Meshless interface tracking for the simulation of dendrite envelope growth

The growth of dendritic grains during solidification is often modelled using the Grain Envelope Model (GEM), in which the envelope of the dendrite is an interface tracked by the Phase Field Interface Capturing (PFIC) method. In the PFIC method, an phase-field equation is solved on a fixed mesh to track the position of the envelope. While being versatile and robust, PFIC introduces certain numerical artefacts. In this work, we present an alternative approach for the solution of the GEM that employs a Meshless (sharp) Interface Tracking (MIT) formulation, which uses direct, artefact-free interface tracking. In the MIT, the envelope (interface) is defined as a moving domain boundary and the interface-tracking nodes are boundary nodes for the diffusion problem solved in the domain. To increase the accuracy of the method for the diffusion-controlled moving-boundary problem, an \h-adaptive spatial discretization is used, thus, the node spacing is refined in the vicinity of the envelope. MIT combines a parametric surface reconstruction, a mesh-free discretization of the parametric surfaces and the space enclosed by them, and a high-order approximation of the partial differential operators and of the solute concentration field using radial basis functions augmented with monomials. The proposed method is demonstrated on a two-dimensional \h-adaptive solution of the diffusive growth of dendrite and evaluated by comparing the results to the PFIC approach. It is shown that MIT can reproduce the results calculated with PFIC, that it is convergent and that it can capture more details in the envelope shape than PFIC with a similar spatial discretization.Comment: Preprint submitted to Journal of Computational Physic

Oscillatory behaviour of the RBF-FD approximation accuracy under increasing stencil size

When solving partial differential equations on scattered nodes using the Radial Basis Function generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the signed approximation error. Based on this observation we are then able to introduce a numerical quantity that indicates whether a given stencil size is locally optimal.Comment: 8 pages, 6 figures, ICCS 2023 Conference Pape

Identification of dynamic systems using deep Gaussian Processes

Zaradi naraščajoče kompleksnosti obravnavanih sistemov in posledično zahtevnega matematičnega modeliranja, v praksi pogosto uporabimo empirične modele ali modele črne škatle, s katerimi modeliramo le povezave med vhodno-izhodnimi vrednostmi, ne pa tudi fizikalnih zakonitosti, ki se jim sistem podreja. Za modeliranje oziroma identifikacijo zveze med vhodnimi in izhodnimi vrednostmi sistema se uporabljajo tudi globoki Gaussovi procesi. Ti za opis kompleksnejših procesov uporabljajo gnezdenje in hierarhično strukturo. Z identificirano zvezo med vhodno-izhodnimi vrednostmi z uporabo Gaussovih procesov lahko za dane vhodne vrednosti napovemo vrednost izhoda in pripadajočo negotovost, kar lahko s pridom uporabimo. V okviru magistrskega dela predstavimo teoretične osnove modeliranja z globokimi Gaussovimi procesi in njihove prednosti. V ta namen v ilustrativnem primeru uspešno identificiramo dinamični sistem nelinearnega nihanja mase, v bolj praktičnem primeru pa obravnavamo bistveno kompleksnejši sistem napovedovanja temperature v prizemni plasti atmosfere.Mathemathical and physical modelling only provide approximate description of the true nature of a dynamic system. The higher the precision of the model the more likely it becomes analytically intractable and, therefore, empirical models or black box models are used. When dynamic systems are considered as black box models, almost no prior knowledge about the system is considered. Deep Gaussian Processes, which use hierarchical structure to provide adequate identification of very complex systems, can be used to identify the mapping between the system input and output values. With the given mapping function we can then provide a one-step ahead prediction of the system output values, together with its uncertainty, which can be advantageously used. In this paper we use deep Gaussian Processes to identify a dynamic system and present its advantages by studying two cases. In the first illustrative case we successfully identify the dynamic properties of a nonlinear oscillating mass, while in the second, more realistic and complex case, we study one-step ahead prediction of air temperature in the atmospheric surface layer