Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

DAMASK – The Düsseldorf Advanced Material Simulation Kit for modeling multi-physics crystal plasticity, thermal, and damage phenomena from the single crystal up to the component scale

A variational free-discontinuity formulation of brittle fracture was given by Francfortand Marigo [1], where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously. The entire evolution of cracksincluding their initiation and branching is determined by this minimization principlerequiring no further criterion. However, a direct numerical discretization of themodel faces considerable difficulties as the displacement field is discontinuous inthe presence of cracks.

A Continuum Phase Field Model for Fracture

Because heterogeneous materials can be designed to
exhibit the best characteristics of their individual
constituents, they are ideally suited to modern
technologies that require materials with an unusual
combination of properties. The study presented in
the current work is aimed to approach effective
electrical conductivity of various random
heterogeneous materials. It starts with the
description of a type of modern material with
specific morphology. Percolation theory and finite
element analysis are used in computations and
comparison with experimental results is provided. To
understand difficulties with classical approaches,
next a two-component composite material on a lattice
is studied, followed by discoveries in the
electrical field properties. Finally, a new
algorithm is suggested, which can be used to find
effective properties for any arbitrary morphology.
Material covered in this book will be of interest to
graduate students and researchers who work on the
front edge of physics, materials science,
engineering and applied mathematics. All necessary
derivations and definitions are included to make the
book self-contained and approachable.

Random Heterogeneous Materials

Summary
In this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT-based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three-dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.

Computational homogenization of elasticity on a staggered grid

In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton---Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton's method using linear fixed point solvers and roughly $$100$$100 times less iterations than the nonlinear fixed point method. However, the Newton---Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant---Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

Efficient fixed point and Newton---Krylov solvers for FFT-based homogenization of elasticity at large deformations

Summary
The FFT-based homogenization method of Moulinec–Suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems. In this work, we generalize the method to problems discretized by trilinear hexahedral elements on Cartesian grids and physically nonlinear elasticity problems. We present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation.

For benchmark problems in linear elasticity, the solver exhibits mesh- and contrast-independent convergence behavior and enables the computational homogenization of complex structures, for instance, arising from computed tomography computed tomography (CT) imaging techniques. There exist 3D microstructures involving pores and defects, for which the original FFT-based homogenization scheme does not converge. In contrast, for the proposed scheme, convergence is ensured. Also, the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes. We demonstrate the power of the approach by computing the elasto-plastic response of a long-fiber reinforced thermoplastic material with 172 × 106 (displacement) degrees of freedom. Copyright © 2016 John Wiley & Sons, Ltd.

FFT-based homogenization for microstructures discretized by linear hexahedral elements

Use of composite voxels in FFT-based homogenization

We present an algorithm for generating volume elements of short fiber reinforced plastic microstructures for prescribed fourth order fiber orientation tensor, fiber aspect ratio and solid volume fraction. The algorithm inserts fibers randomly into an existing microstructure, and removes the resulting overlap systematically based on a gradient descent method. In contrast to existing methods, large fiber aspect ratios (up to 150) and large volume fractions (60 vol% for isotropic orientation and aspect ratio of 33) can be reached. We study the effective linear elastic properties of the resulting microstructures, depending on fiber orientation, volume fraction as well as aspect ratio, and examine the size of a corresponding representative volume element.

Matti Schneider

Papers

Computational homogenization of elasticity on a staggered grid

Efficient fixed point and Newton---Krylov solvers for FFT-based homogenization of elasticity at large deformations

FFT-based homogenization for microstructures discretized by linear hexahedral elements

Use of composite voxels in FFT-based homogenization

The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics