arXiv subjects

Simon Grützner

Publications and source records attributed to Simon Grützner.

3 recordsLinked to original sources

arXiv2022

Identifying Processes Governing Damage Evolution in Quasi-Static Elasticity Part I -- Analysis

We present a quasi-static elasticity model that accounts for damage evolution based on the ideas of Kachanov 1958 and and Rabotnov 1968. We analyze the resulting strongly nonlinear system of differential equations in view of well-posedness. The specific feature is that the displacements are connected to the damage evolution equation via a Nemytskii- or superposition- operator. The novelty in this work is that we present an inverse problem in a parameter identification setting, in which we are able to identify the shape of this Nemytskii-operator. From the material modelling point of view, the relation of material damage and displacements is modelled by the shape of this operator. We establish the Fréchet-derivative of the forward operator as well as the adjoint of the derivative and characterize both via systems of linear differential equations. We prove ill-posedness of the inverse problem and provide a sufficient condition for the classical nonlinear Landweber method to converge.

math.AP↗

arXiv2022

Identifying Processes Governing Damage Evolution in Quasi-Static Elasticity. Part 2 -- Numerical Simulations

We investigate numerically a quasi-static elasticity system of Kachanov-type. To do so we propose an Euler time discretization combined with a suitable finite elements scheme (FEM) to handle the discretization is space. We use ODE-type arguments to prove the consistency of the scheme as well as its convergence rate. We rely on the computational platform FEniCS to perform the FEM discretizations in space needed to compute the model output. The simulation results show a good agreement with both the physics of the problem and with our previous qualitative mathematical analysis results obtained for precisely the same problem setting. Furthermore, our implementation recovers nicely the theoretically expected convergence rate. This is a preliminary study preparing the framework for the rigorous numerical identification of the damage process in Kachanov-type models.

math.NA↗

arXiv2019

Dynamic Inverse Wave Problems - Part I: Regularity for the Direct Problem

For parameter identification problems the Fréchet-derivative of the parameter-to-state map is of particular interest. In many applications, e.g. in seismic tomography, the unknown quantity is modeled as a coefficient in a linear differential equation, therefore computing the derivative of this map involves solving the same equation, but with a different right-hand side. It then remains to show that this right-hand side is regular enough to ensure the existence of a solution. For second-order hyperbolic PDEs with time-dependent parameters the needed results are not as readily available as in the stationary case, especially when working in a variational framework. This complicates for example the reconstruction of a time-dependent density in the wave equation. To overcome this problem we extend the existing regularity results to the time-dependent case.

math.AP↗