# Optimization

Optimization of mechanical structures and systems

### Projects:

Classical continuum approaches do not explicitly consider the specific atomistic or molecular structure of materials. Thus, they are not well suited to describe properly highly multiscale phenomena as for instance crack propagation or interphase effects in polymer materials. To integrate the atomistic level of resolution, the “Capriccio” method has been developed as a novel multiscale technique and is employed to study e.g. the impact of nano-scaled filler particles on the mechanical…

This project targets the formulation and implementation of a method for structural shape and topology optimization within an embedding domain setting. Thereby, the main consideration is to embed the evolving structural component into a uniform finite element mesh which is then used for the structural analyses throughout the course of the optimization. A boundary tracking procedure based on adaptive (or hierarchical) mesh refinement is used to identify interior and exterior elements, as well as such elements that are intersected by the physical domain boundary of the structural component. By this mechanism, we avoid the need to provide an updated finite element mesh that conforms to the boundary of the structural component for every single design iteration. Further, when considering domain variations of the structural component, its material points are not attached to finite element nodal points but rather move through the stationary finite element mesh of the embedding domain such that no mesh distortion is observed. Hence, one circumvents the incorporation of time consuming mesh smoothing operations within the domain update procedure. In order to account for the geometric mismatch between the boundary of the structural component and its non-conforming finite element representation within the embedding domain setting, a selective domain integration procedure is employed for all elements that are intersected by the physical domain boundary. This is to distinguish the respective element area fractions interior and exterior to the structural component. We rely on an explicit geometry description for the structural component, and an adjoint formulation is used for the derivation of the design sensitivities in the continuous setting.

We consider local refinements of finite element triangulations as continuous graph operations, for instance by splitting nodes and inflating edges to elements. This approach allows for the derivation of sensitivities for functionals depending on the finite element solution, which may in turn be used to define local refinement indicators. Thereby, we develop adaptive algorithms exploiting sensitivities for both hierarchical and non-hierarchical mesh changes, and analyze their properties and performance in comparison with established methods.

The mechanical properties and the fracture toughness of polymers can be

increased by adding silica nanoparticles. This increase is

mainly caused by the development of localized shear bands, initiated by

the stress concentrations due to the silica particles. Other mechanisms

responsible for the observed toughening are debonding of the particles

and void growth in the matrix material. The particular mechanisms depend

strongly on the structure and chemistry of the polymers and will be

analysed for two classes of polymer-silica composites, with highly

crosslinked thermosets or with biodegradable nestled fibres (cellulose,

aramid) as matrix materials.

The aim of the project is to study the influence of different mesoscopic

parameters, as particle volume fraction, on the macroscopic fracture

properties of nanoparticle reinforced polymers.

In a continuum the tendency of pre-existing cracks to propagate through

the ambient material is assessed based on the established concept of

configurational forces. In practise crack propagation is

however prominently affected by the presence and properties of either

surfaces and/or interfaces in the material. Here materials exposed to

various surface treatments are mentioned, whereby effects of surface

tension and crack extension can compete. Likewise, surface tension in

inclusion-matrix interfaces can often not be neglected. In a continuum

setting the energetics of surfaces/interfaces is captured by separate

thermodynamic potentials. Surface potentials in general result in

noticeable additions to configurational mechanics. This is

particularly true in the realm of fracture mechanics, however its

comprehensive theoretical/computational analysis is still lacking.

The project aims in a systematic account of the pertinent

surface/interface thermodynamics within the framework of geometrically

nonlinear configurational fracture mechanics. The focus is especially on

a finite element treatment, i.e. the Material Force Method [6]. The

computational consideration of thermodynamic potentials, such as the

free energy, that are distributed within surfaces/interfaces is at the

same time scientifically challenging and technologically relevant when

cracks and their kinetics are studied.

In previous works, the dependence of

failure mechanisms in composite materials like debonding of the

matrix-fibre interface or fibre breakage have been discussed. The

underlying model was based on specific cohesive zone elements, whose

macroscopic properties could be derived from DFT. It has been shown that

the dissipated energy could be increased by appropriate choices of

cohesive parameters of the interface as well as aspects of the fibre.

However due to the numerical complexity of applied simulation methods

the crack path had to be fixed a priori. Only recently models allow

computing the full crack properties at macroscopic scale in a

quasi-static scenario by the solution of a single nonlinear variational

inequality for a

given set of material parameters and thus model based optimization of

the fracture properties can be approached.

The goal of the project is to develop an optimization method, in the

framework of which crack properties (e.g. the crack path) can be

optimized in a mathematically rigorous way. Thereby material properties

of matrix, fibre and interfaces should serve as optimization variables.

### Participating Scientists:

### Publications:

**A non-invasive node-based form finding approach with discretization-independent target configuration**

In:**Advanced Modeling and Simulation in Engineering Sciences**5 (2018), Article No.: 11

ISSN: 2213-7467

DOI: 10.1186/s40323-018-0104-9
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