In the present work, we propose a new approach, the so-called compressed adaptive integration scheme (C-AIS), for the computation of the stiffness and mass matrices in fictitious domain methods requiring the integration of discontinuous functions. The novel approach extends the conventional quadtree-decomposition-based adaptive integration scheme (AIS) by an additional step, in which established image-compression techniques are exploited to decrease the number of integration sub-cells. The benefits of the C-AIS are manifold: First, the compression of the sub-cells inevitably leads to significant savings in terms of computational time required by the numerical integration. Second, the compression procedure, which is executed directly after the quadtree-decomposition algorithm, can be easily included in existing codes. Third, if applied to polynomial integrands, the C-AIS yields exactly the same accuracy as the conventional AIS. Finally, the fourth advantage is seen in the fact that the C-AIS can readily be combined with other approaches seeking a reduction of the number of integration points such as the Boolean-FCM. The efficiency of the C-AIS approach is presented in the context of the FCM based on Cartesian meshes applied to problems of linear elastostatics and modal analysis, while it is also a suitable for the quadrature in other fictitious domain approaches, e.g., CutFEM and cgFEM.

中文翻译：

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基于图像压缩技术的增强型数值积分方案：在虚拟域方法中的应用

在当前的工作中，我们提出了一种新方法，即所谓的压缩自适应积分方案（C-AIS），用于在需要不连续函数积分的虚拟域方法中计算刚度和质量矩阵。这种新颖的方法通过一个额外的步骤扩展了传统的基于四叉树分解的自适应集成方案（AIS），其中利用已建立的图像压缩技术来减少集成子单元的数量。C-AIS的好处是多方面的：首先，子单元的压缩不可避免地导致了数值积分所需的计算时间的大量节省。第二，在四叉树分解算法之后直接执行的压缩过程可以轻松地包含在现有代码中。第三，如果将C-AIS用于多项式整数，则其准确度与常规AIS完全相同。最后，第四个优势体现在以下事实：C-AIS可以很容易地与其他寻求减少积分点数量的方法（例如Boolean-FCM）组合使用。C-AIS方法的效率是在基于笛卡尔网格的FCM上下文中提出的，该笛卡尔网格适用于线性弹性静力学和模态分析问题，同时它也适合于其他虚拟域方法（例如CutFEM和cgFEM）中的正交。