DSpace-CRIS at TU Wienhttps://repositum.tuwien.atThe reposiTUm digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 28 Sep 2021 05:05:46 GMT2021-09-28T05:05:46Z5021A Mass conserving mixed stress formulation for incompressible flowshttp://hdl.handle.net/20.500.12708/5335Title: A Mass conserving mixed stress formulation for incompressible flows
Authors: Lederer, Philip Lukas
Abstract: This work deals with the introduction and the analysis of a new finite element method for the discretization of incompressible flows. The main focus essentially lies on the discussion of the linear incompressible Stokes equations. These equations describe the physical behaviour and the relation -- derived from the fundamental Newtonian laws -- between the fluid velocity and the pressure (-gradient). Where the standard variational formulation of the Stokes equations demand a Sobolev regularity of order one for the velocity, we give an answer to the question if it is possible to define a variational formulation demanding a weaker regularity property of the velocity. With respect to a formally equivalent representation of the Stokes equations, we answer this question by the introduction of a new function space used for the definition of the gradient of the velocity. The resulting variational formulation is well-posed if we assume that the divergence of the velocity is square integrable. Thereby, with respect to the standard formulation, where all partial derivatives have to be square integrable, this is a reduced regularity property. We present certain properties of the new defined function space and discuss a proper continuous trace operator and the density of smooth functions. Motivated by this new variational formulation, we present and analyze a new finite element method in the rest of this work. For the approximation of the velocity we can now choose a conforming discrete space. This results in a (physically correct) incompressibility of the velocity field, thus exact mass conservation is provided. For the approximation of the gradient of the velocity we define new matrix-valued finite element shape functions, which are normal-tangential continuous across element interfaces. We present a detailed stability analysis and prove optimal convergence order of the discretization error.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/20.500.12708/53352019-01-01T00:00:00ZPressure robust discretizations for Navier Stokes equations : divergence-free reconstruction for Taylor-Hood elements and high order hybrid discontinuous Galerkin methodshttp://hdl.handle.net/20.500.12708/2399Title: Pressure robust discretizations for Navier Stokes equations : divergence-free reconstruction for Taylor-Hood elements and high order hybrid discontinuous Galerkin methods
Authors: Lederer, Philip Lukas
Abstract: This thesis focuses on a well-known issue of discretization techniques for solving the incompressible Navier Stokes equations. Due to a weak treatment of the incompressibility constraint there are different disadvantages that appear, which can have a major impact on the convergence and physical behaviour of the solutions. First we approximate the equations with a well-known pair of elements and introduce an operator that creates a reconstruction into a proper space to fix the mentioned problems. \newline Afterwards we use an H(div) conforming method that already handles the incompressibility constraint in a proper way. For a stable high order approximation an estimation for the saddlepoint structure of the Stokes equations is needed, known as the Ladyschenskaja-Babuska-Brezzi (LBB) condition. The independency of the estimation from the order of the polynomial degree is shown in this thesis. For that we introduce an H 2-stable extension that preserves polynomials. All operators and schemes are implemented based on the finite element library Netgen/NGSolve and tested with proper examples.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/20.500.12708/23992016-01-01T00:00:00Z