Direct Iterative Method Research Articles

Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.

Dual-Domain Collaborative Diffusion Sampling for Multi-Source Stationary Computed Tomography Reconstruction.

The multi-source stationary CT, where both the detector and X-ray source are fixed, represents a novel imaging system with high temporal resolution that has garnered significant interest. Limited space within the system restricts the number of X-ray sources, leading to sparse-view CT imaging challenges. Recent diffusion models for reconstructing sparse-view CT have generally focused separately on sinogram or image domains. Sinogram-centric models effectively estimate missing projections but may introduce artifacts, lacking mechanisms to ensure image correctness. Conversely, image-domain models, while capturing detailed image features, often struggle with complex data distribution, leading to inaccuracies in projections. Addressing these issues, the Dual-domain Collaborative Diffusion Sampling (DCDS) model integrates sinogram and image domain diffusion processes for enhanced sparse-view reconstruction. This model combines the strengths of both domains in an optimized mathematical framework. A collaborative diffusion mechanism underpins this model, improving sinogram recovery and image generative capabilities. This mechanism facilitates feedback-driven image generation from the sinogram domain and uses image domain results to complete missing projections. Optimization of the DCDS model is further achieved through the alternative direction iteration method, focusing on data consistency updates. Extensive testing, including numerical simulations, real phantoms, and clinical cardiac datasets, demonstrates the DCDS model's effectiveness. It consistently outperforms various state-of-the-art benchmarks, delivering exceptional reconstruction quality and precise sinogram.

Nonlinear dynamic response of a sandwich plate with negative Poisson’s ratio honeycomb-core layer under low-velocity collision impact

In this paper, a nonlinear analytical model is presented for the low-velocity collision impact of a sandwich plate with auxetic honeycomb-core layer. The auxetic feature of the honeycomb material is realized by mathematically expressing the effective material coefficients in terms of material property and cellular geometric parameters through a homogenization method. The higher order shear deformation theory, the von Kármán nonlinearity theory, and a modified Hertz contact law which accounts for the contact pressure distribution and indentation effect, are employed to establish the kinematic relations. The Newmark time integration scheme in conjunction with the direct iterative method is utilized to establish a solution procedure for the nonlinear dynamic governing equation. The verification of the presented model with the data in published literatures is carried out, followed by a series of numerical analyses for influences of cellular geometric features and impactor’s initial conditions (such as impactor’s initial velocity, mass, and nose curvature radius) on the nonlinear dynamic response. The results of numerical analyses show that the geometric features of the unit cell in the honeycomb-core and impactor’s initial conditions can cause significant influences on the nonlinear collision impact behaviors of the system.

A Controllable Response Power Beam Localization Method and System Implementation

Sound source localization technology has gradually become one of the main methods for fault source target localization in dangerous and difficult-to-discover applications. In order to locate the target fault sound source more intuitively, accurately and in real time in practical applications, an audio-visual positioning system is developed. A 64-channel microphone array is designed, which is evenly distributed in a circular shape, and has a video acquisition target at the same time. The system uses GCC-PHAT to estimate the relative time delay between the signal source arriving at two microphones, and then uses SRP-PHAT algorithm to obtain weighted sum beamforming. A spatial grid search strategy based on spherical coordinate spatial contraction method is proposed. When the output power is maximum, the direction corresponding to the beam is regarded as the sound source direction. Using the same sound source signal, the proposed positioning method is compared with the direct Newton iterative method and the Cartesian coordinate scanning SRP-based maximum method. The simulation results show that the Cartesian coordinate value and spherical coordinate value of the positioning algorithm in this article are closer to the real coordinates of the sound source position than the other two methods, and the positioning operation time is the shortest. In addition, in the actual corona discharge power source positioning experiment, the system can accurately locate the insulator pollution discharge point, which is consistent with the infrared detection and positioning results and has good robustness. Therefore, the system location method can provide a new choice for sound source location applications.

A data-driven framework for designing microstructure of multifunctional composites with deep-learned diffusion-based generative models

This paper introduces a novel integrated microstructure design methodology that replaces the common existing design approaches for multifunctional composites: 1) reconstruction of microstructures, 2) analyzing and quantifying material properties, and 3) inverse design of materials. The problem of microstructure reconstruction is addressed using the diffusion-based generative model (DGM), which is a state-of-the-art generative model formulated with a Markovian diffusion process. Then, the conditional formulation of DGM is introduced for guidance to the embedded desired material properties with a transformer-based attention mechanism, which enables the inverse design of multifunctional composites. Furthermore, a convolutional neural network (CNN)-based surrogate model is utilized to facilitate the prediction of linear/nonlinear material properties for building microstructure-property linkages. Combined, the proposed artificial intelligence-based design framework enables large data processing and database construction that is often not affordable with resource-intensive finite element method (FEM)-based direct numerical simulation (DNS) or iterative reconstruction methods. It is worth noting that the proposed DGM-based methodology is not susceptible to unstable training or mode collapse, which are common issues in generative models that are often difficult to address even with extensive hyperparameter tuning. An example case is presented to demonstrate the effectiveness of the proposed approach, which is designing mechanoluminescence (ML) particulate composites. The results show that the designed ML microstructure samples with the proposed generative and surrogate models meet the multiple design requirements (e.g., volume fraction, elastic constant, and light sensitivity). This assessment demonstrates that the proposed integrated methodology provides an end-to-end solution for practical material design applications.

Nonlinear Vibration of FG-GNPRC Dielectric Beam with Kelvin–Voigt Damping in Thermal Environment

The excellent properties of graphene reinforced composites (GRC) enable them to be promising material candidates for developing high-performance and multifunctional devices and structures. When subjected to thermal environment, temperature can significantly affect the structural behaviors of these components. It is necessary to consider the effects of temperature while conducting structural analysis. This work first attempts to investigate the nonlinear vibration of functionally graded (FG) graphene nanoplatelet (GNP) reinforced composite (FG-GNPRC) dielectric beams with comprehensively considering the effects of FG distribution of the reinforcements, temperature, electrical field and damping. The established governing equations for the FG-GNPRC dielectric beam are discretized and solved by differential quadrature (DQ) and direct iterative methods. The numerical results demonstrate that the application of pre-strain can attenuate the effect of electric fields and temperature on the frequency ratio, resulting in a more stable structure. Among the FG distribution patterns as involved, the FG-GNPRC beam with profile [Formula: see text] exhibits lower frequency ratio and higher stability. This work is envisaged to provide guidelines for the design of FG-GNPRC structures with optimized performances in thermal environment.

Effect of Bending Rigidity and Nonlinear Strains on Free Vibration of Hemi-Ellipsoidal Shells

Abstract This paper focuses on free vibration of hemi-ellipsoidal shells with the consideration of the bending rigidity and nonlinear terms in strain energy. The appropriate form of the energy functional is formulated based on the principle of virtual work and the fundamental form of surfaces. Natural frequencies and their corresponding mode shapes are determined using the modified direct iteration method. The obtained results, which show a close agreement with previous research, are compared with those obtained based on the membrane theory. The effect of the support condition, thickness, size ratio, and volume constraint condition on frequency parameters and mode shapes is demonstrated. With the bending rigidity, shell thickness has a significant impact on the frequency, especially in higher vibration modes and in shells with a considerable thickness but the frequency parameter converges to that determined by using the membrane theory while the reference radius-to-thickness ratio is increasing. In addition, accounting for the bending rigidity solves the issue of determining natural frequencies and mode shapes of the shells using the membrane theory without the volume constraint condition. The obtained results also indicate that the free vibration analysis with bending is essential for the hemi-ellipsoidal shell with a base radius-to-thickness ratio of less than 100, which gives over 2.84% difference compared with that of the shell derived by membrane theory, and this allows engineers to perform the analysis in more applications.

Geometrically nonlinear bending mesh-free analysis of functionally graded porous sandwich beam

In this work, geometrically nonlinear static bending analysis of sandwich beams with a porous core and two skins made of functionally graded materials using a mesh-free method is presented. The material types of the core and face sheets of the beam are chosen so that the material continuity between the layers is guaranteed. The nonlinear governing equation including geometric nonlinearity is established via the principle of virtual work. This equation is discretized into a system of algebraic equations by the mesh-free approach, which is based on the C1 point interpolation method and polynomial basis functions, and then solved by the direct iterative method. The convergence of the mesh-free method is tested to determine the sufficient mesh level for the analysis. Comparative and comprehensive studies are performed to investigate both the correctness and the influences of several important parameters and boundary conditions on the linear and nonlinear deflections of the beam.

Nonlinear static analysis of multi-segmented toroidal shells under external pressure

This paper investigates the nonlinear static responses of multi-segmented toroidal shells with egg-shaped cross sections. The discontinuity effect in terms of displacement and the slope at the toroidal shell edge must be constrained by the Lagrange multiplier technique. The structural shape of the multi-segmented toroidal shells can be described by the differential geometry of the shell with the six-center-combined toroidal shells. The energy functional of the multi-segmented toroidal shells can be written in the appropriate form based on the principle of virtual work. The numerical results of the displacement responses and the volume change can be obtained by the nonlinear finite element method via the fifth-order polynomial shape function and a direct iterative method. In this study, the Lagrange multiplier is the internal force at the toroidal shell junction sustaining the smooth curve of the multi-segmented toroidal shells under several loads. Finally, the results of the volume change and the displacement responses for various geometric parameters are presented and discussed.

Machine learning assisted nonlinear deflection analysis of agglomerated carbon nanotube core smart sandwich plate with three-phase magneto-electro-elastic skin

This research focuses on evaluating the combined effects of the interphase of the three-phase magneto-electro-elastic (TPMEE) composites and carbon nanotubes (CNTs) agglomeration on the nonlinear deflection of the multifunctional sandwich plate, using an artificial neural network (ANN) assisted finite-element (FE) approach. The data points collected from the in-house developed FE computational tool are used to train the ANN model. To this end, a backpropagation-based Levenberg–Marquardt algorithm is used. The core of the plate is made of agglomerated CNTs, and the facesheets are made of TPMEE composites. Two different agglomeration states, partial and complete, are considered for evaluation. Also, three variations of CNTs arrangement are assumed. On the other hand, the interphase effects are incorporated through its volume fractions and compositions. The plate kinematics is based on the higher-order shear deformation theory, and the nonlinearity is assumed to follow von Karman's strain-displacement relation. The equations of motion are derived using the total potential energy principle. Finally, the direct iterative method is used to arrive at the solutions. The numerical examples are also provided to understand the influence of coupling fields associated with the parameters, such as agglomeration, interphase volume fraction, interphase compositions and CNTs arrangement.

Nonlinear deflection characteristics of damaged composite structure theoretical prediction and experimental verification

This research derived a nonlinear computational model for the prediction of deflection responses of the damaged layered composite structures. The model validity was verified using in-house experimentation. The composite panel model has been derived mathematically using the higher-order polynomial kinematics without non-zero shear stress conditions and Green’s strain for the geometrical nonlinearity. The nonlinear deflection responses of combined damage curved shell structures are evaluated numerically utilizing the nonlinear isoparametric finite element technique in conjunction with the direct iterative method. By setting the necessary convergence criteria, the model accuracy has been explored by comparing the deflections of the published literature (the maximum deviation observed is 6.97%). Additionally, the experimentation is done on damaged laminated shell structures, and results are compared to show the proposed model’s efficacy (maximum deviation observed is 2.17%). Finally, a number of numerical examples are solved to assess the effects of the material, geometry and damage-related parameters on the nonlinear deflections of the shell structure.

Numerical investigation on nonlinear vibration of FG-GNPRC dielectric membrane with internal pores

Functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) shows great potential for developing high-performance and multifunctional structures. This paper investigates damped nonlinear vibration of FG-GNPRC dielectric membrane with pores. A two-step hybrid mechanical model is developed to determine the material properties of the multiphase composites. Governing equations involving damping and dielectric properties are derived by utilizing an energy-based approach. To solve the governing equations, Taylor series expansion (TSE) and differential quadrature (DQ) together with direct iterative methods are utilized, respectively. Accuracy and convergence of the presented solution are verified. An extensive numerical study is performed to check the influences of pores, the attributes of the electrical field and GNP on the nonlinear behavior of GNPRC membrane. It is found that the external electric field is evidenced to have more significant effect on regulating the nonlinear vibration of the membrane with smaller aspect ratio pores.

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

Practical analysis of different neutral algorithms for particle simulation of Hall thruster

The modeling of neutral atoms is important for the full-particle simulations of Hall thrusters. In previous studies, researchers have developed various algorithms to model the neutral kinetics. The choice of those algorithms can influence significantly the computational speed, simulation convergence, and physical results. In this work, we perform a full-particle simulation of a typical 1 kW-class SPT-100 Hall thruster using four neutral algorithms, including the fixed-neutral algorithm (FNA), the algorithm of direct simulation of Monte Carlo (DSMC), the collisionless-neutral algorithm (CLNA), and the fluid algorithm (FA), to analyze the effects of different neutral iteration approaches on the simulation results. We found that FNA is sensitive to the initial number density of neutrals, and is difficult to converge properly, while the other algorithms not neglecting the atomic dynamics can get stable results. We count the parameters of the thruster, that is, thrust, specific impulse, and plasma density using different neutral algorithms. The time-averaged results match well with those of the experiment. However, the results differ in the time scale due to the low-frequency oscillations in Hall thrusters. We verify that the oscillations are due to the periodic change of neutrals and establish a zero-dimensional model to analyze the properties of the oscillations in the time scale. It indicates that the ratio of ion migration to neutral migration is the essential factor that significantly affects the calculation results. The model reveals that the direct neutral iteration methods, like DSMC and CLNA, can better simulate the characteristics of discharge fluctuations in Hall thrusters than the quasi-steady-state method, like FA. Finally, we proposed practical suggestions for the selection of the neutral algorithms for the SPT-100 thruster, which can also be generalized to other low- and medium-power Hall thrusters.

Numerical study on nonlinear vibration of FG-GNPRC circular membrane with dielectric properties

Dielectric elastomer membrane has been applied in a wide variety of engineering fields for their excellent dynamic performances. Functionally graded graphene nanoplatelet-reinforced composite (FG-GNPRC) shows great potential for developing high-performance and multifunctional structures. In this work, the damped nonlinear vibration of the FG-GNPRC dielectric membrane is investigated. The effects of damping and dielectric properties are considered in terms of energy while deriving governing equations. Taylor series expansion and differential quadrature together with direct iterative methods are used to discretize and numerically solve the obtained governing equations. The developed model and numerical solution are verified by comparing present results to existing studies. The influences of functionally graded distribution, damping, stretching ratio, dimensions of the membrane, and the attributes of the electrical field and GNP fillers on the nonlinear vibration of the structure are comprehensively investigated. It is found that compared to a uniform distribution, the membrane with functional distribution of GNP demonstrates exhibits more robust performances. The membrane with a smaller thickness-to-radius ratio is more sensitive to the external electrical field. In addition, the direct current voltage is evidenced to have a more significant effect on the nonlinear vibration when the membrane is subject to a relatively small stretching ratio. The present work suggests that the structural behavior of the FG-GNPRC membrane can be actively adjusted by varying the stretching ratio and the properties of the electrical field and the GNP filler. HIGHLIGHTS Effects of damping and dielectric properties are considered for structural behavior in terms of energy. Taylor series expansion and differential quadrature methods are combined to solve governing equations. Two transition regions in frequency ratio are identified due to AC frequency-facilitated polarization and electron tunneling. The effect of the electric field on nonlinear vibration can be regulated by varying the stretching ratio.

Posterior regularization method for phase removal of shale nano-structure imaging in space domain

X-Ray computed tomography is a non-destructive method that is used, among many applications, to study the size, shape, 3D structures and interconnections of pores in shale. We use phase retrieval methods to deal with the “edge enhancement” effect caused by phase shift. The process of phase retrieval can be described by the transport-of-intensity equation (TIE). But this is an ill-posed problem. The existing methods focus on phase retrieval in the frequency domain. To tackle the ill-posedness, we propose a new method whose main idea is to solve this problem in space domain with a regularization technique. We study a synthetic shale model and simulate the projection data. Then we apply three methods to retrieve the phase: conventional method in frequency domain, direct solving method and iterative Tikhonov regularization method in space domain. Finally, we use the standard filtered back-projection (FBP) method to present the outcome. By analyzing the results, we find advantages of the new method: more stability and fewer artifacts under noise perturbations. The study shows that relative errors of the new method are nearly 1% of that of the traditional method based on frequency domain, and hence the new method is promising for the practical data processing.

Nonlinear vibration of annular radially graded plate subjected to temperature at one edge

An annular plate graded in radial direction made of ceramic and metal having intrinsic geometric imperfection subjected to mechanical and thermal loading is studied for nonlinear vibration response. The gradation of material in radial direction is represented by a simple power law. The Hamilton principle along with Von Karman nonlinear strain displacement relation & FSDT employed for developing governing nonlinear equation of motion. The finite element formulation of radially graded imperfect heated plate is solved by displacement control direct iteration method. The analysis reveals a significant reduction in the hardening stiffness of the plate because of the increase of plate-edge temperature. The results also shows that the plate undergoes initial bending deformation due to the plate-edge temperature that substantially influence the dynamic behaviour of the plate. The geometric imperfection shows substantial impact on the dynamic response of the plate when its inner-edge is heated as compared to that for heated outer-edge of the plate. The nonlinear free and forced vibration response for different volume fraction of involved materials is also explored.

Thermo-mechanical large deformation characteristics of cutout borne multilayered curved structure: Numerical prediction and experimental validation

The numerical modeling of static large-deformation behavior of the multilayered flat/curved panels with cutouts subjected to thermo-mechanical load is presented in this work. The numerical model includes third-order displacement polynomials. The large-deformation characteristic of the multilayered panel is incorporated using two nonlinear strains (Green–Lagrange and von-Karman). The governing equation is formulated using Hamilton’s principle, and approximate numerical solutions are obtained using the selective integration scheme (Gauss-Quadrature) associated with Picard’s direct iterative method. A generic computational algorithm is developed in MATLAB using isoparametric finite element (FE) steps. The solution accuracy is verified with the published results and experimental data by utilizing the available/fabricated lab-scale test rig. The role of cutout parameters, including the various geometrical parameters on the nonlinear structural responses, is studied for a clear insight into the damaged structural modeling using the temperature-dependent elastic properties of the laminate. In addition, the effect of nonlinear strain terms on the final structural responses is presented. Based on the observation, the applicability of the full geometrical (Green–Lagrange) strain–displacement relation is established.

Damped vibration analysis of graphene nanoplatelet reinforced dielectric membrane using Taylor series expansion and differential quadrature methods

Membrane has been attracting extensive attention due to their application in various engineering fields. Compared to traditional materials, graphene reinforced composites have demonstrated great potential in developing high-performance and multifunctional membrane structures. It is of great importance to investigate the structural behaviors of such composite membrane with considering damping and their significantly improved dielectric properties. This work studies the damped vibration of graphene nanoplatelets reinforced composite (GNPRC) dielectric membrane. Governing equations are established for the nonlinear damped vibration of the GNPRC dielectric membrane, in which the effects of dielectric properties and damping are incorporated in terms of energy. Taylor series expansion (TSE), differential quadrature (DQ) and direct iterative methods are used to numerically solve the governing equations. The model and the results are validated by comparing present results with previously reported ones. Parametric study is conducted to investigate the effects of the attributes of the electrical field and the GNP fillers, dielectric properties of the composites, damping and the stretching ratio on the nonlinear damped vibration. The result shows that the vibration frequency of the membrane can be actively tuned by changing the attributes of the applied electrical field. When the stretching ratio increases from 1.05 to 1.10, the nonlinear damped frequency can increase by 39.8%, indicating the significant effects of the stretching ratio.

Iterative solution methods for 3D controlled-source electromagnetic forward modelling of geophysical exploration scenarios

We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.