Computational Multiscale Methods

Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientific communities

On the numerical modeling of deformations of pressurized martensitic thin films

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite

Multiscale resolution in the computation of crystalline microstructure

Summary.: This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optima

Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

The power of fast Fourier transform (FFT)-based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory-efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill-conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite-volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite-element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier-type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT-based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass-control parameter can be adjusted to “soften” the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement-based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion

Virtual Element based formulations for computational materials micro-mechanics and homogenization

In this thesis, a computational framework for microstructural modelling of transverse behaviour of heterogeneous materials is presented. The context of this research is part of the broad and active field of Computational Micromechanics, which has emerged as an effective tool both to understand the influence of complex microstructure on the macro-mechanical response of engineering materials and to tailor-design innovative materials for specific applications through a proper modification of their microstructure. While the classical continuum approximation does not account for microstructural details within the material, computational micromechanics allows detailed modelling of a heterogeneous material's internal structural arrangement by treating each constituent as a continuum. Such an approach requires modelling a certain material microstructure by considering most of the microstructure's morphological features. The most common numerical technique used in computational micromechanics analysis is the Finite Element Method (FEM). Its use has been driven by the development of mesh generation programs, which lead to the quasi-automatic discretisation of the artificial microstructure domain and the possibility of implementing appropriate constitutive equations for the different phases and their interfaces. In FEM's applications to computational micromechanics, the phase arrangements are discretised using continuum elements. The mesh is created so that element boundaries and, wherever required, special interface elements are located at all interfaces between material's constituents. This approach can be effective in modelling many microstructures, and it is readily available in commercial codes. However, the need to accurately resolve the kinematic and stress fields related to complex material behaviours may lead to very large models that may need prohibitive processing time despite the increasing modern computers' performance. When rather complex microstructure's morphologies are considered, the quasi-automatic discretisation process stated before might fail to generate high-quality meshes. Time-consuming mesh regularisation techniques, both automatic and operator-driven, may be needed to obtain accurate numeric results. Indeed, the preparation of high-quality meshes is today one of the steps requiring more attention, and time, from the analyst. In this respect, the development of computational techniques to deal with complex and evolving geometries and meshes with accuracy, effectiveness, and robustness attracts relevant interest. The computational framework presented in this thesis is based on the Virtual Element Method (VEM), a recently developed numerical technique that has proven to provide robust numerical results even with highly-distorted mesh. These peculiar features have been exploited to analyse two-dimensional representations of heterogeneous materials' microstructures. Ad-hoc polygonal multi-domain meshing strategies have been developed and tested to exploit the discretisation freedom that VEM allows. To further simplify the preprocessing stage of the analysis and reduce the total computational cost, a novel hybrid formulation for analysing multi-domain problems has been developed by combining the Virtual Element Method with the well-known Boundary Element Method (BEM). The hybrid approach has been used to study both composite material's transverse behaviour in the presence of inclusions with complex geometries and damage and crack propagation in the matrix phase. Numerical results are presented that demonstrate the potential of the developed framework

Numerical Homogenization of Fractal Interface Problems

We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems