Numerical analysis of a bone remodeling model with damage. (English) Zbl 1442.74194

Summary: Bone tissue is a material with a complex structure and mechanical properties. Repetitive loads or diseases can cause microfractures to appear in the bone tissue, which results in a deterioration of its mechanical properties. On the other hand, bone is a constantly evolving tissue, adapting its density to the loading conditions it is subjected to. In this work, we study, from the numerical point of view, a strain-adaptive bone remodeling model coupled with a damage model. The variational problem is written as a coupled system consisting of a nonlinear variational equation for the displacement field and nonlinear parabolic variational inequalities for the apparent density and damage. Then, fully discrete approximations are introduced, using the finite element method and a hybrid combination of Euler schemes. A priori error estimates are proved under adequate regularity conditions, and the linear convergence of the algorithm is deduced. Finally, some one- and two-dimensional numerical simulations of test examples are presented, to demonstrate the accuracy of the approximations and the behavior of the solutions.


74R05 Brittle damage
74S05 Finite element methods applied to problems in solid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74L15 Biomechanical solid mechanics
92C10 Biomechanics
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