The spherical design algorithm in the numerical simulation of biological tissues with statistical fibre-reinforcement

Nowadays, the description of complex physical systems, such as biological tissues, calls for highly detailed and accurate mathematical models. These, in turn, necessitate increasingly elaborate numerical methods as well as dedicated algorithms capable of resolving each detail which they account for. Especially when commercial software is used, the performance of the algorithms coded by the user must be tested and carefully assessed. In Computational Biomechanics, the Spherical Design Algorithm (SDA) is a widely used algorithm to model biological tissues that, like articular cartilage, are described as composites reinforced by statistically oriented collagen fibres. The purpose of the present work is to analyse the performances of the SDA, which we implement in a commercial software for several sets of integration points (referred to as “spherical designs”), and compare the results with those determined by using an appropriate set of points proposed in this manuscript. As terms for comparison we take the results obtained by employing the integration scheme Integral, available in Matlab®. For the numerical simulations, we study a well-documented benchmark test on articular cartilage, known as ‘unconfined compression test’. The reported numerical results highlight the influence of the fibres on the elasticity and permeability of this tissue. Moreover, some technical issues of the SDA (such as the choice of the quadrature points and their position in the integration domain) are proposed and discussed