonlinear fractional differential equations available in the literature.

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.

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.

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.

Application of a lumped Galerkin method to the regularized long wave equation

In this paper, a lumped Galerkin method based on quadratic B-spline finite elements is used to find numerical solutions of the one-dimensional regularized long wave (RLW) equation with a variant of initial and boundary conditions. The obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation. Results are compared with published numerical solutions. A linear stability analysis of the scheme is also investigated.

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.

A numerical solution of the equal width wave equation by a lumped Galerkin method

In this paper, the equal width wave (EW) equation is solved by a numerical technique based on a lumped Galerkin method using quadratic B-spline finite elements to investigate the motion of a single solitary wave, development of two solitary waves interaction and an undular bore. The obtained results are compared with published numerical solutions. A linear stability analysis of the method is also investigated.

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.

A Collocation Solution For Burgers Equation Using Quadratic B-Spline Finite Elements

This paper discusses solving one of the important equations in Fluid Dynamics; which is the one-dimensional Burgers equation. For that purpose, we use quadratic B-spline finite elements within a Collocation method. It is shown that this method is capable of solving Burgers equation accurately for values of viscosity from small to large. Numerical results are obtained for test problems to show the behavior of the solutions of time dependent problems. Computed results are compared with other numerical results.

Numerical solution of RLW equation using linear finite elements within Galerkin's method

The regularised long wave equation is solved by Galerkin's method using linear space finite elements. In the simulations of the migration of a single solitary wave, this algorithm is shown to have good accuracy for small amplitude waves. Moreover, for very small amplitude waves (⩽0.09) it has higher accuracy than an approach using quadratic B-spline finite elements within Galerkin's method. The interaction of two solitary waves is modelled for small amplitude waves.

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.