一种用于计算含V型缺口和夹杂物的应变限制弹性固体响应的高效有限元方法
An efficient finite element method for computing the response of a strain-limiting elastic solid containing a v-notch and inclusions
AI总结：
本文针对含V型缺口和夹杂物的应变限制弹性固体响应问题，提出一种基于光滑点变换和高阶形函数的高效有限元方法，结合Picard线性化处理拟线性椭圆方程的非线性，显著提升了数值精度。
摘要：
区域的精确三角剖分在计算微分算子数值近似中起着关键作用。良好的三角剖分有助于减少离散化误差。在标准配点技术中，光滑曲线区域通常通过在边界上取点并以多边形近似的方式进行网格剖分。然而，这种方法往往会导致几何误差，直接影响数值近似的精度。为了限制此类几何误差，人们引入了等参、亚参和等几何方法，这些方法允许对曲面（或曲线段）进行近似。本文提出了一种高效有限元方法，用于近似椭圆边值问题（BVP）的解，该边值问题控制着含V型缺口和夹杂物的弹性固体的响应。代数非线性本构方程与线性动量平衡相结合，归结为二阶拟线性椭圆偏微分方程。我们的方法允许通过光滑的独特点变换来表示复杂曲线边界。主要思想是获得高阶形函数，使我们能够精确计算有限元矩阵和向量中的元素。采用Picard型线性化来处理控制微分方程中的非线性。测试用例的数值结果表明精度有显著提高。
英文摘要：
Accurate triangulation of the domain plays a pivotal role in computing the numerical approximation of the differential operators. A good triangulation is the one which aids in reducing discretization errors. In a standard collocation technique, the smooth curved domain is typically triangulated with a mesh by taking points on the boundary to approximate them by polygons. However, such an approach often leads to geometrical errors which directly affect the accuracy of the numerical approximation. To restrict such geometrical errors, \textit{isoparametric}, \textit{subparametric}, and \textit{iso-geometric} methods were introduced which allow the approximation of the curved surfaces (or curved line segments). In this paper, we present an efficient finite element method to approximate the solution to the elliptic boundary value problem (BVP), which governs the response of an elastic solid containing a v-notch and inclusions. The algebraically nonlinear constitutive equation along with the balance of linear momentum reduces to second-order quasi-linear elliptic partial differential equation. Our approach allows us to represent the complex curved boundaries by smooth \textit{one-of-its-kind} point transformation. The main idea is to obtain higher-order shape functions which enable us to accurately compute the entries in the finite element matrices and vectors. A Picard-type linearization is utilized to handle the nonlinearities in the governing differential equation. The numerical results for the test cases show considerable improvement in the accuracy.
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