Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method

Abstract

This paper proposes a rapid inverse analysis approach based on the reduced basis method and the Levenberg-Marquardt-Fletcher algorithm to identify the "unknown" material properties: Young's modulus and stiffness-proportional Rayleigh damping coefficient of the interfacial tissue between a dental implant and the surrounding bones. In the forward problem, a finite element approximation for a three-dimensional dental implant-bone model is first built. A reduced basis approximation is then established by using a Proper Orthogonal Decomposition (POD)-Greedy algorithm and the Galerkin projection to enable extremely fast and reliable computation of displacement responses for a range of material properties. In the inverse analysis, the reduced basis approximation for the dental implant-bone model are incorporated in the Levenberg-Marquardt-Fletcher algorithm to enable rapid identification of the unknown material properties. Numerical results are presented to demonstrate the efficiency and robustness of the proposed method.

An efficient goal‐oriented sampling strategy using reduced basis method for parametrized elastodynamic problems

“…In SectionIII, wedescribe various topics related to the RB methodology: approximation, the standard versus goal‐oriented algorithms, error estimations and offline‐online computational procedure. In Section IV, we verify the performance of the proposed algorithm by investigating numerically two problems: a two‐dimensional (2D) linear elastodynamic problem and a three‐dimensional (3D) dental implant simulation problem . Finally, we provide some concluding remarks in Section V.…”

“…Therefore, the relevant system behavior will be described by an implicit input‐output relationship; where its computation requires the solution of the underlying parameter‐PDE (or µ PDE). We pursue model order reduction (MOR) methods (i.e., snapshots‐proper orthogonal decomposition (POD) and reduced basis (RB) ) which permits the efficient and reliable evaluation of this PDE‐induced input‐output relationship in many query and real‐time contexts.…”

“…The RB method was first introduced in the late 1970s for nonlinear analysis of structures and has been investigated and developed more broadly . Recently, the RB method was well developed for various kinds and classes of parametrized PDEs, such as: the eigenvalue problems , the coercive/noncoercive affine/nonaffine linear/nonlinear elliptic PDEs , the coercive/noncoercive affine/nonaffine linear/nonlinear parabolic PDEs , the coercive affine linear hyperbolic PDEs , and several highly nonlinear problems such as Burger's equation and Boussinesq equation . For the linear wave equation, the RB method and associated a posteriori error estimation was developed successfully with some levels ; however, in the RB context none of these works have focused on constructing optimally goal‐oriented RB basis functions.…”

“…In this work, we aim to build an optimally goal‐oriented set of MOR basis functions without computing and storing all the POD snapshots as the aforementioned approaches. The best way to do that is using the RB method with Greedy sampling strategy (see, for instance ). Thanks to the Greedy iterations, the proposed algorithm now looks for the parameter points such that the output error approximation will be minimized.…”

“…Thanks to the Greedy iterations, the proposed algorithm now looks for the parameter points such that the output error approximation will be minimized. For the linear wave equation, this idea is novel and further develops the idea of the standard POD–Greedy sampling procedure currently used , where the algorithm will pick up optimally all parameter points such that the error (or error indicator) of the field variable is minimized. By this way, we expect to improve significantly the accuracy of the RB output functional computations; but consequently, we might lose the rapid convergence rate of the field variable as in the standard POD–Greedy algorithm.…”