Identification of Stresses in Heterogeneous Contact Models

虽然机械负荷和膝关节骨关节炎的发病之间存在明确的相关性，但详细的机制仍然未知。一个障碍是关节软骨内的实际应力分布不能在体内测量。相反，当考虑下肢的多体动力学和软骨的固体力学时，它们可以从步态数据中估计。计算最大后验估计需要解决一个大的，异构的，耦合的优化问题。软骨接触的非穿透条件使得该问题是非光滑的。本项目旨在开发解决此类优化问题的有效方法。接触力学的非光滑性将通过半光滑罚函数方法来解决。所产生的强，局部化的非线性将被视为一个增广拉格朗日方法，一方面和非线性校正步骤内的复合步骤优化算法的另一方面。复合步骤算法已在SPP的第一阶段进行了研究，用于惩罚接触约束的最优控制，并将扩展到利用增广拉格朗日结构和非线性校正步骤。忽略关节软骨中的惯性项将允许在很大程度上在时间范围内并行计算步长，对于切向、法向和非线性校正步长的计算，需要有效的离散化和惩罚接触力学的求解器。关节软骨的复杂几何形状阻碍了标准嵌套网格层次结构的构建，这使得接触和点平滑的砂浆离散化的使用变得困难。相反，我们将研究更简单的对称段与段接触公式，并采用自适应求积和G1连续边界插值进行间隙计算。在接触边界上，重叠块平滑器将在半几何多重网格中使用，为计算切向和法向步长提供高效且与惩罚无关的预处理器。局部能量最小化将导致非线性多重网格为非线性校正步骤提供有效的求解器。在这种情况下，我们将调查水平相关的惩罚，以防止接触锁定粗grids.Finally，算法的发展将被集成到一个有效的优化代码，能够解决运动引起的应力分布在关节软骨的估计问题。代码的效率和有效性将在以标记位置和荧光透视图像形式提供的患者特定步态数据上得到证明。

While a clear correlation between mechanical loading and onset of osteoarthritis in the knee exists, the detailed mechanisms are still unknown. One obstacle is that actual stress distributions within the articular cartilage cannot be measured in vivo. Instead, they can be estimated from gait data when taking multibody dynamics of the lower limb and solid mechanics of the cartilage into account. Computing the maximum posterior estimate requires the solution of a large, heterogeneous, and coupled optimization problem. The non-penetration condition in cartilage contact renders this problem non-smooth.The project aims at developing efficient methods for solving such optimization problems. The nonsmoothness of contact mechanics will be addressed by a semismooth penalty approach. The incurred strong, localized nonlinearity will be treated with an augmented Lagrangian approach on one hand and with nonlinear correction steps within a composite step optimization algorithm on the other hand. The composite step algorithm has been investigated in the first phase of the SPP for optimal control with penalized contact constraints and will be extended towards exploiting augmented Lagrangian structure and nonlinear correction steps. Neglecting inertial terms in the articular cartilage will allow to parallelize step computations over the time horizon to a large extent.For the computation of tangential, normal, and nonlinear correction steps, efficient discretizations and solvers for the penalized contact mechanics are required. The complex geometry of articular cartilage prevents the construction of standard nested grid hierarchies, which makes the use of mortar discretizations of contact and point smoothers difficult. Instead, we will investigate simpler symmetric segment-to-segment contact formulations with adaptive quadrature and G1-continuous boundary interpolation for gap computation. On the contact boundary, overlapping block smoothers will be used within semi-geometric multigrid, providing efficient and penalty-independent preconditioners for computing tangential and normal steps. Local energy minimization will lead to nonlinear multigrid providing efficient solvers for nonlinear correction steps. In that context, we will investigate level-dependent penalization in order to prevent contact locking on coarse grids.Finally, the algorithmic developments will be integrated into an efficient optimization code that is able to solve the estimation problem for motion-induced stress distributions in the articular cartilage. The efficiency and effectivity of the code will be demonstrated on patient-specific gait data available in form of marker positions and fluoroscopy images.