Gradient‐based optimization with B‐splines on sparse grids for solving forward‐dynamics simulations of three‐dimensional, continuum‐mechanical musculoskeletal system models

Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum‐mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation‐driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3‐dimensional, continuum‐mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B‐splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B‐splines allows the use of all existing gradient‐based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2‐muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles. Investigating the interplay between muscular activity and motion is key to improve our understanding of healthy or diseased musculoskeletal systems. This research focuses on reducing the computational effort of continuum‐mechanical musculoskeletal models by computing surrogates based on sparse grids, which enable higher dimensional discretizations. Novel hierarchical B‐splines are used for the first time in this setting to compute surrogates, allowing the use of all existing gradient‐based optimization techniques without further numerical approximation.