Static Tension Research Articles

Uniaxial constitutive model for fiber reinforced concrete: A physics-based data-driven framework

Fiber reinforced concrete (FRC) has improved strength and ductility, making it suitable for a wider range of engineering structures. The development of a constitutive model is crucial for analyzing mechanical behavior of these structures, while this still remains challenging as the complex composition of FRC makes it difficult to formulate explicit relationships among a range of critical material parameters. With the latest development of data and digital technologies, data-driven approaches have emerged as a powerful alternative that are capable of solving advanced and complex engineering problems. Developing data-driven methods based on objective data analysis and decision-making to predict mechanical behavior of engineering structures is an important direction of such development. In this paper, a uniaxial constitutive model of fiber-reinforced concrete is developed by a physics-based data-driven framework. The framework consists of three important parts, including construction of experimental database, parameters calibration of physical models, and implementation of neural network. From the experimental data collected from published literature, an experimental database is built first. A physical model is proposed by modifying the uniaxial constitute model for normal concrete, so that it is more convenient and straightforward to consider the effect of fibers on fiber reinforced concrete subjected to compression, tension and repeat loading conditions. The parameters of the model are calibrated against experimental data by the swarm intelligence optimization algorithms. With the calibrated parameters of the physical model, a Fully Connected Neural Network (FCNN) is trained to be used to predict the physical parameters of fiber reinforced concrete. By comparing with independent experimental data, the proposed fiber-reinforced concrete uniaxial constitutive model constructed using the physics-based data-driven framework can accurately predict the stress–strain relationships of a range of FRC, which suggests that the FRC material model can be used in the numerical simulation and design of FRC components and structures. In addition, the proposed model is applicable to multiple loading conditions.

The strength and elastic property of PLA + graphite composites: experimental and theoretical analyses

Background. The combination of additive technologies with reinforced materials opens up new vistas for creating lightweight and durable products having unique characteristics. Implementing these technologies into the production requires effective evaluation methods of the ultimate limit state of such products. Objective. The article deals with the mechanical properties of samples, manufactured by the method of surfacing FDM with two polylactide-based thermoplastic threads: PLA-Cg+ with 5% layered graphite filling and PLA-CCF with 10% carbon fiber filling. Methods. The impact of 3D printing process parameters, such as print orientation and layer thickness, on specimens' mechanical characteristics under conditions of tension and compression, has been experimentally researched. Results. It is shown that both print orientation and layer thickness substantially influence specimens' mechanical properties of both materials. A comparative analysis of experimental data with calculations by failure criteria has been carried out: Tsai-Hill, Tsai-Wu, Hoffman, Mises, and maximum stresses. Conclusions. The results of the tests proved that there is a significant influence of the studied parameters of the printing process on the mechanical characteristics of PLA + graphite specimens under both tension and compression. For the most part, samples with a smaller thickness have both a higher ultimate strength (limit of proportionality) and a greater relative elongation. It is defined that the best concurrence between experimental and calculated data for both materials can be achieved through using the generalized von Mises criterion.

Effect of accelerated cooling after cross-helical rolling on formation of structure and low-temperature fracture toughness of low-carbon steel

The effect of accelerated cooling after cross-helical rolling of X70 low-carbon steel on the formation of structures and mechanical properties under static tension and impact bending was investigated. The use of interrupted accelerated cooling of steel after cross-helical rolling with exposure at 530 °C (mode I) and continuous accelerated cooling (mode II) leads to the formation of different types and ratios of structures in steel. After rolling according to mode I, the structure is characterized by the presence of ferrite, troostite, granular bainite, and fine Fe3C carbides. After rolling according to mode II, the structure is characterized by the formation of lath bainite and large sections of the martensitic-austenitic (MA) component up to 1 – 2 µm in size. It is shown that a decrease in the fineness of ferrite grains in steel after cross-helical rolling in modes I and II from 12 to 4.6 – 4.3 μm, the formation of a bainitic phase, and hardening of the matrix with carbides led to an increase in the yield strength of steel up to 440 and 490 MPa and tensile strength up to 760 and 880 MPa. Carrying out helical rolling according to mode I makes it possible to significantly increase the low-temperature fracture toughness of steel (KCV–70 °С = 160 J/cm2) compared to the hot-rolled state (KCV–70 °С = 11 J/cm2) and reduce the cold brittleness of steel to the temperatures below –50 °C. The use of continuous accelerated cooling (mode II) does not allow increasing the cold resistance of steel due to the formation of the lath bainite structure and large areas of the MA component.

Appropriateness of hydrostatic pressure-based modeling for strength differential effect in advanced high strength steel

Although numerous models have been proposed to describe the strength differential (SD) effect, their appropriateness and efficiency in the case of advanced high strength steel (AHSSs) have not been investigated. To this end, this study investigates the SD effect in AHSS using a hydrostatic-pressure-dependent plasticity theory. Although pressure model can effectively predict the SD effect in AHSSs with only one coefficient, the original formulation is experimentally inconsistent in the first quadrant of a yield surface. Hence, this study developed a new formulation to remove the inconsistency in a simple manner and validated the pressure model using new experimental data reported for two recently developed AHSSs: DP980 and MART1500Y. The SD effect is characterized based on monotonic compression and tension tests in three different directions with respect to the rolling direction to obtain the stress–strain curves and r-values. Next, the pressure model is used to predict the plastic behavior of both steel sheets. The parallel transition of flow stress and similarity in r-values between the tension and compression states supports the appropriateness of the proposed pressure-based modeling. Furthermore, these insights substantiate the fact that deformation mechanisms cannot explain the SD effect in both steel sheets. Therefore, complex yield functions that describe the SD effect in general materials are not necessary in the case of AHSS. Additionally, the pressure model is incorporated into a distortional plasticity model for applying it to springback in the bending process. The distortional plasticity model with and without the pressure model is calibrated using tension–compression curves, thereby revealing that the SD significantly affects the calibration. Finally, a U-draw bending experiment and the corresponding finite element simulation are conducted to compare the springback; the results show that considering the SD effect improves the springback simulation accuracy.

Effect of adjuvants on physicochemical properties of lime sulfur on flower/paraffin and application on flower thinning.

Adjuvants can effectively enhance the utilization rate of pesticides, but the application of adjuvants in plant growth regulators is rarely studied. This work explored the effects of adjuvants dioctyl sulfosuccinate sodium salt (AOT) and methyl oleate (MO) on lime sulfur (LS), especially the drop behavior on flower and paraffin surface. The results showed that the addition of AOT and AOT+MO can significantly reduce the static and dynamic surface tension of LS from 72mN/m to 28mN/m and 32mN/m respectively, and increase the spreading factor from 0.18 to 1.83 and 3.10 respectively, reduce the bounce factor from 2.72 to 0.37 and 0.27 respectively. The fluorescence tracer test showed that the addition of adjuvants could promote the spreading and permeation of droplets. The field test results revealed that the flower thinning rate of adjuvant and non-adjuvant were 80.55% and 54.4% respectively, and the flower thinning effect of adding adjuvant was the same as that of artificial which the flower thinning rate was 84.77%. The quality of apples treated with adjuvants was similar to that treated with artificial, and the weight of single fruit increased by 24.08% compared with CK (spray water). The application of tank-mixture adjuvant could reduce the dosage of LS for thinning agent application, improve apple's quality, and decrease labor cost and improve the economic benefits of fruit planting and the environmental benefits of plant growth regulators.

A lode-dependent plasticity model for high-strength structural steel

In this study, a new plasticity model was developed to characterize the mechanical behavior of high-strength structural steel under multiaxial stress states. This model utilizes a Lode-dependent yield criterion in conjunction with a non-associated flow rule and isotropic hardening law to determine the evolution of the plastic flow stress. Further, a detailed numerical implementation algorithm for this plasticity model is presented, including the integration algorithm of the constitutive equations using an implicit elastic predictor-return mapping method and formulation of the consistent tangent modulus. To validate the proposed plasticity model, six types of Q690 high-strength steel specimens were tested under monotonic loading. These specimens can be used to obtain the elastoplastic response of the material under uniaxial tension, compression, pure shear, tensile shear, and plane strain state, respectively. Corresponding numerical simulations were performed for each specimen using ABAQUS/Standard with the proposed model being introduced into the analysis via the UMAT user subroutine. Then, the prediction performance of the proposed plasticity model was investigated by comparing the load-displacement curves of each test specimen obtained from both experiments and numerical simulations. It is shown that the proposed plasticity model can accurately predict the mechanical response of the high-strength steel under different stress states and enables to visualize the key stress state parameters and material strength variation at the failure region of each specimen, which confirms the promising prospect of this new model in practical applications.

Mechanical behaviour of fiber-reinforced grout in rock bolt reinforcement

Grouted rock bolts subject to axial loading in the field exhibit various failure modes, among which the most predominant one is the bolt-grout interface failure. Thus, mechanical characterization of the grout is essential for understanding its performance in ground support. To date, few studies have been conducted to characterize the mechanical behaviour of fiber-reinforced grout (FRG) in rock bolt reinforcement. Here we experimentally studied the mechanical behaviour of FRG under uniaxial compression, indirect tension, and direct shear loading conditions. We also conducted a series of pullout tests of rebar bolt encapsulated with different grouts including conventional cementitious grout and FRG. FRG was developed using 15% silica fume (SF) replacement of cement (by weight) and steel fiber to achieve high-strength and crack-resistance to overcome drawbacks of the conventional grout. Two types of steel fibers including straight and wavy steel fibers were further added to enhance the grout quality. The effect of fiber shape and fiber volume proportion on the grout mechanical properties were examined. Our experimental results showed that the addition of SF and steel fiber by 1.5% fiber volume proportion could lead to the highest compressive, tensile, and shear strengths of the grout. The minimum volume of fiber that could improve the mechanical properties of grout was found at 0.5%. The scanning electron microscopy (SEM) analysis demonstrated that steel fibers act as an excellent bridge to prevent the cracks from propagating at the interfacial region and hence to aid in maintaining the integrity of the cementitious grout. Our laboratory pullout tests further confirmed that FRG could prevent the cylindrical grout annulus from radial crack and hence improve the rebar's load carrying capacity. Therefore, FRG has a potential to be utilized in civil and mining applications where high-strength and crack-resistance support is required.

Проектирование конфликтологической готовности курсантов в процессе профессиональной подготовки в военном институте

The article reveals the content of conflict readiness as a form of professional potential of a cadet in a military institute. It is proved that conflict readiness characterizes the cadet’s ability to solve professional tasks in conditions of social tension and conflict. The components of the cadet’s conflict readiness are revealed: 1) knowledge about conflict, skills of working in conflict conditions (cognitive subsystem of readiness); 2) skills of assessment, prevention, and resolution of service and labor conflicts (practical subsystem of readiness); 3) goals, needs, motivation to overcome the conflict (personal subsystem of readiness). Based on the analysis of the research data of the conducted interview (16 military experts, heads of military structural units of the National Guard of the Russian Federation were interviewed), the content of the design of the conflict readiness of cadets was clarified as a process of preliminary development of the basic sections of the project aimed at the formation of conflict potential, clarifying the main features of the joint interaction of teachers, managers, as well as cadets in order to form conflict readiness. The basic sections of this management design are disclosed: 1) the formation of a state project for the growth of the material provision of cadets, maintaining the practices of their material and spiritual expanded consumption, ensuring compliance with the level of physical and intellectual service and labor costs, as well as the monetary allowance of the cadet; 2) formation of cumulative understanding, knowledge, skills of conflict diagnostics, realization of military professional activity in conditions of uncertainty of external social environment, effective actions and decisions in conditions of various types and kinds of conflicts; 3) prevention of emerging conflict practices in the process of professional training of cadets at the military institute (analysis of information flows about conflict risks in the system of service labor actions of students; designing a conflict free culture of communication; maintenance and stimulation of conflict free practices).

Using a vertical three-mass computational model of the vocal folds to match human phonation of three adult males.

Computer models of phonation are used to study various parameters that are difficult to control, measure, and observe in human subjects. Imitating human phonation by varying the prephonatory conditions of computer models offers insight into the variations that occur across human phonatory production. In the present study, a vertical three-mass computer model of phonation [Perrine, Scherer, Fulcher, and Zhai (2020). J. Acoust. Soc. Am. 147, 1727-1737], driven by empirical pressures from a physical model of the vocal folds (model M5), with a vocal tract following the design of Ishizaka and Flanagan [(1972). Bell Sys. Tech. J. 51, 1233-1268] was used to match prolonged vowels produced by three male subjects using various pitch and loudness levels. The prephonatory conditions of tissue mass and tension, subglottal pressure, glottal diameter and angle, posterior glottal gap, false vocal fold gap, and vocal tract cross-sectional areas were varied in the model to match the model output with the fundamental frequency, alternating current airflow, direct current airflow, skewing quotient, open quotient, maximum flow negative derivative, and the first three formant frequencies from the human production. Parameters were matched between the model and human subjects with an average overall percent mismatch of 4.40% (standard deviation = 6.75%), suggesting a reasonable ability of the simple low dimensional model to mimic these variables.

Analysis of the Thermal Effects on the Performance and Reliability of Post‐Tensioned Integral Abutment Bridges

AbstractThere are cases where the common approaches used to assess the temperature loading effects may not be sufficient, especially in the case of integral abutment bridges (IABs). In IABs, the incorrect choice of the design temperature may result in an unwanted tensional state in the bridge decks and additional loads on the abutments. For post‐tensioned bridges, the combined effects of excessive temperature loads and loss of tension in the post‐tensioned cables due to time‐dependent phenomena like relaxation in steel and shrink‐age and creep in concrete may lead to probabilities of exceedance of serviceability (or ultimate) limit states that exceed the code specifications and/or desired levels of safety, and to cascading negative effects on the operation of the road infrastructure. This research aims at assessing the sensitivity of IABs to the long‐term effects of temperature loads and, in a second step, quantifying the probability of exceedance of relevant serviceability limit states.

A new analytical model for force prediction in incremental sheet forming

A new analytical model for forming force prediction in the whole process of incremental sheet forming (ISF) is presented in this work. The modelling of contact region, thickness distribution and strain components are carried out as the basis for the prediction of vertical and horizontal forces. The effect of local contact behaviour and material deformation, as well as the behaviours of global bending, changes of deformation mode and contact condition are taken into account in the modelling of ISF force history. The force prediction model at both plane strain and biaxial tension condition areas is developed based on the differences in contact region and thickness distribution between these two different deformation states. Validated by ISF tests for a truncated cone and pyramid part with varying drawing angles, the prediction of force history captures well the fluctuation of different force components due to the change of deformation conditions and provides an insight into the material deformation mechanisms and the direct correlation to forming force characteristics of the whole ISF process. In a practical case, the model is successfully applied to the force prediction in ISF-based cranial plate manufacture. The model is also validated by a series of ISF tests with accurate force predictions of both the peak and stabilised forces.

Structure and semantics of verbal means of expressing states of emotional tension in english publicistic texts

The aim of the research is to contribute to a deeper understanding of the role and functioning of verbal means of expressing emotional tension in journalistic texts. Methods. The research involves such methods as comparative analysis, transformational analysis, contextual analysis, pragma-semantic analysis. Statistical processing and validation of the obtained data was carried out using Cronbach’s alpha and mean square. Results. The results of the study helped to identify the linguistic features of expressing states of emotional tension. A total of 272 interjections that reflect emotional stress were found in the studied corpus. In general, 589 contexts of the use of modal words, which have the colouring of emotional tension, were also analysed. Conclusions. The study found that emotional stress in journalistic texts is strongly dependent on the context in which it is used. Understanding emotional expressions requires taking into account the sociocultural context, the goals of communication, and the relationship between communicators. Prospects. Further research can expand the scope and diversity of the corpus of journalistic texts, which will enable obtaining more representative results. Adding different genres, time periods, and authors will give a more complete picture of the verbal means of expressing emotional tension.

Optimization of mechanical properties of bio-inspired Voronoi structures by genetic algorithm

Numerous patterns in nature comprise aesthetics structures and intriguing geometries. The geometric factors behind these patterns include area, aspect ratio, circularity, orientation, and regularity. In this study, the quasi-3D porous structures with varying regularities were systematically described and generated by the Voronoi segmentation. Then, genetic algorithms were applied to optimize the mechanical properties of these structures under tension condition. The image-based finite element simulation could calculate each structure's fitness value (strength). Afterward, structures with higher fitness values were selected and underwent the replication process, including crossover and mutation. New structures were generated based on the input structures. The above steps driven by the genetic algorithms were repeated 30 iterations until the tensile strength of the structures was improved and converged. Both simulation and tensile testing results as well as digital image correlation of specimens fabricated by stereolithography 3D printing showed that the Voronoi structures optimized by the genetic algorithm could enhance their stiffness, strength, and toughness values by ∼30%. This research has the potential to be applied in the fields of structural materials and biomimetic micro-aerial vehicles, which require lightweight, strength, and toughness simultaneously.

Calibration and application of damage and rate-dependent constitutive model for SPCE steel

The SPCE steel demonstrates favorable deep-drawing and forming properties, leading to its widespread use in industrial production. However, when subjected to complex loads, it becomes crucial to develop a constitutive model that accurately characterizes its mechanical properties. Such a model serves as the foundation for engineering design and safety assessment. Both quasi-static tension tests and high strain rate tension tests were conducted on SPCE steel samples using static and dynamic tension testing machines at room temperature. The resulting test data was used to introduce three dynamic increase factor indicators (DIFs) that quantify the strain rate effect on SPCE steels. Additionally, the fracture morphology of typical post-test specimens was analyzed using a scanning electron microscope (SEM) to demonstrate the strain rate effect on the mechanical properties of SPCE steels from a microscopic perspective, revealing ductile fracture as the primary fracture mechanism. A double-power plastic constitutive model was developed to describe the material's tensile mechanical behavior prior to necking. Simulation analysis was performed to validate the model, and the parameters of the Ductile damage model were identified based on the simulation data. The proposed constitutive model underwent numerical verification, demonstrating good consistency between the simulation and experimental results. Furthermore, the same constitutive model was applied to process the test data of HC650 steel using the same methodology. The conclusion drawn from this analysis aligned with that of SPCE steel, confirming the suitability of the proposed constitutive model for characterizing the mechanical behavior of steels without yielding a platform under different strain rates.

Stress, a Great Impact on Mental Health

Stress can be defined as a state of worry or mental tension caused by a difficult situation. Every human being has his own understanding of stress. Stress is a natural human response that prompts us to address challenges and threats in our lives. The types of Stress are Acute Stress and Chronic Stress. Acute Stress is the body’s quick reaction to a new and challenging situation whereas chronic stress is Long-term stress experienced for an extended period which can create a negative impact on a person’s health. Things that cause stress are “stressors,” and stress is the personal reaction to that stressors. A number of scientists classified stressors as Biological Stressors, Psychological Stressors, Acute Stressors, Chronic Stressors, Environmental Stressors, Internal Stressors and Psychosocial Stressors. Stress is influenced by internal and external factors of a person. Stress causes different types of impacts on the physical body. Stress has negative consequences on various aspects of quality of life. Chronic stress can lead to or exacerbate mood disorders such as depression and anxiety, bipolar disorder, cognitive (thinking) problems, personality changes, and problem behaviours. Stress management techniques help to reduce stress. Understanding the sources of stress provides opportunities for planning interventions that would reduce the stressors.

Constitutive laws of softened UHPC in biaxial tension–compression: Experimental study using a planar bi-directional element tester

Ultrahigh-performance concrete (UHPC) has been gradually applied in bridge and building engineering to replace traditional reinforced concrete (RC). More realistic simulations are required for the design of structures. However, a two-dimensional constitutive model for UHPC is lacking. Tsinghua University developed a planar bidirectional element tester (PBET) that can perform tests on various structural types of element specimens. In this study, one nonreinforced UHPC element specimen under uniaxial compression and ten nonreinforced UHPC element specimens under biaxial tension and compressive coupling condition were tested using PBET. The compressive stress–strain relationship was established based on the uniaxial compressive test. By analyzing the test results, the softening effect of the principal tensile strain on the peak compressive stress and strain of UHPC was evaluated. Stress and strain softening coefficients were proposed, and the constitutive law of softened UHPC under compressive stress was modified and verified using the test results, which contributes to the improvement and development for the two-dimensional constitutive law for UHPC.

Microstructural evolution and mechanical behaviors of the third–generation automobile QP980 steel under continuous tension and compression loads

As one of the representatives of the third–generation automobile steel, QP980 is a high–quality material for the improvement of automobile lightweight and crash safety. In this paper, tension and compression tests are conducted for the QP980 steel based on a self–designed testing platform. Microstructural evolution and mechanical behavior under monotonic and continuous tension/compression loads are investigated. The results show that the martensitic transformation in QP980 steel is more easily excited in the monotonic tension state than in the monotonic compression state. In the tension–compression–tension (T–C–T) and compression–tension–compression (C–T–C) modes, the Bauschinger effect of QP980 depends on the pre-strain amount and the pre–strain mode. The Bauschinger effect is stronger at a higher pre–strain. It is also stronger in T–C–T than in C–T–C because the transformation of retained austenite after T–C–T is greater than that after C–T–C. By analyzing the hardening and softening behaviors in the forward, reverse and reforward loading branches of T–C–T and C–T–C, the hardening ability during reverse and reforward loading branches is higher than that during forward loading. Permanent softening occurs between reverse loading and forward loading of T–C–T instead of C–T–C. The research result of this paper is favorable for understanding the relation between mechanical behavior and microstructural evolution of QP980 steel under the condition of complex loads, and promote its large–scale application.

THE BALTIC-BLACK SEA UNION AS AN AXIS OF UKRAINE’S SECURITY: REALITIES AND DEVELOPMENT PROSPECTS

The article analyzes the stages of formation of the Baltic-Black Sea community as a geopolitical, economic and security area, the historical and theoretical basis for the emergence of such a community and the changes that took place in the Baltic-Black Sea region after the beginning of the full-scale invasion of the Russian Federation into Ukraine in 2022. In addition, the issue of security cooperation is considered states of the region. The main priorities for the development of cooperation between states in the future have been identified. The main questions regarding the issues of cooperation between the states of the region and the structures of the EU and NATO in conditions of tension and open aggression of the Russian Federation after 2022 have been developed. Today, it is obvious that when specialists and politicians of the region discuss the project of the Baltic-Black Sea Alliance, it is primarily about security aspects and motives. In our opinion, only one security aspect is not enough for a project to be complete and viable. The economic component of project implementation is also extremely important. The West’s delay in accepting Ukraine into the EU and NATO requires Ukraine to find allies to protect its independence and sovereignty from Russian aggression. Such potential allies are primarily Poland, Lithuania, Latvia, Estonia, Romania, Moldova, and Georgia, which, like Ukraine, have more than once been victims of the imperial policy of tsarist, Soviet, and Putin’s Russia. Nowdays we can see that Western Europe is no longer ready to offer new guidelines, then the Baltic-Black Sea Corridor should become not only a territory for the generation of joint economic benefits, but also a territory for the generation of new meanings. The Baltic-Black Sea Union creates new challenges and provides an opportunity to respond to them. This is the true idea of Europeanness. Europe is a search for something new. Ukraine must take on the mission of giving new answers to what Europe can be today.

Fast-response and durable liquid metal wire for wearable electronics

This work presents a strain sensor which is based on high ductility liquid metal coated on traditional commercial lastex yarns with a unique spiral wound structure. The prepared sensor exhibits exceptional flexibility, elasticity, softness of 0.23 N at 100% tensile strain. It is lightweight, weighing only 1 g/m, and boasts a low cost of less than 1 dollar. The sensor has the ability to detect both static and dynamic in-plane tension and normal pressure. It showcases remarkable sensing characteristics of good sensitivity (0.82 for tension, 12.6 N−1 for pressure), extraordinary linearity within 10–100% strain range with goodness-of-fit value over 0.997, rapid response of less than 10 ms, high resolution spanning up to 0.3%, and high durability over 100,000 repetitive cycles. A smart glove based on the proposed strain sensors achieves gesture recognitions using machine learning technique, with an accuracy of 100%. The outstanding performance of this sensor highlights its potential as an affordable and advanced candidate for wearable electronics, which can have wide-ranging applications in sports, healthcare, entertainment, education, and industry.

Wrinkle Prediction of Membrane Structures under Uniform Normal Tensions

Wrinkles always produce to counteract the compressive stress in the thin membrane structures, which has become a popular topic. In this paper, for the inhomogeneous membrane structure with depth-dependent Young’s modulus, a power-law index is introduced. The effect of inhomogeneity on the displacement and stress fields is analyzed. On this foundation, the membrane structures with uniform normal tension loadings are studied to be accorded with practical cases of engineering application. The stress fields are derived by the superposition of the concentrated tension solution based on the Airy stress superposition theory. Compressive stress generally leads to wrinkling, then the wrinkle regions under different tension conditions are predicted accurately. The proposed stress model is simple, and the computation complexity is reduced without using a finite element solution.