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Delaunay refinement is a mesh triangulation method with the goal of generating well-shaped triangles to obtain a valid Delaunay triangulation. In this thesis, an approach of using this method for meshing continuous height field terrains is presented using Perlin noise as the height field. The Delaunay approach is compared to grid-based meshing to verify that the theoretical time complexity O(n log n) holds and how accurately and deterministically the Delaunay approach can represent the height field. However, even though grid-based mesh generation is faster due to an O(n) time complexity, the focus of the report is to find out if Delaunay refinement can be used to generate meshes quick enough for real-time applications. As the available memory for rendering the meshes is limited, a solution for providing a cohesive mesh surface is presented using a hole filling algorithm since the Delaunay approach ends up leaving gaps in the mesh when a chunk division is used to limit the total mesh count present in the application. The methods were implemented in the programming language C++ using the open source library libnoise to generate the Perlin noise and the off-the-shelf solution CGALmesh provided a Delaunay refinement implementation. The video game engine Unity was used to render the output meshes created by the Delaunay and grid approach by interfacing with C++ via a Windows DLL. The time complexity of Delaunay refinement was verified to hold, although it was not possible to draw any conclusions regarding the Delaunay refinement's impact on the mesh's accuracy due to the test parameters used. It was also found that the CGALmesh implementation failed to provide a deterministic generation which is a significant drawback compared to the grid-based approach. Disregarding this, the Delaunay approach was found to be suitable for real-time applications as the generation time took less than 1 second, and is promising for volumetric terrain mesh generation. / Delaunay-raffinemang är en trianguleringsmetod med målet att generera reguljära trianglar för att uppnå en giltig Delaunay-triangulering. I denna avhandling presenteras en metod användandes Delaunay-raffinemang för att skapa polygonytor av kontinuerliga höjdfältsterränger, där Perlin noise används som höjdfält. Delaunay-metoden jämförs med en rutnätsbaserad metod för att verifiera att tidskomplexiteten O(n log n) gäller och hur exakt och deterministiskt som Delaunay-metoden förhåller sig till att representera höjdfältet. Även fast rutnätsmetoden är snabbare på grund av en O(n) tidskomplexitet är rapportens fokus att ta reda på om Delaunay-raffinemang är snabb nog för att användas i realtidsapplikationer för att generera polygonytor. Eftersom det tillgängliga minnet för att rendera polygonytorna är begränsat presenteras en lösning för att få sammanhängande ytor genom en hålutfyllningsalgoritm då Delaunaymetoden lämnar hål i ytan när chunk-uppdelning används för att begränsa det totala antalet polygonytor i applikationen. Metoderna implementerades i programmeringsspråket C++ användades biblioteket libnoise för att generera Perlin noise och den färdiga lösningen CGALmesh användes som implementation av Delaunay-raffinemang. Datorspelsmotorn Unity användes för att rendera polygonytorna som skapades av Delaunay- och rutnätsmetoden genom ett C++-gränssnitt via en Windows DLL. Tidskomplexiteten av Delaunay-raffinemang gällde, men det var inte möjligt att dra några slutsatser gällande hur exakt metoden förhållde sig till höjdfältet på grund av testparametrarna som användes. Ytterligare visade det sig att CGALmesh-implementationen var oförmögen att deterministiskt generera ytorna vilket är en stor nackdel jämfört med rutnätsmetoden. Bortsett från detta så visade sig Delaunay-metoden användbar för realtidsapplikationer då generingstiden tog mindre än 1 sekund, och metoden har dessutom potential för volymetrisk terränggenerering.

In the mid-2000s, a new quantum state of topological insulators was proposed. It deeply refreshed the traditional understanding of electronic band structure, which has been the most fundamental tool to classify metals and insulators. Topological insulators with non-trivial topological charges can host robust surface states or edge states located in the bulk bandgap. To understand this new state, an understanding of the bandgap is not sufficient, and it led to the new field of topological band theory in condensed matter physics. The development of electronic band structure theory also inspired the understanding of topological band theory from the chemical point of view and results in the new topic of topological chemistry. The discovery of topological insulators motivated extensive studies of solid-state materials from topological theory, leading to many topological materials in both insulators and metals. In the last 15 years, various topological materials characterized by different topological electronic structures have been discovered. One of the most important features shared by all different topological materials is the topologically protected non-trivial surface states (TSSs). Such TSSs are essentially different from the dangling bonds because they connect to conduction bands and valence bands in insulators or bulk band crossings in metals. The extra perturbation can only change their detailed shape but not remove them. This characteristic makes TSSs attractive for practical applications in the quantum information process, data storage, and energy conversion. In particular, the robust surface state is an attractive property that benefits energy-related catalysis. The last few years have seen research in this field with a focus on developing efficient topological material catalysts for hydrogen evolution reaction (HER), oxygen evolution reaction (OER), and reduction. To date, the topological catalyst has become a new frontier in both chemistry and materials science. Within the scope of this Ph.D. thesis, several topological semimetals and their HER activity are studied with the help of density functional theory, electrochemical theory, and topological band theory, combined with experimental measurements performed within the workgroup. The spectrum of performed projects ranges from the theoretical design of the high-efficiency hydrogen evolution catalyst with the guidance of topology in close collaboration with experiments and in-depth understanding of the relationship between topological properties and catalysis.