In this paper, a combination of the high-order numerical manifold method with material interpolation is established with the goal of improving the optimization of cracked structures. Complex Fourier shape functions, known for their inherent advantages, are utilized as weight functions in the numerical manifold method. By implementing high-order analysis, the occurrence of checkerboard patterns is effectively avoided. In addition, a modified sensitivity filtering technique is introduced to address the issue of mesh dependency and minimum length scale in the final topology. This technique is based on the notion of a manifold element. Furthermore, a methodology is introduced to represent void regions within the design domain without the need for passive elements. This methodology involves describing an algorithm to accurately determine the manifold element and the corresponding element area within the final topology. Numerical problems of topology optimization are presented to examine all the aforementioned advantages. The findings indicate that the proposed algorithm for topology optimization does not exhibit any instances of checkerboard patterns. Moreover, the final topology of the curved void regions does not show any zigzag patterns. The above method demonstrates a commendable rate of convergence, and the final topology of the cracked regions can be conveniently simulated.

中文翻译：

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改进裂纹结构优化的高阶复数傅立叶数值流形方法


本文建立了高阶数值流形方法与材料插值的结合，旨在提高裂纹结构的优化。复杂的傅里叶形函数以其固有的优点而闻名，在数值流形方法中被用作权函数。通过实施高阶分析，有效避免棋盘图案的出现。此外，还引入了改进的灵敏度过滤技术来解决最终拓扑中的网格依赖性和最小长度尺度问题。该技术基于流形元素的概念。此外，还引入了一种方法来表示设计域内的空隙区域，而不需要无源元件。该方法涉及描述一种算法，以准确确定最终拓扑中的流形元素和相应的元素区域。提出了拓扑优化的数值问题来检验所有上述优点。研究结果表明，所提出的拓扑优化算法没有表现出任何棋盘图案的实例。此外，弯曲空隙区域的最终拓扑没有显示任何之字形图案。上述方法具有值得称赞的收敛速度，并且可以方便地模拟裂纹区域的最终拓扑。