Simulation of the bone remodeling process is extremely important because it makes possible the structure forecast of one or several bones when anomalous situations, such as prosthesis installation, occur. Thus, it is necessary that the mathematical model to simulate the bone remodeling process be reliable; that is, the numerical solution must be stable regardless of initial density field for a phenomenological approach to model the process. For several models found in the literature, this characteristic of stability is not observed, largely due to the discontinuities present in the property values of the models (e.g. Young's modulus and Poisson's ratio). In addition, checkerboard formation and the lazy zone prevent the uniqueness of the solution. To correct these difficulties, this study proposes a set of modifications to guarantee the uniqueness and stability of the solutions, when a phenomenological approach is used. The proposed modifications are: (a) change the rate of remodeling curve in the lazy zone region and (b) create transition functions to guarantee the continuity of the expressions used to describe Young's modulus and Poisson's ratio. Moreover, the stress smoothing process controls the checkerboard formation. Numerical analysis is used to simulate the solution behavior from each proposed modification. The results show that, when all proposed modifications are applied to the three-dimensional models simulated here, it is possible to observe the tendency toward a unique solution.

骨骼重塑过程的模拟非常重要，因为它可以在出现异常情况（例如假肢安装）时预测一个或多个骨骼的结构。因此，模拟骨重建过程的数学模型必须可靠；也就是说，对于模拟过程的现象学方法，无论初始密度场如何，数值解都必须是稳定的。对于文献中发现的几个模型，没有观察到这种稳定性特征，主要是由于模型的属性值（例如杨氏模量和泊松比）中存在不连续性。此外，棋盘格的形成和惰性区域阻碍了解决方案的唯一性。为了纠正这些困难，当使用现象学方法时，本研究提出了一系列修改以保证解的唯一性和稳定性。建议的修改是：(a) 改变惰性区域中的重塑曲线的速率，以及 (b) 创建过渡函数以保证用于描述杨氏模量和泊松比的表达式的连续性。此外，应力平滑过程控制棋盘格的形成。数值分析用于模拟每个建议修改的解决方案行为。结果表明，当所有建议的修改应用于此处模拟的三维模型时，可以观察到趋向于唯一解决方案的趋势。(a) 改变惰性区域中的重塑曲线的速率和 (b) 创建过渡函数以保证用于描述杨氏模量和泊松比的表达式的连续性。此外，应力平滑过程控制棋盘格的形成。数值分析用于模拟每个建议修改的解决方案行为。结果表明，当所有建议的修改应用于此处模拟的三维模型时，可以观察到趋向于唯一解决方案的趋势。(a) 改变惰性区域中的重塑曲线的速率和 (b) 创建过渡函数以保证用于描述杨氏模量和泊松比的表达式的连续性。此外，应力平滑过程控制棋盘格的形成。数值分析用于模拟每个建议修改的解决方案行为。结果表明，当所有建议的修改应用于此处模拟的三维模型时，可以观察到趋向于唯一解决方案的趋势。数值分析用于模拟每个建议修改的解决方案行为。结果表明，当所有建议的修改应用于此处模拟的三维模型时，可以观察到趋向于唯一解决方案的趋势。数值分析用于模拟每个建议修改的解决方案行为。结果表明，当所有建议的修改应用于此处模拟的三维模型时，可以观察到趋向于唯一解决方案的趋势。