Almeida EDGARD S., Spilker ROBERT L.
Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Department of Biomedical Engineering, and Scientific Computation Research Center, Rensselaer Polytechnic Institute, Troy NY 12180-3590.
Comput Methods Biomech Biomed Engin. 1997;1(1):25-46. doi: 10.1080/01495739708936693.
This paper addresses finite element-based computational models for the three-dimensional, (3-D) nonlinear analysis of soft hydrated tissues, such as the articular cartilage in diarthrodial joints, under physiologically relevant loading conditions. A biphasic continuum description is used to represent the soft tissue as a two-phase mixture of incompressible, inviscid fluid and a hyperelastic solid. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the nonlinear biphasic governing equations, including the effects of a strain-dependent permeability and a hyperelastic solid phase under finite deformation. The resulting first-order nonlinear system of equations is discretized in time using an implicit finite difference scheme, and solved using the Newton-Raphson method. Using a discrete divergence operator, an equivalence is shown between the mixed-penalty method and a penalty method previously derived by Suh et al. [1]. In Part II [2], the mixed-penalty and velocity-pressure formulations are used to develop two-dimensional (2-D) quadrilateral and triangular elements and 3-D hexahedral and tetrahedral elements. Numerical examples, including those representative of soft tissue material testing and simple human joints, are used to validate the formulations and to illustrate their applications. A focus of this work is the comparison of alternate formulations for nonlinear problems. While it is demonstrated that both formulations produce a range of converging elements, the velocity-pressure formulation is found to be more efficient computationally.
本文探讨了基于有限元的计算模型，用于在生理相关载荷条件下对诸如动关节中的关节软骨等软湿组织进行三维非线性分析。采用双相连续介质描述将软组织表示为不可压缩、无粘性流体和超弹性固体的两相混合物。使用交替混合罚函数和速度 - 压力有限元公式来求解非线性双相控制方程，包括应变相关渗透率和有限变形下超弹性固相的影响。所得的一阶非线性方程组在时间上使用隐式有限差分格式离散，并使用牛顿 - 拉夫逊方法求解。通过离散散度算子，证明了混合罚函数法与Suh等人[1]先前推导的罚函数法之间的等价性。在第二部分[2]中，混合罚函数和速度 - 压力公式用于开发二维四边形和三角形单元以及三维六面体和四面体单元。数值示例，包括代表软组织材料测试和简单人体关节的示例，用于验证公式并说明其应用。这项工作的一个重点是非线性问题交替公式的比较。虽然证明两种公式都产生了一系列收敛单元，但发现速度 - 压力公式在计算上更有效。
