This study proposes a novel computationally efficient methodology to perform topology optimization (TO) of fourth-order plate structures within the framework of multi-patch isogeometric analysis. This is realized by taking the multifold benefits of isogeometric PHT-Splines to (1) discretize the continuous weak form of plate structures, (2) develop a continuous density field for the material distribution in TO and inherently remove the need for filters, and (3) provide a hierarchical tree structure for the structural mesh to effortlessly implement an adaptive mesh refinement (AMR) strategy. Moreover, to ensure continuity between isogeometric patches, we adopt a strong coupling between the boundaries. This is established by constructing new basis functions, defined as a linear combination of existing functions at the patch interfaces. The density field in TO is further enhanced with a first-neighbourhood smoothening algorithm based on the Shepard function to generate printable topologies and alleviate the post-processing stages after optimization. An element-centre density, based on the control point densities of the isogeometric mesh, is used as the marking scheme for the AMR to determine the subdomains to be refined. Utilizing the Geometry Independent Field approximaTion, the design and adaptive analysis-optimization stages were independently discretized respectively through NURBS and PHT-Splines, allowing easy transfer of multi-patch geometries from industry-standard packages. Multiple numerical examples illustrate the stability of the multi-patch algorithm in optimizing the geometries effectively. The results also show considerable advantages in terms of solution accuracy such as precise field, smooth topology and computational efficiency.

中文翻译：

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板结构自适应等几何拓扑优化的[公式省略]连续多面片框架


本研究提出了一种新颖的计算高效方法，可在多面片等几何分析的框架内对四阶板结构进行拓扑优化 (TO)。这是通过利用等几何 PHT 样条的多重优势来实现的：(1) 离散板结构的连续弱形式，(2) 为 TO 中的材料分布开发连续密度场，并从本质上消除对滤波器的需求，以及（ 3）为结构网格提供分层树结构，以轻松实现自适应网格细化（AMR）策略。此外，为了确保等几何面片之间的连续性，我们在边界之间采用强耦合。这是通过构造新的基函数来建立的，新的基函数定义为面片接口处现有函数的线性组合。 TO 中的密度场通过基于 Shepard 函数的第一邻域平滑算法得到进一步增强，以生成可打印的拓扑并减轻优化后的后处理阶段。基于等几何网格的控制点密度的元素中心密度被用作 AMR 的标记方案，以确定要细化的子域。利用几何独立场近似，设计和自适应分析优化阶段分别通过 NURBS 和 PHT 样条线独立离散，从而可以轻松地从行业标准包传输多面片几何形状。多个数值算例说明了多面片算法在有效优化几何形状方面的稳定性。结果在精确场、平滑拓扑和计算效率等求解精度方面也显示出相当大的优势。