# Solving Stiff Cauchy Problems in Scientific Computing – An Analytic-numeric Approach.

## Abstract

This work presents analytic-numeric approaches for solving Cauchy problems in the context of physical simulations of stiff elastodynamic scenarios. Classical explicit numerical integration schemes have the shortcoming that step sizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to non-physical behavior. To overcome these specific detriments, an appropriate class of integrators that solves the stiff part of the governing equations by employing a closed-form solution is introduced. With these techniques, up to three orders of magnitude greater time steps can be handled compared to conventional methods, and, at the same time, a tremendous increase in overall long-term stability is achieved. This advantageous behavior is demonstrated across a broad spectrum of complex models that include deformable solids, trusses, and textiles, including damping, collision responses, friction, and non-linear material behavior. To realize an efficient and physically accurate simulation of fiber-based systems such as human hair, wool infills, and brushes, an appropriate approach for the physically accurate simulation of densely packed fiber assemblies is presented. This model works on the level of single fibers and considers effects such as adhesion, anisotropic frictional contacts, and Coulomb far-field interactions. Each fiber is modeled as a three-dimensional network of coupled oscillators in which the order of the particles is such that analytical expressions for the mapping of well-known mechanical quantities such as Young’s modulus and shear modulus can be derived. Non-linear material behavior is modeled by introducing support forces. This process allows for large deformations of single fibers without the need for more complicated models such as Kirchhoff rods. The efficiency of this concept in proven with a detailed scientific validation, which demonstrates that this method is capable of correctly capturing all the effects that occur in real fiber assemblies.

## Bibtex

@PHDTHESIS{michels-2014-phd, author = {Michels, Dominik L.}, title = {Solving Stiff Cauchy Problems in Scientific Computing – An Analytic-numeric Approach.}, type = {Dissertation}, year = {2014}, month = mar, school = {Universit{\"a}t Bonn}, abstract = {This work presents analytic-numeric approaches for solving Cauchy problems in the context of physical simulations of stiff elastodynamic scenarios. Classical explicit numerical integration schemes have the shortcoming that step sizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to non-physical behavior. To overcome these specific detriments, an appropriate class of integrators that solves the stiff part of the governing equations by employing a closed-form solution is introduced. With these techniques, up to three orders of magnitude greater time steps can be handled compared to conventional methods, and, at the same time, a tremendous increase in overall long-term stability is achieved. This advantageous behavior is demonstrated across a broad spectrum of complex models that include deformable solids, trusses, and textiles, including damping, collision responses, friction, and non-linear material behavior. To realize an efficient and physically accurate simulation of fiber-based systems such as human hair, wool infills, and brushes, an appropriate approach for the physically accurate simulation of densely packed fiber assemblies is presented. This model works on the level of single fibers and considers effects such as adhesion, anisotropic frictional contacts, and Coulomb far-field interactions. Each fiber is modeled as a three-dimensional network of coupled oscillators in which the order of the particles is such that analytical expressions for the mapping of well-known mechanical quantities such as Young’s modulus and shear modulus can be derived. Non-linear material behavior is modeled by introducing support forces. This process allows for large deformations of single fibers without the need for more complicated models such as Kirchhoff rods. The efficiency of this concept in proven with a detailed scientific validation, which demonstrates that this method is capable of correctly capturing all the effects that occur in real fiber assemblies.}, url = {https://bonnus.ulb.uni-bonn.de/SummonRecord/FETCH-bonn_catalog_39675972} }