Global Stiffness Matrix Research Articles

Improvements for the solution of crack evolution using extended finite element method

It is demonstrated that the eXtended Finite Element Method (XFEM) is of remarkable efficiency in simulating crack evolution by eliminating the need for remeshing and refinement. In this paper, it is shown how to enhance the solution efficiency through a comprehensive mathematical investigation of the solution process using XFEM. A typical example is presented to illustrate the disparities in nodal displacements along the two symmetric faces of the crack resulting from the approximation of XFEM. By analysing the structure and components of the global stiffness matrix, the underlying causes of these discrepancies are identified. Building upon these findings, two improvements of the solution are proposed to gain an acceptable accuracy in computing the nodal displacements. The first improvement consists of the subdivision of the enriched elements depending on the characteristic of the distribution of Gauss points. The second improvement is set by determining the optimal number of Gauss points in each sub-element near the crack tip. To calculate the stress intensity factor of the crack under surface pressure, such improvements are applied in conjunction with the interaction integral method, which significantly reduces computational time and eliminates the influence of surface tractions. The numerical solution is validated by comparing it with the analytical solution and the standard XFEM solution. The proposed improvements can enhance both the accuracy of the solution and the computational efficiency of XFEM.

Full wavefield modeling and dispersion characteristics of underwater MASW

The underwater multichannel analysis of surface wave (UMASW) is becoming an essential tool for surveying subbottom shear wave velocity. Current practice is limited to interpretation based on the fundamental mode. This paper investigates the full dynamic response of the underwater-multilayered structure under two different types of sources (impact and explosion). Fundamental solutions in the transformed domain, after applying both Fourier transform and Fourier–Bessel transform, are derived utilizing the global stiffness matrix method, analogous to a 1-D finite element approach. Solutions in the physical domain are then obtained using the fast and accurate Fourier series and Fourier–Bessel series approach proposed in this paper. The most attractive feature of this novel approach is that the expansion coefficients (or Love numbers) can be pre-calculated, saved, and repeatedly used for other field points. Through numerical analyses, we quantitatively investigate the effect of the water depth and source/receiver types/locations on different wave features (i.e., Scholte, fast-guided, and acoustic modes) from different perspectives (i.e., dispersion curve, Green’s function, frequency-velocity spectrum (FVS), and waveform). We find that (1) unlike the impact source, stronger acoustic waves can be produced by the explosive source, which sometimes causes difficulty in identifying the Scholte waves. (2) The acoustic and interface-guided waves exhibit distinct behaviors depending on the source and receiver locations. Scholte waves and fast-guided waves are weakened when the source and receivers are elevated far from the water/soil interface. Moreover, (3) the response of the Scholte wave can be enhanced by muting the direct acoustic waves. However, (4) the fast-guided wave may also arise in underwater surveys when the Vs of the underlying half-space is much higher than those of the upper layers, exerting a significant influence on the shallow water scenarios and posing challenges for correct mode identification. With the ability to model the entire wavefield (or FVS) that takes into account all propagation modes and the actual survey configuration, inversion can be performed by fitting the entire FVS. This approach eliminates the need to pick dispersion curves and provides a more accurate and stronger model constraint.

Dynamic response of sandwich functionally graded nanoplate under thermal environments and elastic foundations using dynamic stiffness method

The present paper introduces the development of dynamic stiffness method for analyzing small-scale sandwich functionally graded nanoplates resting on elastic foundation in thermal environments. The mathematical formulation is based on classical plate theory in conjunction with nonlocal elasticity theory. The governing equation is derived using Hamilton’s principle. The dynamic stiffness matrix is obtained through the application of the Levy displacement approach and assembled to form the global stiffness matrix. The final matrix is solved for natural frequency of the plates using the Wittrick–Williams algorithm. The proposed methodology is validated against existing literature, demonstrating a strong agreement. Various parametric studies explore the effects of thermal environments, volume fraction index, sandwich configurations, elastic foundation characteristics, nonlocal parameter and boundary conditions. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of sandwich functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Mixed finite element implementation of plane crack problems within strain gradient elasticity

AbstractAn approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method is used, where stresses, strains, and displacements are considered as independent variables. This significantly simplifies the pre‐requirement for approximating functions and allows one to use the simplest triangular finite elements with a linear approximation of displacements. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses in nodes are treated as local unknowns. This discrete problem is solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. The modeling plane crack problem (tension of a square plate with a central crack) is solved. The obtained solution agrees with the results available in the literature. Good convergence is achieved by refining the mesh for all scale parameters.

Parameter identification of beam bridges based on stiffness separation method

The complex structures and extensive computational workloads associated with beam bridges pose challenges in identifying their parameters. This study proposed a stiffness separation method for the parameter identification of beam bridges. Firstly, this method involves decomposing the global stiffness matrix into substructure matrices, thereby reducing the dimensionality of the stiffness matrix and facilitating parameter identification. Secondly, the proposed method can assign the unknown parameters to each substructure which simplifies the parameter identification problem. Thirdly, the obvious local responses can be obtained by applying loads to individual substructures. The accuracy and efficiency of the proposed method are validated through a simply supported and continuous beam bridge. Overall, this method simplifies objective functions and improves the efficiency of parameter identification.

Vectfem: a generalized MATLAB-based vectorized algorithm for the computation of global matrix/force for finite elements of any type and approximation order in linear elasticity

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller.

Remeshing and eigenvalue stabilization in the finite cell method for structures undergoing large elastoplastic deformations

Large strain analysis is a challenging task, especially in fictitious or immersed boundary domain methods, since badly broken elements/cells can lead to an ill-conditioned global tangent stiffness matrix, resulting in convergence problems of the incremental/iterative solution approach. In this work, the finite cell method is employed as a fictitious domain approach, in conjunction with an eigenvalue stabilization technique, to ensure the stability of the solution procedure. Additionally, a remeshing strategy is applied to accommodate highly deformed configurations of the geometry. Radial basis functions and inverse distance weighting interpolation schemes are utilized to map the displacement gradient and internal variables between the old and new meshes during the remeshing process. For the first time, we demonstrate the effectiveness of the remeshing approach using various numerical examples in the context of finite strain elastoplasticity.

An efficient static solver for the lattice discrete particle model

AbstractThe lattice discrete particle model (LDPM) has been proven to be one of the most appealing computational tools to simulate fracture in quasi‐brittle materials. Despite tremendous advancements in the definition and implementation of the method, solution strategies are still limited to dynamic algorithms, resulting in prohibitive computational costs and challenges related to solution accuracy for quasi‐static conditions. This study presents a novel static solver for LDPM, introducing fundamental innovation: (1) LDPM constitutive laws are modified to provide continuous response through all possible strain/stress states; (2) an adaptive arc‐length method is proposed in combination with a criterion to select the sign of the iterative load factor; (3) an adaptive limit‐unloading–reloading path switch algorithm is proposed to restrict oscillations in the global stiffness matrix. Extensive validation of the proposed approach is presented. Numerical results demonstrate that the static solver exhibits satisfactory convergence rates, significantly outperforming available dynamic solutions in computational efficiency.

Fast prediction of concrete equivalent modulus based on the random aggregate model and image quadtree SBFEM

Abstract To evaluate the mechanical property of concrete materials rapidly, a fast prediction model of the concrete equivalent modulus is proposed based on the random aggregate model and scaled boundary finite element method (SBFEM). First, a random aggregate model of meso-concrete is employed to construct the representative volume element (RVE) according to the aggregate content, gradation, shape, etc. Second, the RVE model is transformed to be a grayscale image and stored as a digital matrix. The quadtree mesh is partitioned automatically for simulation by SBFEM. There are only six types of unique subdomains, and the hanging node does not affect the simulation accuracy. The global stiffness matrix can be assembled directly according to the six subdomain stiffness matrices. Finally, the equivalent modulus is predicted by using the numerical homogenization method. Several numerical examples are employed to verify this model, and the results are compared with that of other methods. The result indicates that the proposed model can efficiently determine the equivalent modulus. Furthermore, the effect of the aggregate gradation, shape, porosity, and pore water are studied and analysed in this work. The proposed model is potential and helpful in predicting the mechanical properties of concrete or other composite materials.

Influence of Prestress on Vibration Frequency of Beam String Structures Based on Exact Matrix Stiffness Method

Beam string structures (BSS) are frequently employed in large-span spatial structures due to their high load-carrying efficiency and elegant appearance. Owing to the neglect of the influence of prestress, the natural frequency of the BSS is often overestimated which could be crucial to the dynamic behavior of BSS. This paper presents an exact matrix stiffness analysis method (MSM) for investigating the relationship between prestress and the natural frequency of the planar BSS. A novel dynamic stiffness matrix of beam–columns considering the natural frequency and axial force is used to develop the MSM. The dynamic analysis of the structural vibration stability of BSS is performed by using the global structural stiffness matrix which is assembled from the element stiffness matrices in the global coordinate system. The proposed MSM with the exact dynamic element stiffness matrix for the dynamic analysis of the BSS is verified by comparing with previous results based on the finite element method. The illustrative examples demonstrate that the prestress in the cable has a negative effect on the natural frequency of the BSS and should be considered in the dynamic analysis. As the axial force increases from zero to the buckling load [Formula: see text], the natural frequency of the BSS decreases from the maximum vibration frequency to zero. The influence of prestress on the vibration frequencies of BSS is particularly significant when the prestress to balance the loads is large, especially in the case of large dead and live loads (e.g. floor slabs).

Parameter Identification of Multispan Rigid Frames Using a Stiffness Separation Method.

Identifying the parameters of multispan rigid frames is challenging because of their complex structures and large computational workloads. This paper presents a stiffness separation method for the static response parameter identification of multispan rigid frames. The stiffness separation method segments the global stiffness matrix of the overall structure into the stiffness matrices of its substructures, which are to be computed, thereby reducing the computational workload and improving the efficiency of parameter identification. Loads can be applied individually to each separate substructure, thereby guaranteeing obvious local static responses. The veracity and efficacy of the proposed methodology are substantiated by applying it to three- and eight-span continuous rigid frame structures. The findings indicate that the proposed approach significantly enhances the efficiency of parameter identification for multispan rigid frames.

Plane crack problems within strain gradient elasticity and mixed finite element implementation

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

Quasi-static testing based structural modal parameter estimation

ABSTRACT Structural modal identification is generally implemented within a dynamic framework, requiring multidisciplinary knowledge and adequate operational experience. This preliminary study attempts to explore an easy-to-handle quasi-static alternative method for dynamics-based modal identification of beam-type structures. Acceleration time-history data are replaced by quasi-static deflections induced by slow moving loads. A concept of quasi-static deflection influence surface is proposed based on the theory of influence lines and the principle of virtual work. Its analytical expression is simplified into a deflection matrix by choosing specific points on the surface according to deflection measurement coordinates. The matrix form is divided by external loads to obtain a generalised quasi-static flexibility matrix, which is used to replace the inversion of the global stiffness matrix in the modal eigenvalue equation. The lumped-mass method is also employed to establish the mass matrix. Subsequently, the eigenvalue equation is analytically solved seeking for modal frequencies and mode shapeswithout performing modal tests. The feasibility of the proposed method has been successfully verified against three experimental examples including a continuous box girder with variable cross sections. It was observed that the modal frequencies and mode shapes estimated by the proposed method were very close to those given by the dynamically modal tests.

Natural frequency analysis of gear system with interval shaft misalignment in multidimensions

Due to the existing tolerances, the shaft misalignment of gear system is inevitably uncertain and shows interval property, which will cause the natural frequency to fluctuate in an interval range, therefore the traditional dynamics analysis that takes it as a deterministic parameter cannot reflect the interval characteristics of natural frequency. To overcome this conundrum, an improved interval eigenvalue algorithm based on matrix non-negative factorization and the Chebyshev inclusion function was proposed to find the natural frequency range of gear system under such uncertainty. First, the average meshing stiffness range of the gears with interval shaft misalignment is obtained by the Chebyshev interval approach. Then combined with the consideration of the remaining excitation parameter deviation coefficients, the global interval stiffness matrix and mass matrix of a single-stage gear system are obtained. Afterwards, the non-negative factorizations of these two interval matrices are carried out and parameter vertex theory is finally employed to obtain the upper and lower bounds of the eigenvalues. The feasibility of the applied algorithm is verified through comparison with the scanning method. Numerical results show that the uncertainty of shaft misalignment will enormously widen the system's resonance bandwidth and intensify oscillation, these will provide a new and valuable perspective for the tolerance allocation in gearbox design.

Flexural wave propagation characteristics of metabeam with simultaneous acoustic black hole and local resonator

This paper studies the flexural wave propagation in a metamaterial beam with the coupled acoustic black hole and local resonators (ABHLR). The main objectives are to derive the dynamic stiffness matrix of the coupled ABHLR unit cell and analyse its wave propagation characteristics. Initially, a closed-form solution of Euler–Bernoulli’s beam with an indented acoustic black hole is derived. From this, an exact dynamic stiffness matrix, incorporating local resonator effects, is derived using the spectral element method. Dispersion diagrams are generated by combining Bloch’s theorem with ABHLR transfer matrix. The coupled effects of acoustic black hole (ABH) and local resonator (LR) on the bandgap characteristics are investigated. The coupled ABHLR metamaterial shows either a positive or negative effect on bandgap formations, depending on the parametric values of the unit cell. As a positive effect, for some parametric values, the coupled ABHLR cell simultaneously produces bandgaps at lower frequencies (LR-based) and higher frequency ranges (ABH-based). Similarly, for other parametric value, as a negative effect, the bandgaps of ABHLR cell get merged but the pass bands appear on the overlapping region. This pass-band appears due to the constructive interference of ABH and LR, on each other. However, the attenuation bandgap with a higher attenuation rate is obtained on the overlapping region due to the destructive interference of LR and ABH effect. Finally, the global stiffness matrix of a finite beam with 30 unit cells is developed to analyse the frequency domain response and the results are validated with band gap analysis of the unit cell. The frequency domain response is also validated with finite element simulations using the commercial software. Results show that ABH and LR parameters have to be optimally designed to utilize the attenuation effects and avoid constructive interference due to coupling effects. The coupling of metamaterials can aid in vibration suppression and multiple-frequency filtering in mechanical and aerospace applications.

A mixed finite element/finite volume formulation for granular bentonite mixtures

A mixed finite element/finite volume formulation is presented for characterising the macroscopic behaviour of granular bentonite mixtures. The formulation enables fitting the hydro-mechanical behaviour of a mixture parting from a double porosity numerical model. The granular bentonite mixture is abstracted as a finite set of homogeneous units made with bentonite grains (where microstructure and macrostructure coexist) embedded in megapores. The separate mass balances of macrostructural and microstructural water are solved for each unit using the finite volume method. The water mass balance in the megaporosity is calculated for the entire domain using the finite element method. To solve the equilibrium equation, a displacement-based finite element method is applied after defining a global stiffness matrix that integrates the effect of megaporosity. After implementing these approaches in a double porosity numerical model, satisfactory modelling results were obtained in agreement with the experimental tests. The results of the proposed model effectively fit those of a triple porosity bentonite unit model that uses a contact model for displacement compatibility of the units. This contributes to the qualification of the presented model’s ability to produce compatible strains of the units using the proposed global stiffness.

Seismogram Synthesis of Multi-Scale Layered Transversely Isotropic Saturated Half-Space Using a Revised Stiffness Matrix Method

Seismogram synthesis for a multi-scale transversely isotropic (TI) saturated layered half-space subjected to seismic dislocation source is studied by a revised stiffness matrix method in this paper. First, based on Biot’s theory of wave propagation in a fluid-saturated porous solid, the governing equations of motion in TI saturated medium are transformed by Fourier–Hankel integral transform. Next, the contribution of the seismic dislocation source can be imposed in the form of body forces. Then, the global stiffness matrix with well-conditioned properties is obtained by assembling each layer’s exact stiffness matrix, which can avoid accumulating errors in the matrix transmission. By extracting the divergence exponent terms of the stiffness matrix concerning thickness and wavenumber, the bottleneck of simulating the wave propagation process in multi-scale half-space, where the layer thickness and velocity vary sharply from the near-surface region to the crustal zone, could be overcome, and a fast and accurate seismogram synthesis could be achieved. The accuracy of the proposed method is verified by comparing the calculated results with those of published literature. Finally, the method is applied to the multi-scale system with realistic superficial fine layers. Numerical examples regarding thickness, TI properties and porosity are presented to illustrate the effects of superficial layers on ground motion. The numerical results show that the surface properties affect the wave propagation and energy distribution, leading to variable impacts on the surface response in different directions; this phenomenon is more pronounced in the case of high-frequency ground motions.

Optimal bracing system design for funicular twin arches against out-of-plane buckling

Twin arch bridges are often designed with a bracing member system that can significantly enhance the out-of-plane stability. This study presents an exact matrix stiffness method (MSM) for the out-of-plane buckling analysis of funicular twin arch bridges and investigates the optimal bracing system design. A novel 14 × 14 s-order element stiffness matrix of three-dimensional beam–columns was used in the MSM and the out-of-plane buckling analysis was performed after assembling the exact global structural stability stiffness matrix. The influences of the bracing location, number, length, and flexural rigidity of both the transverse bracing and X-bracing systems on the buckling capacity of the twin arch system were investigated. The in-plane stiffness of the bracing system (i.e., the horizontal flexural rigidity in a transverse bracing system) was found to be the critical factor in suppressing the out-of-plane buckling of the twin arch system. In addition, the twin arch system achieved the maximum out-of-plane buckling capacity with the bracing location at S/4, where S is the total arc length. For optimal bracing system design of funicular twin arches, it is recommended that one uses the more economical X-bracing system or a transverse bracing system with adequately rigid bracing members.

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

Insight into numerical characteristics of embedded finite elements for pile‐type structures employing an enhanced formulation

AbstractA number of problems encountered in the field of computational geotechnics requires the modelling of numerous pile‐type structures embedded in a three‐dimensional soil continuum. Traditional modelling approaches to this problem result in computational costs that must be regarded as unbearable for most practical purposes. As an attractive alternative, we propose an enhanced embedded FE model with implicit interaction surface (EB‐I) where coupling between the contacting domains is realized by an implicit surface‐to‐volume (2D‐to‐3D) coupling scheme. As a novelty, the latter implements a non‐linear interface constitutive model that allows for explicit consideration of endpoint interaction, but does not represent a constraint for the solid mesh generation. As the slender structure is discretized employing the Timoshenko beam theory, shear deformability is explicitly considered, as opposed to earlier EB‐I‐type models reassessed in this paper. The credibility of the proposed EB‐I is numerically validated on the basis of comparative studies. It is found that the 2D‐to‐3D coupling scheme generally improves the well‐posedness of the resultant global stiffness matrix, making the proposed EB‐I computationally competitive to geometrically simplified line‐to‐volume coupling schemes. Future lines of research are carefully addressed throughout this work and include the normal stress recovery technique as well as applications to large‐scale simulations.