Adaptive bone remodelling simulations which use finite element analysis can potentially aid in the design of orthopedic implants and can provide examples which test specific bone remodelling hypotheses in a quantitative manner. By concentrating on remodelling algorithms in which the geometry is fixed but the tissue stiffness changes based on strain energy density, we have predicted stability conditions for bone remodelling and we have tested the applicability of these conditions using numerical simulations. The stability requirements arrived at using a finite element formulation are similar to the requirements arrived at in an earlier analytical study. In order to test the stability conditions, we have developed an Euler backward time stepping technique which uses the derivation for stability. These simulations arrived at solutions which were impossible using Euler forward time stepping as applied in this study. Cases in which a simplified version of the derived Euler backward method are unstable or marginally stable have also been seen, but when the Euler backward method is applied using the full derived matrices, no instabilities are apparent. The results of the stability tests indicate that the converged density distributions in the examples studied are stable. Although a priori conditions which ensure stability are not found, a test for stability is provided, given an assumed density distribution.

Finite element simulation of adaptive bone remodelling: A stability criterion and a time stepping method

Presented is a review of shape optimal design and shape design sensitivity analysis with an emphasis on techniques dealing with shape of the boundaries of two- and three-dimensional bodies. Attention is focused on the continuum structural shape optimization based on numerical models by either finite elements or boundary elements. This requires sophisticated design sensitivity analysis techniques and a careful choice of design variables.

A review on shape optimal design and sensitivity analysis

Shape sensitivity expressions are derived for linear piezoelectric structures with coupled mechanical and elastic fields. By adopting the quasielectrostatic approximation for these inherently anisotropic materials, the adjoint variable method of optimization and the material derivative formulation of shape variations are used in a systematic procedure to evaluate the total variation of a general performance criterion with respect to shape variations. The material (total) derivative of the adopted integral functional is found in terms of primary and adjoint quantities, as well as the deformation velocity field. Since the structure is assumed to undergo dynamic response, domain integrations evaluated at the initial time are also needed in this formulation.

Shape sensitivity analysis of piezoelectric structures by the adjoint variable method

Design sensitivity analysis in dynamic thermoviscoelasticity with implicit integration

Design sensitivity analysis of coupled thermoviscoelastic systems

A continuous approach is adopted in order to find the total variation of a general performance criterion. The present analysis may find important physical applications in the simultaneous shape optimization and control of large space structures in time by applied thermal and/or mechanical loads. The adjoint variable method and the material derivative concept are used to find the sensitivity expressions

R. A. Meric

Papers

Simultaneous material/load/shape variations of thermoelastic structures