Finite Element Method Solution Research Articles

Investigation into Vibration Excitation and Mode Selection of Thin Rectangular Plates with Multiple Bolts and Stand-Off Supports

The flexural vibrations of a thin rectangular plate with multiple rigid-point supports were examined theoretically, numerically, and experimentally. The analytical resonant frequencies and mode shapes obtained via the superposition method were compared with the finite element method (FEM) solutions. Experimental measurements were obtained using an amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI). The investigations considered two boundary conditions: (1) a completely free rectangular plate with one bolt and stand-off support at the center and (2) a rectangular plate with five bolts and stand-off supports along the short edge. In both cases, the plate was actuated using four macro-fiber composites (MFCs). The vibration characteristics of the plate varied significantly with the number and position of the rigid-point supports and the configuration of the MFC driving voltage. An analytical solution is proposed for frequency selection and mode control through the strategic placement of the rigid-point supports. The proposed solution provides a simple and efficient approach for controlling the vibration mode of the plate in various applications, such as vibration suppression, energy harvesting, and structural health monitoring.

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

The main aim of this study is the dynamic analysis of isotropic homogeneous beams using the method of initial functions (MIFs) and comparison with classical beam theories and FEM. Also, this research employs the state space methodology, with a special emphasis on isotropy, to analyse simply supported beam systems. A mathematical model for the dynamic response of beams is constructed using the method of initial functions. The novelty of this study lies in its approach to dynamic analysis, where isotropic homogeneous beams are explored without making assumptions, thus ensuring increased precision using the method of initial functions. Importantly, the approach remains free from restrictive assumptions and relies solely on mathematical formulations, yielding results superior to classical beam theories such as Euler–Bernoulli, Timoshenko, and Rayleigh beam theories. In this work, the application of MIFs of various orders (4th, 6th, 8th, and 10th) to calculate natural frequencies is explored, enabling a thorough examination of the beam’s dynamic characteristics. In addition, parameters such as normal stresses, shear stresses, and deflections in different directions are considered to provide a comprehensive understanding of beam behaviour. To validate the findings, a detailed comparison with a finite element method (FEM) is conducted, achieving excellent agreement between the analytical results and FEM solutions. Furthermore, the influence of Poisson’s ratio (μ) on natural frequencies is investigated by varying its value from 0.18 to 0.30. The research also explores the deviation of plane stress values of the beam section from those estimated using the FEM for the corresponding components.

Vacuum insulated glazing (VIG) units under wind load—part 1: global deformation and stresses on the outer glass surfaces

This paper presents the first part of a study on the effect of wind loads on Vacuum Insulated Glass (VIG) units. The study provides background information on VIG and relevant Standards, explains the numerical modelling process, and discusses the implications of the results in relation to European and North American codes and Standards. The focus of the study is on vertical windows and façade installations in low-rise buildings (< 25 m) commonly found in residential buildings. The mechanical behaviour of VIG was analysed using the Finite Element Method (FEM) with respect to various design parameters such as glass pane size, glass thickness, surface pressure magnitude, and edge boundary conditions. The study analysed global deformation as well as the stresses on the outer glass surfaces. The VIG features such as pillar geometry and contact dynamics, and material non-linear effects, were explicitly modelled. In addition, monolithic glass plates were also modelled, and the FEM results of the monolithic cases were in reasonable agreement with an analytical solution obtained from linear thin plate theory. These results highlight the limit of linear behaviour in monolithic plate bending. The centre-of-pane deflection of the VIG was in good agreement with the FEM and analytical solutions of the equivalent thickness monolithic pane, for sample sizes below 800 × 800 mm. However, for larger glass sizes, a deviation was found, and the VIG exhibited a higher plate stiffness than the equivalent thickness monolithic pane. The simulations also showed that the stresses in the glass panes are highly dependent on the edge of glass boundary condition. Finally, the results demonstrated that with appropriate design choices, the VIG can satisfy the Standards requirements for wind load and glass design in both Europe and North America.

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

Investigation of acoustic radiation from a sphere vibrating on the free surface of a finite depth water using a boundary element method

In this study, a boundary element method (BEM) is applied to investigate acoustic radiation from a sphere vibrating in pulsating mode on the free surface of finite or infinite depth water. Effect of the free surface is introduced by employing a half-space Green’s function. A modified version of the Helmholtz integral equation (HIE) is used to calculate acoustic radiation from the sphere vibrating in pulsating mode on the free surface. Free-terms of the HIE are calculated using two different forms of integrals and “dummy” boundary elements. Moreover, to simulate finite depth fluid medium, a chain image-source method is used to derive a waveguide Green’s function. To demonstrate applicability of the method presented, calculated acoustic pressures are compared with those by finite element method (FEM) and analytical calculations. Additionally, the effects of submergence depth and vibration frequency on acoustic radiation are investigated for infinitely deep water together with those of water depth and field point distance on acoustic radiation for finite water depth medium. The calculations show that there is a good agreement between BEM, FEM and analytical solutions. Also, it is observed that field point distance significantly affects the convergence behavior of waveguide Green’s function. Furthermore, it is noted that submergence depth, domain depth and vibration frequency have pronounce influence on radiated pressure amplitude and pressure field pattern.

Applying New Algorithms for Numerical Integration on the Sphere in the Far Field of Sound Pressure

For some sound sources, the function of the square of sound pressure amplitudes on the sphere in the far field is an integrable function or can be integrated with geometrical simplifications, so an exact or approximated analytical expression for the sound power can be calculated. However, often the sound pressure on the sphere in the far field can only be defined in discrete points, for which a numerical integration is required for the calculation of the sound power. In this paper, two new algorithms, Anchored Radially Projected Integration on Spherical Triangles (ARPIST) and Spherical Quadrature Radial Basis Function (SQRBF), for surface numerical integration are used to calculate the sound power from the sound pressures on the sphere surface in the far field, and their solutions are compared with the analytical and the finite element method solution. If function values are available at any location on a sphere, ARPIST has a greater accuracy and stability than SQRBF while being faster and easier to implement. If function values are available only at user-prescribed locations, SQRBF can directly calculate weights while ARPIST needs data interpolation to obtain function values at predefined node locations, which reduces the accuracy and increases the calculation time.

3D Stabilized FEM Solution of the MHD Equations in an External Medium and Around a Solid

In this study, we consider 3-D MagnetoHydroDynamic (MHD) flow problems with different configurations which are mathematically expressed by system of coupled partial differential equation with coupled boundary conditions. These equations are solved numerically using one of the most popular schemes named as the finite element method (FEM) with SUPG type stabilized version in order to obtain accurate and stable solutions especially for the high values of the problem parameters. Obtained numerical solutions are visualized in terms of figures by taking the 2-D slices of the 3-D data in order to emphasize the accuracy of the proposed formulation.

Analyzing of continuous and discontinuous contact problems of a functionally graded layer: theory of elasticity and finite element method

Contact mechanics analysis is crucial because such problems often arise in engineering practice. When examining contact mechanics, the material property of the contacting components is a crucially significant aspect. It is more complex to solve the contact mechanics of systems that are composed of materials that do not have a homogenous structure compared to materials that have homogeneous qualities throughout. While many studies on contact problems with homogeneous materials exist, those involving non-homogeneous materials are scarce in the literature. As material technology improves fast, there will be a greater need to solve such problems. In this respect, analytical and finite element method (FEM) solutions of the continuous and discontinuous contact problems of a functionally graded (FG) layer are carried out in this article. The FG layer in the problem rests on a rigid foundation and is pressed with a rigid punch. From the solutions, the contact length, contact stress, initial separation distance, and beginning and ending points of separation were determined, and the results were compared. It has been concluded that the FEM findings are consistent with the analytical results to a satisfactory degree. This study analyzes contact problem using different approaches and accounts for the influence of body force in a contact geometry that has yet to be reported.

A Mode-Matching-Based Formulation for the Analysis of Curved Rectangular Waveguides

This article presents a mode-matching formulation for the scattering analysis of curved rectangular waveguides. We use exponentially scaled Hankel functions with complex-valued orders to solve the electromagnetic fields in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> -and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -plane bend configurations, and then, we obtain a generalized scattering matrix (GSM) representation for characterizing U-and S-shaped waveguides. Numerical results show that our method is accurate and computationally efficient compared with a finite-element method (FEM) and experimental reference solutions.

Analytical solution for the temperature field of cold region tunnels that considers thermal resistance

The temperature field is a crucial factor in the design and research of cold region tunnels. However, current research has focused on the effects of insulation, ventilation and heating on heat transfer, while disregarding the effect of thermal resistance from embedded components on heat conduction. In this study, a new analytical solution for transient heat transfer that considers interfacial thermal resistance is developed. This solution employs the shifting function method and finite integral transform technique. The proposed solution is highly versatile and can accommodate arbitrary time-varying ambient temperature, the convective heat transfer coefficient at the air-solid interface of the tunnel and the heat source power of embedded heating components. Furthermore, the consistency and accuracy of the proposed solution are validated by comparing with the finite element method solutions for two tunnel examples: (1) only concrete linings are applied; and (2) concrete linings are applied, along with an insulation layer and an embedded heat source. The new solution can provide valuable insights into the design and optimization of anti-frost systems in cold region tunnels and exhibits high potential for engineering applications.

Finite Element Modeling of an Interior Permanent Magnet Synchronous Motor for Calibration and Testing Purposes

This research focuses on the utilization of Finite Element Analysis (FEA) for the modeling of an interior permanent magnet synchronous machine (IPMSM). The long term goal of the project is to model the given motor and simulate its operating conditions using the Finite Element Method (FEM). The FEM solutions will also be used to extract important motor operating conditions and parameters.

The geoelectric field coast effect: Results from two dimensional finite element method and analytic solutions

The geoelectric fields induced during geomagnetic disturbances (GMD) may affect the normal operation of technological systems at the Earth’s surface. The non-uniform Earth conductivity structure, especially large lateral conductivity contrasts, can lead to the enhancement of geoelectric fields near boundaries such as a coastline. In this paper, we give an overview of the applicability of a two-dimensional finite element method (2D-FEM) for studying lateral variations of the Earth’s conductivity, which can be considered as one of magnetotelluric sounding forward modelling methods, including both the E and the H polarization cases. Then the spatial distribution of the geoelectric field near a continent-ocean boundary is calculated with different models. This is compared with analytic solutions for simple models to verify the accuracy of the FEM modelling, which is calculated by using the ANSOFT software. More realistic scenarios are then developed to emphasize the advantages of FEM. The factors influencing the coast effect are discussed. Results show that frequency, the shape of seafloor boundaries, and the depth of ocean have a significant impact on the geoelectric fields. Therefore, these factors should attract more attention when assessing risk for man-made conducting systems located within the coastal influence range.

Closed-form solution of Euler–Bernoulli frames in the frequency domain

This paper presents the frequency domain formulation of the Green’s Functions Stiffness Method (GFSM) for plane Euler–Bernoulli frames subjected to arbitrary external loads (forces and bending moments). The GFSM is a generalization of the Stiffness Method (SM) or the Finite Element Method (FEM) that computes the closed-form analytical structural response by decomposing it into a homogeneous and a fixed or particular responses. The homogeneous response is obtained from the displacements at the element ends in the absence of external loads, while the fixed response is calculated using the Green’s functions of fixed elements through a superposition integral that includes the external loads. The GFSM can be categorized as a mesh reduction method that generalizes the Spectral Element Method (SEM), in which the structural response is first obtained in the frequency domain and then converted to the time domain using the Fast Fourier Transform algorithm. Compared to the SEM, the significant advantage of the GFSM is that the frequency domain structural response is exact and avoids the need for dense meshes when external loads have complex patterns. Three examples are presented to demonstrate the effectiveness of the GFSM, each of which shows excellent agreement when compared to the FEM solution even using far fewer elements.

The elastoplastic large deformation analysis based on meshless radial basis reproducing kernel particle method

In this paper, the radial basis reproducing kernel particle method (RB-RKPM) is extended to solve the elastoplastic large deformation problem. The Galerkin weak form of the elastoplastic large deformation problem based on the full Lagrange scheme is used to establish the solution equation, which is then solved using the Newton–Raphson iterative method. The RB-RKPM is established for the elastoplastic large deformation problem. The effect of the penalty factor, the scale parameter, and the shape parameter on the calculation accuracy are introduced, and the optimal parameters are obtained. Several numerical examples were used in the simulation. The RB-RKPM solution is compared with the reproducing kernel particle method (RKPM) solution and the finite element method (FEM) solution, the effectiveness of the RB-RKPM in solving elastoplastic large deformation problem is verified, and the RB-RKPM has high computational accuracy.

A unified vibration modeling of open cylindrical shell-rectangular plate coupling structures based on the dynamic stiffness method

This paper presents a unified vibration modeling for free and forced vibration analysis of coupled open cylindrical shell-plate structures (COSPS). In the model, the coupled structure is first split into several open cylindrical shells and rectangular plates, and then based on Kirchoff's thin plate theory and Flügge's thin shell theory, the dynamic stiffness (DS) matrix of each substructure is separately established by applying the generalized superposition method and the projection method. Subsequently, according to the continuity and equilibrium conditions of the coupled boundary, the coordinate transformation matrix of each substructure is derived. After obtaining fundamental DS matrices and their coordinate transformation matrices, global DS matrices of various COSPS are assembled using a strategy similar to the finite element method (FEM) without repeating the theoretical derivation. To verify the convergence and reliability of the current formulation, free vibration and forced vibration analysis of three types of coupled structures are carried out, and the results are compared with those from published works and FEM solutions. In addition, an experimental model of a coupled structure is established and an experimental test is performed. The comparison results show that the proposed model is reliable and effective, and its modeling process is more direct and convenient. This work not only greatly expands the application scope of the DSM but also provides a new idea for the vibration analysis of COSPS.

The nut-and-bolt motion of a bacteriophage sliding along a bacterial flagellum: a complete hydrodynamics model

The ‘nut-and-bolt’ mechanism of a bacteriophage-bacteria flagellum translocation motion is modelled by numerically integrating the 3D Stokes equations using a Finite-Element Method (FEM). Following the works by Katsamba and Lauga (Phys Rev Fluids 4(1): 013101, 2019), two mechanical models of the flagellum-phage complex are considered. In the first model, the phage fiber wraps around the smooth flagellum surface separated by some distance. In the second model, the phage fiber is partly immersed in the flagellum volume via a helical groove imprinted in the flagellum and replicating the fiber shape. In both cases, the results of the Stokes solution for the translocation speed are compared with the Resistive Force Theory (RFT) solutions (obtained in Katsamba and Lauga Phys Rev Fluids 4(1): 013101, 2019) and the asymptotic theory in a limiting case. The previous RFT solutions of the same mechanical models of the flagellum-phage complex showed opposite trends for how the phage translocation speed depends on the phage tail length. The current work uses complete hydrodynamics solutions, which are free from the RFT assumptions to understand the divergence of the two mechanical models of the same biological system. A parametric investigation is performed by changing pertinent geometrical parameters of the flagellum-phage complex and computing the resulting phage translocation speed. The FEM solutions are compared with the RFT results using insights provided from the velocity field visualisation in the fluid domain.

Construction of generalized shape functions over arbitrary polytopes based on scaled boundary finite element method's solution of Poisson's equation

SummaryA general technique to develop arbitrary‐sided polygonal elements based on the scaled boundary finite element method is presented. Shape functions are derived from the solution of the Poisson's equation in contrast to the well‐known Laplace shape functions that are only linearly complete. The application of the Poisson shape functions can be complete up to any specific order. The shape functions retain the advantage of the scaled boundary finite element method allowing direct formulation on polygons with arbitrary number of sides and quadtree meshes. The resulting formulation is similar to the finite element method where each field variable is interpolated by the same set of shape functions in parametric space and differs only in the integration of the stiffness and mass matrices. Well‐established finite element procedures can be applied with the developed shape functions, to solve a variety of engineering problems including, for example, coupled field problems, phase field fracture, and addressing volumetric locking in the near‐incompressibility limit by adopting a mixed formulation. Application of the formulation is demonstrated in several engineering problems. Optimal convergence rates are observed.

Phase change heat transfer analysis of anisotropic structures with moving heat source using the element-free Galerkin method

The computational model of phase change heat transfer analysis for anisotropic structures with moving heat sources is established using the element-free Galerkin method (EFGM), and the accuracy and superiority of the proposed model are verified by experimental results and two butt welding examples of anisotropic plates. The influence of off-angle and thermal conductivity factors on the temperature field, welding pool, and weld seam is investigated. The results show that EFGM temperature and welding pool have higher accuracy compared with the finite element method solutions. It is demonstrated that EFGM has a distinct advantage in the tracing of moving heat sources and dynamic phase interfaces during the butt welding process, whether along straight or complex paths. There is a high-quality welding pool and a uniform temperature in butt welding of anisotropic structures compared with isotropic structures. The increase of thermal conductivity factor can accelerate the phase change process and its reasonable range is suggested to be 2–4. It can reduce the temperature concentration and accelerate the solidification speed of weld seam when the off-angle is enlarged, and its recommended range is 30°–45°.

A hydro‐mechanically‐coupled XFEM model for the injection‐induced evolution of multiple fractures

AbstractIn this paper, a two‐dimensional eXtended Finite Element Method (XFEM) solution is presented for the hydro‐mechanically coupled hydro‐frac‐induced propagation of multiple fractures in rocks. Fractures are considered one‐dimensional objects, and the rock matrix is regarded as a two‐dimensional medium. Rules for improving the accuracy of the solution are included to deal with the transfer of variables between fractures and matrix in the coupling process. The main variables include fluid pressure and fracture aperture. The following assumptions are made: (1) the fluid pressure inside a fracture is applied as a net pressure; (2) fluid leak‐off is ignored; (3) the fluid front is regarded as a fracture front provided that the area of fluid lag has a small effect on fracture propagation compared to the area at which the fluid pressure is applied. The fluid flow is governed by the lubrication equation, while the XFEM is used to describe the behaviour of the elastic medium. The simulation results are compared with analytical solutions for a single hydraulic fracture model to verify and validate the proposed algorithm. A simulation with multiple hydraulic fractures is also performed to investigate the interference effect of multiple fractures and fracture propagation. The shadow effect caused by multiple fractures is analysed to manifest the stress variation along the propagation path.

Efficient prediction of airborne noise propagation in a non-turbulent urban environment using Gaussian beam tracing method

This paper presents a noise propagation approach based on the Gaussian beam tracing (GBT) method that accounts for multiple reflections over three-dimensional terrain topology and atmospheric refraction due to horizontal and vertical variability in wind velocity. A semi-empirical formulation is derived to reduce truncation error in the beam summation for receivers on the terrain surfaces. The reliability of the present GBT approach is assessed with an acoustic solver based on the finite element method (FEM) solutions of the convected wave equation. The predicted wavefields with the two methods are compared for different source-receiver geometries, urban settings, and wind conditions. When the beam summation is performed without the empirical formulation, the maximum difference is more than 40 dB; it drops below 8 dB with the empirical formulation. In the presence of wind, the direct and reflected waves can have different ray paths than those in a quiescent atmosphere, which results in less apparent diffraction patterns. A 17-fold reduction in computation time is achieved compared to the FEM solver. The results suggest that the present GBT acoustic propagation model can be applied to high-frequency noise propagation in urban environments with acceptable accuracy and better computational efficiency than full-wave solutions.