Modelling of muscle behaviour by the finite element method using Hill's three-element model

We present a numerical algorithm for the determination of muscle response by the finite element method. Hill's three-element model is used as a basis for our analysis. The model consists of one linear elastic element, coupled in parallel with one non-linear elastic element, and one non-linear contractile element connected in series. An activation function is defined for the model in order to describe a time-dependent character of the contractile element with respect to stimulation. Complex mechanical response of muscle, accounting for non-linear force–displacement relation and change of geometrical shape, is possible by the finite element method. In an incremental-iterative scheme of calculation of equilibrium configurations of a muscle, the key step is determination of stresses corresponding to a strain increment. We present here the stress calculation for Hill's model which is reduced to the solution of one non-linear equation with respect to the stretch increment of the serial elastic element. The muscle fibers can be arbitrarily oriented in space and we give a corresponding computational procedure of calculation of nodal forces and stiffness of finite elements. The proposed computational scheme is built in our FE package PAK, so that real muscles of complex three-dimensional shapes can be modelled. In numerical examples we illustrate the main characteristic of the developed numerical model and the possibilities of solution of real problems in muscle functioning.