Power-law Variation Research Articles

Contribution of blowing-snow sublimation to the surface mass balance of Antarctica

Abstract. Blowing-snow sublimation is a key boundary layer process in polar regions and is the major ablation term in the surface mass balance (SMB) of the Antarctic ice sheet. This study updates the blowing-snow model in the Regional Atmospheric Climate Model (RACMO), version 2.3p3, incorporating blowing-snow sublimation into the prognostic equations for temperature and water vapour. These updates address numerical artefacts in the previous model version by replacing the uniformly discretised ice particle radius distribution, which limited the maximum ice particle radius to ≤ 50 µm, with a non-uniform distribution covering radii from 2 to 300 µm without additional computational overhead. The improved model is validated against meteorological observations from site D47 in Adélie Land, East Antarctica. The updates fix the numerical artefacts, successfully predicting the power-law variation in the blowing-snow flux with wind speed while improving the prediction of its magnitude. Additionally, a qualitative comparison with CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation) satellite data shows that RACMO accurately forecasts the spatial pattern of monthly blowing-snow frequencies. The model also yields an average blowing-snow layer depth of 230±116 m at D47, matching typical satellite observation values. Results reveal that, without blowing snow, sublimation in Antarctica mainly occurs in summer (October–March), with minimal surface sublimation in winter (April–September). Introducing the blowing-snow model creates an additional sublimation mechanism primarily contributing in winter. From 2000–2012, model-integrated blowing-snow sublimation averaged 175±7 Gt yr−1, a 52 % increase from the previous version. Total sublimation, summing blowing-snow and surface sublimation, reached 234±10 Gt yr−1, 47 % higher than in simulations without the blowing-snow model. This increase leads to a 1.2 % reduction in the Antarctic ice sheet's integrated SMB. Additionally, changes in sublimation in coastal and lower escarpment zones underscore the importance of the model updates for Antarctic climatology.

Assessment of the near-Sun magnetic field of the 10 March 2022 coronal mass ejection observed by Solar Orbiter

Aims. We estimate the near-Sun axial magnetic field of a coronal mass ejection (CME) on 10 March 2022. Solar Orbiter’s in situ measurements, 7.8 degrees east of the Sun-Earth line at 0.43 AU, provided a unique vantage point, along with the WIND measurements at 0.99 AU. We determine a single power-law index from near-Sun to L1, including in situ measurements from both vantage points. Methods. We tracked the temporal evolution of the instantaneous relative magnetic helicity of the source active region (AR), NOAA AR 12962. By estimating the helicity budget of the pre-and post-eruption AR, we estimated the helicity transported to the CME. Assuming a Lundquist flux-rope model and geometrical parameters obtained through the graduated cylindrical shell (GCS) CME forward modelling, we determined the CME axial magnetic field at a GCS-fitted height. Assuming a power-law variation of the axial magnetic field with heliocentric distance, we extrapolated the estimated near-Sun axial magnetic field to in situ measurements at 0.43 AU and 0.99 AU. Results. The net helicity difference between the post-and pre-eruption AR is ( − 7.1 ± 1.2)×1041 Mx2, which is assumed to be bodily transported to the CME. The estimated CME axial magnetic field at a near-Sun heliocentric distance of 0.03 AU is 2067 ± 405 nT. From 0.03 AU to L1, a single power-law falloff, including both vantage points at 0.43 AU and 0.99 AU, gives an index −1.23 ± 0.18. Conclusions. We observed a significant decrease in the pre-eruptive AR helicity budget. Extending previous studies on inner-heliospheric intervals from 0.3 AU to ∼1 AU, referring to estimates from 0.03 AU to measurements at ∼1 AU. Our findings indicate a less steep decline in the magnetic field strength with distance compared to previous studies, but they align with studies that include near-Sun in situ magnetic field measurements, such as from Parker Solar Probe.

Broadband dielectric spectroscopy studies of Co3O4 and Co2TiO4 two-phase spinel composites

Broadband dielectric spectroscopy studies of the two-phase composites, (1 – x) Co3O4 + x Co2TiO4, have been reported over a wide range of compositions (0 ≤ x ≤ 1) and temperatures (80 K ≤ T ≤ 573 K). Normal spinel Co3O4 and inverse spinel Co2TiO4 phases are well distinguishable with ∼4.6 % variation in the lattice parameter of the Co3O4 phase with an increase of x from 0 to x = 1. The ac-conductivity (σ) of this composite follows Johnscher's power law (∼Aωs) variation with frequency exponent factor ‘s’ both greater than as well as less than unity. The temperature dependence of ‘σ’ reveals the Arrhenius-like behavior instead of variable range hopping of charge carriers with activation energy in the 2–1800 meV range. Systematic analysis of such conductivity mechanisms for all the compositions is compared with the pristine Co3O4 system.

Experimental study of discharge current oscillations with dust particles

We present a detailed experimental study of discharge current oscillations in a planar cathode plasma with poly-dispersed alumina dust particles. The dominant frequency of oscillation depends on the discharge voltage, operating pressure, and amount of dust particles placed on the cathode. The power-law variation in the dominant frequency with different external operating parameters is presented. Experimental observations suggest that the dominant mechanism behind the generation of these oscillations is the cathode spot injection of sub-micron-sized dust particles. The cathode spots also aid in the generation of fine dust particles. The threshold limit on dust particle density dispersed on the cathode suggests that below the threshold limit, the fine particles depleting the electrons play an important role and lead to the generation of self-excited oscillations. Operating above the threshold limit, a stable dust cloud was observed together with the suppression of self-excited oscillations.

Assessment and control of the mine tremor disaster induced by the energy accumulation and dispersion of thick-hard roofs

In order to solve the problem that current theory models cannot accurately describe thick-hard roof (THR) elastic energy and assess the mine tremor disasters, a theoretical method, a Timoshenko beam theory on Winkler foundation was adopted to establish the THR’s periodic breaking model. The superposition principle was used for this complex model to derive the calculation formulas of the elastic energy and impact load on hydraulic supports. Then, the influence of roof thickness h, cantilever length L1, and load q on THR’s elastic energy and impact load was analyzed. And, the effect of mine tremor disasters was assessed. Finally, it is revealed that: (1) The THR’s elastic energy U exhibits power-law variations, with the fitted relationships U=0.0096L13.5866, U=5943.9h−1.935, and U=21.049q2. (2) The impact load on hydraulic supports FZJ increases linearly with an increase in the cantilever length, thickness, and applied load. The fitted relationships are FZJ=1067.3L1+6361.1, FZJ=125.89h+15100, and FZJ=10420q+3912.6. (3) Ground hydraulic fracturing and liquid explosive deep-hole blasting techniques effectively reduce the THR’s cantilever length at periodic breakages, thus eliminating mine tremor disasters.

Mixed series solution for vibration and stability of porous bi-directional functionally graded beams

A new analytical solution based on the Ritz method is presented in this paper for analyzing the free vibration and buckling behavior of porous bi-directional functionally graded (2D-FG) beams under various boundary conditions. The solution is based on first-order shear deformation theory (FSDT). The selection of solution functions used in Ritz methods distinguishes the methods from each other and determines the accuracy of the analytical solution. To accurately capture the system's behavior and achieve the desired results, these functions have been carefully selected as a combination of polynomial and trigonometric expressions tailored as mixed series functions for each boundary condition. The study considers three types of porosity, namely PFG-1, PFG-2, and PFG-3. The equations of motion are derived using Lagrange's principle, taking into account the power-law variation of the beam material components throughout the volume. The non-dimensional fundamental frequencies and critical buckling loads are calculated for different boundary conditions, gradation exponents in the x and z directions (px, pz), slenderness (L/h), porosity coefficient (e), and porosity types. Initially, the accuracy of the mixed series functions is investigated for non-porous bi-directional functionally graded beams, and the numerical results are compared with existing literature to validate the proposed solution. Subsequently, the paper focuses on analyzing the influence of porosity on the free vibration and buckling behavior of bi-directional functionally graded beams using the developed solution method.

An enhanced strain gradient model for temperature-dependent sound transmission loss in double-walled functionally graded microplates

With the growing use of functionally graded (FG) microplates in structural acoustic metamaterials for aerospace and automotive applications, accurate modeling of sound transmission loss is critical for effective vibration and noise control engineering. In this study, a theoretical model is formulated based on the first-order shear deformation theory (FSDT) to estimate sound transmission loss (STL) through air-filled rectangular double-walled FG microplates with simply supported boundary conditions. The microplates are subjected to a nonlinear thermal environment, and the modified strain gradient theory (MSGT) is employed, incorporating three material length scale parameters to account for the size effect. The material properties are temperature-dependent and vary across the thickness following a power-law distribution. Size-dependent coupled vibroacoustic equations are derived utilizing the sound velocity potential, normal velocity continuity conditions, and Hamilton's principle, and are subsequently solved using the Galerkin method. Comparative analysis is performed by contrasting the results obtained from the MSGT model with those from the modified couple stress theory (MCST) and classical continuum theory (CCT) models, allowing for the assessment of the accuracy and precision of the proposed solution. Additionally, the influence of various parameters, such as gradient index, length scale parameters, temperature variation, incident angles, and acoustic cavity depth, on the STL is investigated. Key findings demonstrate that in the stiffness-controlled region, length scale parameters significantly enhance STL, while power-law index and temperature variation reduce it.

Free vibration behavior in functionally graded material plates with dual efficient quadrilateral finite elements

Functionally graded materials represent advanced composites with typically two materials that exhibit gradual variations in their mechanical properties throughout the structure’s thickness in general, as determined by a presumed mathematical model. The current investigation focuses on analyzing free vibrations of functionally graded material plate structures with two efficient isoparametric quadrilateral finite elements (Lagrange and Serendipity) together for the first time and different material combinations based on first-order shear deformation theory. The analysis was carried out on relatively thin as well as medium thick functionally graded plates with varying power law indices and at varied boundary conditions. The framework equation was formed with Hamilton’s principle, and solutions were done with finite element method, considering a shear correction factor for the behavioral study of varying material properties. The findings align well with the information presented in prior literature. After validating the present model, parametric analysis was conveyed to show the impact of length-to-thickness ratio, aspect ratio, volume fractions, and skew angles for two novel materials on the fundamental frequencies and mode shapes. The results thus obtained with the two quadrilateral elements have a significant effect on the changes in these parameters. Also, the natural frequencies of different functionally graded plates, including two novel materials, were compared and discussed.

Computation of hydromagnetic tangent hyperbolic non-Newtonian flow from a rotating non-isothermal cone to a non-Darcy porous medium with thermal radiative flux

A theoretical and numerical study is conducted on nonlinear, steady-state thermal convection boundary layer flow of a magnetized incompressible Tangent Hyperbolic non-Newtonian fluid from a rotating cone to a non-Darcy porous medium. Power-law variation in temperature on the cone surface is considered and thermal radiation heat transfer is also present. The Brinkman-Darcy-Forchheimer model is deployed for the porous medium. The study is motivated by rotational (spin) coating with new emerging magnetic rheological polymers, a process which often utilizes filtration media and high temperatures. The transformed non-dimensional conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. Extensive visualization of axial, tangential velocity components and temperature distributions with variation in key parameters including Rosseland radiative number, Darcy number, Forchheimer number (non-Darcy inertial parameter), magnetic interaction parameter, tangent-hyperbolic non-Newtonian power-law index and Weissenberg (non-Newtonian) number is included. Additionally, axial and tangential (circumferential) skin friction and Nusselt number values are tabulated for variation in key control parameters. With increasing Weissenberg number, axial velocity is depleted near the cone surface, tangential velocity is suppressed throughout the boundary layer regime and temperature is strongly enhanced. Axial flow is strongly decelerated further from the cone surface with increasing tangent-hyperbolic power-law index and there is also a significant depletion in tangential (swirl) velocity. Temperature is however boosted throughout the boundary layer transverse to the cone surface with a rise in tangent-hyperbolic power-law index. Both tangential and axial velocity are suppressed with increment in magnetic interaction parameter whereas temperature and thermal boundary layer thickness are enhanced. With larger Darcy number (i.e. greater permeability), axial velocity is strongly increased near the cone surface with no tangible modification further from the cone; However tangential velocity is consistently elevated throughout the boundary layer with greater Darcy number whereas temperature is depleted. An increase in Forchheimer number substantially damps both axial and tangential velocity whereas it elevates temperature. Increasing radiative flux strongly energizes the magnetic polymer and elevates temperature but suppresses the axial and tangential velocities. With elevation in non-isothermal wall exponent, axial skin friction is suppressed whereas Nusselt number are elevated at the cone vertex. Further along the cone surface a similar response is observed but there is also a reduction in magnitudes of tangential skin friction.

Kinetic inductance in superconducting CoSi2 coplanar microwave transmission lines

We have looked into cobalt disilicide (CoSi2) as a potential building block for superconducting quantum circuits. In order to achieve this, we annealed a thin layer of Co to create microwave cavities with thickness of d = 10–105 nm from CoSi2 embedded in the silicon substrate. The cavity properties were measured as a function of temperature and power. In the films measuring 10 and 25 nm, we find a significant kinetic inductance LK with a non-BCS power-law variation δLK ∝ T4.3±0.2 at low temperatures. The quality factor of the studied microwave resonances varied from 3 × 103 (d = 10 nm) to ∼5 × 104 (d = 105 nm) and increased as d(A − log d) with thickness, with two-level systems having very little effect. The power dependence of kinetic inductance was analyzed in terms of heat flow due to electron–phonon coupling, which was found to be stronger than estimated for heat relaxation by regular quasiparticles.

Towards urban wind utilization: The spatial characteristics of wind energy in urban areas

Urban wind energy is gaining increasing recognition for its environmentally friendly nature, particularly in the context of energy shortages. Predicting wind speed in urban environments is challenging due to the varying roughness and drag caused by obstacles on the ground. The complexity of urban morphology significantly disrupts the wind field and hampers the utilization of wind energy. This study aimed to investigate the diverse impacts of highly heterogeneous urban environment on the urban wind energy by innovatively establishing spatially varying power law wind profiles. Firstly, long-term LIDAR observations were conducted to measure the vertical wind profile in the study area. Then, a high-resolution LCZ map was created and integrated into the Weather Research and Forecasting (WRF) model to reproduce the urban wind field, which was validated using the observational data. Finally, the spatial characteristics of the urban wind field and wind energy were analyzed based on 3 representative points and the urban LCZ categories. The results show that the average power law exponents of each LCZ classification ranges from 0.29 to 0.75, with the average power law exponent of the high-volume building area being larger than 0.6. The highest wind power density in the study area can reach 343.3W/m2 at 200m height. Further, significant negative correlation coefficients were found between wind power density and building height, as well as building density, with values of approximately −0.6 and −0.35, respectively. Overall, the high-volume-ratio built-up areas with higher roughness had a notable impact on the wind speed by increasing the power law exponents. These spatially varying wind profiles are expected to provide technical support for the utilization of urban wind energy.

Power Curve Characterization I: Improving the Bin Method.

Neural network approaches have been used to try and characterise the power performance of wind turbines in a wind farm. Stochastic techniques are often dismissed as being inferior to those which use neural networks but those stochastic techniques proposed are often overly simplistic. We propose modifications to the IEC 61400-12 bin method in order to provide a more accurate characterisation of wind turbine performance in complex terrain taking account of direction and using a varying power law dependence of power output on wind speed. The method is validated using data from a real wind farm

Characterization of hardness and elastic modulus using nanoindentation and correlation with wear behavior of UHMWPE during uniaxial tension

UHMWPE is the material of choice for bearing surfaces in total joint arthroplasty making its wear and mechanical properties important factors of contribution in longevity of prosthetic hip/knee implants. In this study, the variation of hardness and elastic modulus with applied load in textured UHMWPE has been investigated. Texture has been induced through uniaxial tension of UHMWPE modifying its microstructure which in turn influences the wear resistance and hence the mechanical properties of the material. Previous studies have shown hardness to be a major factor influencing wear resistance. However, recently, the ratio of hardness (H) to elastic modulus (E) has been recognized as a more influential parameter of wear resistance. The validity of predicting wear resistance using H/E ratio has been examined in this work. Power law variation with load for the bioimplant material UHMWPE has been investigated at different strain levels. It has been observed that power law exponent of 2 can only be achieved at higher load levels. Overall, this work provides an insight into influencing the properties of bioimplant material UHMWPE by modifying the microstructure of the material through inducing texture which ultimately affects the longevity of the prosthetic implants.

A modified decomposition method for the analysis of porous triangular fin with a power exponent of thermal properties and magnetic effect

A closed-form solution of the triangular porous fin with a simultaneous variation of power law-dependent heat transfer coefficient, internal heat generation, and surface emissivity parameters under the influence of external magnetic and electric fields is carried out. Darcy’s model has been used to simulate flow in the porous triangular fin with insulated boundary conditions. The governing singular value equation is nondimensionalized and solved by modified Adomian decomposition method (MADM) and the results of MADM are compared with the numerical solution obtained from the finite difference method (FDM) in the limiting conditions. The graphical analysis of the significant power law variation of thermophysical parameters, Hartmann number and important design parameters such as the half-thickness parameter of the triangular fin are performed and physically interpreted. A comparative study has been carried out with multiple power law parameters at different values while other thermophysical parameters were kept at a fixed level and it has been found that fin temperature is highest at higher values of power index parameters. From this study, it has been found that with the increasing value of the Hartmann number as well as the porosity parameter, the efficiency of the triangular porous fin increases rapidly.

A Molecular Dynamics Study of Nucleation and Grain Growth of Novel Al-15Mg<sub>2</sub>Si-4.5Si Composite during Rapid Cooling Based Semi Solid Slurry Preparation

Owing to their several attractive features such as high hardness, high elastic modulus, light weight, high strength to weight ratio, high thermal conductivity, and high temperature strength, composites from Al-Mg2Si family offers promise towards deployment in several industries such as automobile, aerospace, marine, defence and electronic. The present molecular dynamics (employing LAMMPS) based simulation study is one of the first attempt to investigate the nucleation and grain growth mechanisms of Mg2Si phase at atomic level in case of novel Al-15Mg2Si-4.5Si composite, during semi-solid processing. Modified embedded atom method (MEAM) potential has been used to study the atomic interactions in the composite. Reaching the melt state at 1000 K, the temperature of the system is first decreased from 1000 K to 853 K and then the system is held at 853 K for 100 ps. The simulations are performed with three different cooling rates. With lowering of temperature, randomly distributed Mg and Si atoms form atomic clusters at arbitrary locations within the system, which is the nucleation stage for Mg2Si phase formation. Cluster size, radial distribution function has been used to investigate the structural evolution of Mg-Si clusters. Cooling rate significantly influences the grain size as well as the grain growth kinetics. The information about the thermodynamic state of the system has been revealed by extracting the values of internal energy, enthalpy, specific heat. during the slurry preparation and isothermal holding stages. The growth mechanism of Mg2Si nucleus has been characterized from the temporal variation of (Mg + Si) atoms taking part in the cluster formation. Power-law variation is observed in the cooling stage whereas a linear variation is observed in the isothermal stage.

Mechanistic Insights into Hyaluronic Acid Induced Peptide Nanofiber Organization in Supramolecular Hydrogels.

Composite hydrogels composed of low-molecular-weight peptide self-assemblies and polysaccharides are gaining great interest as new types of biomaterials. Interactions between polysaccharides and peptide self-assemblies are well reported, but a molecular picture of their impact on the resulting material is still missing. Using the phosphorylated tripeptide precursor Fmoc-FFpY (Fmoc, fluorenylmethyloxycarbonyl; F, phenylalanine; Y, tyrosine; p, phosphate group), we investigated how hyaluronic acid (HA) influences the enzyme-assisted self-assembly of Fmoc-FFY generated in situ in the presence of alkaline phosphatase (AP). In the absence of HA, Fmoc-FFY peptides are known to self-assemble in nanometer thick and micrometer long fibers. The presence of HA leads to the spontaneous formation of bundles of several micrometers thickness. Using fluorescence recovery after photobleaching (FRAP), we find that in the bundles both (i) HA colocalizes with the peptide self-assemblies and (ii) its presence in the bundles is highly dynamic. The attractive interaction between negatively charged peptide fibers and negatively charged HA chains is explained through molecular dynamic simulations that show the existence of hydrogen bonds. Whereas the Fmoc-FFY peptide self-assembly itself is not affected by the presence of HA, this polysaccharide organizes the peptide nanofibers in a nematic phase visible by small-angle X-ray scattering (SAXS). The mean distance d between the nanofibers decreases by increasing the HA concentration c, but remains always larger than the diameter of the peptide nanofibers, indicating that they do not interact directly with each other. At a high enough HA concentration, the nematic organization transforms into an ordered 2D hexagonal columnar phase with a nanofiber distance d of 117 Å. Depletion interaction generated by the polysaccharides can explain the experimental power law variation and is responsible for the bundle formation and organization. Such behavior is thus suggested for the first time on nano-objects using polymers partially adsorbing on self-assembled peptide nanofibers.

Soil–pile interaction in vertical vibration in inhomogeneous soils

AbstractThis paper develops a novel reference analytical solution for axially loaded piles in inhomogeneous soils, extending the pioneering elastodynamic model of Nogami and Novak (1976) to piles embedded in vertically inhomogeneous soils. Following the classical earlier model, the pile is modelled as a rod, using the strength‐of‐materials solution, and the soil layer as an approximate continuum, which rest on rigid rock. The approximation lies in reducing the number of dependent variables by eliminating certain stresses and displacements in the governing elastodynamic equations: the vertical normal and vertical shear stresses in the soil are controlled exclusively by the vertical component of the soil displacement. Soil inhomogeneity is introduced via a power law variation of shear modulus with depth, and perfect bonding is assumed at the soil–pile interface. The proposed generalized formulation treats two types of inhomogeneity by employing pertinent eigen expansions of the dependent variables over the vertical coordinate. The response is expressed in terms of generalized Fourier series and includes: (i) displacements and stresses along the pile and the pile–soil interface; and (ii) displacement and stress in the soil. Contrary to available models for homogeneous soils, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered.

Criticality in transient behavior of coupled oscillator system toward chimera and synchronization.

Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, while how the system is steering toward different final destinies upon spatially localized perturbation is still unknown. Through a systematic numerical analysis of the evolution of the spatiotemporal patterns of multi-chimera states, we uncover a critical behavior of the system in transient time toward either chimera or synchronization as the final stable state. We measure the critical values and the transient time of chimeras with different numbers of clusters. Then, based on an adequate verification, we fit and analyze the distribution of the transient time, which obeys power-law variation process with the increase in perturbation strengths. Moreover, the comparison between different clusters exhibits an interesting phenomenon, thus we find that the critical value of odd and even clusters will alternatively converge into a certain value from two sides, respectively, implying that this critical behavior can be modeled and enabling the articulation of a phenomenological model.

Finite Element Based Static and Dynamic Vibration Analysis of a Beam with Axial Variation of Material Properties

Abstract: The study examines the vibration analysis of a damped cantilever beam with nonuniform variation in material properties using finite element methods. With power modification of material parameters in the axial direction, the static and dynamic response of a prismatic rectangular beam with damping has been examined. For the finite element approach, two noded beam elements with two degrees of freedom at each node were evaluated. The power law variation of material properties was used throughout the investigation. This research investigated the effects of proportional damping on displacement, velocity, and acceleration responses. The proposed beam's mass, stiffness, and damping matrices were calculated using Hamilton's concept. In the temporal domain, the Newmark Method has been utilized

Rigorous plane-stress solution and buckling analysis of rectangular functionally graded plates subjected to non-uniform edge loads

In this article, analytical solutions for the buckling analysis of simply supported rectangular functionally graded (FG) plates subjected to various forms of non-uniform in-plane loads are presented. Both perfect and porous FG plates with power-law material property variation are considered. The analytical solution is obtained by adopting a two-step approach: First, a rigorous plane-stress solution is developed based on the superposition of Airy’s stress functions, and later, these stress solutions are used in the buckling load computation of the FG plate based on Galerkin’s technique. One key aspect of the proposed solution is that it can consider different non-uniform waveforms of the in-plane loads at two opposite sides of the plate. Results show that the nature of the edge load has a profound effect on the stress solution of the FG plates. Several tables and graphs are presented to obtain a qualitative and quantitative understanding of the buckling behavior of FG plates influenced by various plate design parameters which include plate aspect ratios, power-law exponent, and porosity index. Comparative studies are also presented to highlight some erroneous results published in the open literature. It can be emphasized that the analytical results reported here can be used as a reference to judge the accuracy of other numerical solutions such as the finite element method.