Search

In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...

Checkpointing, i.e. saving and reading results of finite element computation is crucial, especially for long-time running simulations where execution is interrupted and user would like to restart the process from last saved time step. On the other hand, visualization of results in thid-party software such as ParaView is inevitable. In the previous DOLFIN versions (2017.1.0 and older) these two functionalities were strictly separated. Results could have been saved via HDF5File interface for later computations and/or stored in a format understood by ParaView - VTK’s .pvd (File interface) or XDMF (XDMFFile interface). This led to data redundancy and error-prone workflow. The problem essentially originated from incompatibilities between both libraries, DOLFIN and ParaView (VTK). DOLFIN’s internal representation of finite element function is based on vector of values of degrees of freedom (dofs) and their ordering within cells (dofmap). VTK’s representation of a function is given by it’s values at some points in cell, while ordering and geometric position of these points is fixed and standardised within VTK specification. For nodal (iso- and super-parametric) Lagrange finite elements (Pk , dPk ) both representations coincide up to an ordering. This allows to extend XDMF specification and introduce intermediate way of storing finite element function - intrinsic to both, ParaView and DOLFIN. The necessary work was done as a part of Google Summer of Code 2017 project Develop XDMF for- mat for visualisation and checkpointing, see https://github.com/michalhabera/gsoc-summary. New checkpointing functionality is exposed via write checkpoint() and read checkpoint() methods.

We provide a novel analysis of the PCD (pressure convection-diffusion) preconditioner based on infinite-dimensional interpretation of the underlying operators, which is the basis of operator preconditioning theory. Our results help to answer open questions concerning practical implementation and performance. In particular, we clarify the choice of boundary conditions which is, to our knowledge, a problem missing rigorous analysis, although some heuristic treatment has appeared; see [1, 2]. Moreover, our approach provides a unified theory for both variants (order of the commuted operators) which have thus far only been treated separately in the literature. Last but not least, our infinite-dimensional analysis reveals Fredholm structure in the preconditioner which must be respected in the discretization; failure to do so results in significant deterioration of performance, which has previously been observed but not correctly identified and resolved.[1] Howard C. Elman, David J. Silvester, and Andrew J. Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. 2nd. Oxford University Press, 2014, Section 9.3.2[2] Maxim A. Olshanskii and Yuri V. Vassilevski. “Pressure Schur Complement Preconditioners for the Discrete Oseen Problem”. In: SIAM Journal on Scientific Computing 29.6 (2007), pp. 2686–2704. DOI: 10.1137/070679776, Section 2.1

The talk provides a novel analysis of performance and boundary conditions issues for the PCD preconditioner for Navier-Stokes equations. Then it gives an advertisement to FENaPack, an open-source implementation built on top of the FEniCS project.

The FEniCS project aims to provide a high productivity environment for development of finite element based simulation software. Techniques applied to achieve this goal include mixed language programming and code generation, which enables writing high performance programs in a high level language. End-user productivity is a high priority goal in our software designs. To sustain the productivity of the multinational team of part-time developers (mainly researchers and students) is paramount to the long term survival of the project. To minimize the developer workload while making the process open and accessible to new contributors and users, we regularly question which tools are the best available for our needs. On this poster we will present our current tool choices and work flows for developers and the wider FEniCS community. This list includes version control, build systems, testing, release management, team communication, documentation, and end user support. The most recent addition to our toolbox are developer curated Docker images. We are investigating their usefulness in testing infrastructure, end user deployment, HPC cluster deployment, and as reproducible software environments to accompany journal publications. We welcome discussion on alternatives that can simplify our lives.Poster presented at SIAM CSE17 PP108 Minisymposterium: Software Productivity and Sustainability for CSE and Data Science.

The talk provides a novel analysis of performance and boundary conditions issues for the PCD preconditioner for Navier-Stokes equations. Then it gives an advertisement to FENaPack, an open-source implementation built on top of the FEniCS project.Proceedings at: https://doi.org/10.6084/m9.figshare.5086369.v2.

The talk presents a posteriori error estimation technique called the method of equilibrated fluxes and gives an implementation using the FEniCS project.

Poster presented at 2015 SIAM Conference on Computational Science and Engineering, Salt Lake City.

The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5., Archive of Numerical Software, Vol 3, No 100 (2015), Starting Point and Frequency: Year: 2013

We present a numerical model of Bridgman crystal growth. Pseudo-incompressibility constraint is used to handle jumps in density during phase change. ALE formulation is employed to account for moving parts of the system. Field equations and movement of material interfaces are decoupled in fractional step manner. Naviér-Stokes problem is extended to solid phase where no flow is enforced by Darcy-like forcing. Latent heat of phase change is added to effective heat capacity as approximate Dirac-$\delta$. Backward Euler discretization in space and P2/P1/P1 in space are used. Transient and stationary solutions are being found and compared to temperatures measured directly inside a steady system. Influence of pull-rates on growth process and shape of phase interface are being examined. Powered by TCPDF (www.tcpdf.org)