Smoothed Particle Hydrodynamics and Magnetohydrodynamics

This paper presents an overview and introduction to Smoothed Particle Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we give a basic grounding in the fundamentals of SPH, showing how the equations of motion and energy can be self-consistently derived from the density estimate. We then show how to interpret these equations using the basic SPH interpolation formulae and highlight the subtle difference in approach between SPH and other particle methods. In doing so, we also critique several `urban myths' regarding SPH, in particular the idea that one can simply increase the `neighbour number' more slowly than the total number of particles in order to obtain convergence. We also discuss the origin of numerical instabilities such as the pairing and tensile instabilities. Finally, we give practical advice on how to resolve three of the main issues with SPMHD: removing the tensile instability, formulating dissipative terms for MHD shocks and enforcing the divergence constraint on the particles, and we give the current status of developments in this area. Accompanying the paper is the first public release of the NDSPMHD SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD algorithms that can be used to test many of the ideas and used to run all of the numerical examples contained in the paper.Comment: 44 pages, 14 figures, accepted to special edition of J. Comp. Phys. on "Computational Plasma Physics". The ndspmhd code is available for download from http://users.monash.edu.au/~dprice/ndspmhd

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

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Isogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation

This thesis is concerned with the development of a computational model and simulation technique capable of capturing the complex physics behind the intriguing phenomena of Elasto-capillarity. Elastocapillarity refers to the ability of capillary forces or surface tensions to deform elastic solids through a complex interplay between the energy of the surfaces (interfaces) and the elastic strain energy in the solid bulk. The described configuration gives rise to a three-phase system featuring a fluid-fluid interface (for instance the interface of a liquid and an ambient fluid such as air) and two additional interfaces separating the elastic solid from the first and second fluids, respectively. This setup is encountered in the wetting of soft substrates which is an emerging young field of research with many potential applications in micro- and nanotechnology and biomechanics. By virtue of the fact that a lot of physical phenomena under the umbrella of the wetting of soft substrates (e.g. Stick-slip motion, Durotaxis, Shuttleworth effect, etc.) are not yet fully understood, numerical analysis and simulation tools may yield invaluable insights when it comes to understanding the underlying processes. The problem tackled in this work – dubbed Elasto-Capillary Fluid-Structure Interaction or Binary-Fluid-Structure Interaction (BFSI) – is of multiphysics nature and poses a tremendous and challenging complexity when it comes to its numerical treatment. The complexity is given by the individual difficulties of the involved Two-phase Flow and Fluid-Structure Interaction (FSI) subproblems and the additional complexity emerging from their complex interplay. The two-phase flow problems considered in this work are immiscible two-component incompressible flow problems which we address with a Cahn-Hilliard phase field-based two-phase flow model through the Navier-Stokes-Cahn-Hilliard (NSCH) equations. The phase field method – also known as the diffuse interface method – is based on models of fluid free energy and has a solid theoretical foundation in thermodynamics and statistical mechanics. It may therefore be perceived as a physically motivated extension of the level-set or volume-of-fluid methods. It differs from other Eulerian interface motion techniques by virtue of the fact that it does not feature a sharp, but a diffuse interface of finite width whose dynamics are governed by the joint minimization of a double well chemical energy and a gradientsquared surface energy – both being constituents of the fluid free energy. Particularly for two-phase flows, diffuse interface models have gained a lot of attention due to their ability to handle complex interface dynamics such moving contact lines on wetted surfaces, and droplet coalescence or segregation without any special procedures. Our computational model for the FSI subproblem is based on a hyperelastic material model for the solid. When modeling the coupled dynamics of FSI, one is confronted with the dilemma that the fluid model is naturally based on an Eulerian perspective while it is very natural to express the solid problem in Lagrangian formulation. The monolithic approach we take, uses a fully coupled Arbitrary Lagrangian– Eulerian (ALE) variational formulation of the FSI problem and applies Galerkin-based Isogeometric Analysis for the discretization of the partial differential equations involved. This approach solves the difficulty of a common variational description and facilitates a consistent Galerkin discretization of the FSI problem. Besides, the monolithic approach avoids any instability issues that are associated with partitioned FSI approaches when the fluid and solid densities approach each other. The BFSI computational model presented in this work is obtained through the combination of the above described phase field-based two-phase flow and the monolithic fluid-structure interaction models and yields a very robust and powerful method for the simulation of elasto-capillary fluid-structure interaction problems. In addition, we also show that it may also be used for the modeling of FSI with free surfaces, involving totally or partially submerged solids. Our BFSI model may be classified as “quasi monolithic” as we employ a two-step solution algorithm, where in the first step we solve the pure Cahn-Hilliard phase field problem and use its results in a second step in which the binary-fluid-flow, the solid deformation and the mesh regularization problems are solved monolithically

Simulation of pore-scale flow using finite element-methods

I present a new finite element (FE) simulation method to simulate pore-scale flow. Within the pore-space, I solve a simplified form of the incompressible Navier-Stoke’s equation, yielding the velocity field in a two-step solution approach. First, Poisson’s equation is solved with homogeneous boundary conditions, and then the pore pressure is computed and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field I estimate the effective permeability of porous media samples characterized by thin section micrographs, micro-CT scans and synthetically generated grain packings. This two-step process is much simpler than solving the full Navier Stokes equation and therefore provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier-Stoke’s equation. My numerical model is verified with an analytical solution and validated on samples whose permeabilities and porosities had been measured in laboratory experiments (Akanji and Matthai, 2010). Comparisons were also made with Stokes solver, published experimental, approximate and exact permeability data. Starting with a numerically constructed synthetic grain packings, I also investigated the extent to which the details of pore micro-structure affect the hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic anisotropy of unconsolidated granular packings. With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace equation (Akanji and Matthai,2010