Linear Elements Research Articles

Enriched Finite Element Method Based on Interpolation Covers for Structural Dynamics Analysis

This work proposes a novel enriched finite element method (E-FEM) for structural dynamics analysis. We developed the enriched 3-node triangular and 4-node tetrahedral displacement-based elements (T-elements). The standard linear shape functions of these T-elements were enriched using interpolation cover functions over each patch of elements. We also introduced and compared different orders of cover functions; higher-order functions obtained higher computational performance. Subsequently, the forced and free vibration analyses were performed on various typical numerical examples. The proposed enriched finite element method generated more precise numerical results and ensured faster convergence than the original linear elements.

Integration of locating baseline-free nonlinear elements and identifying model-free nonlinear restoring forces in structures

Structural local nonlinearity often occurs when a structure is subject to strong excitation. Therefore, the identification of structural nonlinearity localizations and the characteristics of nonlinear restoring forces is of great significance in ensuring structural safety and reliability. Among the existing nonlinearity identification methods, locating structural nonlinear elements under seismic excitations is still a challenging problem and identifying model-free nonlinear restoring forces requires prior knowledge of nonlinearity position. To overcome these limitations, a novel two-step nonlinearity localization and identification algorithm is proposed in this paper. First, by considering the wavelet energy distribution in the nonlinear element become different from those of other linear elements, relative wavelet entropy which is able to quantify the difference or similarity between two wavelet energy distributions is employed to localize structural nonlinear elements based on the sparsity of structural nonlinearity. In the localization process, structural response data of the baseline linear structure are not required. Then, nonlinear restoring forces from nonlinear elements are regarded as ‘unknown virtual forces’ to the underlying linear structure. The generalized Kalman filter with unknown input, which was recently proposed by the authors, is used to identify the ‘unknown virtual forces’ and reconstruct the characteristics of nonlinear restoring forces without the models of nonlinear restoring forces. The proposed method is verified by numerical simulations of different types of nonlinear elements in different structures. Furthermore, it is tested by locating the MR damper and identifying its model-free nonlinear restoring forces in a laboratory four-story shear-type frame.

Non-conforming and conforming five-node quadrilateral graded finite elements

This study evaluates the performance of conforming and non-conforming transition graded finite elements dealing with graded materials. Two-dimensional 5-node non-conforming quadrilateral element (Q5) is used to transition from one large 4-node linear quadrilateral to two small 4-node linear quadrilateral elements (Q4). In addition, a conforming Q5 element is used in between one Q4 and one 8-node quadratic elements (Q8). A single element test of Q5 graded element is done. Then, the accuracy and convergence study of Q5 graded element are examined. We found that the use of Q5 transition elements improved computational efficiency while maintaining the accuracy of the solutions.

The Effect of the Position Determination Error for Flexible Linear Array Elements on the Tomogram Focusing.

In the article, the study of the quality of tomogram focusing during the inspection of objects with curved surfaces by flexible acoustic array was described. The main goal of the study was theoretically and experimentally define the acceptable deviation limits of the elements' coordinates values. The tomogram reconstruction was performed by the total focusing method. The Strehl ratio was chosen as a criterion for assessing the quality of tomogram focusing. The ultrasonic inspection procedure were simulated and validated experimentally by means of convex and concave curved arrays. In the study, it was proven that the elements coordinates of the flexible acoustic array were determined with an error of no more than 0.18λ and the tomogram image was obtained in sharp focus.

Structural Design of the Substructure of a 10 MW Floating Offshore Wind Turbine System Using Dominant Load Parameters

Fully coupled integrated load analyses (ILAs) to evaluate not only the load response but also the structural integrity are required to design a floating offshore wind turbine, since there has been no firmly established approach for obtaining the structural responses of a FOWT substructure in the time domain. This study aimed to explore if a direct strength analysis (DSA) technique that has been widely used for ships and offshore structures can adequately evaluate the FOWT substructure. In this study, acceleration and nacelle thrust were used for the dominant load parameters for DSA. The turbine thrust corresponding to the 50-year return period was taken from the literature. The acceleration response amplitude operator (RAO) was obtained through frequency response hydrodynamic analysis. The short-term sea states defined by the wave scatter diagram (WSD) of the expected installation area was represented by the JONSWAP wave spectrum. To account for the multi-directionality of the short-crested waves, the 0th order moments of the wave spectrum were corrected. The probabilities of each short-term sea state and each wave incidence angle were applied to derive the long-term acceleration for each return period. DSA cases were generated by combining the long-term acceleration and nacelle thrust to maximize the forces in the surge, sway, and heave directions. Linear spring elements were placed under the three outer columns of the substructure to provide soft constraints for hive, roll, and pitch motions. Nonlinear spring elements with initial tension were placed on the three fairlead chain stoppers (FCSs) to simulate the station-keeping ability of the mooring lines; they provided initial tension in the slacked position and an increased tension in the taut position. The structural strength evaluation of the coarse mesh finite element model with an element size same as the stiffener spacing showed that high stresses exceeding the permissible stresses occurred in the unstable members of the substructure. The high stress areas were re-evaluated using a fine mesh finite element model with an element size of 50 mm × 50 mm. The scope of structural reinforcement was identified from the fine mesh analyses. It was found that the DSA can be properly utilized for the substructure strength assessment of a FOWT.

A kinematically stabilized linear tetrahedral finite element for compressible and nearly incompressible finite elasticity

We propose a stabilized linear tetrahedral finite element method for static, finite elasticity problems involving compressible and nearly incompressible materials. Our approach relies on a mixed formulation, in which the nodal displacement unknown filed is complemented by a nodal Jacobian determinant unknown field. This approach is simple to implement in practical applications (e.g., in commercial software), since it only requires information already available when computing the Newton–Raphson tangent matrix associated with irreducible (i.e., displacement-based) finite element formulations. By nature, the proposed method is easily extensible to nonlinear models involving visco-plastic flow. An extensive suite of numerical tests in two and three dimensions is presented, to demonstrate the performance of the method.

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

Improving the convergence rate of Newman’s battery model using 2nd order finite element method

While battery cycling experiments last for years, battery modelling can save time and is environment friendly, to study the degradation mechanisms of lithium-ion batteries. However, battery models are being challenged by issues of long-time calculation, poor accuracy and bad convergence. In this work, a Julia-based framework (Jubat) has been developed for Newman’s battery model. This framework benefits from the high execution efficiency of Julia language, so the mean computational time is lower than its counterpart PyBaMM written in Python. Governing equations are solved by the finite element method, and good accuracy is obtained. The convergence rate is improved by 2nd order elements, which capture better flux variation within elements than the finite volume method and the finite element method using linear elements, with slightly higher computation cost. Numerical examples show easy use of this framework and ability to solve practical problems like drive cycles. This work provides a compact, readable and fast framework for battery modelling and is useful for the rapid development of advanced battery management system. The code is open-source at https://github.com/weilongai/JuBat.

Development of linear-element boundary element method for inverse solution from induced far-field displacements to reservoir loading source

An inverse solution strategy using linear element of Boundary Element Method(BEM)was developed, which will become a novel application base to potentially reflect the equivalent mechanical effect around reservoir corresponding to CO2 spreading during its injection process. Firstly, a linear-element BEM code was studied to improve the accuracy in analyzing the connection between induced ground minute deformations and underground inner loading. Secondly, the space regularization was adopted to deal with the ill-posed problem during the optimization to inverse problem solution. The boundary conditions of a near-field structure as active source were reversely figured out in an efficient numerical way based on its passive deformation that was sampled at a few specific positions at far field. Meanwhile, the effectiveness of linear-element BEM in solving inverse problems was demonstrated. Finally, based on the implementation of hydraulic fracturing in a coal seam in Alberta, Canada, the monitored data of its induced surface displacements was input to invert the equivalent internal pressure onto are-opened fracture within reservoir. The results show that the proposed inverse solution workflow and programmed algorithm are applicable to effectively guiding hydraulic fracturing interpretation so as to further facilitating coal bed methane recovery.

Implementation of photon partial distinguishability in a quantum optical circuit simulation

We are concerned with numerical simulations of quantum optical circuits under certain realistic conditions, specifically that photon quantum states are not perfectly indistinguishable. The partial photon distinguishability presents a serious limitation in implementation of optical quantum information processing. In order to properly assess its effect on quantum information protocols, accurate numerical simulations, which closely emulate quantum circuit operations, are essential. Our specific objective is to provide a computer implementation of the partial photon distinguishability which is in principle applicable to existing simulation techniques used for ideal quantum circuits and which avoids a need for their significant modification. Our approach is based on the Gram-Schmidt orthonormalization process, which is well suited for our purpose. Photonic quantum states are represented by wavepackets which contain information on their time and frequency distributions. In order to account for the partial photon distinguishability, we expand the number of degrees of freedom associated with the circuit operation extending the definition of the photon channels to incorporate wavepacket degrees of freedom. This strategy allows to define delay operations in the same footing as the linear optical elements.

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

AbstractThis article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history‐dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time‐independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non‐intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.

Numerical simulation of wind-structure-soil interaction effects on the CAARC tall building model using hybrid CUDA-OpenMP parallelization

In the present work, the dynamic effects of the wind action over the Commonwealth Advisory Aeronautical Council (CAARC) tall building model are numerically evaluated considering the influence of soil-foundation interaction. A numerical model is developed considering a partitioned coupling scheme for fluid-structure-soil interaction, in which the fluid, structure and soil subsystems are solved sequentially using independent algorithms. The Finite Element Method (FEM) is adopted for spatial discretization, where linear hexahedral elements with underintegration techniques are used. The fundamental flow equations are kinematically described using an arbitrary lagrangian-eulerian (ALE) formulation and numerically solved using the explicit two-step Taylor-Galerkin scheme, while Large Scale Simulation (LES) is employed for the treatment of turbulent flows. Structure and soil are considered as deformable elastoplastic media, using a corotational approach to deal with physical and geometric nonlinearities. Dynamic equilibrium equation is solved using the generalized-α method and infinite elements are utilized in the soil domain to avoid elastic wave reflections. Load transfer between soil and structure is performed using a three-dimensional contact algorithm based on the penalty method that allows separation and sliding along soil-structure interfaces. A hybrid parallelization strategy based on CUDA-OpenMP techniques is proposed to accelerate the processing time associated with the present simulations. Numerical results obtained here are compared with numerical and experimental results reported by other authors, where one can observe that soil-structure interaction may significantly influence the building response due to wind loading and effects of aeroelastic instability may be considerably reduced in the presence of soil support.

Distributed Adaptive Fuzzy Modeling for Industrial Gas–Electricity Networks With Multioperation Modes

Industrial gas-electricity networks typically operate in several modes due to large demand variations. In this study, the nonlinear spatio-temporal transient phenomenon of gas networks (GN) is analyzed by a novel modeling approach, named Takagi-Sugeno fuzzy linear graph (TSFLG), and then a parameter space decomposition (PSD)-based adaptive parameter estimation algorithm is developed for greater efficiency. By merging TS fuzzy theory, the nonlinear terms are replaced by a set of linear elements interpolated by membership functions, which are then connected in series to construct the TSFLG structure. The GN is partitioned into several subnetworks for parallel estimation, and the coupling parameters are determined by the covariance intersection fusion rule. Within each subnetwork, efficiency is further improved by estimating alternately linear and nonlinear parameters. The proposal is assessed in an industrial GN with 12 nodes. Test results indicate that the proposed improves efficiency by 38.64<inline-formula><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> while maintaining accuracy.

An efficient hybrid direct-iterative solver for three-dimensional higher-order edge-based finite element simulation for magnetotelluric data in anisotropic media

This work investigates the use of the high-order finite element method (FEM) for 3D magnetotelluric (MT) modeling in anisotropic media. The high-order FE is more accurate but also more difficult than the linear elements. Numerically, the resulting linear system is larger and denser than the linear case with the same grid. In 3D cases, finite element discretization of Maxwell's equations often leads to large but sparse linear systems of equations, requiring effective methods to solve them. However, for the domain-frequency Maxwell's equations, the discrete rotor operator with a large kernel hampers the convergence of common iterative solvers. Robust and effective solvers are needed to solve 3D electromagnetic (EM) problems. In this study, we presented a robust direct-iterative solver based on a block-diagonal preconditioner. The proposed solver is efficient for high-order FE modeling with arbitrary anisotropy. We implemented the algorithm in a high-level Julia language, which is easy to read, maintain, and extend. Numerical examples show that the high-order elements produce a more accurate solution on a coarse grid than the linear case with the same grid. Compared with the direct solver, the hybrid direct-iterative solver reduces the computation time and memory usage. In addition, the solver only needs a few iterations to converge compared with the traditional BCISGTAB solver with an SSOR preconditioner. Our modeling results suggested the FGMRES solver with the block-diagonal preconditioner is quite effective for 3D MT high-order FE modeling.

The non-Lefschetz locus of vector bundles of rank 2 over [formula omitted

A finite length graded R-module M has the Weak Lefschetz Property if there is a linear element ℓ in R such that the multiplication map ×ℓ:Mt→Mt+1 has maximal rank for every integer t. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. In this paper we study the non-Lefschetz locus for the first cohomology module H⁎1(P2,E) of a vector bundle E of rank 2 over P2. The main result is to show that this non-Lefschetz locus has the expected codimension under the assumption that E is general.

A mixed FEM for the coupled Brinkman–Forchheimer/Darcy problem

This paper develops the a priori analysis of a mixed finite element method for the filtration of an incompressible fluid through a non-deformable saturated porous medium with heterogeneous permeability. Flows are governed by the Brinkman–Forchheimer and Darcy equations in the more and less permeable regions, respectively, and the corresponding transmission conditions are given by mass conservation and continuity of momentum. We consider the standard mixed formulation in the Brinkman–Forchheimer domain and the dual-mixed one in the Darcy region, and we impose the continuity of the normal velocities by introducing suitable Lagrange multiplier. The finite element discretization involves Bernardi–Raugel and Raviart–Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

A linear elastic finite element approach to fatigue life estimation for defect laden materials

This work assumed that the stress-life behaviors of defect-laden materials converge at a certain low life cycle and attempted to estimate fatigue life by applying relative knock-downs to the endurance limit of a reference stress-life data. Linear elastic finite element analysis on small cracks initiated from defects was performed to evaluate the effect of defect size, aspect ratio, and location on endurance limit. While for internal defects size and shape are both influential, size and depth were found to be more important for surface defects. These findings were validated against the fatigue lives of laser powder bed fused Ti-6Al-4V specimens.

Jacobian weighted element differential method for solid mechanics

In this paper, a novel collocation method is proposed to solve the problems in solid mechanics. In the proposed method, the gradients across the element boundaries are evaluated by a weight-averaging algorithm to achieve higher order continuity. Then, the collocation method is employed to discretize the governing equations for all domain nodes, including the nodes at the interfaces of elements for which the earlier collocation methods have to use the traction-equilibrium equations. As a result of this difference, linear and serendipity elements can be used in the proposed method to reduce the total number of freedoms and easily used for very complex geometries. Moreover, the proposed method significantly improves the accuracy compared with the earlier collocation methods. The provided numerical test cases show that the method can also avoid the degeneration of accuracy when odd-order elements are employed, which further demonstrates the advantages of the proposed method. The proposed method are also leads to better convergency in nonlinear cases and iterative matrix solver. By analyzing the coefficient matrix, we found that the proposed method is better conditioned. The proposed method also yields promising advantages in transient problems for explicit time marching because of the advantage of the collocation methods.

Simultaneous shape and topology optimization of wings

This paper presents a method for simultaneous optimization of the outer shape and internal topology of aircraft wings, with the objective of minimizing drag subject to lift and compliance constraints for multiple load cases. The physics are evaluated by the means of a source-doublet panel method for the aerodynamic response and linear elastic finite elements for the structural response, which are one-way coupled. At each design iteration, a mapping procedure is applied to map the current wing shape and corresponding pressure loads to the unfitted finite element mesh covering the design domain. Wings of small fixed-wing airplanes both with and without a stiffening strut are optimized. The resulting wings show internal topologies with struts and wall-truss combinations, depending on the design freedom of the shape optimization. The lift distributions of the optimized wings show patterns like the ones obtained when performing optimization of wing shapes with constraints on the bending moment at the root.

Optimal Analysis of Non-Uniform Galerkin-Mixed Finite Element Approximations to the Ginzburg–Landau Equations in Superconductivity

This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg–Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart–Thomas face element are used for the order parameter , the magnetic field , and the magnetic potential , respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg–Landau model, the order parameter is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of influences the accuracy of . The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for and the corresponding more precise estimates, including the -norm. With the quasi-projection, we prove that the lower-order approximation to does not pollute the accuracy of . Our numerical experiments confirm the optimal convergence of . The approach can be extended to many other multiphysics models.