A novel algorithm to resolve lack of convergence and checkerboard instability in bone adaptation simulations using non‐local averaging,International Journal for Numerical Methods in Biomedical Engineering

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A novel algorithm to resolve lack of convergence and checkerboard instability in bone adaptation simulations using non‐local averaging
International Journal for Numerical Methods in Biomedical Engineering ( IF 2.2 ) Pub Date : 2020-11-22 , DOI: 10.1002/cnm.3419
José Luis Calvo-Gallego ₁ , Peter Pivonka ₂ , José Manuel García-Aznar ₃ , Javier Martínez-Reina ₁

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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.

中文翻译：

一种使用非局部平均解决骨骼适应模拟中缺乏收敛和棋盘不稳定性的新算法

棋盘格是骨骼适应和拓扑优化的有限元 (FE) 模拟中的典型不稳定性。它由具有交替刚度元素的拼凑图案组成，导致预测的骨密度缺乏收敛性和不稳定性。已经提出了平均技术来解决这个问题。最广为人知的技术之一（基于节点的公式）具有严重的缺点，例如：对网格密度和元素集成类型（完全与减少）的高度敏感，更重要的是，振荡解决方案也会导致缺乏收敛。我们提出了一种新的解决方案，包括非局部平滑技术。它定义为在网格的某个积分点控制骨骼适应的机械刺激，在相邻积分点中获得的刺激的平均值。该平均值使用到邻里中心距离的衰减函数进行加权。新技术已被证明可以克服所有提到的问题并以稳健的方式执行。它是在一个空心圆柱体上进行测试的，类似于长骨的骨干，经受弯曲或扭转。与其他平均技术和没有控制棋盘不稳定性的模型相比，棋盘不稳定性被消除，骨骼适应的局部收敛迅速实现。新算法也在相同的几何形状上进行了测试，结果良好，但在包含空隙的模型中，会产生应力集中，这通常会导致棋盘不稳定，就像在其他应用中一样，例如骨植入物界面的模拟。

更新日期：2020-11-22

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