arXiv subjects

Christian Hesch

Publications and source records attributed to Christian Hesch.

6 recordsLinked to original sources

arXiv2023

Variational formulation and monolithic solution of computational homogenization methods

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE2 and IGA2 approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null-space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE2 and IGA2 formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general non-linear problems.

cs.CE↗

arXiv2023

Computational homogenization of phase-field fracture

In this contribution we investigate the application of phase-field fracture models on non-linear multiscale computational homogenization schemes. In particular, we introduce different phase-fields on a two-scale problem and develop a thermodynamically consistent model. This allows on the one hand for the prediction of local micro-fracture patterns, which effectively acts as an anisotropic damage model on the macroscale. On the other and, the macro-fracture phase-field model allows to predict complex fracture pattern with regard to local microstructures. Both phase-fields are introduced in a common framework, such that a joint consistent linearization for the Newton-Raphson iteration can be developed. Finally, the limits of both models as well as the applicability are shown in different numerical examples.

cs.CE↗

arXiv2022

Computational homogenization of higher-order continua

We introduce a novel computational framework for the multiscale simulation of higher-order continua that allows for the consideration of first-, second- and third- order effects at both micro- and macro-level. In line with classical two-scale approaches, we describe the microstructure via representative volume elements (RVE) that are attached at each integration point of the macroscopic problem. To take account of the extended continuity requirements of independent fields at micro- and macro-level, we discretize both scales via isogeometric analysis (IGA). As a result, we obtain an IGA2-method that is conceptually similar to the well-known FE2-method. We demonstrate the functionality and accuracy of this novel multiscale method by means of a series of multiscale simulations involving different kinds of higher-order continua.

cs.CE↗

arXiv2021

A strain-gradient formulation for fiber reinforced polymers: Hybrid phase-field model for porous-ductile fracture

A novel numerical approach to analyze the mechanical behavior within composite materials including the inelastic regime up to final failure is presented. Therefore, a second-gradient theory is combined with phase-field methods to fracture. In particular, we assume that the polymeric matrix material undergoes ductile fracture, whereas continuously embedded fibers undergo brittle fracture as it is typical e.g. for roving glass reinforced thermoplastics. A hybrid phase-field approach is developed and applied along with a modified Gurson-Tvergaard-Needelman GTN-type plasticity model accounting for a temperature-dependent growth of voids on microscale. The mechanical response of the arising microstructure of the woven fabric gives rise to additional higher-order terms, representing homogenized bending contributions of the fibers. Eventually, a series of tests is conducted for this physically comprehensive multifield formulation to investigate different kinds and sequences of failure within long fiber reinforced polymers.

cs.CE↗

arXiv2021

Multidimensional coupling: A variationally consistent approach to fiber-reinforced materials

A novel mathematical model for fiber-reinforced materials is proposed. It is based on a 1-dimensional beam model for the thin fiber structures, a flexible and general 3-dimensional elasticity model for the matrix and an overlapping domain decomposition approach. From a computational point of view, this is motivated by the fact that matrix and fibers can easily meshed independently. Our main interest is in fiber reinforce polymers where the Young's modulus are quite different. Thus the modeling error from the overlapping approach is of no significance. The coupling conditions acknowledge both, the forces and the moments of the beam model and transfer them to the background material. A suitable static condensation procedure is applied to remove the beam balance equations. The condensed system then forms our starting point for a numerical approximation in terms of isogeometric analysis. The choice of our discrete basis functions of higher regularity is motivated by the fact, that as a result of the static condensation, we obtain second gradient terms in fiber direction. Eventually, a series of benchmark tests demonstrate the flexibility and robustness of the proposed methodology. As a proof-of-concept, we show that our new model is able to capture bending, torsion and shear dominated situations.

cs.CE↗

arXiv2020

Frontiers in Mortar Methods for Isogeometric Analysis

Complex geometries as common in industrial applications consist of multiple patches, if spline based parametrizations are used. The requirements for the generation of analysis-suitable models are increasing dramatically since isogeometric analysis is directly based on the spline parametrization and nowadays used for the calculation of higher-order partial differential equations. The computational, or more general, the engineering analysis necessitates suitable coupling techniques between the different patches. Mortar methods have been successfully applied for coupling of patches and for contact mechanics in recent years to resolve the arising issues within the interface. We present here current achievements in the design of mortar technologies in isogeometric analysis within the Priority Program SPP 1748, Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.

cs.CE↗