Calvo-Gallego José Luis, Pivonka Peter, García-Aznar José Manuel, Martínez-Reina Javier
Departmento de Ingeniería Mecánica y Fabricación, Universidad de Sevilla, Seville, Spain.
School of Mechanical, Medical and Process Engineering, Queensland University of Technology, QLD, Australia.
Int J Numer Method Biomed Eng. 2021 Feb;37(2):e3419. doi: 10.1002/cnm.3419. Epub 2020 Dec 9.
Checkerboard is a typical instability in finite element (FE) simulations of bone adaptation and topology optimization in general. It consists in a patchwork pattern with elements of alternating stiffness, producing lack of convergence and instabilities in the predicted bone density. Averaging techniques have been proposed to solve this problem. One of the most acknowledged techniques (node based formulation) has severe drawbacks such as: high sensitivity to mesh density and type of element integration (full vs reduced) and, more importantly, oscillatory solutions also leading to lack of convergence. We propose a new solution consisting in a non-local smoothing technique. It defines, as the mechanical stimulus governing bone adaptation in a certain integration point of the mesh, the average of the stimuli obtained in the neighbour integration points. That average is weighted with a decay function of the distance to the centre of the neighbourhood. The new technique has been shown to overcome all the referred problems and perform in a robust way. It was tested on a hollow cylinder, resembling the diaphysis of a long bone, subjected to bending or torsion. Checkerboard instability was eliminated and local convergence of bone adaptation was achieved rapidly, in contrast to the other averaging technique and to the model without control of checkerboard instability. The new algorithm was also tested with good results on the same geometry but in a model containing a void, which produces a stress concentration that usually leads to checkerboard instability, like in other applications such as simulations of bone-implant interfaces.
棋盘格现象是骨适应性有限元（FE）模拟以及一般拓扑优化中典型的不稳定性。它表现为一种具有交替刚度单元的拼凑图案，会导致预测骨密度缺乏收敛性和出现不稳定性。人们已经提出了平均技术来解决这个问题。其中一种最受认可的技术（基于节点的公式）存在严重缺陷，例如：对网格密度和单元积分类型（完全积分与缩减积分）高度敏感，更重要的是，振荡解也会导致缺乏收敛性。我们提出了一种新的解决方案，即非局部平滑技术。它将在网格的某个积分点处获得的相邻积分点的刺激平均值定义为控制骨适应性的机械刺激。该平均值由到邻域中心距离的衰减函数加权。新技术已被证明能克服所有上述问题并稳健运行。它在一个类似于长骨干骺端的空心圆柱体上进行了测试，该圆柱体承受弯曲或扭转。与其他平均技术以及未控制棋盘格不稳定性的模型相比，棋盘格不稳定性被消除，并且骨适应性实现了快速局部收敛。新算法在相同几何形状但包含一个空洞的模型上也进行了测试，结果良好，空洞会产生应力集中，通常会导致棋盘格不稳定性，就像在其他应用中，如骨植入界面模拟中一样。
