Articles (NTIS)

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Challenges in Automatic Differentiation and Numerical Integration in Physics-Informed Neural Networks Modelling
(2025) Daněk, Josef; Pospíšil, Jan
In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is often critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary, but using higher precision arithmetic with neural networks does not have to bring an improvement at all. Three problematic use-cases for solving differential equations using PINNs are analyzed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.
Dihedral tilings of the sphere by regular polygons and quadrilaterals I: squares and rhombi
(2025) Luk, Hoi Ping
We classify the edge-to-edge dihedral tilings of the sphere by squares and rhombi with a systematic procedure applicable to similar problems.
Some Bounds on the Threshold Dimension of Graphs
(2025) Francis, Mathew C.; Majumder, Atrayee; Mathew, Rogers
The threshold dimension of a graph G is the minimum number of threshold graphs whose intersection yields G. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of various graph parameters including treewidth, maximum degree, degeneracy, number of vertices, and vertex cover number. We also study threshold dimension of random graphs and graphs with high girth.
WO3/CuWO4 nanocomposite thin films for humidity resilient acetone gas sensing
(2025) Kumar, Nirmal; Kumar, Akash; Čapek, Jiří; Comini, Elisabetta; Haviar, Stanislav
We demonstrate a high-performing and selective acetone gas sensor based on WO3/CuWO4 nanocomposites produced by sequentially sputter-deposited WO3 thin films and CuO nanoparticles, engineered to reduce the humidity interference. By optimizing deposition order and layer thicknesses, we harnessed synergistic p-n/n-n heterojunctions and the formation of a catalytic CuWO4 ternary phase. The best performing configuration (20 nm of tungsten oxide film on top of nanoparticles) exhibits a high response (S = 23) to 10 ppm acetone at 300 °C, fast response in dry/humid conditions (38 s/58 s), and low detection limit (0.6 ppm). More importantly, the sensor exhibited > 95 % retention of its response in 90 % relative humidity compared to a loss of >50 % for pristine WO3. The reduced humidity interference is assigned to heterojunction formation at the WO3/CuWO4 interface, Lewis acid sites that allow for acetone selective adsorption, and bulk-dominated conduction. This noble-metal-free acetone sensor overcomes a known shortcoming of metal oxide-based sensors, enabling accurate acetone detection in humid environments for breath-based disease diagnosis (e.g., diabetes) and industrial safety monitoring.
Ultra-low-resistivity nitrogen-doped p-type Cu2O thin films fabricated by reactive HiPIMS
(2025) Rezek, Jiří; Koloros, Jan; Houška, Jiří; Čerstvý, Radomír; Haviar, Stanislav; Kolenatý, David; Damte, Jemal Yimer; Baroch, Pavel
We have successfully fabricated the nitrogen-doped cuprous oxide thin films on the amorphous standard soda-lime glass by reactive high-power impulse magnetron sputtering. The energy of film-forming particles was controlled by the value of pulse-averaged target power density, which has a significant impact on the elemental composition, structure and optoelectrical properties of the films. We have shown that the high-energy regime is more suitable for preserving Cu2O structure and leads to continuous substitution of oxygen by nitrogen compared with the low-energy regime. Moreover, in the high-energy regime, it is possible, to some extent, to independently control the electrical resistivity and optical properties. The electrical resistivity decreases down to ≈ 0.05 ohm.cm (upper bound of the hole mobility of 0.08 ± 0.05 cm²/Vs) at the optical band gap 2.0–2.3 eV. Special attention is paid to the formation of nitrogen molecules and their ability to form shallow acceptor states. Experimental results supported by our Density Functional Theory calculations indicate that N2 replacing Cu in the Cu2O lattice is one possible (but not the only possible) acceptor. We have also found that the formation of nitrogen molecules is preferred in a high-energy regime.
Strongly thermochromic W-doped VO2 films with a large temperature coefficient of electrical resistance near room temperature
(2025) Farrukh, Sadoon; Vlček, Jaroslav; Rezek, Jiří; Houška, Jiří; Čerstvý, Radomír; Kozák, Tomáš
We report the crystal structure, surface morphology, electronic band structure, optical and electrical properties, and semiconductor-metal transition characteristics of strongly thermochromic W-doped VO2 films with a large (up to −16 % 1/K) temperature coefficient of electrical resistance at a small hysteresis width of electrical resistivity (down to 3 °C) near room temperature, and with a wide temperature operation range at a high detection sensitivity (≥ 8 % 1/K) and low values of the electrical resistivity. They were deposited at a reduced substrate temperature of 350 °C onto soda-lime glass (SLG) with two versions of yttria-stabilized zirconia (YSZ) interlayers possessing different cubic crystal orientations, and onto bare SLG and monocrystalline YSZ and Al2O3 substrates for comparison. The W-doped VO2 depositions were performed using reactive deep oscillation magnetron sputtering with feedback pulsed O2 flow control, allowing us to increase the deposition rate of films up to 20–30 nm/min for a target-substrate distance of 100 mm. The results are important for further improvement of the thermochromic performance of VO2-based coatings for energy-saving smart windows and for a new design of high-performance infrared detectors and temperature sensors prepared by a fast, low-temperature, scalable synthesis.
The radial integral of the geopotential
(2025) Tenzer, Robert; Novák, Pavel; Eshagh, Mehdi
In Newtonian theory of gravitation, used in Earth’s and planetary sciences, gravitational acceleration is standardly regarded as the most fundamental parameter that describes any vectorial gravitational field. Considering only conservative gravitational field, the vectorial field can be described by a scalar function of 3D position called the gravitational potential from which other parameters (particularly the gravitational attraction and the gravitational gradient) are derived by applying the gradient operators. Gradients of the Earth’s gravity potential are nowadays measured with high accuracy and applied in various geodetic and geophysical applications. In geodesy, the gravity and gravity gradient measurements are used to determine the Earth’s gravity potential (i.e., the geopotential) that is related to geometry of equipotential surfaces, most notably the geoid approximating globally the mean sea surface. Reversely to the application of gradient operator, the application of radial integral to gravity yields the gravity potential differences and the same application to gravity gradient yields the gravity differences. This procedure was implemented in definitions of rigorous orthometric heights and differences between normal and orthometric heights (i.e., the geoid-to-quasigeoid separation). Following this concept, we introduce the radially integrated gravity potential (i.e., the geopotential), and provide mathematical definitions of this functional in spatial and spectral domains. We also define its relationship with other parameters of the Earth’s gravity field via Poisson, Hotine, and Stokes integrals. We then discuss prospects of using this functional in gravimetric geophysics in the context of interpreting the Earth’s inner structure. In numerical examples, we demonstrate that the indefinite radial integral of the disturbing potential (i.e., difference between actual and normal gravity potentials) has a spatial pattern that better exhibits a long-wavelength signature of deep mantle than the global geoidal geometry. This finding is explained by the fact that a more detailed spatial pattern attributed mainly to a lithospheric structure is filtered out proportionally with increasing degree of spherical harmonics in this functional. The global geoidal geometry, on the other hand, comprises not only a deep mantle signature but eventually also a gravitational signature of lithosphere, most notably across large orogens, even after applying spectral decompensation or filtering.
Nonlinear dynamics of ball vibration absorber considering stability, stationarity and rolling-condition boundaries
(2025) Dyk, Štěpán; Bulín, Radek; Rendl, Jan
The paper presents a detailed nonlinear analysis of a ball vibration absorber (BVA), which consists of a harmonic oscillator with a spherical cavity and a rolling ball as an absorber. Frequency response curves are calculated using the harmonic balance method and pseudo arc length continuation, and stability is assessed using stability analysis applied to modulation equations. This is particularly important at higher excitation amplitudes where the modulation equations provide information on the presence of strongly modulated response regimes. Codimension-2 continuation is used to identify the onset of instability and non-stationary regions with respect to all key design parameters. The study highlights the critical role of the rolling and contact conditions in maintaining the validity of the solution and provides conditions for their satisfaction. The results provide valuable insights into the non-linear dynamic behaviour of the BVA, revealing its effectiveness in vibration reduction and its limitations due to parameter selection and design constraints.
Convolution neural network for fluid flow simulations in cascade with oscillating blades
(2025) Bublík, Ondřej; Heidler, Václav; Vimmr, Jan
This paper aims to design a computational model for simulating the unsteady flow field in a cascade of oscillating blades. The core of the new model is a convolutional neural network, which is trained on a simplified cascade consisting of three blades. The primary advantage lies in significantly reducing the computational cost, as the new model is several orders of magnitude faster than traditional CFD methods for evaluations, though training the model remains computationally intensive. The convolutional neural network can accurately predict the unsteady flow field, as demonstrated in validation examples. In the next step, a composition algorithm is proposed to combine several simplified cases, enabling the solution of a cascade with any number of blades.
Thermally-induced microstructural evolution in nanoparticle-based CuO, WO3 and CuO–WO3 thin films for hydrogen gas sensing
(2025) Shaji, Kalyani; Haviar, Stanislav; Zeman, Petr; Procházka, Michal; Čerstvý, Radomír; Kumar, Nirmal; Čapek, Jiří
This study systematically investigates the microstructural evolution of nanoparticle-based CuO, WO3, and composite CuO–WO3 thin films induced by their post-deposition annealing. The films were reactively deposited using a magnetron-based gas aggregation technique, with the composite films consisting of alternating monolayers of CuO and WO3 nanoparticles. After deposition, the films were annealed in synthetic air at temperatures ranging from 200 to 400 °C and characterized using scanning electron microscopy, X-ray diffraction, Raman spectroscopy, and X-ray photoelectron spectroscopy. Annealing of the CuO films led to the most pronounced changes associated with a gradual enhancement of crystallinity accompanied by significant particle growth with increasing annealing temperature, while the WO3 and CuO–WO3 films were more thermally stable to crystallization and particle growth. Notably, at 400 °C, the CuO–WO3 films crystallized into a novel 𝛾-CuWO4 phase. The annealed films were further evaluated for their gas-sensing performance upon H2 exposure, and the obtained results were analysed in relation to film properties and the microstructural evolution induced by annealing.
Homogenization of the acoustic streaming in periodic rigid porous structures
(2025) Rohan, Eduard; Moravcová, Fanny
The paper presents a new model of the acoustic streaming (AS) in rigid porous media. The modelling is based on the classical perturbation approach combined with the periodic homogenization. The first one enables to linearize the Navier–Stokes equations for a barotropic fluid using the decomposition into the first and the second order subproblems governing the fluid dynamics in the rigid period scaffolds. The acoustic wave captured by the first order problem provides the Reynolds stress which appears in the second order problem as the streaming source term. Both the subproblems are treated by the homogenization resulting in the dynamic Darcy flow equations. Using spectral analysis of the characteristic microscopic dynamic Stokes flow and the associated spectral decomposition of the responses, the dynamic permeability is derived and also the driving force for the time-averaged permanent flow is evaluated. The AS can be observed at both the macroscopic and the microscopic levels. While the acoustics-driven microflows are observed for any microstructure, the macroscopic AS depends on the porous microstructure geometry, its nonsymmetry and boundary conditions. For porous particulate structures, the forces and force moments acting on the suspended particles are computed and the influence of the wave frequency and geometrical features is examined. All these effects are illustrated using 2D examples of periodic porous microstructures.
Consistent determination of the gravimetric geoid and orthometric height
(2025) Tenzer, Robert; Novák, Pavel
Various computational methods have been developed and applied to determine regional gravimetric geoid models with high accuracy using surface gravity and terrain data, while also often taking into consideration topographic mass density information. Helmert’s orthometric height is, on the other hand, until now solely used for practical realization of vertical geodetic controls in countries where the orthometric height is adopted for the definition of official height systems. Whereas small errors (at the level of a few centimeters) are reported for accurately determined regional gravimetric geoid models, errors in Helmert’s orthometric height reach several centimeters and decimeters already at levelling networks realized in lowlands and regions with moderately elevated topography. In mountainous regions with extremely elevated topography, these errors reach several meters. In Helmert’s definition of the orthometric height, the mean value of gravity within topographic masses is computed approximately from observed surface gravity by applying the Poincaré-Prey gravity gradient reduction, without applying complex computational methods that are used in the gravimetric geoid modelling. This approximation introduces errors due to assuming a constant topographic mass density and disregarding terrain geometry and mass density heterogeneities inside the geoid. Consequently, values of Helmert’s orthometric heights are not consistent with accurately determined regional gravimetric geoid models and should not be fitted or combined with GNSS/levelling data. To address this theoretical inconsistency, we propose a computational scheme based on applying developed methods for consistent determination of the regional gravimetric geoid and orthometric height to achieve their full compatibility by means of improving the accuracy of the orthometric height. We demonstrate that computational methods applied in the regional gravimetric geoid modelling can be modified to determine also the accurate orthometric height, so that both quantities are computed consistently and simultaneously. We also show that the proposed computational scheme can be also used for an accurate conversion of normal to orthometric heights by means of applying the geoid-to-quasigeoid separation. This allows an independent validation of regional gravimetric geoid models.
Unveiling effects of Zr alloying on structure and properties of nanocrystalline Cu–Zr films
(2025) Zhadko, Mariia; Benediktová, Anna; Čerstvý, Radomír; Houška, Jiří; Čapek, Jiří; Kolenatý, David; Minár, Jan; Baroch, Pavel; Zeman, Petr
Nanocrystalline Cu–Zr films with Zr content in the range of 0.3–2.7 at.% were deposited by direct current magnetron sputter deposition. Effects of Zr alloying on the structure, surface, mechanical, and electrical properties were systematically investigated using X-ray diffraction, electron microscopy, atomic force microscopy, indentation, and the four-point probe method. The experimental results revealed that the Zr content significantly affects the structural and functional characteristics of the films, with the most notable changes observed between 0.3 and ≈1.5 at.% Zr. Beyond this range, further increase in the Zr content results in only minor changes in the microstructure and mechanical properties, while the solubility, electrical resistivity, and surface roughness continue to rise. The alloyed Cu–Zr films exhibit hardness values between 3.2 and 4.2 GPa, exceeding 2.5 GPa measured for the unalloyed Cu film, which is attributed to the combined effect of grain boundary strengthening due to structural refinement and Zr segregation, along with solid solution strengthening.
Multilayer design of sustainable multifunctional Zr–Cu–N coatings: A route for enhanced mechanical and antibacterial performance
(2025) Daniel, Rostislav; Ziegelwanger, Tobias; Zítek, Michal; Červená, Michaela; Haviar, Stanislav; Meindlhumer, Michael; Baroch, Pavel; Keckes, Jozef; Zeman, Petr
Wear-resistant protective coatings with antimicrobial activity are essential for durability and hygiene in healthcare, public spaces, food industry, consumer products, and industrial environments. This study developed sustainable multifunctional Zr–Cu–N coatings with exceptional damage tolerance, and antibacterial properties using non-reactive and reactive sputtering of only two elemental Zr and Cu targets without external heating. The coatings’ superior performance stems from a sophisticated multilayer architecture combining elastic ZrCu metallic glass, hard and stiff ZrN ceramic, and hard and tough ZrN–Cu nanocomposite coatings. Each constituent was optimized for composition and mechanical properties before integration into multilayer structures to provide high damage tolerance and antibacterial functionality. Antibacterial efficacy was tested in a high-traffic environment over 60 days, showing consistent antimicrobial performance. Fracture stress and toughness were assessed through in situ bending experiments on microcantilever beams fabricated by focused ion beam milling. Results revealed that optimizing the thicknesses of ductile and stiff sublayers significantly enhances damage tolerance while maintaining high hardness and wear resistance. The incorporation of Cu in an unbonded state within the ZrN–Cu nanocomposite facilitates sustainable and scalable production of these multifunctional coatings with antibacterial properties, making them ideal for large surface applications in high-traffic environments like hospitals, office buildings, and public transport.
Automorphisms and Isomorphisms of Maps in Linear Time
(2025) Kawarabayashi, Ken-ichi; Mohar, Bojan; Nedela, Roman; Zeman, Peter
A map is a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices, which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus g not equal 0, parametrized by the genus g . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex v does not depend on the choice of v . The algorithm applies a sequence of local reductions and produces a uniform map while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.
Review of the problem of the Earth shape
(2025) Vaníček, Petr; Novák, Pavel; Santos, Marcelo
The determination of the shape of the Earth has been one of the fundamental problems geodesy was supposed to solve; it has been and possibly still is the main geodetic problem. It is thus appropriate for geodesists to look at this problem periodically, and this is what the authors of this paper aim to do. About 50 years ago, geodesists started using satellites as a new and very powerful tool. Many problems that were either impossible to solve or that presented almost unsurmountable hurdles to solutions have now been solved relatively simply, so much so that in the eyes of some people, satellites can solve all geodetic problems, and attempts are being made to show that this is indeed the case. We feel that the time has come to show that even satellites have their limitations, the main one being that for them to remain in their orbit, they must fly quite high, typically at several hundred kilometres. The gravitational field of the Earth (and that of any celestial body) smoother as one gets higher and higher. In other words, the gravitational field at the satellite orbit altitude loses detailed information that one can see at the surface of the Earth. In this contribution, we shall try to explain what satellites have contributed to the study of the shape of the Earth and what issues remain to be sorted out.
Cubic graphs with colouring defect 3
(2024) Nedela, Roman; Karabáš, Ján; Máčajová, Edita; Škoviera, Martin
The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While 3-edge-colourable graphs have defect 0, those that cannot be 3-edge-coloured (that is, snarks) are known to have defect at least 3. In this paper we focus on the structure and properties of snarks with defect 3. For such snarks we develop a theory of reductions similar to standard reductions of short cycles and small cuts in general snarks. We prove that every snark with defect 3 can be reduced to a snark with defect 3 which is either nontrivial (cyclically 4-edge-connected and of girth at least 5) or to one that arises from a nontrivial snark of defect greater than 3 by inflating a vertex lying on a suitable 5-cycle to a triangle. The proofs rely on a detailed analysis of Fano flows associated with triples of perfect matchings leaving exactly three uncovered edges. In the final part of the paper we discuss application of our results to the conjectures of Berge and Fulkerson, which provide the main motivation for our research.
Homogenization of fluid saturated double porosity media with a new type of the contrast in the Biot mesoscopic model
(2024) Rohan, Eduard; Nguyen, Vu-Hieu; Naili, Salah
This paper presents a mathematical model of a double poro-elastic medium derived by the homogenization of a two-component periodically heterogeneous Biot continuum characterized by strong heterogeneities in the permeability and poroelastic coefficients of mesoscopic structure. A new scaling of the mesoscopic material parameters with respect to the small parameter which is involved in the asymptotic analysis is introduced to capture this high contrast quality of the mesoscopic model. It is shown that such a scaling ansatz of the mesoscopic material parameters can be justified using different micromodels associated with the two mesoscopic components. While the “matrix” is a little permeable stiff phase, the “conductive channels” are made of a very soft fibrous structure equivalently represented by an network of helical springs. The unfolding method of the homogenization is employed to derive the macroscopic model involving the frequency-dependent effective parameters. Semipermeable interfaces between hard dual porosity and soft primary porosity are considered. The wave dispersion of two shear wave modes S1, S2, and two pressure wave modes P 1, P 2, is illustrated in an numerical example and validated, in a part, using the reference dispersion analysis based on the Bloch wave decomposition.
Implementace a testování softwarové knihovny pro výpočet efektů vzdálené zóny u sférických integrálních transformací
(2024) Belinger, Jiří; Šprlák, Michal; Pitoňák, Martin; Trnka, Petr; Novák, Pavel
Integrální transformace jsou užitečným matematickým aparátem a základem pro formulaci odhadů veličin tíhového pole včetně šíření chyb. Jedním z předpokladů integrálních transformací je globální datové pokrytí. Dostupnost pozemních měření je však obecně limitována. V praxi se globální integrál rozkládá na dvě oblasti – blízké a vzdálené zóny. Nezanedbatelný příspěvek dat ve vzdálené zóně vyžaduje přesné vyhodnocení. Za tímto účelem je v prostředí MATLAB vytvářena knihovna pro výpočet efektů vzdálené zóny u integrálních transformací pro derivace tíhového potenciálu až do třetího řádu. Příspěvek popisuje implementaci teoretických odvození efektů vzdálených zón a numerické testování knihoven.
Identifying internal resonance regimes in free-vibrating systems with multiple autoparametric couplings
(2024) Dyk, Štěpán; Bulín, Radek; Rendl, Jan; Smolík, Luboš
This paper introduces a method for identifying internal resonance regimes in free-vibrating systems with multiple nonlinear couplings, illustrated using a chain of two Rott’s pendula. The method based on decomposing Hamiltonian and subsequent treatment of the coupling energies allows for a comprehensive understanding of autoparametric system behaviour. A novel signal-processing-based coupling energy evaluation significantly enhances the effectiveness of the method, revealing internal resonances with slow energy exchange. In the theoretical part, a general step-by-step procedure applicable to any conservative coupled oscillators is introduced. We demonstrate that concerning such free-vibrating systems, the proposed method effectively uncovers internal resonances influenced by system parameters, initial conditions, and the initial energy of the system. Through an illustrative example of the system with multiple quadratic couplings, we also show that slow energy exchange occurs for sum and difference internal resonances.

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