A multi-level finite element nodal ordering using algebraic graph theory

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

Grouping the nodes and form a cluster with less energy consumption by that maximizing the life time is an challenging task in WSNs. The two important steps in clustering are Cluster formation and Cluster Head (CH) selection. The novel and efficient clustering called Clustering using Eigen Values (CEV) is proposed in this paper with the increased lifetime of the sensor nodes using the spectral graph theory. This work uses the Laplacian matrix of spectral theory for clustering. The Eigen values of Laplacian Matrix and its corresponding eigenvector are used to group the nodes of WSN. CH is selected using fuzzy logic and constraints on energy and distance. This work is evaluated and compared with LEACH and HEED for performance comparison. The results obtained in this work show that the proposed work yields better performance when compared to other existing cluster based techniques based on the parameters such as network lifetime and energy consumption.

Multi-agent systems (MASs) are ubiquitous in natural and artificial systems. This paper aims to establish the finite-time adaptive consensus criterion for a class of MASs with nonlinear dynamics. Traditionally, the finite-time consensus criterion is often established based on the prior information on Lipschitz constants and the eigenvalues of Laplacian matrix. However, it is difficult to acquire the above prior information for most real-world engineering systems. To overcome the above difficulty, this paper develops the finite-time consensus criteria for a class of MASs with nonlinear dynamics via adaptive technique. In detail, we design the finite-time distributed node-based and edge-based adaptive consensus protocols for a class of MASs with fixed and switching topologies. Numerical simulations are also given to validate the proposed finite-time adaptive consensus criterion.

In this note, we investigate cooperative robust consensus problem for a class of linear heterogeneous multi-agent systems (MASs) with parameter uncertainty in agents. All the agents can have nonidentical dynamic dimensions. By combining the communication network, a distributed dynamic compensator is established based on only the measurable output information of the exosystem, which provides an estimated value of the state of the exosystem for each agent. The distributed dynamic output feedback protocol (DDOFP) is proposed by utilizing the exosystem compensator, through which the cooperative robust consensus is achieved under switching topology by following output regulation framework. Here, we introduce the stabilization scheme for a class of liner switched systems to analyse the closed-loop stability, which avoids the dependence on the eigenvalues of Laplacian matrix, such that it is a fully distributed control strategy without any global information. Finally, the simulation example is illustrated to verify the feasibility and effectiveness of the theory results.

There has been a growing demand for nonlinear model updating procedures in the structural dynamics community as advanced vehicles have started to incorporate nonlinearity into their designs. Finite element (FE) model updating is difficult for a nonlinear structure for several reasons: there may be many unknown parameters in the FE model so that multiple solutions may exist, and it is very expensive to compute the structure’s nonlinear normal modes from a full FE model that are an excellent metric to use for nonlinear updating. A recent work showed that some of these challenges can be overcome by updating a nonlinear reduced order model (ROM) rather than the full FE model. The updated ROM can be used to compute response statistics, stresses and ultimately life of the structure. However, one drawback of this approach is that one does not gain physical insight into which parameters in the FE model were in error, and so it is difficult to transfer the lessons learned to future FE models. This work explores the feasibility of updating a FE model to correlate with a ROM with a set of known parameters. The target ROM parameters consist of the linear stiffness and the nonlinear stiffness coefficients which have been identified previously by updating the ROM to correlate with experimental measurements. In this process an optimization routine is setup in which the free parameters are those of the FE model, such as boundary stiffness springs, imperfections, pre-stress, etc. The optimization procedure is wrapped around a ROM creation algorithm that iteratively tunes the FE parameters, creates a ROM, and evaluates the ROM parameters with respect to the target ROM in order to minimize the objective function. This work will demonstrate the procedure on a numerical example of a curved beam in order to verify its effectiveness on the nonlinear model updating.

Skull stiffness of children is of great significance to protect brain tissues during traffic accidents. In this paper, the effects of skull stiffness on traumatic brain injury (TBI) are investigated by using a validated 6-year-old child head finite element (FE) model according to Chinese standard GB/T 24550-2009 Results showed that the HIC value, intracranial pressure, Von Mises stress and shear strain of brain tissue increased with the decrease of skull stiffness during the impact of FE model with engine hood. Therefore, the injury risk of the child head decreases with the increase of skull stiffness. And the skull with higher stiffness can provide a better protection for brain tissues during traffic accidents.

There are numerous applications of graph theory and algebraic graph theory in combinatorial optimization and optimal structural analysis. In this paper, a new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented. Furthermore, the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied. This block diagonalization leads to an efficient method for the eigensolution of adjacency and Laplacian matrices of special graphs. The eigenvalues and eigenvectors are used for efficient nodal ordering and partitioning of large structural models. The present method is far more simple than any existing general approach.

Introduction: Three-dimensional (3D)-printed custom pelvic implants have become a clinically viable option for patients undergoing pelvic cancer surgery with resection of the hip joint. However, increased clinical utilization has also necessitated improved implant durability, especially with regard to the compression screws used to secure the implant to remaining pelvic bone. This study evaluated six different finite element (FE) screw modeling methods for predicting compression screw pullout and fatigue failure in a custom pelvic implant secured to bone using nine compression screws. Methods: Three modeling methods (tied constraints (TIE), bolt load with constant force (BL-CF), and bolt load with constant length (BL-CL)) generated screw axial forces using functionality built into Abaqus FE software; while the remaining three modeling methods (isotropic pseudo-thermal field (ISO), orthotropic pseudo-thermal field (ORT), and equal-and-opposite force field (FOR)) generated screw axial forces using iterative physics-based relationships that can be implemented in any FE software. The ability of all six modeling methods to match specified screw pretension forces and predict screw pullout and fatigue failure was evaluated using an FE model of a custom pelvic implant with total hip replacement. The applied hip contact forces in the FE model were estimated at two locations in a gait cycle. For each of the nine screws in the custom implant FE model, likelihood of screw pullout failure was predicted using maximum screw axial force, while likelihood of screw fatigue failure was predicted using maximum von Mises stress. Results: The three iterative physics-based modeling methods and the non-iterative Abaqus BL-CL method produced nearly identical predictions for likelihood of screw pullout and fatigue failure, while the other two built-in Abaqus modeling methods yielded vastly different predictions. However, the Abaqus BL-CL method required the least computation time, largely because an iterative process was not needed to induce specified screw pretension forces. Of the three iterative methods, FOR required the fewest iterations and thus the least computation time. Discussion: These findings suggest that the BL-CL screw modeling method is the best option when Abaqus is used for predicting screw pullout and fatigue failure in custom pelvis prostheses, while the iterative physics-based FOR method is the best option if FE software other than Abaqus is used.

Background Accurate biomechanical investigation of human intervertebral discs (IVDs) is difficult because of their complicated structural and material features. Aim To investigate probabilistic distributions of the biomechanical responses of the IVD by considering varying nonlinear structural and material properties using a stochastic finite element (FE) model. Methods A FE model of a L3-4 disc was reconstructed, including the nucleus pulposus (NP), annular matrix and fibers. A Monte Carlo method was used to randomly generate 500 sets of the nonlinear material properties and fiber orientations of the disc that were implemented into the FE model. The FE model was analyzed under seven loading conditions: a 500 N compressive force, a 7.5Nm moment simulating flexion, extension, left-right lateral bending, and left-right axial rotation, respectively. The distributions of the ranges of motion (ROMs), intradiscal pressures (IDP), fiber stresses and matrix strains of the disc were analyzed. Results Under the compressive load, the displacement varied between 0.29 mm and 0.76 mm. Under the 7.5Nm moment, the ROMs varied between 3.0° and 6.0° in primary rotations. The IDPs varied within 0.3 MPa under all the loading conditions. The maximal fiber stress (3.22 ± 0.64 MPa) and matrix strain (0.27 ± 0.12%) were observed under the flexion and extension moments, respectively. Conclusion The IVD biomechanics could be dramatically affected by the structural and material parameters used to construct the FE model. The stochastic FE model that includes the probabilistic distributions of the structural and material parameters provides a useful approach to analyze the statistical ranges of the biomechanical responses of the IVDs.

A method for continuous box-girder bridge structure finite element (FE) model updating based on static-load testing is proposed. The procedure includes field static-load testing, FE modeling, FE model (parameter) updating, etc. Based on this procedure, the structure FE model updating for an expressway 5-span pre-stressed concrete continuous box-girder bridge is conducted. And the bridge structure static FE model, which is consistent with the static-load testing actual measure response, is gained. The updated FE model (parameter) is foundation of structural performance evaluation for the bridge.

Static and dynamic FE analysis of an RC protective structure dedicated to snow avalanche mitigation

Partial element equivalent circuit (PEEC) is a convenient method for modeling transient behavior of high temperature superconductors (HTS) magnets made of metal-as insulation windings. In HTS coils, especially those made of tapes, screening currents are known to lower the magnetic field, to generate losses, and to contribute to the mechanical stress. Including screening current calculation in PEEC method can be achieved by subdividing the superconducting tape along its width, leading to larger models and much longer computation time. Herein, we propose a finite element (FE) model for screening currents calculation, which can be combined with PEEC model for transport current addition. The FE model is defined as thin shell using formulation with normal component of electric vector potential T as nodal variable, calculated by an iterative procedure. Source term in the FE model is obtained from variation of radial flux density. Critical current densities calculated in the FE model are fed back to the PEEC model at each step of transport current calculation. This work presents in detail the FE model and its coupling with PEEC model. The model is validated by comparison with other 3D FE models and models with axial symmetry. The effect of reducing the size of FE model, by averaging screening currents in adjacent turns, is also presented in this study. Finally, we present simulation results for current ramps and a quench of a double-pancake magnet, obtained by PEEC with and without adding FE screening current model.

Probabilistic finite element predictions of the human lower limb model in total knee replacement

We invert InSAR from ascending and descending passes of ENVISAT/ASAR data to estimate the distribution of coseismic slip for the 2008 Mw 7.2 Yutian earthquake by separately using uniform finite element models (FEMs), non-uniform FEMs, and analytical elastic half-space (Okada) models. We use the Steepest Descent method and Laplacian smoothing to regularize and estimate the slip distribution for each of these different models. Fault surface ruptures interpolated from Quickbird optical images constrain the strike of the fault. The uniform FEM and Okada models assume a fault embedded in a homogeneous and elastic finite domain or half-space, respectively, while the non-uniform FEM describes a domain with spatially variable elastic properties according to the geological structure of Tibet plateau and Tarim. We find that the estimated slip distribution of the non-uniform FEM is similar to that of the uniform FEM but has a slightly larger moment (~2 %). However, more noticeable discrepancies occur between the slip distributions of the uniform FEM and Okada model, where the slip pattern estimated with the FEM looks more scattered and locates at a lower depth. We conclude that, for the case of the Yutian earthquake, the improvement brought by simulating the distribution of geologic material properties is insignificant. This suggests that the uniform geologic structure of Okada models may be sufficient for simulating large intra-continental earthquakes.

Electrical Discharge Machining (EDM) is a machining method commonly used to produce complex shapes and deep holes by eroding hard metals with an electric arc. There is a growing demand for process simulation using finite element models in order to improve the quality and efficiency of EDM, to reduce costs, to improve resource efficiency, and to facilitate its application in critical areas such as aerospace and mechanical engineering. Finite element models have greatly improved the prediction accuracy of EDM processes, simulated complex hybrid machining processes, and provided important guidance for the optimization of EDM processes. This paper systematically reviews the research progress of finite element modeling for EDM. Finite element method modeling is evaluated mainly in terms of four indicators: material removal rate, surface roughness, tool wear ratio, and recast layer thickness. Firstly, the importance and application of EDM are described, and the EDM finite element method modeling and its advantages are summarized. Then, the single-spark simulation model and the multi-spark simulation model of EDM are compared and discussed. Among the mainstream finite element models, the prediction error of the material removal rate for single-spark simulation ranges from 8.2% to 14.75%, while the prediction error of the recast layer thickness for multi-spark simulation can be as low as 1.98%. Finally, the applications of finite element modeling in EDM hybrid machining processes’ performance prediction and new material machining are summarized, and future research directions and trends in EDM finite element modeling are predicted.