A novel contact formulation for micro finite element analysis of articular joints: implementation, verification and performance analysis

Micro finite element analysis (µFE) uses Cartesian (or voxel) meshes based on high resolution computed tomography images to non-invasively investigate bone biomechanics. Although specialised μFE solvers are capable of efficiently solving problems with billions of degrees of freedom, they poorly predict contact interactions due to the artificially jagged surface of voxel meshes. The simulated smoothed surface, sliding contact (SS-SC) formulation has been shown to be able to reduce the error associated with voxel meshes. However, this formulation has not as yet been implemented in a μFE solver such as ParOSol, and its accuracy advantage over conventional approaches when both bodies are deformable remains untested. In the present study, the SS-SC formulation was implemented within ParOSol, resulting in the new version called ParOSol Contact. This required new multigrid operators to be defined and an incremental solution process to be implemented. Comparative analysis using a sphere–block benchmark problem demonstrates that ParOSol Contact outperforms conventional contact methods in predictive accuracy of displacements, strains and stresses, including in problems where both bodies have unequal elastic moduli. A new 3D manufactured solution including contact was developed for rigorous code verification. The results confirm that native ParOSol and ParOSol Contact approach the theoretical convergence rates for higher mesh refinement and higher contact stiffness respectively, indicating a low likelihood of coding errors that could impact their accuracy. Based on the tests conducted here, ParOSol Contact achieved a reduction of computation time by 81.7% going from one to two multigrid levels, and by 22.3% going from two levels to three multigrid levels. We conclude that ParOSol Contact can accurately and efficiently predict solutions to contact problems using high resolution voxel meshes. This can potentially make it feasible to perform μFE analysis of fracture initiation at bone joints, biomechanics of osteoarthritic articular surfaces and integrity of bone implant-interfaces.