Identification of Stresses in Heterogeneous Contact Models

• Applicants: Professor Dr.-Ing. Georg Duda; Dr. Martin Weiser, since 11/2022
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 423768832

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
• Ehemaliger Antragsteller: Professor Dr. Anton Schiela, until 11/2022