Galerkin Approach Research Articles

Dynamic characteristics of cracked simply supported bidirectional functionally graded Rayleigh beam

This paper investigates the dynamic behavior of cracked Rayleigh beams constructed from bidirectional functionally graded (BDFG) materials under simple boundary conditions. A torsional massless spring is employed to model the beam's open crack type. The vibration equations are obtained using Hamilton's principle. The graded beam material properties are varied throughout the thickness based on the power-law distribution and in the longitudinal direction using the exponential material distribution. To solve the dynamic equations, Galerkin's approach is employed. The paper evaluates the impacts of the axial index, gradient property index, beam modulus ratio, and crack parameters on the natural frequencies of the FG beam. The results indicate that the dimensionless natural frequencies of intact graded beams decrease with increased gradient index k. In contrast, they increase with a rise in the modulus ratio. Additionally, the results demonstrate that an increase in the crack depth ratio decreases dimensionless natural frequencies.

Application of Galerkin Approach for the Determination of the Optimum Buckling Value of a Prismatic Bar

Application of Galerkin Approach for the Determination of the Optimum Buckling Value of a Prismatic Bar

Regularized Perona & Malik model involving Caputo time-fractional derivative with application to image denoising

This work tackles a novel partial differential equation based on a time-fractional order derivative for image denoising and restoration. We ameliorate the classical Perona–Malik model by considering Caputo time-fractional order derivative with a regularized diffusion. We start by investigating the existence of the solution to the proposed model. As a first result, we use the Faedo–Galerkin approach to prove the existence and uniqueness of a weak solution to an auxiliary fractional problem. Secondly, we establish the existence and uniqueness of a weak solution to our model via Schauder fixed point method. To validate the efficiency of our model, we present some numerical experiments on images with different features and various noise levels. We begin by introducing an adaptive discrete scheme to the proposed model. We illustrate the sensitivity of the time-fractional order derivative with respect to some well-known criteria. Finally, the obtained numerical results show a great efficiency and robustness against noise compared to the competitive models.

Acoustics in Flameholding Solid-Fuel-Ramjet Fuel Grains

Experiments are conducted to determine how acoustic perturbations affect the performance and flameholding of solid-fuel ramjets with nonstandard combustion chambers. The focus is on the effect of wall cavities carved in the fuel grain using additive manufacturing. An improved understanding of how the wall geometry contributes to the establishment of acoustic modes is sought. A novel combustion mechanism was developed using a counterflow burner to study the combustion and regression of solid model fuel polymethyl methacrylate. The diffusion flame between the fuel and oxidizer was studied numerically using a solid-fuel decomposition and melt layer model to simulate convection and pyrolysis of the material. This model was validated using new experimental data as well as previously published works. The foam layer parameters are critical to the success of the validation, showing that the increased residence time of the gas in the bubbles facilitates the fuel breakdown. Fourth-order computational simulations of ramjet combustion without regressing fuel walls using a novel discontinuous Galerkin approach are performed with a fully conjugate solution for the thermal wave in the solid. Turbulent transport strongly affects the heat feedback to the walls, and low-frequency vortical modes (e.g., with a vortical wavemaker) associated with a recirculation region at the injector upstream wall are linked to an increase in chamber pressure and fuel mass flux.

Nonlinear energy harvesting via an axially moving piezoelectric beam with both d 31 and d 33 modes

Piezoelectric energy harvesters (PEHs) in the literature typically operate with a single conversion mechanism (either d 31 or d 33); the output power, therefore, is limited, and not sufficient to sustainably energize low-power electronics. In this study, a nonlinear PEH with coupled d 31 and d 33 modes is designed and evaluated. An axially moving piezoelectric beam (AMPB) was applied to investigate the contribution of d 31 and d 33 to the output, and the critical parameters of the configuration were determined. A distributed parametric electromechanical model was established to characterize the non-linear dynamics of AMPB with d 31 and d 33 modes. The Galerkin approach and the harmonic-balance approach were employed conjointly to investigate the forced response of the energy harvesting system. The axial velocity’s effects upon energy harvesting were as well discussed. Comparison of the frequency response functions (FRFs) for voltage and power output between energy structures of d 31 and d 33 modes revealed several discrepancies. For instance, the voltage and power output of the d 33 mode were greater than those of d 31 mode for low frequencies, and the difference between the two modes decreased as the frequency increased. For the composite mode d 31 and d 33, under the same parameter conditions, the voltage and power output were greater than the output of any single mode. The analytical results were supported by a numerical method through the finite difference method. Both analytical and numerical results indicated the FRF could be increased by increasing the excitation amplitude, reducing the damping coefficient, or increasing the electrode spacing. The present study showed the efficiency of the use of the FRF using nonlinear transverse vibration of AMPB for d 31 and d 33 modes.

A IETI-DP method for discontinuous Galerkin discretizations in isogeometric analysis with inexact local solvers

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting.

Thermoconvective instability in a ferrofluid saturated porous layer

The Forchheimer-extended Brinkman’s Darcy-flow model was used to investigate the initiation of ferroconvection in a flat porous layer while accounting for effective viscosity. The rigid ferromagnetic, rigid paramagnetic and stress-free isothermal boundary conditions are the three categories. The eigenvalue issue can be properly addressed for stress-free boundaries; the Galerkin approach is utilized to find the critical stability constraints quantitatively for other barriers. It was discovered that the boundary types had a strong influence on the system’s stabilization. Ferromagnetic boundaries are less preferred than paramagnetic boundaries in control of convection. The dependence of many physical limitations on the linear stability of the system is intentionally given, and it is demonstrated that increasing the value of the viscosity ratio delays the beginning of convection.

Numerical method to solve generalized nonlinear system of second order boundary value problems: Galerkin approach

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of 
 different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

Nonlinear free vibrations analysis of truncated conical shells made of bidirectional functionally graded materials

For the first time, present paper addresses the nonlinear vibrations analysis of a truncated conical shell made of bidirectional functionally graded (BDFG) materials whose mechanical properties varied continuously through the thickness and the length directions of the conical shell based on the power law distribution. Accordingly, the set of nonlinear equation of motions of the conical shell are derived based on first-order shear deformation theory (FSDT) and von Karman strain-displacement relations using Hamilton’s principle. System of partial differential equations of the BDFG conical shell are transformed into time-dependent ordinary differential equations using Galerkin approach. Afterward, the nonlinear equations are analytically solved by means of modified Poincaré–Lindstedt method, and the nonlinear frequency of the BDFG conical shell is obtained. For verification purpose, the present outcomes are compared with those available in previous researches. Eventually, the effect of longitudinal and transverse gradient indexes, vibration amplitude, and geometrical parameters on the nonlinear frequency of the BDFG conical shell is investigated. The present research may give benchmark solutions and some guidelines for designing BDFG truncated conical shells.

Multiple-Frequency Preconditioned Iterative Inverse Source Solutions

Inverse source formulations relate unknown equivalent radiation sources to given field observations, which are commonly obtained from measurements. After discretization, the linear operator equation is advantageously solved in the form of an appropriate positive semidefinite normal system of equations. Several preconditioning strategies for the iterative solution of such linear systems of equations are investigated in this article. Observation point density-based preconditioners balance the effect of the constraint equations and speed up the solution process for globally irregular sample location distributions. Reduced-accuracy approximations of the radiation operator based on propagating plane-wave expansions of the pertinent Green’s functions can speed up the operator evaluation considerably. They are used within inner–outer iterative solvers or, even more efficiently, within a two-step iterative solution of the inverse source problem. Multiple-frequency inverse source solutions are considerably accelerated by start vector estimations from already available solutions for other nearby frequencies, where in particular a modified Galerkin approach is found to be very effective. The presented accelerated iterative solution approaches achieve the same accuracy levels as its not accelerated counterparts. They are validated and evaluated for a variety of near-field (NF) and far-field (FF) observation data of different antennas, whereby the benefits of the introduced solution approaches are of particular importance for large antennas with many unknowns.

Dynamic response of a thin-walled curved beam with a mono-symmetric cross-section under a moving mass

This paper presents analytical solutions for the dynamic response of thin-walled curved beams induced by moving vehicles. The solutions encompass vertical, torsional, radial, and axial motions. A comprehensive set of governing equations for the dynamic response of thin-walled curved beams is established by considering the mass inertia of moving vehicles, rotary inertia, and warping resistance. The solutions are obtained for the four-directional response of curved beams, involving vertical, torsional, radial, and axial motions, based on the Fourier finite integral transformation, the Laplace–Carson transformation, their inverse transformations, and the Galerkin approach. A comparison of the results calculated by the analytical solutions in this study are compared with those calculated by the moving-force solutions using the Galerkin approach reported in a related literature, demonstrates the reliability and superiority of the analytical solutions. Hence, an extensive parametric study is carried out to investigate the effect of the mass inertia of moving vehicles, velocities, higher-order modes of vibrations, and radii of curvature on the dynamic responses of beams. As per the results, the mass and velocity of the moving vehicles play important roles in the dynamic response of curved beams. Furthermore, the dynamic response for thin-walled curved beams increases significantly with heavier vehicles and higher velocities, confirming that it is necessary to consider the mass inertia effect to study the dynamic response of curved beams.

Nonlinear dynamic study of non-uniform microscale CNTR composite beams based on a modified couple stress theory

This study aims at investigating the nonlinear dynamic behavior of microscale carbon nanotube reinforced (CNTR) composite Euler–Bernoulli beams with a non-uniform cross-section, based on a modified couple stress theory (MCST). The nonlinear partial differential equations (PDEs) of motion are established based on the Von-Karman nonlinear strain–displacement relationship and Hamiltonian principle. The coupled PDEs are reduced to a single PDE, by neglecting the effects of the axial inertia and considering two different types of boundary conditions (i.e. clamped–clamped and clamped–free). At the same time, the single PDE is reverted to a nonlinear ordinary differential equation (ODE) by means of the Galerkin approach, and it is solved by using a semi-inverse method and the method of multiple time scales (MTS) for a free and forced vibration analysis, respectively. A large systematic numerical analysis is here performed to check for the sensitivity of the nonlinear response of CNTR composite beams to different boundary conditions and reinforcement parameters, with useful scientific insights for further computational investigations on the topic.

A Dynamic Analysis of Porous Coated Functionally Graded Nanoshells Rested on Viscoelastic Medium

Theoretical research has numerous challenges, particularly about modeling structures, unlike experimental analysis, which explores the mechanical behavior of complex structures. Therefore, this study suggests a new model for functionally graded shell structures called “Tri-coated FGM” using a spatial variation of material properties to investigate the free vibration response incorporating the porosities and microstructure-dependent effects. Two types of tri-coated FG shells are investigated, hardcore and softcore FG shells, and five distribution patterns are proposed. A novel modified field of displacement is proposed by reducing the number of variables from five to four by considering the shear deformation effect. The shell is rested on a viscoelastic Winkler/Pasternak foundation. An analytical solution based on the Galerkin approach is developed to solve the equations of motion derived by applying the principle of Hamilton. The proposed solution is addressed to study different boundary conditions. The current study conducts an extensive parametric analysis to investigate the influence of several factors, including coated FG nanoshell types and distribution patterns, gradient material distribution, porosities, length scale parameter (nonlocal), material scale parameter (gradient), nanoshell geometry, and elastic foundation variation on the fundamental frequencies. The provided results show the accuracy of the developed technique using different boundary conditions.

Investigation of aspect ratio effects on flow characteristics and vorticity generation in shock-induced rectangular bubble

In shock-induced Richtmyer–Meshkov instability, the polygonal bubbles, including the rectangular interface can offer favorable circumstances for shock refraction research. Also, the aspect ratio is crucial in characterizing the physics behind this instability. Therefore, the aspect ratio effects on the flow characteristics and vorticity generation in the shock-induced rectangular light gas bubbles are investigated numerically in this study. This investigation is performed for a wide range of aspect ratios varying from 0.25 to 3.0. These effects are highlighted in the interface morphology, wave paradigms, vorticity generation, enstrophy evolution, and a qualitative examination of interface deformation. For simulating two-component gas flows, a high-order explicit modal discontinuous Galerkin approach is employed to solve a two-dimensional system of compressible inviscid Euler equations. Existing experimental results for shock-induced square and cylindrical light bubbles are used to validate the numerical model and solver. The results indicate that the effects of aspect ratio are critical in characterizing the interface morphology during the interaction between the shock wave and rectangular bubble. The interface morphology evolves faster as the aspect ratio increases, and the process of transverse jet formation is observed to be significantly different. The aspect ratio effects result in a substantial change in the flow field with bubble deformation, vortex development, vorticity generation, and transverse jet formation. The effects of aspect ratio are further investigated in detail through the mechanism of vorticity generation and enstrophy evolution. Finally, these effects on shock trajectories and interface characteristics over time are thoroughly explored.

A Wave Based Method for the analysis of a fully and partially saturated halfspace under harmonic loading

The dynamic response of a fully or partially saturated halfspace due to vibrations from traffic or rotating machinery can significantly be influenced by water and air phases within the pores. An extension of the Wave Based Method (WBM) is presented to consider these influences and to propose an efficient numerical method for the dynamic analysis of fully and partially saturated soil layers. Biot’s theory for the low frequency range and the Berryman-Thigpen-Chin model (BTC model) are introduced and a transmitting boundary condition is implemented to approximate the Sommerfeld radiation condition. Moreover, coupling conditions for dry, fully and partially saturated soil layers are formulated and involved into a weighted Galerkin approach. The accuracy of the WBM is evaluated for a fully saturated and a layered halfspace with different degrees of saturation. For this, numerical examples are compared with reference solutions from literature. It is shown that the WBM delivers adequate field responses and succeeds in modeling different types of system behaviour, which are characteristic for a fully saturated and layered halfspace.

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. A concise proposal on how to overcome these shortcomings has been raised during the last decade by the Discontinuous Petrov–Galerkin (DPG) methodology. However, DPG has also some limitations and difficulties: the method requires ultraweak variational formulations, obtained through a hybridization process, which is not trivial to implement at the discrete level.Our motivation is to offer a simpler alternative for the case of parametric PDEs, which can be used with any variational formulation. Indeed, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently in an online stage, for a given range of parameters. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE parameter, to the matrix of coefficients of optimal test functions (in some basis expansion) associated with that PDE parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE parameters). When solving online the resulting (compressed) Petrov–Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix–vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure is as fast as an (unstable) Galerkin approach. We illustrate our findings by means of 2D–3D Eriksson–Johnson problems, together with 2D Helmholtz equation.

Numerical development for freezing of phase change material loading nanoparticles for improving water treatment

The container with a wavy insulated wall and one branch shaped fin has been applied in this study. The process of freezing was simulated based on implicit way via Galerkin approach. The testing material is a mixture of H2O and nano-powders of alumina. The fraction of additives is less than the critical value for assumption of single phase mixture. Verification test has demonstrated the correctness of the modeling procedure. The utilizing denser mesh close the region of ice front makes the accuracy of modeling to increase. Owing to the negligible speed of liquid water, the associated terms in governing equations have been discarded. Regarding the utilized diameter of alumina, dp = 40 nm is the best option which leads to a lower period of freezing. With dispersing alumina with optimum size, the solidification time declines about 41.22 %.

Generalized thermoelastic dissipation in micro/nano-beams with two-dimensional heat conduction

The precise prediction of quality factor based on thermoelastic dissipation (TED) is of great importance for the optimal design of micro/nano-resonators. In this work, employing the nonlocal-dual-phase-lag (NDPL) model and the modified-couple-stress (MCS) model, analytical TED expressions are developed for the first time for micro/nano-beam resonators with two-dimensional (2D) heat conduction along the thickness and length directions. Initially, the governing equation of NDPL coupled thermoelasticity is established, and then the solutions of 2D fluctuation temperature are obtained by utilizing the Galerkin approach. Subsequently, the natural frequency involving the MCS size-dependence length-scale parameter is achieved based on the governing equation of vibration. Finally, analytical generalized TED models are attained according to the energy-definition approach. Three representative types of constraint boundary conditions for the beam resonator, namely, clamped-clamped, clamped-free, and simply supported, are considered in this study. And the present developed TED models are confirmed to possess perfect generality and excellent convergence performance. Moreover, an expression of equivalent thermal relaxation time in the 2D case is introduced and expressed in terms of the length- and thickness-directional thermal relaxation times. Two typical materials, silicon and gold, are selected in the simulation part. The exploration of fluctuation temperature, TED spectra, and thermal relaxation times considering multiple physical effects is conducted as an emphasis. Results reveal that the effect of heat-conduction dimension is crucial for micro/nano-beams with a small ratio of length to thickness, while the significant NDPL effect is dependent upon the ultra-high vibration frequency. Additionally, the MCS size-dependence effect can enhance the natural frequency and improve the quality factor of micro/nano-beam resonators.

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

Dynamic Snap-Through and nonlinear vibrations of bistable asymmetric Cross-Ply composite laminated cantilever shell under external excitation

The bistable asymmetric composite laminated structures have the broad applications in the design of the deformable aircrafts and energy harvester due to the unique nonlinear dynamic snap-through characteristics. This paper studies the dynamic snap-through phenomena and nonlinear vibrations of the bistable asymmetric cross-ply composite laminated cantilever shell under the external excitation. Reddy’s third-order shear deformation theory and Hamilton principle are applied to establish the differential dynamic governing equation of the bistable asymmetric composite laminated cantilever shell. Two-degree-of-freedom nonlinear dynamic system is obtained by employing Galerkin approach. The curvature-displacement relationship is given for the bistable asymmetric composite laminated cantilever shell. Chebyshev-Ritz is utilized to obtain both natural frequencies and mode shapes of the bistable asymmetric composite laminated cantilever shell. Considering the cantilever boundary condition, the free vibrations are analyzed by considering different factors of the bistable asymmetric composite laminated cantilever shell, such as the curing temperature, numbers of the layers and geometry parameters. It is observed that the dynamic snap-through behaviors and chaotic vibrations are found under varying the frequency and amplitude of the external excitation. Several validations are carried out among the obtained results with these of other references. The results of the numerical simulation and finite element software ABAQUS are in good agreement with these of the vibration experiment.