arXiv subjects

Gregor Kosec

Publications and source records attributed to Gregor Kosec.

At least 37 records · Page 2Linked to original sources

arXiv2024

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.

math.NA↗

arXiv2023

Spatially-varying meshless approximation method for enhanced computational efficiency

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatial variation of computational node regularity. Rather than covering the entire domain with scattered nodes, only regions with geometric details are covered with scattered nodes, while the rest of the domain is discretised with regular nodes. Consequently, in regions covered with regular nodes the approximation using solely the monomial basis can be performed, effectively reducing the required stencil size compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour. The performance of the proposed hybrid scattered-regular approximation approach, in terms of computational efficiency and accuracy of the numerical solution, is studied on natural convection driven fluid flow problems. We start with the solution of the de Vahl Davis benchmark case, defined on square domain, and continue with two- and three-dimensional irregularly shaped domains. We show that the spatial variation of the two approximation methods can significantly reduce the computational complexity, with only a minor impact on the solution accuracy.

math.NA↗

arXiv2023

Strong form mesh-free $hp$-adaptive solution of linear elasticity problem

We present an algorithm for $hp$-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve-estimate-mark-refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for $h$-, $p$- or $hp$-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced $hp$-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed $hp$-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed $hp$-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the user intervention.

math.NA↗

arXiv2022

A Meshless Solution of a Small-Strain Plasticity Problem

When the deformations of a solid body are sufficiently large, parts of the body undergo a permanent deformation commonly refereed to as plastic deformation. Several plasticity models describing such phenomenon have been proposed, e.g. von Mises, Tresca, etc. Traditionally, the finite element method (FEM) is the numerical tool of choice for engineers who are solving such problems. In this work, however, we present the implementation of the von Mises plasticity model with non-linear isotropic hardening in our in-house developed MEDUSA library, utilizing a variant of meshless methods -- namely the radial basis function-generated finite differences (RBF-FD). We define a simple plane stress case, where a 2D block is fixed at one edge, and a tensile force, which causes the block to deform, is applied to it at the opposite edge. We show that results are in good agreement with the numerical solution obtained by Abaqus FEA, a commercial FEM solver.

math.NA↗

arXiv2022

Stability analysis of RBF-FD and WLS based local strong form meshless methods on scattered nodes

The popularity of local meshless methods in the field of numerical simulations has increased greatly in recent years. This is mainly due to the fact that they can operate on scattered nodes and that they allow a direct control over the approximation order and basis functions. In this paper we analyse two popular variants of local strong form meshless methods, namely the radial basis function-generated finite differences (RBF-FD) using polyharmonic splines (PHS) augmented with monomials, and the weighted least squares (WLS) approach using only monomials. Our analysis focuses on the accuracy and stability of the numerical solution computed on scattered nodes in a two- and three-dimensional domain. We show that while the WLS variant is a better choice when lower order approximations are sufficient, the RBF-FD variant exhibits a more stable behavior and a higher accuracy of the numerical solution for higher order approximations, but at the cost of higher computational complexity.

math.NA↗

arXiv2022

Implicit-Explicit Error Indicator based on Approximation Order

With the immense computing power at our disposal, the numerical solution of partial differential equations (PDEs) is becoming a day-to-day task for modern computational scientists. However, the complexity of real-life problems is such that tractable solutions do not exist. This makes it difficult to validate the numerically obtained solution, so good error estimation is crucial in such cases. It allows the user to identify problematic areas in the computational domain that may affect the stability and accuracy of the numerical method. Such areas can then be remedied by either \textit{h}- or \textit{p}-adaptive procedures. In this paper, we propose to estimate the error of the numerical solution by solving the same governing problem implicitly and explicitly, using a different approximation order in each case. We demonstrate the newly proposed error indicator on the solution of a synthetic two-dimensional Poisson problem with tractable solution for easier validation. We show that the proposed error indicator has good potential for locating areas of high error.

math.NA↗

arXiv2022

Dynamic mode decomposition as an analysis tool for time-dependent partial differential equations

The time-dependent fields obtained by solving partial differential equations in two and more dimensions quickly overwhelm the analytical capabilities of the human brain. A meaningful insight into the temporal behaviour can be obtained by using scalar reductions, which, however, come with a loss of spatial detail. Dynamic Mode Decomposition is a data-driven analysis method that solves this problem by identifying oscillating spatial structures and their corresponding frequencies. This paper presents the algorithm and provides a physical interpretation of the results by applying the decomposition method to a series of increasingly complex examples.

math.NA↗

arXiv2022

A hybrid RBF-FD and WLS mesh-free strong-form approximation method

Since the advent of mesh-free methods as a tool for the numerical analysis of systems of Partial Differential Equations (PDEs), many variants of differential operator approximation have been proposed. In this work, we propose a local mesh-free strong-form method that combines the stability of Radial Basis Function-Generated Finite Differences (RBF-FD) with the computational effectiveness of Diffuse Approximation Method (DAM), forming a so-called hybrid method. To demonstrate the advantages of a hybrid method, we evaluate its computational complexity and accuracy of the obtained numerical solution by solving a two-dimensional Poisson problem with an exponentially strong source in the computational domain. Finally, we employ the hybrid method to solve a three-dimensional Boussinesq's problem on an isotropic half-space and show that the implementation overhead can be justified.

math.NA↗

arXiv2022

A sharp-interface mesoscopic model for dendritic growth

The grain envelope model (GEM) describes the growth of envelopes of dendritic crystal grains during solidification. Numerically the growing envelopes are usually tracked using an interface capturing method employing a phase field equation on a fixed grid. Such an approach describes the envelope as a diffuse interface, which can lead to numerical artefacts that are possibly detrimental. In this work, we present a sharp-interface formulation of the GEM that eliminates such artefacts and can thus track the envelope with high accuracy. The new formulation uses an adaptive meshless discretization method to solve the diffusion in the liquid around the grains. We use the ability of the meshless method to operate on scattered nodes to accurately describe the interface, i.e., the envelope. The proposed algorithm combines parametric surface reconstruction, meshless discretization of parametric surfaces, global solution construction procedure and partial differential operator approximation using monomials as basis functions. The approach is demonstrated on a two-dimensional h-adaptive solution of diffusive growth of dendrites and assessed by comparison to a conventional diffuse-interface fixed-grid GEM.

math.NA↗

arXiv2022

Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs

In this paper, we present a novel parallel dimension-independent node positioning algorithm that is capable of generating nodes with variable density, suitable for meshless numerical analysis. A very efficient sequential algorithm based on Poisson disc sampling is parallelized for use on shared-memory computers, such as modern workstations with multi-core processors. The parallel algorithm uses a global spatial indexing method with its data divided into two levels, which allows for an efficient multi-threaded implementation. The addition of bootstrapping enables the algorithm to use any number of parallel threads while remaining as general as its sequential variant. We demonstrate the algorithm performance on six complex 2- and 3-dimensional domains, which are either of non-rectangular shape or have varying nodal spacing or both. We perform a run-time analysis of the algorithm, to demonstrate its ability to reach high speedups regardless of the domain and to show how well it scales on the experimental hardware with 16 processor cores. We also analyse the algorithm in terms of the effects of domain shape, quality of point placement, and various parallelization overheads.

cs.DC↗

arXiv2022

Uncertainty Assessment of Dynamic Thermal Line Rating for Operational Use at Transmission System Operators

Transmission system operators (TSOs) in recent years have faced challenges in order to ensure maximum transmission capacity of the system to satisfy market needs, while maintaining operational safety and permissible impact on the environment. A great help in the decision-making process was introduced with the Dynamic Thermal Rating (DTR) - an instrument to monitor and predict the maximal allowed ampacity of the power grid based on weather measurements and forecast. However, the introduction of DTR raises a number of questions related to the accuracy and uncertainty of the results of thermal assessment and the level of acceptable risk and its management. In this paper, we present a solution for estimating DTR uncertainty, appropriate for operational use at TSOs. With the help of conductor surface temperature measurements, weather measurements and predicted weather data, we also estimate the error of the weather forecast and the DTR itself. Following the results of the data analyses, we build an operative solution for estimating the ampacity uncertainty based on Monte Carlo random simulations and integrate it into the operational environment of ELES - the operator of the Slovenian electric power transmission network.

math.NA↗

arXiv2021

GPU accelerated RBF-FD solution of Poisson's equation

The Radial Basis Function-generated finite differences became a popular variant of local meshless strong form methods due to its robustness regarding the position of nodes and its controllable order of accuracy. In this paper, we present a GPU accelerated numerical solution of Poisson's equation on scattered nodes in 2D for orders from 2 up to 6. We specifically study the effect of using different orders on GPU acceleration efficiency.

math.NA↗

arXiv2021

Discretized boundary surface reconstruction

Domain discretization is an essential part of the solution procedure in numerical simulations. Meshless methods simplify the domain discretization to positioning of nodes in the interior and on the boundary of the domain. However, generally speaking, the shape of the boundary is often undefined and thus needs to be constructed before it can be discretized with a desired internodal spacing. Domain shape construction is far from trivial and is the main challenge of this paper. We tackle the simulation of moving boundary problems where the lack of domain shape information can introduce difficulties. We present a solution for 2D surface reconstruction from discretization points using cubic splines and thus providing a surface description anywhere in the domain. We also demonstrate the presented algorithm in a simulation of phase-change-like problem.

math.NA↗

arXiv2021

p-refined RBF-FD solution of a Poisson problem

Local meshless methods obtain higher convergence rates when RBF approximations are augmented with monomials up to a given order. If the order of the approximation method is spatially variable, the numerical solution is said to be p-refined. In this work, we employ RBF-FD approximation method with polyharmonic splines augmented with monomials and study the numerical properties of p-refined solutions, such as convergence orders and execution time. To fully exploit the refinement advantages, the numerical performance is studied on a Poisson problem with a strong source within the domain.

math.NA↗

arXiv2021

RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous media

Radial Basis Function-generated Finite Differences (RBF-FD) is a popular variant of local strong-form meshless methods that do not require a predefined connection between the nodes, making it easier to adapt node-distribution to the problem under consideration. This paper investigates an RBF-FD solution of time-domain acoustic wave propagation in the context of seismic modeling in the Earth's subsurface. Through a number of numerical tests, ranging from homogeneous to highly-heterogeneous velocity models including non-smooth irregular topography, we demonstrate that the present approach can be further generalized to solve large-scale seismic modeling and full waveform inversion problems in arbitrarily complex models enabling more robust interpretations of geophysical observations

cs.CE↗

arXiv2021

Monomial augmentation guidelines for RBF-FD from accuracy vs. computational time perspective

Local meshless methods using RBFs augmented with monomials have become increasingly popular, due to the fact that they can be used to solve PDEs on scattered node sets in a dimension-independent way, with the ability to easily control the order of the method, but at a greater cost to execution time. We analyze this ability on a Poisson problem with mixed boundary conditions in 1D, 2D and 3D, and reproduce theoretical convergence orders practically, also in a dimension-independent manner, as demonstrated with a solution of Poisson's equation in an irregular 4D domain. The results are further combined with theoretical complexity analyses and with conforming execution time measurements, into a study of accuracy vs. execution time trade-off for each dimension. Optimal regimes of order for given target accuracy ranges are extracted and presented, along with guidelines for generalization.

math.NA↗

arXiv2020

Fast variable density node generation on parametric surfaces with application to mesh-free methods

Domain discretization is considered a dominant part of solution procedures for solving partial differential equations. It is widely accepted that mesh generation is among the most cumbersome parts of the FEM analysis and often requires human assistance, especially in complex 3D geometries. When using alternative mesh-free approaches, the problem of mesh generation is simplified to the problem of positioning nodes, a much simpler task, though still not trivial. In this paper we present an algorithm for generation of nodes on arbitrary $d$-dimensional surfaces. This algorithm complements a recently published algorithm for generation of nodes in domain interiors, and represents another step towards a fully automated dimension-independent solution procedure for solving partial differential equations. The proposed algorithm generates nodes with variable density on surfaces parameterized over arbitrary parametric domains in a dimension-independent way in $O(N\log N)$ time. It is also compared with existing algorithms for generation of surface nodes for mesh-free methods in terms of quality and execution time.

math.NA↗

arXiv2019

Medusa: A C++ Library for solving PDEs using Strong Form Mesh-Free methods

Medusa, a novel library for implementation of strong form mesh-free methods, is described. We identify and present common parts and patterns among many such methods reported in the literature, such as node positioning, stencil selection and stencil weight computation. Many different algorithms exist for each part and the possible combinations offer a plethora of possibilities for improvements of solution procedures that are far from fully understood. As a consequence there are still many unanswered questions in mesh-free community resulting in vivid ongoing research in the field. Medusa implements the core mesh-free elements as independent blocks, which offers users great flexibility in experimenting with the method they are developing, as well as easily comparing it with other existing methods. The paper describes the chosen abstractions and their usage, illustrates aspects of the philosophy and design, offers some executions time benchmarks and demonstrates the application of the library on cases from linear elasticity and fluid flow in irregular 2D and 3D domains.

cs.MS↗