arXiv subjects

Mitja Jančič

Publications and source records attributed to Mitja Jančič.

18 recordsLinked to original sources

arXiv2026

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.

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arXiv2026

Some observations regarding the RBF-FD approximation accuracy dependence on 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 able to introduce a numerical quantity that could indicate whether a given stencil size is locally optimal. This work is an extension of our ICCS 2023 conference paper.

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arXiv2026

A superconvergence result in the RBF-FD method

Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is well known that the truncation error of the differential operator approximation is determined by the degree of monomial augmentation. Naively, one might think that the solution error will have the same order of convergence. We present a superconvergence result that shows otherwise - for some augmentation degrees, order of convergence is higher than expected.

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arXiv2024

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

This paper deals with a numerical analysis of plastic deformation under various conditions, utilizing Radial Basis Function (RBF) approximation. The focus is on the elasto-plastic von Mises problem under plane-strain assumption. Elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress surpasses the yield stress, corrections are applied locally through a return mapping algorithm. The non-linear deformation problem in the plastic domain is solved using the Picard iteration. The solutions for the Navier-Cauchy equation are computed using the Radial Basis Function-Generated Finite Differences (RBF-FD) meshless method using only scattered nodes in a strong form. Verification of the method is performed through the analysis of an internally pressurized thick-walled cylinder subjected to varying loading conditions. These conditions induce states of elastic expansion, perfectly-plastic yielding, and plastic yielding with linear hardening. The results are benchmarked against analytical solutions and traditional Finite Element Method (FEM) solutions. The paper also showcases the robustness of this approach by solving case of thick-walled cylinder with cut-outs. The results affirm that the RBF-FD method produces results comparable to those obtained through FEM, while offering substantial benefits in managing complex geometries without the necessity for conventional meshing, along with other benefits of meshless methods.

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arXiv2024

Spatially dependent node regularity in meshless approximation of partial differential equations

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatially varying 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 discretized with regular nodes. A simpler approximation can be used in regions covered by regular nodes, effectively reducing the required stencil size and computational cost compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour. This paper is an extended version of conference paper entitled "Spatially-varying meshless approximation method for enhanced computational efficiency" [arXiv:2303.01760] presented at "International Conference on Computational Science (ICCS) 2023". The paper is extended with discussion on development and implementation of a hybrid regular-scattered node positioning algorithm (HyNP). The performance of the proposed HyNP algorithm is analysed in terms of separation distance and maximal empty sphere radius. Furthermore, it is demonstrated that HyNP nodes can be used for solving problems from fluid flow and linear elasticity, both in 2D and 3D, using meshless methods. The extension also provides additional analyses of computational efficiency and accuracy of the numerical solution obtained on the spatially-variable regularity of discretization nodes. In particular, different levels of refinement aggressiveness and scattered layer widths are considered to exploit the computational efficiency gains offered by such solution procedure.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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