- Higher order space-time finite element methods on evolving and fixed domains: applied to the Navier-Stokes equations and wave problems

# Higher order space-time finite element methods on evolving and fixed domains: applied to the Navier-Stokes equations and wave problems

Publication date

2022

Document type

PhD thesis (dissertation)

Author

Anselmann, Mathias

Advisor

Referee

Richter, Thomas

Granting institution

Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg

Exam date

2022-05-11

Organisational unit

DDC Class

510 Mathematik

Keyword

Discontinuous Galerkin in time method

Higher order space-time finite element method

Immersed boundary method

Navier-Stokes equation

Galerkin-collocation method

Geometric Multigrid Method

Abstract

In this work, various aspects of space-time finite element discretizations for the Navier-Stokes equations and wave problems are investigated.

First, an overview of the current state of the science is given in Chapter 1 and the notation is introduced in Chapter 2.

Then, in Chapter 3, flow problems on time-dependent domains are studied.

These play a major role in engineering practice, for example in the simulation of fluid-structure interaction problems, such as the interaction of a fluid with a rotor.

In this work, rigid bodies moving in a viscous fluid described by the time-dependent Navier-Stokes equations are considered as an example problem for this purpose.

This means that the flow area changes depending on the position of the rigid body.

A classical way to simulate such problems is the Arbitrary-Lagrangian-Eulerian method (ALE method), which is based on the transformation of the flow domain to a reference mesh.

This transformation has the disadvantage that it is computationally intensive, can induce nonlinearities, and also, for larger deformations of the flow domain, an update of the underlying computational grid is inevitable.

Further, such a remeshing is again computationally expensive and usually means that all matrices and vectors of the underlying simulation have to be rebuilt.

In this work, a method is presented in which all computations are performed on a fixed background grid that can be arbitrarily cut by the rigid body.

This method is called the Cut Finite Elmenent Method (CutFEM).

A prerequisite for an implementation is to be able to integrate functions over arbitrarily cut cells in order to be able to assemble the underlying system matrices and vectors.

To accomplish this, a flexible method was developed, based on the iterated application of 1d integrals.

In the case of cells that become very small, if they are cut by the rigid body, unphysical steep gradients of the solution arise in these cells.

These result in poor conditioning of the system matrix and numerical instabilities.

To circumvent the problem, a so-called "ghost penalty" stabilization was implemented, which constrains the gradients and makes the method robust.

Furthermore, the stabilization operator implicitly propagates the flow solution into the entire rigid-body domain.

This allows all calculations to be performed on a fixed, time-independent background grid.

All aspects mentioned so far have been verified in numerical convergence tests and simulations.

Then, in Chapter 4, a geometric multigrid method for higher order finite element space-time discretizations of the Navier-Stokes equations is then implemented for time-independent domains.

The method is based on a Vanka smoother, which dampens high-frequency error components over all space-time degrees of freedom in each time step.

All major parts of our code are implemented in parallel using the Message Passing Interface (MPI).

The multigrid method, used in this work, is verified by simulating the 3d DFG benchmark "flow around a cylinder", with more than 96 million space-time degrees of freedom per time step.

Adapting the method to problems with time-dependent domains in conjunction with CutFEM results in increased iteration counts, depending on the size of the rigid body.

The causes of this are analyzed and investigated numerically in this work.

In Chapter 5 classical continuous finite element methods for time discretization are combined with collocation methods to obtain efficient higher order methods in time, which also result in a discrete solution of higher regularity.

Such Galerkin-Collocation methods are developed and applied first for the wave equation and then for the Navier-Stokes equations.

Compared to traditional continuous finite element time discretizations, we could significantly increase the time step size with these methods, while maintaining the same quality of the solution.

Especially in coupled problems, methods of this type could be advantageous if coefficients of the derivative of one problem are included in the problem description of the other problem.

First, an overview of the current state of the science is given in Chapter 1 and the notation is introduced in Chapter 2.

Then, in Chapter 3, flow problems on time-dependent domains are studied.

These play a major role in engineering practice, for example in the simulation of fluid-structure interaction problems, such as the interaction of a fluid with a rotor.

In this work, rigid bodies moving in a viscous fluid described by the time-dependent Navier-Stokes equations are considered as an example problem for this purpose.

This means that the flow area changes depending on the position of the rigid body.

A classical way to simulate such problems is the Arbitrary-Lagrangian-Eulerian method (ALE method), which is based on the transformation of the flow domain to a reference mesh.

This transformation has the disadvantage that it is computationally intensive, can induce nonlinearities, and also, for larger deformations of the flow domain, an update of the underlying computational grid is inevitable.

Further, such a remeshing is again computationally expensive and usually means that all matrices and vectors of the underlying simulation have to be rebuilt.

In this work, a method is presented in which all computations are performed on a fixed background grid that can be arbitrarily cut by the rigid body.

This method is called the Cut Finite Elmenent Method (CutFEM).

A prerequisite for an implementation is to be able to integrate functions over arbitrarily cut cells in order to be able to assemble the underlying system matrices and vectors.

To accomplish this, a flexible method was developed, based on the iterated application of 1d integrals.

In the case of cells that become very small, if they are cut by the rigid body, unphysical steep gradients of the solution arise in these cells.

These result in poor conditioning of the system matrix and numerical instabilities.

To circumvent the problem, a so-called "ghost penalty" stabilization was implemented, which constrains the gradients and makes the method robust.

Furthermore, the stabilization operator implicitly propagates the flow solution into the entire rigid-body domain.

This allows all calculations to be performed on a fixed, time-independent background grid.

All aspects mentioned so far have been verified in numerical convergence tests and simulations.

Then, in Chapter 4, a geometric multigrid method for higher order finite element space-time discretizations of the Navier-Stokes equations is then implemented for time-independent domains.

The method is based on a Vanka smoother, which dampens high-frequency error components over all space-time degrees of freedom in each time step.

All major parts of our code are implemented in parallel using the Message Passing Interface (MPI).

The multigrid method, used in this work, is verified by simulating the 3d DFG benchmark "flow around a cylinder", with more than 96 million space-time degrees of freedom per time step.

Adapting the method to problems with time-dependent domains in conjunction with CutFEM results in increased iteration counts, depending on the size of the rigid body.

The causes of this are analyzed and investigated numerically in this work.

In Chapter 5 classical continuous finite element methods for time discretization are combined with collocation methods to obtain efficient higher order methods in time, which also result in a discrete solution of higher regularity.

Such Galerkin-Collocation methods are developed and applied first for the wave equation and then for the Navier-Stokes equations.

Compared to traditional continuous finite element time discretizations, we could significantly increase the time step size with these methods, while maintaining the same quality of the solution.

Especially in coupled problems, methods of this type could be advantageous if coefficients of the derivative of one problem are included in the problem description of the other problem.

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Access Right

Open access