arXiv subjects

Łukasz Kaczmarczyk

Publications and source records attributed to Łukasz Kaczmarczyk.

4 recordsLinked to original sources

arXiv2026

The maximal dimensions of path and graph algebras

We consider the class of acyclic connected directed graphs with $N\geq 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.

math.CO↗

arXiv2025

Conservative data-driven finite element framework

This paper presents a new data-driven finite element framework that is applicable to a broad range of engineering simulation problems. In the data-driven approach, the conservation laws and boundary conditions are satisfied by means of the finite element method, while the experimental data is used directly in numerical simulations, avoiding material models. Critically, we introduce a "weaker'" mixed finite element formulation, which relaxes the regularity requirements on the approximation space for the primary field. At the same time, the continuity of the normal flux component is enforced across inner boundaries, allowing the conservation law to be satisfied in the strong sense. The relaxed regularity of the approximation spaces makes it easier to observe how imperfections in the datasets, such as missing or noisy data, result in non-uniqueness of the solution. This can be quantified to predict the uncertainty of the results using methods such as Markov chain Monte Carlo. Furthermore, this formulation provides a posteriori error indicators tailored for the data-driven approach, providing confidence in the results and enabling efficient solution schemes via adaptive hp-refinement. The capabilities of the formulation are demonstrated on an example of the nonlinear heat transfer in nuclear graphite using synthetically generated material datasets. This work provides an essential component for numerical frameworks for complex engineering systems such as digital twins.

cs.CE↗

arXiv2020

Numerical investigation into fracture resistance of bone following adaptation

Bone adapts in response to its mechanical environment. This evolution of bone density is one of the most important mechanisms for developing fracture resistance. A finite element framework for simulating bone adaptation, commonly called bone remodelling, is presented. This is followed by a novel method to both quantify fracture resistance and to simulate fracture propagation. The authors' previous work on the application of configurational mechanics for modelling fracture is extended to include the influence of heterogeneous bone density distribution. The main advantage of this approach is that configurational forces, and fracture energy release rate, are expressed exclusively in terms of nodal quantities. This approach avoids the need for post-processing and enables a fully implicit formulation for modelling the evolving crack front. In this paper density fields are generated from both (a) bone adaptation analysis and (b) subject-specific geometry and material properties obtained from CT scans. It is shown that, in order to correctly evaluate the configurational forces at the crack front, it is necessary to have a spatially smooth density field with higher regularity than if the field is directly approximated on the finite element mesh. Therefore, discrete density data is approximated as a smooth density field using a Moving Weighted Least Squares method. Performance of the framework is demonstrated using numerical simulations for bone adaptation and subsequent crack propagation, including consideration of an equine 3rd metacarpal bone. The degree of bone adaption is shown to influence both fracture resistance and the resulting crack path.

cs.CE↗

arXiv2013

Corotational formulation for 3d solids. An analysis of geometrically nonlinear foam deformation

This paper presents theory for the Lagrange co-rotational (CR) formulation of finite elements in the geometrically nonlinear analysis of 3D structures. In this paper strains are assumed to be small while the magnitude of rotations from the reference configuration is not restricted. A new best fit rotator and consistent spin filter are derived. Lagrange CR formulation is applied with Hybrid Trefftz Stress elements, although presented methodology can be applied to arbitrary problem formulation and discretization technique, f.e. finite volume methods and lattice models, discreet element methods. Efficiency of CR formulation can be utilized in post-buckling stability analysis, damage and fracture mechanics, modelling of dynamic fragmentation of bodies made from quasi-brittle materials, solid fluid interactions and analysis of post-stressed structures, discreet body dynamics.

math.NA↗