A. van Keulen

Level Set Based Topology Optimization with Stress Constraints and Consistent Sensitivity Analysis

For aeronautical applications of topology optimization, it is of importance to develop topology optimization techniques, that can handle stress constraints in an efficient and accurate manner. The development of such topology optimization techniques is a challenging task due to the local nature of the stress constraints, their highly non-linear behaviour with respect to the design variables and the so-called singularity phenomenon. An accurate sensitivity analysis is essential for these type of problems with multiple constraints. In this paper, we propose a methodology of dealing with stress constraints in a level set based framework. In this framework, the level set function nodal values are related to element densities by an exact Heaviside projection. Stress relaxation and constraint aggregation techniques are used to deal with the singularity phenomenon and the local nature of the stress, respectively. A constrained optimization problem is then solved, in which the design variables (the level set nodal values) are updated in the projected steepest-descent direction, which is determined using a consistent sensitivity analysis.We demonstrate the effectiveness of this technique on two numerical examples. The results show that the level set method with a consistent sensitivity analysis allows for the treatment of multiple constraints by using constrained optimization techniques.

A Suite of Mathematical Models for Bone Ingrowth, Bone Fracture Healing and Intra-Osseous Wound Healing

In this paper, some modeling aspects with respect to bone ingrowth, frac- ture healing and intra-osseous wound healing are described. We consider a finite el- ement method for a model of bone ingrowth into a prosthesis. Such a model can be used as a tool for a surgeon to investigate the bone ingrowth kinetics when position- ing a prosthesis. The overall model consists of two coupled models: the biological part that consists of non-linear diffusion-reaction equations for the various cell den- sities and the mechanical part that contains the equations for poro-elasticity. The two models are coupled and in this paper the model is presented with some preliminary academic results. The model is used to carry out a parameter sensitivity analysis of ingrowth kinetics with respect to the parameters involved. Further, we consider a Finite Element model due to Bailon-Plaza and Van der Meulen for fracture heal- ing in bone. This model is based on a set of coupled convection-diffusion-reaction equations and mechanical issues have not been incorporated. A parameter sensitiv- ity analysis has been carried out. Finally, we consider a simplified model due to Adam to simulate intra-osseous wound healing. This model treats the wound edge as a moving boundary. To solve the moving boundary problem, the level set method is used. For the mesh points in the vicinity of the wound edge, a local adaptive mesh refinement is applied.

Chaos: The speed limiting phenomenon in dynamic atomic force microscopy

This paper investigates the closed-loop dynamics of the Tapping Mode Atomic Force Microscopy using a new mathematical model based on the averaging method in Cartesian coordinates. Experimental and numerical observations show that the emergence of chaos in conventional tapping mode AFM strictly limits the imaging speed. We show that, if the controller of AFM is tuned to be faster than a certain threshold, the closed-loop system exhibits a chaotic behavior. The presence of chaos in the closed-loop dynamics is confirmed via bifurcation diagrams, Poincare sections, and Lyapunov exponents. Unlike the previously detected chaos due to attractive forces in the AFM, which can be circumvented via simple changes in operation parameters, this newly identified chaos is seemingly inevitable and imposes an upper limit for the closed-loop bandwidth of the AFM.