Model order reduction of dynamic skeletal muscle models

Abstract

Forward-dynamics simulations of three-dimensional continuum-mechanical skeletal muscle models are a complex and computationally expensive problem. Considering a fully dynamic modelling framework based on the theory of finite elasticity is challenging as the muscles' mechanical behaviour requires to consider a highly nonlinear, viscoelastic and incompressible material behaviour. The governing equations yield a nonlinear second-order differential algebraic equation (DAE), which represents a challenge to model order reduction (MOR) techniques. This contribution shows the results of the offline phase that could be obtained so far by applying a combination of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). MotivationModelling and simulation of the human movement to predict, for example, joint loading, has been a topic of interest for many years (c.f. [1]). State of the art multi-body models include skeletal muscles as one-dimensional objects, which of course do not properly represent reality. Consequently, more realistic three-dimensional continuum-mechanical skeletal muscle models based on realistic muscle geometries are developed. Such models are able to simulate the muscle deformation and forces due to a stimulation (forward simulations, c.f. [2]). Needless to say, that the continuum-mechanical approach requires substantially more computational effort if compared to the one-dimensional model context. Particularly if multi-muscle systems are considered or within the many-query context, the three-dimensional muscle models become prohibitively expensive. This is where model order reduction (MOR) offers an effective remedy. Skeletal muscle modelThe motion and deformation of a continuous muscle, whose shape corresponds to the reference domain Ω 0 ⊂ R 3 , is described by means of the balance of momentumTherein, ρ 0 is the muscle density, v is the velocity field, P is the first Piola-Kirchhoff stress tensor and B denotes the body forces. Additionally, this governing equation is subject to an incompressibility constraint equation, which restricts the set of admissible states, to ensure that no volumetric change occurs. With F being the deformation gradient and J := detF the Jacobian or volume ratio, the incompressibility constraint readsThe nonlinear constitutive equation used in this work follows the general setup of [3,4]. Therein, muscle tissue, is modelled as hyperelastic, homogeneous, transversely isotropic and incompressible material. Furthermore, owing to the electrically active muscle fibres, the muscle tissue does not only exhibit passive but also active material behaviour. These assumptions yield an additive split of the strain energy function, which is inherited by the first Piola-Kirchhoff stress tensorwith p(X, t) denoting the hydrostatic pressure. Introducing a parameter µ ∈ P ⊂ R p comprising any property of the muscle model that shall be varied and using the finite element method to discretize Equations (1) and (2), leads the following second-order paramet...