摘要
人们普遍将描述骨再造过程的微分方程看作有多个自由度的非线性动力系统 ,它相对优化目标不断变化 ,可以导致很多组可能解。因此关于理论解稳定性条件的分析是非常必要的。本文的目的是阐述非线性动力系统分析的数学方法应用于一种骨自优化控制方程 ,研究该骨再造方程的全局稳定性。分析了两个理论模型 :两单元模型和 2× 2单元模型 ,该模型特点是其中各个单元的应力、应变都是相关的 ,这样与实际有限元模型更加接近。以朱兴华等[1] 提出的高阶非线性骨再造速率方程作为控制方程 ,重点考察其中再造率系数B(t)取为指数形式和引入非线性再造方程的阶数时 ,骨自优化方程获得稳定解的条件。并以两个经典的二维平面问题作为算例 ,与上述两个模型的理论分析结果进行对比 ,使分析得到的结果得以确认。
From the character of the differential equation describing the bone-remodeling process, it can be considered as a nonlinear dynamic system with many degrees of freedom, which behaves divergently relative to the objective, leading to many possible solutions. So it is necessary to analyze the stationary states related to this theory and their stability conditions. The aim of this paper was to illustrate the application of mathematical tools for the analysis of non-linear dynamical systems to the study of global stability of a kind of bone remodeling scheme. Using the high-order nonlinear speed equation of bone-remodeling suggested by Zhu Xinghua et al as controlling equation, in which the exponent form was chosen for the remodeling coefficient B(t) and the order of nonlinear remodeling equation introduced, the stability conditions were investigated. Two models were analyzed: the two-unit model and 2×2 elements model. Their characteristics lied in that both stresses and strains in each element were all dependent upon each other, which dosely resembled the practical FEM models. Then two 2D typical examples were used to compare with the analytical results to verify the latter. This kind of mathematically analytical method plays critical roles in investigating the effects of some parameters in the complicated nonlinear dynamical system and is used to study the controlling process of bone-remodeling.
出处
《中国生物医学工程学报》
CAS
CSCD
北大核心
2002年第4期289-297,共9页
Chinese Journal of Biomedical Engineering
基金
国家自然科学基金资助 ( 39970 191)
吉林省科技发展计划项目 ( 19980 5 5 4 0 1)
关键词
骨自优化方程
稳定解
非线性分析方法
定态
Bone remodeling equation
Nonlinear dynamical system
Stable solutions
Stationary state
