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.

中文翻译：

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在混合 Web 样条有限元方法的上下文中处理由于标准边界条件引起的约束

目的

本文旨在提出一种新的边界条件和有限元空间逼近的 web-spline 基，以解决由于齐次和非齐次引起的约束问题；Dirichlet 边界条件。本文考虑了条件为 CAB 的 Navier-Lamé 二维线弹性方程。后者允许在获得的线性系统中完全插入基本边界条件；不使用数值方法作为拉格朗日乘数。本研究开发了混合有限元；使用 B 样条 Web 样条空间的方法。这些提供了同构的精确实现；Dirichlet 边界条件，去除了标准造成的约束；状况。本文证明了弱解的存在性和唯一性，证明了二次情况下数值解的收敛性。加权扩展 B 样条；方法已成为一种更加熟练的解决方案。

设计/方法/方法

在本文中，本研究使用加权有限元方法的实施来求解具有新边界条件 CA、B 的 Navier-Lamé 系统（Koubaiti等人，2020），它概括了众所周知的基础，尤其是 Dirichlet 和诺依曼条件。新提出的边界条件允许使用单个 Matlab 代码，它总结了系统中遇到的所有类型的边界条件。通过使用此模型，可以在单个目录中获取多个程序的同时节省时间和编程资源。

发现

结果表明，基于 Web 样条的二次线性有限元满足 inf-sup 条件，这是解的存在唯一性所必需的。离散解的存在证明了这一点。使用二次情况的数值解建立完全收敛。由于 Navier-Lamé 问题的规律性有限，它不会通过增加 Web-spline 的次数而改变。计算出的相对误差及其比率表明它们的数量级为 1/H。因此，它为数值解稳定性提供了理论有效性。这个问题使用CA、B边界条件的优点是降低了Matlab编程的复杂度。

原创性/价值

混合有限元方法是一种强大的技术，可以使用偏微分方程解决工程和物理科学中的难题。一些重要的应用包括主要基于拉格朗日近似的结构力学、流体流动、热力学和电磁场（Zienkiewicz 和 Taylor，2000）。然而，这种近似在领域建模的层次上受到了很大的限制，特别是在复杂边界的情况下，例如曲线图形式的边界。最近，研究界试图开发一种基于所谓的 B 样条的新近似方法，该方法似乎在解决工程问题方面具有卓越的效果。