The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline- (WEB-spline-) based discrete solution for the 2D Navier-Lamé equation of linear elasticity with a different type of mixed boundary condition called CA,B boundary condition. Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

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Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

In this article we are intending to solve the Navier-Lame problem in 2D with Dirichlet and Neumann boundaries, using the mixed finite element P 1 -bubble P 1 . We want to introduce a new weak formulation of this problem with help of another new unknown which is equal to divergence of the displacement. We do the necessary calculations of this problem in order to come up with a Matlab program that visualizes the numerical solution.Some numerical results which are shown, prove that our method is more efficient than the ordinary finite element.

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