Geodesic
An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver
Theoretical and applied mechanics, Tome 49 (2022) no. 2.

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Numerical simulations of the dynamics of soft biological tissues are highly non-trivial because tissues generally exhibit complex biological response to external and internal actions, including large deformations and remodeling. Combining the advantages of globally implicit approach (GIA) solvers with the general applicability of the semi-implicit \emph{General Plasticity Algorithm} (GPA), introduced by some of us some years ago, we present a new, efficient plasticity algorithm, which we call \emph{Bio Mechanics Basis Plasticity Algorithm} (BMBPA). This is fully implicit, based on a nested Newton solver, and naturally suited for massively parallel computations. The Bilby-Kr{ö}ner-Lee (BKL) multiplicative decomposition of the deformation gradient tensor is employed to introduce the unknowns of our model. We distinguish between global and local unknowns, associated with local and global equations, which are connected by means of a resolution function. The BMBPA asks for very few conditions to be applied and thus can be easily employed to solve several types of biological and biomechanical problems. We demonstrate the efficacy of BMBPA by performing two numerical experiments of a monophasic model of fiber-reinforced tissues. In one case, we consider the shear-compression test of a cubic specimen of tissue, while, in the other case, we focus on the unconfined compression test of a cylinder. The BMBPA is capable of solving the deformation and the remodeling of anisotropic biological tissues by employing a computation time of hours, while the GPA, applied to the same problems as the BMBPA, needs a substantially longer amount of time. All computations were performed in parallel and, within all tests, the performance of the BMBPA displayed substantially higher than the one of the GPA. The results of our simulations permit to study the overall mechanical behavior of the considered tissue and enable further investigations in the field of tissue biomechanics.
Keywords: nested Newton based algorithm, fiber-reinforced biological tissues, plastic-like distortions, computational biomechanics 
@article{TAM_2022_49_2_a5,
     author = {Markus M. Knodel and Salvatore Di Stefano and Arne N\"agel and Alfio Grillo},
     title = {An efficient algorithm for biomechanical problems based on a fully implicit nested {Newton} solver},
     journal = {Theoretical and applied mechanics},
     pages = {183 - 221},
     publisher = {mathdoc},
     volume = {49},
     number = {2},
     year = {2022},
     url = {http://geodesic.mathdoc.fr/item/TAM_2022_49_2_a5/}
}
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Markus M. Knodel; Salvatore Di Stefano; Arne Nägel; Alfio Grillo. An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver. Theoretical and applied mechanics, Tome 49 (2022) no. 2. http://geodesic.mathdoc.fr/item/TAM_2022_49_2_a5/