Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations

Tailoring large pores of porphyrin networks on Ag(111) by metal-organic coordination

The engineering of nanoarchitectures to achieve tailored properties relevant for macroscopic devices is a key motivation of organometallic surface science. To this end, understanding the role of molecular functionalities in structure formation and adatom coordination is of great importance. In this study, the differences in formation of Cu-mediated metal–organic coordination networks based on two pyridyl- and cyano-bearing free-base porphyrins on Ag(111) are elucidated by use of low-temperature scanning tunneling microscopy (STM). Distinct coordination networks evolve via different pathways upon codeposition of Cu adatoms. The cyano-terminated module directly forms 2D porous networks featuring fourfold-coordinated Cu nodes. By contrast, the pyridyl species engage in twofold coordination with Cu and a fully reticulated 2D network featuring a pore size exceeding 3 nm2 only evolves via an intermediate structure based on 1D coordination chains. The STM data and complementary Monte Carlo simulations reveal that these distinct network architectures originate from spatial constraints at the coordination centers. Cu adatoms are also shown to form two- and fourfold monoatomic coordination nodes with monotopic nitrogen-terminated linkers on the very same metal substrate—a versatility that is not achieved by other 3d transition metal centers but consistent with 3D coordination chemistry. This study discloses how specific molecular functionalities can be applied to tailor coordination architectures and highlights the potential of Cu as coordination center in such low-dimensional structures on surfaces

Object Oriented Finite Element Analysis for Structural Optimization using p-Elements

The optimization of continuous structures requires careful attention to discretization errors. Compared to ordinary low order formulation (h-elements) in conjunction with an adaptive mesh refinement in each optimization step, the use of high order finite elements (so called p-elements) has several advantages. However, compared to the h-method a higher order finite element analysis program poses higher demands from a software engineering point of view. In this article the basics of an object oriented higher order finite element system especially tailored to the use in structural optimization is presented. Besides the design of the system, aspects related to the employed implementation language Java are discussed

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry