In der vorliegenden Arbeit wird ein Beitrag zur numerischen Behandlung der inversen Kräfteberechnung für Aufgabenstellungen aus der Biomechanik vorgestellt. Die Behandlung inverser Probleme ist insbesondere aus mathematischer Sicht bezüglich der Existenz und Regularität von Lösungen nicht trivial. Auf die theoretischen Aspekte soll hier nicht eingegangen werden.
Vielmehr soll das entwickelte algorithmische Lösungskonzept zur Behandlung inverser Probleme im Vordergrund stehen.
Diese Arbeit beschränkt sich auf die Untersuchungen am menschlichen Femur. Dabei wird das komplexe dynamische Problem der Lasten am Femur auf statisch äquivalente Lasten zurückgeführt. Die Idee besteht darin, aus einer bekannten bzw. mittels CT1-Aufnahmen erfassten optimalen Dichteverteilung die Lasten zu ermitteln. Dadurch k¨onnen z.B. individuelle Therapien für Patienten entwickelt werden. Es wird festgestellt, dass diese Problemstellung um ein vielfaches komplexer als die Problemstellung des Knochenumbaus ist. Beim Knochenumbau
wird aus einer vektoriellen Grösse, nämlich der vorgegebenen Kraft, eine skalare Grösse also die Dichte ermittelt. Das inverse Problem stellt den umgekehrten Fall dar. Aus dieser Problemstellung resultiert eine nichtlineare Optimierungsaufgabe, die mit Hilfe der Algorithmen der mathematischen Optimierung behandelt wird. Zur Lösung der Optimierungsaufgabe werden sowohl ableitungsfreie als auch Gradienten-Verfahren eingesetzt, wobei die gradientenbasierten Verfahren im Vordergrund stehen. Um mit Gradienten-Verfahren die resultierende Optimierungsaufgabe lösen zu können, wird eine Sensitivitätsanalyse implementiert. Im Kapitel 6 wird gezeigt, dass für das inverse Problem mehrere gleichwertige Lösungen vorhanden sein können.
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