One-dimensional Finite Element Method Research Articles

The carbon release triggered by 1.32 Ga sill emplacement and its potential environmental implications

Magmatic activity is one of the important pathways for the delivery of deep Earth carbon to the surface. The massive carbon release in this process can have significant impacts on atmospheric-oceanic environment. Previous studies have done a lot of work on the relationship between Phanerozoic magmatic activity and carbon release, but there is relatively limited attention has been given to investigating the association between Precambrian magmatic activity and carbon release. The Yanliao Large Igneous Province at 1.32 Ga exhibits extensive development of sills, and the reaction between sills and surrounding rocks triggers the release of carbon. Simultaneously, the magmatic activity during this period is considered as the final response to the breakup of the Columbia supercontinent, coinciding with the occurrence of the Mesoproterozoic Oxygenation Event To explore the connection between this magmatic activity and global carbon cycling, environmental changes and planetary evolution, nine representative stratigraphic columns are selected from the Yanliao area. We use the SILLi 1.0 1D model by utilizing the one-dimensional finite element method (FEM) to simulate and estimate the amount of carbon release triggered by sill emplacement. The simulation results indicate that the emplacement of sills increased the surrounding rock temperature and vitrinite reflectance, leading to a decrease in the total organic carbon (TOC) content. A large amount of organic carbon and inorganic carbon was released, which was initiated by the reaction between sills and surrounding rocks, with a total carbon release up to 1.24 × 1013 tons. The estimated CO2 equivalent released during this magma activity episode is expected to be greater than 4.58 × 1013. In Mesoproterozoic strata, the emplacement of sills activates carbon within the lithosphere could have implications for the global environment. Further work needs to be done in other ancient cratons that possess Lower Riphean strata to find additional evidence of the impact of this magmatic event on the Earth system. From this study, it is evident that magmatic activity during the Precambrian period could promote the activation of carbon in crustal sediments and influence global environment, which can a reference for people to understand the planetary evolution process.

Vibration energy accumulation and absorption characteristics of pseudo acoustic black hole wedge

The Acoustic Black Hole (ABH) is an effective structure for harvesting and dissipating energy from flexural waves. It works by reducing the thickness of the ABH wedge in a power law relationship with m ≥ 2, which rapidly decreases the local phase velocity of flexural waves and leads to an energy convergence effect. However, the vibration performance may differ if the tapered wedge does not meet the requirements of a typical ABH. This article introduces and discusses two types of “pseudo-ABH” wedges: linear-degenerate ABH and step-degenerate ABH. These wedges do not follow the power-law and continuity conditions of typical ABH, respectively. To establish the dynamic models of pseudo-ABH wedges, the lumped-mass transfer matrix method (LTMM) and traveling wave analysis are introduced and validated using analytic solutions of the non-uniform one-dimensional Euler-Bernoulli beam and finite element method. Meanwhile, a good match between the results of the Timoshenko and Euler-Bernoulli beam theory also indicates the correctness of our model. The energy ratio and reflection coefficient are defined to characterize the energy accumulation effect and vibration absorption performance of the pseudo-ABH wedges. The vibration performance of the linear-degenerate ABH is closer to the typical ABH than the step-degenerate ABH. The step-degenerate ABH has more effective energy accumulation and reduction performance for individual frequencies. However, as the number of segments in the step-degenerate ABH increases, its dynamic performance becomes closer to that of the typical ABH. The novelty of this article is to study vibration energy accumulation and absorption characteristics of pseudo acoustic black hole wedges using LTMM model, disclose the differences of linear and step degenerate ABH for their performance, and recommend the step degenerate ABH in place of the typical ABH of a power law thickness relationship.

Exact artificial boundary condition and 1D time-domain method for seismic response analysis of layered saturated poroelastic half-space

This study proposed a one-dimensional (1D) time-domain finite element method (FEM) for seismic response analysis of a layered saturated soil half-space under obliquely incident plane waves, based on Biot's u-w wave equations. The spatially two-dimensional problem was transformed into an equivalent 1D time-domain problem along the depth according to Snell's law. An exact artificial boundary condition for saturated poroelastic media was developed to simulate the wave absorption and input effects of the truncated half-space. Using the displacement and stress continuity conditions between the adjacent saturated poroelastic-medium layers and combining all finite elements and the exact artificial boundary condition, the total dynamic equation for the spatially 1D problem was obtained. The 1D dynamic equation was solved using the FEM with explicit time integration. The proposed method was verified by analyzing two engineering sites in the time domain and a comparison with the frequency-domain transfer matrix method with fast Fourier transform.

Modeling the Combined Effects of Cyclable Lithium Loss and Electrolyte Depletion on the Capacity and Power Fades of a Lithium-Ion Battery

In this study, we present a modeling approach to estimate the combined effects of cyclable lithium loss and electrolyte depletion on the capacity and discharge power fades of lithium-ion batteries (LIBs). The LIB cell based on LiNi0.6Co0.2Mn0.2O2 (NCM622) was used to model the discharge behavior in the multiple degradation modes. The discharge voltages for nine different levels of cyclable lithium loss and electrolyte depletion were measured experimentally. When there was no cyclable lithium loss, the 50% of electrolyte depletion brought about 5% reduction in discharge capacity at 0.05 C discharge rate, while it resulted in 46% reduction when it was coupled with 30% of cyclable lithium loss. The 50% of electrolyte depletion with no cyclable lithium loss caused 1% reduction in discharge power during 0.5 C discharge at the state of charge (SOC) level of 0.8, while it resulted in 13% reduction when it was coupled with 30% of cyclable lithium loss. The modeling results obtained by using the one-dimensional finite element method were compared with the experimental data. The justification of the modeling methods is demonstrated by the high degree of concordance between the predicted and experimental values. Using the validated modeling methodology, the discharge capacity and usable discharge power can be estimated effectively under various combined degradation modes of cyclable lithium loss and electrolyte depletion in the LIB cell.

Estimation of the maximum permissible intrusive force for intrusion of a canine tooth: one-dimensional finite element study

The aim of the present research study is to estimate the maximum permissible intrusive force mathematically with one-dimensional finite element method and to achieve the safe intrusion of a canine tooth without any damage to periodontal ligament (PDL). For this purpose, one-dimensional finite element (FE) model of a canine tooth was developed consisting of tooth (crown and root) and PDL structure. The maximum permissible intrusive force was computed mathematically. For this purpose, the maximum permissible stress was considered as ‘EPDL/1.5’, where ‘EPDL’ is the Young’s Modulus of PDL and factor of safety (FOS) is 1.5. The maximum permissible intrusive force was computed to be approximately 10 g-force (gf). For verification of this computed force value, finite element analysis was carried out on 3-dimensional canine tooth, PDL and bone structures for better verification and it is proved that 10 g-force (gf) value is safe without any damage to PDL tissues.

Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT12) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

Cost-Performance Trade-off in thermoelectric air conditioning system with graded and constant material properties

Thermoelectric (TE) air cooling is a solid-state technology that has the potential to replace conventional vapor compression-based air conditioning. In this paper, we present a detailed system-level modeling for thermoelectric air conditioning system with position-dependent (graded) and constant material properties. Strategies for design optimization of the system are provided in terms of cost-performance trade-off. Realistic convection heat transfer at both sides of the system are taken into account in our modeling. Effects of convection heat transfer coefficients, air flowrate, and thermoelectric material properties are investigated with varying key parameters such as TE leg thickness, module fill factor, and input current. Both constant material properties and graded properties are considered for the TE materials, and they are compared in terms of the degree of cooling, coefficient of performance (COP), and power consumption. For graded materials, we employ one-dimensional finite element methods to solve the coupled electrical and thermal current equations with arbitrary profile of material properties varying with position along the TE legs. We find that graded materials can enhance the degree of cooling, but only at the expense of COP, compared to the case of constant property materials. With constant material properties of ZT = 1 and relatively low electric current, the power consumption of TE coolers can be lower than those of conventional air conditioners at an equivalent cooling capacity. Considering additional advantages such as demand-flexible operation, low noise, and high scalability, thermoelectric cooling could be a competitive technology for future air conditioning applications..

A simple finite element procedure for free vibration of rectangular thin and thick plates

A simple finite element procedure is introduced and developed to solve the free vibration problem of rectangular plates. The proposed method first reduces the original plate problem to two simple beam problems. Each beam problem is then discretized separately with the help of one-dimensional finite element method (FEM). Since only the one-dimensional FEMs are dealt with in the proposed approach, the implementation of the proposed approach is much easier than the case where the two-dimensional FEM is applied to the problem. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the vibration problems of thin and thick rectangular plates. The simulations have been done for plates with different boundary conditions including the simply supported, clamped, and free boundary conditions. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

A shape sensing methodology for beams with generic cross-sections: Application to airfoil beams

This work presents a shape sensing method capable of handling some geometrical complexities commonly observed in aerospace structures. The method presented is based on the one-dimensional inverse Finite Element Method (1D-iFEM), which is capable of accurately reconstructing structural displacements of beam structures using surface strain measurements. The effects of cross-sectional variation in shear strains due to transverse or torsional loads for any general beam profile is accounted for in this 1D-iFEM formulation. The introduction of these effects allows the use of iFEM for the shape sensing of solid or thin-walled prismatic beams with any general beam profile. The performance of the new method is demonstrated through some example problems of prismatic beams under various static loading scenarios.

Modification of a Projection Method for Analysis of Radiation of a Radial Dipole in the Presence of an Inhomogeneous Body of Revolution

The problem of radiation of a radial electric dipole located on the axis of rotation of an axisymmetric body consisting of a homogeneous dielectric sphere and an external inhomogeneous dielectric layer is considered. The new numerical algorithm developed to solve the problem is based on a projection method that includes the projection of fields to transverse spherical harmonics in combination with a one-dimensional finite element method in a projection form along the radial coordinate. The algorithm is also generalized to the case in which the inner sphere is perfectly conducting. Numerical results, which characterize both the efficiency of the method and the effect of various parameters of the body on the directional pattern of the dipole, are presented.

Modifications of the hybrid projection method for analysis of electromagnetic scattering by inhomogeneous bodies of revolution.

A numerical analysis of wave scattering by a body of revolution composed of a homogeneous dielectric sphere and an external inhomogeneous dielectric layer with arbitrary generatrix is carried out. A modification of the hybrid projection method developed in the paper includes expansion of the fields in terms of transverse vector spherical functions, projection of the Maxwell's equations in the inhomogeneous region on the indicated functions, and use of the one-dimensional method of finite elements in projection form over the radial coordinate. Two more modifications correspond to the case of a perfectly conducting internal sphere and for the case of the absence of the internal sphere. The indicated modifications are realized in MATLAB codes for the case of excitation of the body by a circularly polarized plane wave propagating along the axis of revolution. Numerical results presented and discussed in the paper include the cases of wave scattering by a homogeneous sphere displaced from the origin of coordinates, a perfectly conducting sphere placed inside an inhomogeneous spheroidal layer, and a half-spherical Maxwell's fish-eye lens.

Forced vibrations and wave propagation in multilayered solid spheres using a one-dimensional semi-analytical finite element method

A numerical model is proposed to compute the eigenmodes and the forced response of multilayered elastic spheres. The main idea is to describe analytically the problem along the angular coordinates with spherical harmonics and to discretize the radial direction with one-dimensional finite elements. The proper test function must be carefully chosen so that both vector and tensor spherical harmonics orthogonality relationships can be used. The proposed approach yields a general one-dimensional formulation with a fully analytical description of the angular behaviour, suitable for any interpolating technique. A linear eigenvalue problem, simple and fast to solve, is then obtained. The eigensolutions are the spheroidal and torsional modes. They are favourably compared with literature results for a homogeneous sphere. The eigensolutions are superposed to compute explicitly the forced response. The latter is used to reconstruct the propagation of surfaces waves. In particular, the collimation of a Rayleigh wave (non-diffracted surface wave propagating with a quasi-constant width) excited by a line source in a homogeneous sphere is recovered with the model. Based on the vibration eigenmodes, a modal analysis shows that such a wave is a superposition of fundamental spheroidal modes with a displacement confined at the equator of the sphere. These modes are the so-called Rayleigh modes, of sectoral type and high polar wavenumbers. When a thin viscoelastic coating is added to the sphere, the Rayleigh mode behaviour is recovered in a limited frequency range, allowing the generation of a collimating wave at the interface between the sphere and the coating.

Characteristics of Electromagnetic Wave Absorption in Cylindrical Black Holes with Positive and Negative Refractive Indexes

The problem of plane electromagnetic wave scattering by cylindrical “black holes” with radial profile of dielectric permittivity and magnetic permability in the shell chosen in the form e(ρ) = μ(ρ) ~ 1/ρ2 and with constant permittivity and permeability in the central region is considered. The problem in rigorous formulation is solved both analytically for a basic model of the absorber and numerically using the one-dimensional method of finite elements for a modified model. The results obtained for absorption efficiency, radar cross-section, and field distribution in the cylinder are presented and discussed. It is shown that the absorption efficiency of the cylinder with negative refractive index at realistic values of parameters of the cylinder remains higher than the absorption efficiency of a similar cylinder with positive refractive index. However the indicated advantage is not so great as the advantage compared to the absorption efficiency of the black body predicted in the previous publications for the case of extremely low losses at the surface of the cylinder and extremely high magnitudes of negative permittivity and permeability and losses in the central region of radius approaching zero.

Shear locking in one-dimensional finite element methods

The accuracy of the results obtained with Finite Element Methods depends on the basis functions employed to approximate the displacement fields. If the beam is very thin, there can be shear locking and the numerical approximation leads to erroneous solutions. In this work, shear locking is analyzed by calculating expressly the curves of the transverse deflection, the cross-section rotation, and the shear strain in different approaches. The influence of the ratio between the shear stiffness and the bending stiffness is explicitly shown when force and moment loads are applied. It is concluded that the locking behavior at nodes is not the decisive factor to assess the quality of the solution. The important point is to analyze the entire curves when the ratio between the stiffnesses is varied. It is verified that the mixed interpolation and the discrete shear gap approaches are superior to the pure displacement and to the field consistency approaches.

Dynamic soil structure interaction model for beams on viscoelastic foundations subjected to oscillatory and moving loads

A new method for dynamic analysis of Euler-Bernoulli beams resting on multi-layered viscoelastic soil is presented. The governing differential equations for beam and soil displacements are obtained using the extended Hamilton’s principle, and these equations are solved using one-dimensional finite element method and analytical solutions. An iterative solution algorithm is used in conjunction with an implicit time integration scheme to obtain beam and soil responses as functions of time. The developed model resembles the traditional two-parameter spring foundation model but the analysis shows that the traditional foundation spring parameters cannot be prescribed a priori but determined as part of the solution. The model produces accurate beam response as illustrated by comparisons with equivalent two-dimensional finite element analysis. A few problems are analyzed that illustrate the novel features of the dynamic foundation model.

A Comparison of a One-Dimensional Finite Element Method and the Transfer Matrix Method for the Computation of Wind Music Instrument Impedance

This work presents a computation tool for the calculation of wind instrument input impedance in the context of linear planar wave propagation with visco-thermal losses. The originality of the approach lies in the usage of a specific and simple 1D finite element method (FEM). The popular Transfer Matrix Method (TMM) is also recalled and a seamless formulation is proposed which unifies the cases cylinders vs. cones. Visco-thermal losses, which are natural dissipation in the system, are not exactly taken into account by this method when arbitrary shapes are considered. The introduction of an equivalent radius leads to an approximation that we quantify using the FEM method. The equation actually solved by the TMM in this case is exhibited. The accuracy of the two methods (FEM and TMM) and the associated computation times are assessed and compared. Although the TMM is more efficient in lossless cases and for lossy cylinders, the FEM is shown to be more efficient when targeting a specific precision in the realistic case of a lossy trumpet. Some additional features also exhibit the robustness and flexibility of the FEM over the TMM. All the results of this article are computed using the open-source python toolbox OpenWind.

Performant and Simple Numerical Modeling of District Heating Pipes with Heat Accumulation

This paper compares approaches for accurate numerical modeling of transients in the pipe element of district heating systems. The distribution grid itself affects the heat flow dynamics of a district heating network, which subsequently governs the heat delays and entire efficiency of the distribution. For an efficient control of the network, a control system must be able to predict how “temperature waves” move through the network. This prediction must be sufficiently accurate for real-time computations of operational parameters. Future control systems may also benefit from the accumulation capabilities of pipes. In this article, the key physical phenomena affecting the transients in pipes were identified, and an efficient numerical model of aboveground district heating pipe with heat accumulation was developed. The model used analytical methods for the evaluation of source terms. Physics of heat transfer in the pipe shells was captured by one-dimensional finite element method that is based on the steady-state solution. Simple advection scheme was used for discretization of the fluid region. Method of lines and time integration was used for marching. The complexity of simulated physical phenomena was highly flexible and allowed to trade accuracy for computational time. In comparison with the very finely discretized model, highly comparable transients were obtained even for the thick accumulation wall.

Land subsidence prediction for a new urban mass rapid transit line in Hanoi

The rapidly increasing population, urbanization, and industrialization of Hanoi city in the last few decades have necessitated the urgent development of a new underground public transport infrastructure such as the urban mass rapid transit (UMRT) system that is now under construction. Simultaneously, concerns on the possible effects of land subsidence on the to-be-constructed and operated UMRT lines must be addressed. One of the difficulties while assessing land subsidence in Hanoi is the insufficient subsidence monitoring data. Therefore, a good geotechnical characterization of the subsoil profile and proper numerical analysis can help predict groundwater pumping-induced subsidence adequately. The UMRT line No. 2 that was selected for this study is special as it runs across the old quarter downtown, located at the very heart of Hanoi city. In this paper, the Hanoi aquifer system, groundwater extraction, and subsidence monitoring network of Hanoi are reviewed comprehensively. Soil data from 30 investigation boreholes were analyzed to characterize the subsoil profile underlying a segment of the UMRT line No. 2 that consists of both elevated and underground sections. Seven soil layers were identified up to a depth of 50 m, including a compressible soft clay layer 10–15 m thick that is supposed to be most vulnerable to subsidence by groundwater pumping. Based on the long-term groundwater monitoring data, one-dimensional finite element method (FEM) consolidation analyses were carried out for three drawdown boundary conditions of 5, 10, and 15 m and yielded a total subsidence of 0.15, 0.35, and 0.5 m, respectively, after 25 years with 1994 as the reference year. The results of this study confirmed that land subsidence in particular should be considered in the construction and maintenance of the UMRT lines in Hanoi, in particular for underground sections that are founded on soft to medium clay layers.

Finite Element Analysis and Experimental Validation of Transfer Function of Rotating Shaft System With Both an Open Crack and Anisotropic Support Stiffness

In this paper, transfer function of rotating shaft system for detecting transverse open crack is developed. Rotating shaft system is modeled using one-dimensional finite element method (1D-FEM), and quantitative analysis is performed. Open crack is modeled as weak asymmetry rotating with shaft's rotation. It is known that, when both open crack and support stiffness anisotropy coexist, various frequency components of shaft's vibration are generated through their successive interaction. This paper evaluates the order of these components, and concludes that first five main components are enough to investigate interaction of open crack and support stiffness anisotropy. Then, five sets of transfer functions for these components are derived. The validity of this set of transfer functions is confirmed by numerical simulation. Moreover, excitation experiment utilizing active magnetic bearing (AMB) is performed, and the validity of derived transfer function was verified experimentally.

Radiant heat recovery by thermoelectric generators: A theoretical case-study on hot steel casting

We present a detailed numerical analysis to quantify the power generation performance of a thermoelectric module in radiant heat recovery application. Due to the large temperature difference typically involved in such a system, temperature-dependent material properties of thermoelectric elements are taken into account for accurate performance prediction by employing an iterative algorithm based on the one-dimensional finite element method. Careful analysis on the radiation heat transfer with optical parameters such as surface emissivity and view factor is performed to precisely quantify the heat input to the thermoelectric system. Parasitic heat losses such as air convection loss at the hot surface and conduction through the substrates and gap fillers are also taken into account to analyze their impacts on the power output. A case study on the radiant waste heat recovery from hot steel casting slabs in steel industry is discussed in detail to theoretically estimate the power output performances and optimize the module design. We find that a power density as high as ∼1.5 kW/m2 and a system efficiency as high as ∼4.6% can be achieved at a 2 m distance from the 1200 K hot steel slab using the state-of-the-art Bi2Te3 alloys with a relatively small leg thickness of 3 mm and a 20% fill factor. This optimal design with small form factors ensures a reduced material cost while keeping the power output near the maximum, so that an estimated power cost remains as low as ∼0.2 $/Watt.