Mathematical Development of a Novel Discrete Hip Deformation Algorithm for the In Silico Elasto-Hydrodynamic Lubrication Modelling of Total Hip Replacements

Abstract

In this paper, the procedure to achieve an accurate deformation model of a total hip replacement (THR) was proposed with the aim to obtain a numerical tool to be simply merged into THR elasto-hydrodynamic computational synovial lubrication algorithms. The approach was based on the Finite Element Method (FEM) and was developed in a Matlab code, allowing the definition of the influence matrix and of a boundary conditions vector. It works with linear tetrahedra and performs the displacement calculation for both the acetabular cup and the femoral head, taking into account the anatomical hip relative motion, by coupling them with a cubic interpolation matrix. Two simulations were conducted in order to validate the algorithm and the results were compared with the ones obtained by the commercial software Ansys. The comparison provides a satisfactory agreement in terms of surface deformation, Von Mises stress and strain energy, proving the reliability of the model and the possibility to use the model in the in silico prostheses tribological simulations, avoiding the complexity and the high computational resource requirement coming from the coupling between complex lubrication algorithms and FEM commercial software, and with the possibility to directly act on many key parameter characteristics of the investigated problem.

Implementation of a Finite Element Deformation Model Within an Elasto-Hydrodynamic Lubrication Numerical Solver for a Ball in Socket Tribopair

“…The purpose of this work is to implement a deformation model based on the discretization of the ball in socket joint in linear tetrahedra finite elements (Ruggiero and Sicilia, 2021), in order to assemble an influence matrix which relates the surfaces' radial deformation vector to the pressure one generated by the fluid meatus within the joint gap. The objective is to fully merge the finite element model with the lubrication algorithm into a comprehensive MATLAB code which works to solve the lubrication problem at the interface between the ball and socket surfaces, keeping information about the whole bodies composing the kinematical couple, in order to obtain a fluid-solid interaction system in which both the computational fluid dynamics and computational solid model are implemented.…”

Video1.MP4

“…With reference to a previous work of the same authors (Ruggiero and Sicilia, 2021), the linear tetrahedra mesh was created for both the bodies and is shown in Figure 1, where the fixed nodes are highlighted by black triangles and the pressure loaded nodes are identified by red arrows oriented toward the radial loading direction.…”

“…2 referring to the socket does not need further transformations, the one referring to the ball has to be rotated. The rotation is obtained through the cubic interpolation matrices J bs and J sb [described in (Ruggiero and Sicilia, 2021)] which, respectively, convert a surface quantity vector defined in the ball surface grid u b into its version defined in the socket surface grid u s and vice versa, as written in Eq. 3.…”

“…The surfaces' separation h field is obtained in Eq. 8 by the composition of: 1) the geometrical approach h g due to the eccentricity vector e of the ball center with respect to the socket one, which is projected on the radial unit vector r and subtracted from the radial clearance c (Ruggiero and Sicilia, 2020b); 2) the total surfaces' radial deformation field δ, which is directly obtained from its vector version δ coming from the finite element deformation model described above (Ruggiero and Sicilia, 2021); 3) the penetration wear depth field u w cumulated until the analyzed time instant t, which is obtained through the Archard wear model by integrating over time the linear wear rate, i.e., the product between the wear factor function k w (scaling the nominal wear factor k w0 by a power α w of the ratio between the separation h and the surface arithmetic roughness R a , in order to take advantage of the Archard wear model over the whole analyzed domain, even in that lubricated zones in which the film thickness is much larger than the roughness), the lubricant pressure p, and the sliding velocity v sl (Gao, et al, 2017;Ruggiero and Sicilia, 2020b).…”

“…1) The proportional models, which assume a proportionality between the local deformation and the local pressure, such that each nodal deformation on the analyzed lubrication grid does not depend on other nodes [e.g., the constraint column model or independent spring model Jalali-Vahid, et al, 2003;Ruggiero and Sicilia, 2020b), the elastic foundation model (Fregly, et al, 2003;Halloran, et al, 2005;Pèrez-Gonzàlez, et al, 2008;Mukras, et al, 2010;Askari and Andersen, 2018;Srivastava, et al, 2021), etc. ]; it can be used when the two materials composing the couple are characterized by very different stiffness, such as to consider only the deformation of the softer body; 2) The convolution method, which consists in solving the mechanical problem associated with the evaluation of the surface deformation of an elastic half-space due to a distributed surface load (i.e., the pressure field) by applying the discrete Fourier transform or the boundary element method (Evans and Hughes, 2000;Wang, et al, 2003;Wang and Jin, 2004b;Wang and Jin, 2004a;Wang and Jin, 2008;Gao, et al, 2009;Wang, et al, 2009;Shettar, et al, 2018); in this case, each lubrication grid point communicates with the other nodes through the definition of an influence matrix which relates the radial surface deformation field to the pressure one; 3) The finite element method, aims to elaborate displacements and constraint reactions of the whole analyzed bodies, by discretizing them in several finite elements and solving the weak formulation of the mechanical equilibrium problem (Jagatia and Jin, 2001;Halloran, et al, 2005;Ruggiero, et al, 2018;Ruggiero, et al, 2019;Ruggiero and Sicilia, 2021); this approach ensures accurate deformation results, however, it can be very expensive in terms of computational time and could require an additional dedicated software with respect to the one used to solve the lubrication problem.…”