# PhD defence by Morten Eggert Nielsen on Computational Methods for Wave-structure Interaction - Numerical Analysis of a RBF-based Method

## Time

09.11.2020 kl. 13.00 - 17.00

## Description

Morten Eggert Nielsen, Department of Energy Technology, will defend the thesis "Computational Methods for Wave-structure Interaction - Numerical Analysis of a RBF-based Method"

### TITLE

Computational Methods for Wave-structure Interaction - Numerical Analysis of a RBF-based Method

### PHD DEFENDANT

Morten Eggert Nielsen

### SUPERVISOR

Professor Lars Damkilde, Department of the Built Environment

### MODERATOR

Associate Professor Jannie Sønderkær Nielsen, Department of the Built Environment

### OPPONENTS

Associate Professor Peter Frigaard, Department of the Built Environment, Aalborg University (Chairman)

Professor Peter Troch, Ghent University

Professor Harry B. Bingham, Technical University of Denmark

### ABSTRACT

This project aims to advance the state of the art within computational methods for wave-structure interaction problems. This is attempted by developing a new method for non-linear potential flows, which is based on radial basis function-generated finite differences (RBF-FD). The new RBF-based method is investigated due to its high-order accuracy and mesh-free nature, which makes it possible to discretize the governing partial differential equations on unstructured node sets that conform with the time-dependent free surface and other moving boundaries.

Unstructured node sets and nearest neighbor stencil selections will in general result in asymmetric stencils. These asymmetric stencils give rise to temporal instabilities as the eigenvalues of the discrete gradient operator will not be purely imaginary. This numerical issue can be remedied by adding dissipative terms to the free surface conditions. In this thesis, the dissipative terms are based on hyperviscosity, which is a high-order Laplace operator that seeks to stabilize the system without deteriorating the accuracy.

Hyperviscosity works by shifting the spurious eigenvalues, i.e. eigenvalues related to highly oscillatory eigenvectors, towards the left half of the complex plane, while leaving the physical eigenvalues intact. The amount by which the eigenvalues are shifted depends on the scaling of the hyperviscosity operator. This scaling parameter is in general problem dependent, but it is shown that a heuristic scaling law can be derived based on the Nyquist frequency in combination with an approximate l1 normalization of the hyperviscosity operator.

Near boundaries the heuristic scaling law may give rise to instabilities. However, these instabilities are related to irregularities in the scaling parameters rather than the asymmetric stencils. Thus, the temporal stability can be improved by smoothing out the irregularities, e.g. by applying a moving median filter, when boundaries are intersecting the free surface. This smoothing operation leaves the eigenspectrum globally intact, while shifting the eigenvalues with large real parts even further towards the left half of the complex plane.

The proposed stabilization technique in combination with the developed node generation and update strategy, which enables node refinements towards the free surface and other moving boundaries, results in a stable computational method suitable for non-linear wave-structure interaction problems. Preliminary two-dimensional test cases show promising results, which illustrate the potential of the proposed method and further investigations are encouraged.

### The defence will be in english - all are welcome

### Due to the Covid-19 situation, the PhD defence will be carried out

via Zoom. Please send an email to Linda Vabbersgaard Andersen no

later than 6 November 2020 and you will get an invite for the event

and, if requested, a copy of the thesis.