Computational Zooming in Near Unilateral Cracks by Schwarz Method with Total Overlap

IF 0.8 4区 数学 数学研究 Pub Date : 2019-06-01 DOI:10.4208/jms.v52n4.19.02
F. B. Belgacem
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Abstract

We focus on the numerical solver of unilateral cracks by the Schwarz Method with Total Overlap. The aim is to isolate the treatment at the vicinity of the cracks from other regions of the computational domain. This avoids any direct interaction between specific approximations one may use around the singularities born at the tips of the cracks and more standard methods employed away from the cracks. We apply an iterative sub-structuring technique to capture the small structures by insulating the cracks into patches and making a zoom around each of them. The macro-problem is in turn set on the whole domain. As for the classical Schwarz method, the communication between the micro (local) and macro (global) levels is achieved iteratively through some suitable boundary conditions. The micro problem is fed by Dirichlet data along the (outer) boundary of the patches. The specificity of our approach is that the macro problem inherits transmission conditions. Although they are expressed across the cracks, the final algebraic system to invert is blind to the discontinuities of the solution. In fact, the stiffness matrix turns out to be the one related to a safe domain, as if cracks were closed or the unilateral singularities were switched off. Only the right hand side is affected by what happens at the vicinity of the cracks. This enables users to run one of many efficient algorithms found in the literature to solve the linear macro-problem. In the other hand side, in spite of the still bad conditioning and the non-linearity of the unilateral micro problems, they are reduced in size and may be inverted properly by convex optimization algorithms. A successful convergence analysis of this variant of the Schwarz Method is performed after adapting to the unilateral non-linearity the variational tools developed by P. L. Lions. AMS subject classifications: 35N86, 65N55 ∗Corresponding author. Email addresses: faker.ben-belgacem@utc.fr (F. Ben Belgacem), faten.jelassi@ utc.f (F. Jelassi), nmgmati@iau.edu.sa (N. Gmati) F. Ben. Belgacem, N. Gmati and F. Jelassi / J. Math. Study, 52 (2019), pp. 378-393 379
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全重叠条件下近单边裂纹的Schwarz方法计算放大
重点研究了具有全重叠的单边裂纹的Schwarz方法的数值求解。目的是将裂缝附近的处理与计算域的其他区域隔离开来。这避免了在裂缝尖端产生的奇点周围可能使用的特定近似与在裂缝之外使用的更标准的方法之间的任何直接相互作用。我们应用迭代子结构技术通过将裂缝绝缘成块并对每个裂缝进行缩放来捕获小结构。宏观问题反过来又涉及到整个领域。在经典的Schwarz方法中,微观(局部)和宏观(全局)之间的通信是通过一些合适的边界条件来迭代实现的。微问题由沿斑块(外)边界的狄利克雷数据馈送。我们方法的特殊性在于宏观问题继承了传导条件。尽管它们是在裂缝中表示的,但最终要反转的代数系统对解的不连续是视而不见的。事实上,刚度矩阵是与安全域相关的矩阵,就好像裂缝是闭合的或单边奇点被关闭了一样。只有右手边受到裂缝附近发生的情况的影响。这使用户能够运行文献中发现的许多有效算法之一来解决线性宏观问题。另一方面,尽管单侧微问题仍然具有较差的条件和非线性,但通过凸优化算法可以减小其规模,并可以适当地反转。在适应p.l. Lions开发的变分工具的单边非线性后,对这种施瓦茨方法的变体进行了成功的收敛分析。AMS学科分类:35N86, 65N55 *通讯作者。电子邮件地址:faker.ben-belgacem@utc.fr (F. Ben Belgacem), faten。jelassi@ utc。f (f . Jelassi), nmgmati@iau.edu.sa (N. Gmati) f . Ben。belgem, N. Gmati和F. Jelassi / J. Math。研究,52 (2019),pp. 378-393 379
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数学研究
数学研究 MATHEMATICS-
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