# Prosjekt- og masteroppgaver for Trond Kvamsdal

## General background

**Vision:** *Mathematics for a better society!*

**My position at NTNU are within Computational Mathematics**, i.e., development of new theories/methods within Applied Mathematics and Numerical Analysis to make robust and efficient numerical software programs for challenging applications in Science and Technology.

**Main areas of research:** Adaptive Isogeometric Finite Element Methods, Reduced Order Modelling

**Main area of application:** Computational Mechanics, i.e. both Solid/Structural and Fluid Mechanics relevant for Civil, Mechanical, Marine, and Petroleum Engineering as well as Biomechanics, Geophysics and Renewable Energy.

**External collaborators:** I am collaborating with the research group on Computational Science and Engineering (located in Trondheim) at Department of Mathematics and Cybernetics within SINTEF Digital.

## Isogeometric Analysis

### Definition

*Isogeometric Analysis is characterised by use of common spline basis for geometric modelling and finite element analysis of a given object.
*

### Introduction

The engineering process starts with designers encapsulating their perception using Computer Aided Design (CAD) tools. Thus, the geometry described in CAD systems is to be considered exact in the sense that it represents the projection from the designers perception of the desired object onto an electronic description. Today most CAD systems use spline basis function and often Non-Uniform Rational B-Splines (NURBS) of different polynomial order.

During and after the design stages different levels of numerical solid and fluid simulations are performed on the object. Very often this involves using the Finite Element Method (FEM) where the geometry is represented by piecewise low order polynomials. This practise introduce either significant approximation errors or fine FEM models with a large number of finite elements that make the numerical simulation computational costly. Furthermore, a huge amount of man-hours (** in some applications about 80% of total time is spent on mesh generation**, see Hughes et al. 2005) have to be spent in order to remodel the object into a suitable finite element mesh. This information transfer between models suitable for design (CAD) and analysis (FEM) is considered being a severe bottleneck in industry today.

To address this issue Hughes et al. introduced in [1] a analysis framework which is based on NURBS (Non-Uniform Rational B-Splines), which is standard technology employed in CAD systems. They propose to match the exact CAD geometry by NURBS surfaces, then construct a coarse mesh of Spline Elements. Throughout, the isoparametric philosophy is invoked, that is, the solution space for dependent variables is represented in terms of the same functions which represent the geometry. For this reason, they denote it ** isogeometric analysis (IGA)**.

We at NTNU and SINTEF Digital in Trondheim have been the pioneers in development of adaptive methods for isogeometric analysis using locally refined B-splines (LR B-splines), see [2]-[5]. We have also pursued development of high-fidelity solvers for challenging problems as high Reynolds number flow around wind turbine blades ([6) and [7]) as well as acoustic scattering by a submarine ([8] and [9]).

[1] T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", *Computational Methods in Applied Mechanics and Engineering*, 194: 4135-4195 (2005).

[2] K. A. Johannessen, T. Kvamsdal and T. Dokken, "Isogeometric analysis using LR B-splines", *Computational Methods in Applied Mechanics and Engineering*, 269: 471-514 (2014).

[3] M. Kumar, T. Kvamsdal and K. A. Johannessen, "Simple a posteriori error estimators in adaptive
isogeometric analysis", *Computers and Mathematics with Applications*, 70: 1555-1582 (2015).

[4] M. Kumar, T. Kvamsdal and K. A. Johannessen, "Superconvergent patch recovery and a posteriori error estimation
technique in adaptive isogeometric analysis", *Computational Methods in Applied Mechanics and Engineering*, 316: 1086-1156 (2017).

[5] A. Stahl, T. Kvamsdal and C. Schellewald, "Post-processing and visualization techniques for isogeometric
analysis results", *Computational Methods in Applied Mechanics and Engineering*, 316: 880-943 (2017).

[6] K. Nordanger, R. Holdahl, A. M. Kvarving, A. Rasheed and T. Kvamsdal, "Implementation and comparison of three isogeometric Navier–Stokes solvers applied to simulation of flow past a fixed 2D NACA0012 airfoil at high Reynolds number", *Computational Methods in Applied Mechanics and Engineering*, 284: 664-688 (2015).

[7] K. Nordanger, R. Holdahl, T. Kvamsdal, A. M. Kvarving, and A. Rasheed, "Simulation of airflow past a 2D NACA0015 airfoil using an isogeometric incompressible Navier–Stokes solver with the Spalart–Allmaras turbulence model", *Computational Methods in Applied Mechanics and Engineering*, 290: 183-208 (2015).

[8] J. V. Venås, T. Kvamsdal and T. Jenserud, "Isogeometric analysis of acoustic scattering using infinite elements", *Computational Methods in Applied Mechanics and Engineering*, 335: 152-193 (2018).

[9] J. V. Venås and T. Kvamsdal, "Isogeometric boundary element method for acoustic scattering by a submarine", *Computational Methods in Applied Mechanics and Engineering*, In press (2019).

### Project topics

Relevant topics for a project or a master thesis may be:

- Efficient numerical integration
- Selection of collocation points
- Adaptive methods for IGA
- Applications within structural/solid mechanics
- Applications within fluid mechanics
- Applications within acoustic scattering
- Applications within geoscience

## Reduced Order Modelling

### Definition

*Reduced Order Modelling (ROM) is a technique for reducing the computational complexity of mathematical models in numerical simulations.
*

### Introduction

High fidelity simulations of flow can be quite demanding, involving up to O(10^6) to O(10^9) degrees of freedom, and several hours or days of computational time, even on powerful parallel architectures. These techniques become prohibitive when expected to deal quickly and efficiently with repetitive solutions of partial differential equations. One set of PDE encountered on a regular basis is the Navier-Stokes equation, used to simulate flow around complex geometries, e.g., offshore structures. To address the issues associated with computational efficiency, the field of Reduced Order Modelling (ROM) is evolving quickly.

For flow simulations, the governing equations are typically either Stokes the Navier-Stokes equations, which are written in terms of certain input parameters whose effect one might want to investigate. For this sort of problem, reduced order modelling (ROM) is a generic term used to identify any approach aimed at replacing the high fidelity problem with one featuring a much lower numerical complexity. The key to the success of any ROM is the ability to evaluate the solution to this reduced problem at a cost (usually in terms of computational time) that is independent of the dimension of the original high- fidelity problem.

Reduced basis methods represent one notable instance of ROM techniques. They exploit the parametric dependence of the solution by combining a handful of high fidelity simulations (snapshots) computed a priori for a small set of parameter values. This way, a large linear system is replaced by a much smaller one, whose dimension is related to the number of snapshots. The key, then, is to construct such reduced bases. One method that can be used, is proper orthogonal decomposition (POD).

### Project topics

Relevant topics for a project or a master thesis may be:

- Reduced basis methods for parametric structural/solid problems
- Reduced basis methods for parametric Stokes problems
- Reduced basis methods for shape optimization (structural/solid or Stokes problems)
- Adaptive methods for reduced basis methods
- A posteriori error estimation of error induced by the reduced basis methods