Calculation Scheme Research Articles

DETERMINATION OF OPERATING PARAMETERS OF A CRYOGENIC TANK FOR LOW-TONNAGE STORAGE OF LNG

In the modern world, the gas industry plays a key role in the fuel and energy complex of any country, ensuring energy security and economic development. With the constant consumption of natural gas, there is a need for efficient methods of its transportation and storage. One of the ways of transportation and storage of natural gas is its liquefaction.In the case of low-tonnage liquefied natural gas (LNG) production complexes, where we are talking about small volumes of production of the order of several thousand tons of LNG per year, in comparison with large-tonnage production, where productivity amounts to millions of tons of LNG per year, the efficiency of storage processes comes to the fore, as this allows for reliable and economically feasible operation of small-scale LNG complexes.In this context, determining the operating parameters of cryogenic tanks for low-tonnage storage of liquefied natural gas is becoming an urgent research topic. The method of storing cryogenic products in semi-insulated tanks has a number of advantages, such as low energy consumption, high efficiency, and long storage time.This article is devoted to determining the parameters of the cryogenic tank operation during low-tonnage storage of liquefied natural gas. The study is aimed at determining the parameters of the cryogenic tank operation based on the calculated heat flow to the stored LNG. In this paper, the authors developed and compared various calculation schemes of heat inflow, which considered such factors as heat transfer by radiation, convection and conduction, including heat inflows to LNG through thermal bridges and from the combined-cycle phase of stored liquefied gas.The result of the study is the revealed method for determining the heat flow to LNG stored in a cryogenic tank, which is optimal in terms of the accuracy of the results obtained and the simplicity of calculations. The results of this study can be used in the further development of storage technologies for both liquefied natural gas and other industrial gases.

A Multilevel Approach for Direct Core Calculation Schemes

A method of dynamic homogenization was recently proposed as an alternative technique for three-dimensional (3D) core calculations using a direct approach. This technique allows for producing homogenization parameters for a subdomain within the core at its actual state, and offers the advantage of avoiding off-line calculations, cross-section interpolations, and expensive 3D transport calculations. The methodology has shown a remarkable reduction in computational cost compared to 3D transport. However, the application of direct calculation schemes in multiphysics and multicycle problems still demands an extensive use of computational resources. Within the framework of traditional two-step calculations, several multilevel schemes have been developed to accelerate lattice calculations for the generation of the multiparameter cross-section libraries. With a similar objective, this work investigates how the multilevel approach can be applied within the framework of direct calculation schemes and discusses how depletion calculations may be performed. The objective is to enhance their performance and reduce memory requirements in favor of a high-fidelity model of the matter behavior in the reactor. We have then identified suitable flux solvers for each level and explored various homogenization options. We tested the methodology in a 3D pressurized water reactor core problem inspired by the TVA Watts Bar Unit 1 Multi-Physics Multi-Cycle OECD/NEA Benchmark, and performed a comparative analysis to assess the accuracy and computational efficiency of these new calculation schemes against direct single-level calculations and a two-step calculation based on pin-by-pin homogenization, as well as Monte Carlo simulations. Our analysis proves that the computational resources required to solve the full-core problem using a multilevel scheme are significantly reduced, and the best computational features are provided by multilevel dynamic homogenization. Furthermore, we observed that the multilevel approach not only allows for speedup, but can also be advantageously applied in order to improve the accuracy of the final solution of a calculation scheme when the core solver relies on an approximate transport operator.

Research of the Wear Rate of Elements of the Flow Part of Centrifugal and Axial Pumping Units

The paper presents the results of a theoretical and experimental study of the wear rate of the elements of the flow part of centrifugal and axial pumps. Theoretical formulas recommended by various authors, obtained for models with flat samples based on energy theory, do not take into account the features of hydraulic machines. Considering the movement of solid particles in the inter-blade channels of the impellers of centrifugal and axial pumps, calculation schemes were chosen that correspond to the hydraulic and physical wear processes. The analysis shows that the influence of centrifugal and inertial forces in the inter-blade channel of the impellers of centrifugal and axial pumps causes separation and redistribution of solid particles in the flow. As a result, in centrifugal pumps in the end part of the blade and axial pumps in the end gap of the impeller, the local concentration of solid particles increases compared to the average. The work also provides dependencies for calculating the intensity of water-abrasive wear of pump working parts.

Vector form intrinsic finite element analysis for nonlinear parametric resonances of planar beam structures

The parametric resonance of a beam structure may lead to a disastrous consequence. To assess the safety of a structure, structural engineers need a straightforward and practical method to calculate the large-amplitude nonlinear parametric resonance responses of the beam structure. This paper proposes a newly developed vector form intrinsic finite element (VFIFE) calculation scheme to analyse the nonlinear parametric resonance of planar beam structures. By taking into account the geometric stiffness effect of the internal axial force for the frame element, a new relationship between internal nodal forces and deformation components is first established based on the VFIFE principle and the parametric resonance mechanism of frame structures. A numerical example of a portal frame is presented to demonstrate the efficacy of this new calculation scheme. A parametric resonance test of the cantilever beam is conducted to verify the applicability of the VFIFE method. The numerical results show that the proposed scheme can well simulate the nonlinear phenomena of parametric resonances. The numerical predictions are consistent with the experimental observations. A filtering-based finite-time Lyapunov exponent approach is presented to determine the stability boundary of the structure. The numerical stability boundary of parametric resonance agrees well with the experimental boundary. Possible causes of deviations between the numerical and experimental results are discussed. The proposed VFIFE scheme is a simple and effective means of analysing the stability and nonlinear responses of the parametric resonance of planar beam structures.

Deep learning-based frequency-multiplexing composite-fringe projection profilometry technique for one-shot 3D shape measurement

The pursuit of precise and efficient 3D shape measurement has long been a focal point within the fringe projection profilometry (FPP) domain. However, achieving precise 3D reconstruction for isolated objects from a single fringe image remains a challenging task in the field. In this paper, a deep learning-based frequency-multiplexing (FM) composite-fringe projection profilometry (DFCP) is proposed, where an end-to-end absolute phase retrieval network (APR-Net) is trained to directly recover the absolute phase map from a FM composite fringe pattern. The obtained absolute phase map exhibits exceptional precision and is devoid of spectrum crosstalk or leakage disturbance typically encountered in traditional FM techniques. APR-Net is intricately crafted, incorporating a nested strategy along with the concept of centralized information interaction. A diverse dataset encompassing various scenarios and free from spectrum aliasing is assembled to guide the training process. A seven-map loss calculation scheme is employed to guide the training process, of which the efficacy is proved through ablation experiments. In the first qualitative experiment, DFCP demonstrates comparable phase accuracy to ground truths with 47 fewer projected images, outperforming other three methods with mean absolute phase errors of 0.0052 rad, 0.1761 rad, 0.0169 rad, and 0.0139 rad. The second qualitative experiment and the quantitative evaluation respectively prove DFCP’s capability in high dynamic range 3D measurement and in precise 3D measurement, with sphere diameter errors of 0.0780 mm and 0.0726 mm, and a spherical centroid distance error of 0.0555 mm.

Influence of Calculation Parameters on the Slope Stability of the Historical Rasos Cemetery in Vilnius (Lithuania)

Many objects of cultural and religious importance are located on naturally or artificially shaped hills. In such cases, the stability of the slopes determines both the safety of users and the facilities themselves and thus the preservation of cultural heritage. The analysis of the slope stability requires the assessment of the soil shear strength, defined as the ratio of resistance forces to driving forces. An important issue in the slope stability analysis is the proper determination of soil mechanical parameters, including their changes due to soil moisture. This paper presents an assessment of the slope stability of the Rasos Cemetery in Vilnius (Lithuania), where, due to a partial slope sliding, some of the tombstones were destroyed by being covered with soil. An analysis of the slope stability (factor of safety) was performed for two cross-sections using six calculation schemes. In each of them, calculations were carried out for three groups of soil parameters and for four design approaches (characteristic values, DA2, DA3a, and DA3b in accordance with the geotechnical standard applicable in the European Union—Eurocode 7: Geotechnical design). For the calculation approach (DA3a) for different groups of soil parameters, the values of the slope stability factors of safety were practically below 1.0. If the value of the factor of safety is less than 1.0, the slope is unstable. In the range of values 1.0–1.3, there is a serious risk of slope instability, and only when the factor of safety value reaches above 1.3 is the slope classified as stable. The comparison of the slope stability factors obtained with different design approaches (in relation to the DA3a) shows that the most unfavorable stability factor is approximately 43% lower than the result obtained from the characteristic values, approximately 21% lower than the result obtained using the DA2 design approach, and approximately 24% lower than the result obtained using the DA3b design approach. The analytical and numerical method (GEO5) analysis showed that, regardless of the adopted calculation approach, the required slope stability factor was not obtained. The article shows that selecting the correct geological cross-section to determine the critical slope slip surface and adopting the relevant design approach is crucial for slope sustainability. Based on the analysis, a proprietary solution for the construction work was proposed, which will ensure both the full stability of the slope and the safety of all graves and consequently landslide disaster prevention.

НЕСУЩАЯ СПОСОБНОСТЬ БАЛОК ТАВРОВОГО ПРОФИЛЯ С ПОЛКОЙ ИЗ ВЫСОКОПРОЧНОГО БЕТОНА И ПРОДОЛЬНОЙ АРМАТУРОЙ КЛАССА А500С

Experimental and numerical studies of the bearing capacity and deformation of T-beams with a flange made of high-strength concrete and longitudinal reinforcement of class A500C with a web made of concrete of ordinary strength were carried out on prototypes of beams and numerical models. The beams being tested are considered at different percentages of reinforcement. The calculation scheme of the beams corresponded to a four-point bend. As a result of experimental and numerical studies, the nature of the destruction of beams, their deformation and the nature of cracking are determined. Experimental and numerical relationships of relative deflections on relative bending moments are obtained. A comparative analysis of the values of the bearing capacity and the relative height of the compressed zone of the beams, obtained experimentally, numerically and according to SP 63.13330.2018, was carried out. The nature of the distribution of deformations in concrete of normal section of beams at different levels of loading is determined.

Modern approaches to determining the standardised parameters of vibration state of NPP steam pipelines

During the operation of NPP steam pipelines the main damaging factors are not only corrosion and erosion leading to thinning of the pipe wall but also fatigue damage resulting not only from temperature self-compensation efforts, but also high vibration load. As practice shows, the most promising direction of ensuring the vibration resistance of steam pipelines at present is the computational and experimental study of the stress-strain state of pipelines under vibration loading. This work shows that the strength of the most commonly used standard sizes of steel pipelines under the influence of vibration loads is determined in accordance with Russian State Standard R 59115.9-2021. Taking into account the stress concentration, the amplitudes of the conditional elastic reduced stresses were determined for the 4 variants of the calculation schemes accepted for consideration. The permissible amplitude of stresses from an operating condition of 60 years is determined. It is shown that the permissible voltage amplitude for the design temperature t = 350°C will be no more than 46.4 Mpa for a pipe made of 12X18H10T. As for NPP pipelines, the normalized vibration parameters in the regulatory documentation were not established until 2022, when the following vibration velocity limits were adopted for pipelines of nuclear power plants in Russian State Standard R 59115.11-2021, vibration resistance testing is not required: nmax up to 15 mm/s, nmsv up to 7 mm/s. Thus, currently in the Russian regulatory documentation there are no normalized values of vibration parameters (as a rule, vibration velocity) for pipelines selected depending on the frequency of vibration exposure. It is proposed to develop and introduce into the regulatory documentation frequency-dependent criteria for limiting vibration parameters of NPP pipelines, specified in accordance with the actual operating conditions of these pipelines.

Имитационное моделирование несущей способности составной деревянной шпалы с учетом поездной нагрузки

In this article the process of a composite wooden sleeper 3D-model creating is described, taking into account the physical and virtual components of pine wood. A strength analysis of the sleeper was carried out at the assembly stage. A calculation scheme of pressure in composite wooden sleeper due to the impact of rolling stock is proposed. The influence of train load on the connection strength of sleeper components was modelled. As a result of the calculation, it was established that the strength limits were not exceeded, which indicates the ability of a composite wooden sleeper to withstand a load from rolling stock wheels of up to 50 kN without loss of reliability in the connections between the elements

ОПРЕДЕЛЕНИЕ КОЛИЧЕСТВА ПЕРЕМЕЩАЕМОГО МАТЕРИАЛА ШНЕКОВЫМ ДОЗАТОРОМ В РЕЖИМЕ ТОРМОЖЕНИЯ

Screw conveyors have found application in the food, processing and chemical industries. Depending on the characteristics of the transported raw materials, conveyors of various designs are used. Due to a number of advantages in the production of dry bulk feed mixtures for farm animals, it is rational to use horizontal screw dispensers of a closed type driven by asynchronous electric motors. To increase the life of asynchronous motors, their operating mode provides for smooth braking, which affects the accuracy of dosing feed components. In this regard, the purpose of the research was formulated – to determine the influence of the structural and technological parameters of screw dispensers on the amount of material to be moved during the braking of the screw. Calculations were performed for a standard DSh-60 auger using well-known laws of mechanics. When considering the calculation scheme, it was revealed that in the screw braking mode, the mass of the dosed materials is influenced by the material supply and the screw rotation frequency. The permissible ranges of the screw rotation frequency have been experimentally established when it changes in the braking mode, which are in the range from 94.8 min-1 to 18.7 min-1. As a result of the calculations carried out, it was revealed that as the screw rotation frequency increases in the specified range, the amount of material moved by it increases by 3.7 times. The lowest feed value for the screw in question is set, which is 0.8 kg/min (at = 18.7 min-1), and the minimum mass moved by the DSh-60 screw in braking mode is 19 g. Calculated values of the mass of the dosed component can be entered into the screw control program to adjust the dosing accuracy.

The colourful antenna subtraction method

We present a general subtraction scheme for NNLO calculations in massless QCD: the colourful antenna subtraction method. It is a reformulation of the antenna subtraction approach designed to address some of the limitations of the traditional framework, especially aiming at high-multiplicity processes. In the context of the new formalism, structures needed to locally subtract the infrared-divergent behaviour of real emission corrections are systematically inferred from virtual subtraction terms, relying on the cancellation of infrared singularities and on the correspondence between integrated and unintegrated antenna functions. We illustrate in detail how the colourful antenna subtraction method works up to NNLO. The algorithm is particularly suited to be fully automated for the generation of NNLO subtraction terms for generic processes. We employ the new formalism to assemble the subtraction terms required for the calculation of the NNLO correction to hadronic three-jet production and describe their validation procedure.

Development of uncertainty quantification module for VVER analysis in STREAM/RAST-V two-step method

This paper introduces the creation of a module for Uncertainty Quantification (UQ) specifically designed for VVER analysis through the implementation of the STREAM/RAST-V two-step approach. The aim was to expand the range of use by developing a UQ module tailored for analyzing VVER. This research presents two innovative computational functionalities: (1) development of a library for the pin-based pointwise energy slowing down method (PSM), and (2) extension of the analysis area to study hexagonal-geometry fuel assemblies. The proposed UQ scheme was evaluated through verification using UAM benchmark, and comparative analysis between codes using SCALE 6.2.2 for. STREAM provides an accuracy comparable to that of SCALE 6.2.2. Additionally, a PSM covariance library was utilized in the calculations, achieving 0.7941% and 0.7907% accuracies in the hot full power and hot zero power calculations, respectively. To assess the UQ sequences in the two-step method, the STREAM/RAST-V calculation scheme was verified using the STREAM lattice code. To conclude, this study furnishes comprehensive insights into the development of the UQ module within the two-step method for VVER analysis, and it validates its performance through utilization of the UAM benchmark.

A verifiable EVM-based cross-language smart contract implementation scheme for matrix calculation

The wide application of smart contracts allows industry companies to implement some complex distributed collaborative businesses, which involve the calculation of complex functions, such as matrix operations. However, complex functions such as matrix operations are difficult to implement on Ethereum Virtual Machine (EVM)-based smart contract platforms due to their distributed security environment limitations. Existing off-chain methods often result in a significant reduction in contract execution efficiency, thus a platform software development kit interface implementation method has become a feasible way to reduce overheads, but this method cannot verify operation correctness and may leak sensitive user data. To solve the above problems, we propose a verifiable EVM-based smart contract cross-language implementation scheme for complex operations, especially matrix operations, which can guarantee operation correctness and user privacy while ensuring computational efficiency. In this scheme, a verifiable interaction process is designed to verify the computation process and results, and a matrix blinding technology is introduced to protect sensitive user data in the calculation process. The security analysis and performance tests show that the proposed scheme can satisfy the correctness and privacy of the cross-language implementation of smart contracts at a small additional efficiency cost.

Inflationsberechnung: Zinspolitik der EZB wird als Inflationstreiber statistisch nicht erfasst

Abstract After ten consecutive increases in the prime rate by the European Central Bank’s (ECB), the question arises: What significance do these measures have, both indirectly for the interest rates increased at all value-added levels on the corporate side, and for households? While the prices for rents, being the largest component, are factored into inflation calculations, the dramatically increased interest rates for real estate loans – specifically for owner-occupied housing – are not considered at all. In this article, an extension of the calculation scheme is demonstrated to make transparent the direct effect of the prime rate increase on real estate owners’ expenses.

Research on thermal deformation of cylinder bore with orthogonal experiment method of aluminum alloy cylinder block in diesel engine

Using aluminum alloy materials provides an efficient and direct method for achieving engine weight reduction. In this study, a thermo-mechanical coupling simulation model of the diesel engine was established to investigate the deformation of cylinder bores in both cast iron and aluminum alloy cylinder blocks. The reliability of the simulation model is validated by the temperature field test of the cast iron cylinder. Four evaluation indexes, namely average roundness, cylindricity, coaxiality, and light leakage, were used to assess the deformation of the cylinder bore. The results indicate that the temperature is the primary factor contributing to the deformation in both cast iron and aluminum alloy cylinder bores. Under thermo-assembly condition, the average roundness, cylindricity, coaxiality, and light leakage rate of aluminum alloy cylinder bore were observed to be higher by 36.71 %, 12.69 %, 14.77 %, and 18.85 %, respectively, compared to those of cast iron cylinder bore. To decrease the deformation of the cylinder bore, the orthogonal experiment method was used to study the three factors affecting the heat dissipation of the water jacket. It was discovered that extending the water jacket extends upward was the most effective way to reduce the temperature of the cylinder block. By selecting the optimal combination scheme for simulation calculations, an optimized cylinder temperature of 162 °C, with the maximum deformation of the cylinder bore decreasing by 13.64 %. Additionally, the average roundness, cylindricity, coaxiality, and light leakage rate of the cylinder bore showed reductions of 17.48 %, 21.8 %, 23.00 %, and 14.10 %, respectively, when compared to the original aluminum alloy cylinder block.

Generalized Substrate Screening GW for Covalently Bonded Interfaces.

An accurate description of the interfacial quasiparticle electronic structure is key to the design of heterogeneous materials. While the first-principles GW approach is state-of-the-art, the computational cost is high for large interface systems. This has led to the substrate screening GW approach for weakly coupled interfaces, which breaks down for covalently bonded interfaces. In this work, we present the generalized substrate screening GW approach, based on the following two considerations: (i) the contribution of the interfacial covalent bond to the polarizability can be efficiently calculated with a low energy cutoff; (ii) the contribution of the deprotonated adsorbate to the interface polarizability can be well approximated by that of the protonated molecule. Our approach is exemplified using interfaces formed between benzene-1,4-dithiol (BDT) and Au(111), which feature the widely used Au-S bonds in experiments. Our work provides a robust and simple scheme for accurate and efficient GW calculations of covalently bonded interfaces.

Research and application of relay protection setting calculation for SFC system of pumped storage power station

Pumped storage is a key way to meet the regulatory needs of the power system. Currently, SFC starting has become the main starting method for large pumped storage power plants. This article introduces the typical structure and operation principle of SFC in pumped storage power plants. Based on existing guidelines, the relay protection configuration and setting principles of the SFC system in pumped storage power plants are elaborated. Starting from engineering practice, a relay protection setting calculation scheme for SFC input and output transformers is proposed and put into operation on site, ensuring the reliable and stable operation of the SFC system in pumped storage power plants.

Optimized Design of Scoring Criteria for Large-Scale Innovation Competitions

The reasonable and reliable evaluation score calculation scheme wins the top priority in large-scale innovation-competitions currently. Due to the disparity among individual experts, and the existing standard score calculation model merely based on the experts' own circumstance, it cannot fully reflect the comprehensive level of players, resulting in a certain degree of error in the evaluation results. From the perspective of individual and decision-making of a group, this paper improves existing models and introduces the concept of modified scores. To verify the feasibility of the model, four schemes were designed based on the analysis of data distribution. Consistency and difference factors were applied to compare. Come to a conclusion: scheme adopted the new standard score calculation model ranked the top, indicating that the new calculation model is more reliable. Next, considering that the volatile correction factor affected by data, there is a possibility of some poor actual results with high correction scores. To rectify this case, a power exponent is added to the correction factor and a sign function is adopted to specify the positive or negative of correction scores. To verify the feasibility of the model, four sets of controlled experiments were designed with the introduction of two factors: power exponent and reconsideration bonus, as well as the ranking consistency test based on the condition that the award order of the expert agreement was consistent. In the end, it was found that the scheme that introduced both power exponent and reconsideration bonus points had a sorting consistency rate of 74%, which increased by 30% compared to the original model, indicating the rationality of the modification. The expert evaluation standard score calculation model established in this paper comprehensively considers individual and group decision-making, and provides reasonable correction for the differential scoring between experts. At the same time, this scheme provides a reference basis for further optimizing large-scale innovation competition plans in the future.

A comprehensive thermodynamic modeling of the solubility of sugar alcohols in ionic liquids

Environmental regulations have recently drawn attention from fossil sources to green materials, among which sugar alcohols (SA) have particular importance due to their vast use in the nutrition, pharmaceutical, and chemical industries. Their properties and equilibrium conditions, including the extraction and solubility behavior utilizing novel solvents such as ionic liquids (ILs), are essential for developing further processes and their optimization. On account of this, the present study aims to develop a comprehensive thermodynamic approach based on simple and accurate tools such as activity coefficient models (ACMs) and equations of state (EoSs), which have not been hitherto addressed. To this end, eleven calculation schemes based on these two approaches were applied to the largest databank investigated thus far (13 SAs, 21 ILs, and 653 data points), and the performance of the models was analyzed. NRTL, free-volume Flory-Huggins (FVFH), and Redlich-Kister (RK) ACMs with different schemes, as well as athermal and ideal solution models, were evaluated and compared against well-known cubic EoSs such as Peng-Robinson (PR) and Valderrama-Patel-Teja (VPT). Benefitting from three adjustable parameters and a simple form, the three-parameter RK model yielded the best results with a 0.82 % (2.77 K) deviation in the entire databank. Moreover, employing temperature-dependent interaction parameters with FVFH ACM (3.55 K, 1.05 %), PR EoS (3.60 K, 1.07 %), and VPT EoS (3.66 K, 1.08 %) improved the accuracy of the calculations remarkably. The results of this contribution prove that thermodynamic models based on NRTL, FVFH, and RK ACMs are the best options for calculating solid–liquid conditions equilibrium of SA-IL systems with a simple and precise calculation procedure, and can therefore be employed for the simulation and optimization tools. This study investigates the most extensive SA-IL databank, a large number of models from different routes, and the nature of SA-IL pairs employing the obtained parameters.

Modelling the piezoelectric effect in a composite polymer film filled with disperse piezoelectric

Mesoscopic modelling of a composite material constituted by an electroneutral polymer (matrix) film filled with micropowder of piezoelectric ceramics (filler) is presented. The calculation scheme resembles that of the RVE (Representative Volume Element) method. The representative element (cell) is a right-angle prism with square cross-section, the height of which is equal to the film thickness. Around the midsection of the prism, there are a few (from 2 to 4) spherical piezoelectric particles positioned close to one another. The length of the base side of the prism is determined from the assumed solid phase content of the composite. To simulate the film, the cells are arranged in a continuous flat layer, inside which they are coupled by means of periodic boundary conditions imposed on their lateral surfaces. To reduce the artifacts of the model, the position of the center of each particle is chosen randomly within the area of the prism cross-section. In the framework of this approach, a low-density polyethylene film with embedded barium titanate particles is considered as an example. One of the surfaces of the film is fixed (no displacements), and the other one is left free. For the characteristics of the particles close to those used in the experiment and the typical matrix material parameters (elasticity moduli, Poisson coefficients, dielectric permittivity), the voltage output (piezoeffect) induced in the film in response to the uniform pressure applied to its free surface is evaluated. It is shown that the electric potential along the thickness of the film is nonuniform: it grows inside the particles and falls down in the interparticle gap. In order to present the results in the form applicative for comparison with the experimental data, for each variant of the system (number of the particles, weight fraction of the solid phase), the calculated values are averaged over a few tens of realizations of the particle positions inside the element.