Keywords: structural analysis, optimisation, topology, contact analysis, finite elements, biomechanics.

Generalized topology optimisation of structures has been an area of intense research work for the last decade. In the early years several two-dimensional models considering different aspects of the problem were analyzed. Different types of microstructures were tested, topology design models for vibration and buckling loads control were developed and shell and plate structures considered. Recently, three-dimensional models have been proposed, extending the concepts developed for two-dimensional problems, and launching the topology optimization as an important design tool (see Bendsøe and Sigmund [1]).

Although the high level of development of the field, models for topology optimization focusing particular aspects and problems have still a special interest. It is the case of the model proposed in this paper. This work presents a development of a three-dimensional computational model for topology optimization with contact conditions. The model uses a material distribution approach and the optimisation criterion is the structural compliance under multiple loads, subjected to an isoperimetric constraint on volume.

The model is an extension of the model developed in Fernandes et al. [2], with the equilibrium equation expressed for contact problems. The necessary conditions for optimum are derived analytically based on the Augmented Lagrangian associated with the problem. The methodology is similar to the one proposed by Rodrigues [3] for shape optimisation of mechanical components.

An important issue of the problem is that it is not self-adjoint. The stationarity of the Lagrangian with respect to the design variable yields a linear elastic problem with additional displacement constraints on the contact boundary, obtained from the solution of the state problem. The model is approximated numerically through a suitable finite element discretization and solved by a first order optimisation method. The computational model developed was tested in several numerical applications with successful results.

Besides the application on structural optimisation, the model has a special interest for biomechanics applications, particularly orthopaedic implants. The model is adapted to study the bone remodelling after a non cemented hip arthroplasty. In this problem the interface bone/implant requires a very careful modelling, and considering the two bodies in contact is the correct approach. Since some bone remodelling models proposed recently are based on topology optimisation (see Fernandes et al. [4]), the contact model proposed in this work as special interest in this field. Numerical examples in this area are also presented.

1

M. Bendsøe, O. Sigmund, "Topology Optimization, Theory, Methods and Applications", Springer-Verlag, 2003.

2

P.R. Fernandes, J.M. Guedes, H. Rodrigues, "Topology Optimization of 3D Linear Elastic Structures", Computers & Structures, 73, 583-594, 1999. doi:10.1016/S0045-7949(98)00312-5

3

H. Rodrigues, "A Mixed Variational Formulation for Shape Optimization of Solids with Contact Conditions", Structural Optimization, 6, 19-28, 1993. doi:10.1007/BF01743171

4

P. Fernandes, H. Rodrigues, C. Jacobs, "A Model of Bone Adaptation Using a Global Optimization Criterion Based on the Trajectorial Theory of Wolff", Computer Methods in Biomechanics & Biomedical Engineering, 2, 125-138, 1999. doi:10.1080/10255849908907982

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