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In an aging society where the number of joint replacements rises, it is important to also increase the longevity of implants.
In particular hip implants have a lifetime of at most 15 years. This derives primarily from
pain due to implant migration, wear, inflammation, and dislocation, which is affected by
the positioning of the implant during the surgery. Current joint replacement practice uses
2D software tools and relies on the experience of surgeons. Especially the 2D tools fail to
take the patients’ natural range of motion as well as stress distribution in the 3D joint
induced by different daily motions into account.
Optimizing the hip joint implant position for all possible parametrized motions under the
constraint of a contact problem is prohibitively expensive as there are too many motions
and every position change demands a recalculation of the contact problem. For the
reduction of the computational effort, we use adaptive refinement on the parameter
domain coupled with the interpolation method of Kriging. A coarse initial grid is to be
locally refined using goal-oriented error estimation, reducing locally high variances. This
approach will be combined with multi-grid optimization such that numerical errors are
reduced.

In this thesis, adaptive algorithms in optimization under PDE constraints have been inves-
tigated. In its application, the aim of optimization is to increase the longevity of implants,
namely the hip joint implant, and in doing so to minimize stress shielding and simultaneously minimize the influence of locally high stresses, that, above a threshold value, are malign to the bone structure. Under the constraint of the equilibrium of forces, describing an elastodynamic setup, coupled with a contact inequality condition, a computationally expensive problem formulation is given.
The first step to make the solution of the given problem possible and efficient was to change over to the spatial equilibrium equation, thus rendering an elastostatic setup. Subsequently the intrinsically dynamic motions – trajectories in the load domain – were converted to the static setup. Thus, the trajectories are marginalized to the load domain and characterized with probability distributions. Therefore the solving of the PDE constraint, the contact problem, is simplified.
Yet in the whole optimization process, the solving of the PDE, the spatial equilibrium equation together with the contact condition has the most expensive contribution still and hence needed further reduction. This was achieved by application of Kriging interpolation to the load responses of the integrated distribution of stress difference and the maximum stresses. The interpolation of the two response surfaces only needs comparatively few PDE solves to set up the models. Moreover, the Kriging models can be adaptively extended by sequentially adding sample-response pairs. For this the Kriging inherent variance is used to estimate ideal new sample locations with maximum variance values. In doing so, the overall interpolation variance and therefore the interpolation error is reduced.
For the integration of the integrated stress differences and penalty values on the relative high dimensional load domain Monte Carlo integration was implemented, averting the curse of dimension. Here, the motion’s probability distribution combined with patient specific data of motion frequencies is taken advantage of, making obsolete the use of the otherwise necessary importance sampling.
Throughout the optimization, the FE-discretization error and the subsequently attached errors entering the solution process via PDE discretization and approximative
solving of the PDE, Kriging interpolation and Monte Carlo integration need to decrease. While the FE-discretization error and the solution of the elastostatic contact problem were assumed precise enough, numerics showed, that the interpolation and integration errors can be controlled by adaptive refinement of the respective methods. For this purpose comparable error quantities for the particular algorithms were introduced and effectively put to use.
For the implant position’s optimization, the derivative of the objective function was derived using the implicit function theorem. As the FE-discretization changes with implant position modifications big enough, a special line search had to be used to deal with the discontinuities in the objective function.
The interplay and performance of the subalgorithms was demonstrated numerically on a reduced 2D setup of a hip joint with and without the implant. Consequently the load domain and the control variable were also limited to the 2D case.

Following axon pathfinding, growth cones transition from stochastic filopodial exploration to the formation of a limited number of synapses. How the interplay of filopodia and synapse assembly ensures robust connectivity in the brain has remained a challenging problem. Here, we developed a new 4D analysis method for filopodial dynamics and a data-driven computational model of synapse formation for R7 photoreceptor axons in developing Drosophila brains. Our live data support a 'serial synapse formation' model, where at any time point only a single 'synaptogenic' filopodium suppresses the synaptic competence of other filopodia through competition for synaptic seeding factors. Loss of the synaptic seeding factors Syd-1 and Liprin-α leads to a loss of this suppression, filopodial destabilization and reduced synapse formation, which is sufficient to cause the destabilization of entire axon terminals. Our model provides a filopodial 'winner-takes-all' mechanism that ensures the formation of an appropriate number of synapses.