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We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15] . A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ω h ⁎ is chosen for some given partition Ω h and the exact solution u ∈ C 1 ( Ω ) , and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.

In this paper, we focus on solving a class of two-by-two block complex system of linear equations arising from finite element discretization of a distributed optimal control constrained by a time-harmonic parabolic equations. We propose a new iterative method for solving the obtained system. In each iteration of the method a two-by-two block system of linear equations with real coefficient matrix should be solved. We solve this system inexactly using the generalized minimal residual (GMRES) and the Chebyshev acceleration methods in conjunction with the real-valued preconditioned square block (PRESB) preconditioner. The convergence and spectral properties of the method are discussed. Numerical results in 2-dimensional case are presented to demonstrate the efficiency of the method.

Frequency variations are normally employed for damage detection as frequencies possess the advantages of the high accuracy and high anti-noise capability. However, the validity and applicability of such methods are restricted due to the limited order of frequencies. A damage localization and quantification approach is proposed for beam bridges by using the frequency variation of parked vehicle-bridge systems. Firstly, the phenomenon of frequency variation of parked vehicle-bridge systems is briefly illustrated. Secondly, a damage location index (DLI) is defined by the frequency change rate (FCR) of the vehicle-bridge systems with the vehicle parked at different positions before and after damage. Then the relationship between the frequency-parameter change rate (FPCR) and damage severity is established via finite element model (FEM) and further employed to formulate a damage equation(s) for severity estimation. The dimension of coefficient matrix in damage equations is reduced to large extent as the damage locations are determined by DLI in advance, which is meaningful in enhancing the effectiveness of severity estimation. Results of numerical and experimental examples with different damage scenarios indicate that the proposed approach can localize and quantify damages with high accuracy and efficiency by using frequency information only. The effects of measurement noise and FEM error are discussed to examine the robustness of the proposed approach. The cases when damage occurs in neighbor elements are analyzed to investigate the feasibility and robustness of the proposed approach.

We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O ( M N log 3 ⁡ N ) computational complexity and O ( M N log 2 ⁡ N ) memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

The paper presents a method for building matrices in the QTT format, the columns and rows of which are reordered in a special way, by z-permutation. To obtain a matrix in this permutation, a new operation in the QTT (Quantized Tensor Train) format, z-kron, is introduced. This reordering allows one to reduce the QTT ranks of the approximation of the stiffness matrix, which makes it possible to accelerate the convergence of the numerical solution of the system. For example, when solving the Dirichlet problem for Poisson’s equation by the finite element method (FEM), where the QTT format is used to store the coefficient matrix, reordering the rows and columns in a coefficients matrix with dimensions $$n \times n$$ , where $$n = {{4}^{d}}$$ , makes it possible to prevent the exponential in $$d$$ growth of ranks.

Hole drilling or contour milling for the large and complex workpieces such as automobile panels and aircraft fuselages makes a high combined demand on machining accuracy, stiffness and workspace of machining equipment. Therefore, a 5-DOF (degrees of freedom) parallel kinematic machine (PKM) with redundant constraints is proposed. Based on the kinematics analysis of the parallel mechanism using intermediate variables, the kinematics problems of the PKM are solved through equivalent kinematics model. The structural stiffness matrix method is adopted to model the stiffness of the parallel mechanism of the PKM, where the stiffness of each joint and branch component is obtained by stiffness formula and finite element analysis. And the stiffness model of the parallel mechanism is improved by correction coefficient matrix, each element of which is constructed as a polynomial function of three independent end variables of the parallel mechanism. The terminal stiffness matrices obtained by simulation result are used to determine the coefficients of polynomial function by least square fitting to describe the correction coefficient over the workspace of the parallel mechanism quantitatively. The experiment results prove that the modification method can greatly improve the stiffness model of the parallel mechanism. To enhance the machining accuracy of the PKM, the proposed kinematics model and the improved stiffness model are utilized to optimize the working stiffness of parallel machine by searching the best relative position of parallel machine and workpiece. A plate workpiece taken as example is examined in the case study section, which demonstrates the effectiveness of optimization method.

In this paper, the element differential method (EDM), a new numerical method proposed recently, is coupled with the multi-domain boundary element method (MDBEM), an improved Boundary Element Method (BEM), for solving general multi-scale heat conduction and elasticity problems. The basic algebraic equations in MDBEM are formulated in terms of displacements/temperatures and surface tractions/heat fluxes, which are the same as those in EDM. Therefore, when coupling these two methods, we don't need to transform the variables such as the equivalent nodal forces into the surface tractions as done in the Finite Element Method (FEM). The key task in the proposed coupled method is to use the displacement/temperature consistency conditions and the surface traction/heat flux equilibrium equations at interface nodes to eliminate all BEM nodes except for those on the interfaces, rather than to iterate. After elimination, the coefficient matrix we get is sparse although a small part is dense. The coupled method inherits the advantages of EDM in flexibility and computational efficiency, and the advantage of BEM in the robustness of treating multi-scale problems. Three numerical examples of general heat conduction and mechanical problems are given to demonstrate the correctness and efficiency of this coupled method.

A combined method is provided for simulating the compressible wormhole propagation, in which, the splitting mixed finite element (SMFE) method is applied for the parabolic-type pressure equation, the modified mass-conservative characteristic (MCC) method is presented for the convection-diffusion type solute transport equation, and the traditional finite element method is applied for the porosity equation. The application of the splitting mixed element method results in a symmetric positive definite coefficient matrix of the separated mixed element system and the mass-conservative characteristic finite element method not only does well in handling convection-dominant diffusion problem but also keeps mass balance globally. The convergence of this method is considered and the optimal L 2 -norm error estimate is also derived. Finally, some numerical examples are provided to confirm theoretical analysis and effectiveness of the proposed method.

A block-inverse preconditioner (BIP) is proposed to accelerate solving implicit time integration in the context of Newton–Krylov approach used in micromagnetic simulations for solving the Landau–Lifshitz–Gilbert equation. A coefficient matrix is generated and stored for the linear system of Newton method. BIP is formulated by subdividing the coefficient matrix into small blocks and directly inverting them. The cost of preconditioning is low since inverting and multiplying small blocks is fast, which can be further minimized by parallel computing on multicore CPUs or GPUs. The effectiveness, speed, robustness, and scalability of BIP is demonstrated by numerical simulation experiments. Comparisons to incomplete LU preconditioning methods are conducted to demonstrate the effectiveness of BIP.

Advances in micromachining have led to the development of microelectromechanical systems (MEMS) devices with suspended structures used in a variety of sensors. Of note for this work are sensor types where two elements exist on the suspended membrane, including examples like air flow and differential pressure detectors, gas detection, and differential scanning calorimetry sensors. Intuitively one would argue that some thermal loss exists between the two elements. However, surprisingly little is documented about this electrothermal interaction. The work presented here addresses this shortcoming by defining a new parameter set, a matrix of thermal coupling coefficients. They are used within our novel two-port electrothermal model based on the heat flow equation adapted as a linear system of equations. However, the model is only effective with knowledge of these coefficients. We introduce an approach to extract the coefficients using finite-element method (FEM)-based multiphysics simulation tools and revisit and extend our previous method of non-ideal power coupling, this time to extract the coefficient matrix from measured data. Both specialist simulation tools and device manufacturing are very expensive. However, they are the only choices in the absence of an analytic model. A major contribution of this work is the derivation of a model to predict the coefficients by analytic means from the device dimensions and material properties. The research contribution and paper culminate in a comparison of analytic, simulated, and experimentally extracted values of two different devices to verify and demonstrate the effectiveness of the proposed models. The values compare well and show that the best results achieved are approximately 90 % and 70 % thermal linkage respectively for vacuum and atmospheric pressure conditions.

We present a wavelet finite-element method (WFEM) based on B-spline wavelets on the interval (BSWI) for three-dimensional (3D) frequency-domain airborne EM modeling using a secondary coupled-potential formulation. The BSWI, which is constructed on the interval (0, 1) by joining piecewise B-spline polynomials between nodes together, has proved to have excellent numerical properties of multiresolution and sparsity and thus is utilized as the basis function in our WFEM. Compared to conventional basis functions, the BSWI is able to provide higher interpolating accuracy and boundary stability. Furthermore, due to the sparsity of the wavelet, the coefficient matrix obtained by BSWI-based WFEM is sparser than that formed by general FEM, which can lead to shorter solution time for the linear equations system. To verify the accuracy and efficiency of our proposed method, we ran numerical experiments on a half-space model and a layered model and compared the results with one-dimensional (1D) semi-analytic solutions and those obtained from conventional FEM. We then studied a synthetic 3D model using different meshes and BSWI basis at different scales. The results show that our method depends less on the mesh, and the accuracy can be improved by both mesh refinement and scale enhancement.

We embed the fractional Allen-Cahn equation into a Galerkin variational framework and thus develop its corresponding finite element procedure and then prove rigorously its mathematical and physical properties for the finite element solution. Combining the merits of the conjugate gradient (CG) algorithm and the Toeplitz structure of the coefficient matrix, we design a fast CG for the linearized finite element scheme to reduce the computation cost and the storage to O(M log⁡ M ) and O(M), respectively. Numerical experiments confirm that the proposed fast CG algorithm recognizes accurately the mass and energy dissipation, the phase separation through a very clear coarse graining process, and the influences of different indices r of fractional Laplacian and different coefficients K,η on the width of the interfaces.

This paper utilizes the 3-array variation of the compressed sparse row (CSR) format to store upper triangular elements of coefficient matrix, and applies the Intel MKL PARDISO solver to solve large sparse linear systems of equations, when calculating the 3-D magnetotelluric (MT) fields using edge-based finite element (FE). By default, Intel MKL PARDISO stores matrices in RAM, namely in-core (IC) mode, however, it is quite memory consumption. One can specify that Intel MKL PARDISO stores matrices on disk, it is referred to as out-of-core (OOC) mode. But the OOC mode consumes longer time than the IC mode. We compare both time and memory consumption of IC and OOC mode via COMMEMI 3D-1A and 3D-2A models. As shown in the examples, the OOC mode can solve very large problems since the amount of main memory required is significantly reduced, in the acceptable computing time, which makes frequency loop parallelization of 3-D MT modeling possible on a small memory PC.

This paper proposes a novel green element method (GEM) by mixing the idea of finite difference method (FDM). In the novel method, We come to the original formula when boundary integral equation is applied to an element, and use difference quotient of the central nodal value on two sides of the shared edge of adjoining elements to approximate the boundary integration ∫ΓG∇p · nds. This treatment is similar to FDM, and the integral operator relevant to element size controls the estimated error. The novel GEM makes the numerical solution correspond to the actual physical meaning, and the coefficient matrix of the global matrix is a banded sparse matrix with larger bandwidth than previous GEMs. Meanwhile, the instability of the original GEM is illuminated. We have proven it by theoretical error analysis and five numerical examples that, the accuracy of the novel GEM is three-order higher than the original GEM, and the novel GEM has a good convergence and stability, which is the property that the original GEM does not have. Indeed, the novel GEM proposed in this paper is essentially a new numerical method mixed with the idea of boundary element method (BEM), finite element method (FEM), and FDM. In contrast with BEM, FEM, FDM and previous GEM, the characteristics of our novel GEM include: (i) Compared with FEM and FDM, the novel GEM has the accuracy of BEM and can better accord with material balance. (ii) Compared with BEM, the novel GEM can solve nonlinear problems with heterogeneous media, which are hard to be handled by BEM. (iii) Compared with previous GEMs, the novel GEM has a three-order accuracy, and has a better convergence that the calculation error can be well controlled by the element size.

Damage detection of concrete structures based on finite element model and measured response parameters has been an important research topic in recent years. It is well known that test data of mechanical behavior of concrete show great scatterness. As a result, the measured response parameters of concrete structures sometimes have gross errors. The gross error is a physical quantity that is much larger than data noise, which may lead to serious distortion of calculation results. To this end, a new robust estimate method termed as the augmented inverse estimate is proposed in this work for damage detection of concrete structures to resist gross errors in data. It has the advantages of very simple programming, convenient utilization, high computational accuracy, and broad prospect of application. Central to the augmented inverse estimate are the augmentation of coefficient matrix and the multiple computations based on feedback evaluation. A reinforced concrete beam structure is used as an example to verify the proposed method. It was found that the proposed method can successfully identify the location and extent of structural damage even if the used data have gross errors.

Due to the pollution errors in acoustic analysis, the requirement of very refined meshes at high wavenumbers limits the practical application of the conventional finite element method (FEM). In this work, the three-node linear triangular element with linear interpolation cover functions (FEwC) is applied to solve acoustic radiation problems in two dimensions. The present FEwC has considerable generality since the polynomial interpolation cover functions can be easily implemented without changing the mesh topology. As the original linear approximation space is enriched by the cover functions, the FEwC can significantly alleviate the pollution errors and increase the convergence rate of the original low-order element in acoustic computation. We show the proper imposition of the Dirichlet boundary conditions in acoustic problems and illustrate an effective iterative algorithm to resolve the system equations with a singular coefficient matrix. The Dirichlet-to-Neumann mapping technique is used to truncate the infinite problem domain into a bounded computational domain for the exterior problems and the Sommerfeld radiation condition then can be satisfied at infinity. Several numerical experiments demonstrate that the present scheme is a promising numerical tool for acoustic radiation problems.

In this paper, we construct a new implementation of stochastic finite element methods for partial differential equations with random inputs. The basis functions of generalized polynomial chaos are determined not by the usual notion of degree of a multivariate polynomial, but by the Euclidean degree. Then the corresponding linear combination of basis from the stochastic finite element methods is obtained, where the coefficient matrix is sparse and symmetric. In numerical experiments considering stochastic diffusion and Helmholtz equations, our approach with Euclidean degree of gPC basis achieves a better convergence rate than ones with total degree.

Fast and high precision numerical simulation under multi-source condition is the key to two and a half dimensional (2.5D) high efficiency and high precision inversion of DC resistivity method. In this paper, a 2.5D multi-source fast and high-precision numerical simulation method based on DC resistivity method is proposed, which can be applied to arbitrary scale and complex geoelectric model. Firstly, the boundary conditions are transferred to make the coefficient matrix independent of the position of the source. On the premise of using the direct solution method to solve the linear equations, the fast back-substitution solution is realized under the multi-source condition. Secondly, a set of wave number selection method that can be adapted to arbitrary scale model is proposed, and the inverse Fourier transform is calculated by using the finite element method. And also, the potential value in the spatial domain of any lateral section is calculated. Finally, an effective solution to the singularity of the field source is proposed. The numerical simulation algorithm shows that the 2.5D numerical simulation of DC resistivity method under multi-source condition proposed in this paper has the characteristics of fast calculation efficiency, high numerical calculation accuracy and high numerical stability.

A new signal processing method called complex continuous wavelet transform (CCWT) is introduced in this paper to localize pile damage because it clearly reveals inherent characteristics of response signals. In this method, CCWT is first performed on the response signal to obtain the wavelet coefficient matrix. The resultant coefficients are then employed to calculate phase angles at different frequency bands with an aim of pile damage localization. However, the CCWT method is only demonstrated via laboratory tests on pile specimens, and its application on actual piles has not been examined. Moreover, various factors such as pile-soil interaction need to be considered when the CCWT method is applied on actual piles. To address these issues, a numerical example of 3D finite element pile model followed by a parameter analysis and an experimental verification on an actual pile are investigated. The results demonstrate that the CCWT method is capable of localizing pile damage under different damage scenarios. However, there are still some interference points in the grayscale images of phase angles and the reduction of interference points needs to be addressed by mutual verification with other pile damage detection methods and engineering experience.

A new combined characteristic mixed finite element method is developed for solving compressible miscible displacement in porous media. In this algorithm, the splitting mixed finite element (SMFE) method is applied for solving the parabolic-type pressure equation, and the mass-conservative characteristic (MCC) finite element method is applied for solving the convection-diffusion type concentration equation. The application of the splitting mixed element method results in a symmetric positive definite coefficient matrix of the separated mixed element system, while the mass-conservative characteristic finite element method not only does well in handling convection-dominant diffusion problem but also keeps mass balance. The convergence of this method is considered and the optimal L2-norm error estimate is also derived. Finally, some numerical examples are provided to confirm our theoretical analysis.

This paper presents the formulation of exact stiffness matrices applied in linear generalized beam theory (GBT) under constant and/or linear loading distribution in the longitudinal direction. Also, the assortment of the correct exact stiffness matrix and the corresponding shape function are presented based on main transversal deformation mode, which can be divided into: (1) dominant distortion mode; (2) dominant torsion mode; (3) and critical distortion–torsion mode. Special attention is given to the hyperbolic–trigonometric shape functions, which are organized in a system of vector in function of longitudinal direction and a coefficient matrix obtained from the completeness requirement. This approach has the benefit of compacting the terms of the stiffness matrix and systematizing the boundary conditions of an element by applying the completeness coefficient matrix as a transformation matrix. As a result, in linear analysis, a single element can represent the stress and displacement fields. Moreover, due to the higher-order continuous derivatives properties of hyperbolic–trigonometric shape functions, the generalized internal shear is obtained without the typical discontinuity of Hermitian shape functions. A full and detailed example, applied in a thin-walled circular hollow cross section, provides not only an illustration of the presented approach, but also a quick introduction point in GBT.

Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.

The weak form quadrature element method is a high-order algorithm which has been applied successfully to consolidation analysis of saturated and unsaturated soils. Its superiority over the conventional finite element method has been verified. However, in consolidation analysis, pore pressure oscillations will appear when small time increments are used due to the very small permeability and near-incompressibility of the pore water. This can produce almost zero values of main diagonal elements in the coefficient matrix. To overcome the pressure oscillations, we propose a coupled composite quadrature element in which different orders of integration are employed for the pore pressure and the displacement. Its performance is evaluated and compared with that of the standard element through 1D and 2D numerical tests. Our results show that pressure oscillations can be effectively alleviated and stability and accuracy can be significantly enhanced by using the proposed element.

The surface electric field analysis of the converter valve shield system is a large-scale electrostatic field problem, which is difficult to analyse. The fast multipole boundary element method (FMBEM), which is suitable for solving large-scale problems, can accelerate the computation speed and conserve memory. However, the coefficient matrix implicitly formed by using the FMBEM is sometimes ill-conditioned, especially for large-scale problems; thus, the convergence of iteration is poor. In this paper, a fast solver is proposed to improve efficiency. First, an adaptive GMRES( m ) with variant restart parameter is adjusted for the Galerkin FMBEM. In addition, the sparse approximate inverse preconditioner is improved, and a new sparsity pattern is proposed for the multiscale problem derived from the converter valve shield system. The numerical results show that the accuracy can meet the engineering requirements compared with the finite element method. Compared with other solvers and preconditioners, the algorithm can achieve a satisfactory convergence rate and reduce the computation time. In addition, a single bridge shield system of ±160 kV converter valve is successfully analysed using the proposed method.

This paper presented a novel Green element method (GEM) based on two sets of nodes, one of which contains all vertices of the polygon representing the value of the pressure, called pressure nodes, the other contains all midpoints of the edges of the polygon representing the value of the normal flux, called flux nodes. The novel method considers both the pressure and the normal flux explicitly, and utilizes the physical fact that, the algebraic sum of normal fluxes at the flux nodes which are on the edge shared by adjacent elements is zero. Therefore, the normal flux term at each internal flux node can be offset in the final global equations, and the order of the global coefficient matrix is only the number of pressure nodes. Compared to previous GEMs, the novel method has actual second-order accuracy. Moreover, because we selected the flux node at the midpoints of edges, the novel method made the solution process more in accordance with physical meaning, and the numerical solution has a better continuity in the computational domain than previous GEMs. It is worth mentioning that, to the authors’ knowledge, this is the first time to utilize two sets of nodes when estimating the numerical solution to the problem described by only one differential equation. Therefore, the novel method may bring new ideas to other numerical methods, such as finite difference or finite element method, etc.

In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrodinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L 2 -norm and H α / 2 -norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L 2 -norm and L ∞ -norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O ( M 2 ) to O ( M ) and the computational complexity from O ( M 3 ) to O ( M log ⁡ M ) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques.

In this paper, for the Helmholtz transmission eigenvalue problem, we propose a two-grid discretization scheme of non-conforming finite elements. With this scheme, the solution of the transmission eigenvalue problem on a fine grid π h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid π H and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid π h . We prove the resulting solution still maintains an asymptotically optimal accuracy, and we report some numerical examples in two dimension and three dimension on the modified-Zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues.

The Finite Element method can be implemented to model geophysical electromagnetic data using one of two methodologies called Nodal and Vector Finite Elements. This paper presents a comparison between the two approaches, emphasizing memory usage and processing time, when simulating Marine Controlled Source Electromagnetic (MCSEM) data in three-dimensional models. The study is carried out using unstructured meshes and a direct solver. Computational cost information from both methodologies are gathered from four different 3D models, each emphasizing a different aspect of the problem. The results indicate that the Vector Finite Element methodology requires less memory and processing time to calculate the same data using the same mesh. Although the nodal method generates a smaller linear system than the vector method, the vector coefficient matrix is significantly more sparse than the nodal one. The greater sparsity makes the vector approach more computationally efficient, requiring less memory and running in less time than the nodal method to generate results with the same level of accuracy.

The inverse finite element method (iFEM) is a mechanics-based algorithm for deformed-shape estimation of structures. In this paper, an enhanced inverse beam element named iEBT2 is developed based on classical beam theory. The element formulation is derived by minimizing a weighted-errors functional that consists of experimental and numerical section strains. The improved coefficient matrix KR is always non-singular, assuring that the solution of iFEM formulation exists. Location-independent feature of matrix KR simplifies the inverse finite element modeling, especially for complicated structures. Weighting constants are utilized to define error functional, aiming to penalize the contributions from “measure-less” stations. Numerical and experimental cases have been performed and demonstrated excellent predictive capability when iEBT2 model has complete strain measures. In the case of missing strain components, iEBT2 enables deformation estimation and its accuracy is acceptable in general. The enhanced inverse element extends the practical usefulness of iFEM in shape-sensing analysis of civil infrastructures.

The Absolute Nodal Coordinate Formulation (ANCF) can be used to model a flexible multibody system subject to both overall motions and large deformations. With an increasing number of finite elements of ANCF to mesh flexible bodies, however, the computation cost will become extremely high. To improve the computational efficiency of ANCF for a large-scaled flexible multibody system, a systematic method is proposed in this study for the model order reduction based on the proper orthogonal decomposition and the Galerkin projection. At first, an approach for the selection of reduced constraint equations is proposed to deal with the singularity of the coefficient matrix of the Reduced-Order Model (ROM). Then, the computation of the reduced stiffness matrix and generalized force vector of the ROM are parallelized via the OpenMP directives. Afterwards, two parametric approaches are presented to make the ROM be adaptive to the change of system parameters. One is the interpolation approach of reduced-order basis vectors on a manifold to obtain the parametric ROM, and the other is the fast-to-slow ROM approach via the reduced-order basis vectors extracted from the fast dynamic responses to simulate slow dynamic responses. Finally, four numerical examples are given to validate the efficacy of the proposed method for the dynamic simulations of both rigid and flexible multibody systems.

In this paper, the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an $N$-component mixture are studied. For the general choice of phase variables, the unisolvent property of the coefficient matrix involved in the $N$-phase models based on the pairwise surface tensions is established. Moreover, the symmetric positive-definite property of the coefficient matrix on an $(N-1)$-dimensional hyperplane --- which is of fundamental importance to the well-posedness of the models --- can be proved equivalent to some physical condition for pairwise surface tensions. The $N$-phase Allen-Cahn and $N$-phase Cahn-Hilliard equations can then be derived from the free-energy functional. A natural property is that the resulting dynamics of concentrations are independent of phase variables chosen. Finite element discretizations for $N$-phase models can be obtained as a natural extension of the existing discretizations for the two-phase model. The discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size. Numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions., Comment: 31 pages

Without applying any stable element techniques in the mixed methods, two simple generalized mixed element (GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner (H–R) variational principle. The main features of the GME formulations are that the common $$C_{0}$$ -continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions.

In this paper, a method has been presented to study the free vibration of a rotating shell consisting of p coupled conical shells. This method considers the Coriolis and centrifugal forces as well as the initial hoop tension resulting from rotation of the shell. Matrix transform and power series methods were employed to solve the equations. The advantage of using the matrix transform approach is that the dimension of the coefficient matrix, which is finally constructed to obtain the natural frequencies of a shell made of p cones, remains 8×8. The obtained results were validated with the help of existing special cases of the studied problem reported in the literature, and by means of FEM software programs. By verifying the presented equations and their solution method, the fundamental natural frequencies of a rotating shell made of two conical shells were obtained for different conditions. Every thin-walled axisymmetric shell can be represented by a number of joined conical shells. So, as a practical example, the presented method was employed to analyze the vibrations of an automobile tire ring as a rotating thin-walled shell with a relatively complex geometry.

Unstructured tetrahedral grids with local refinement facilitate the use of total-field solution approaches to geophysical electromagnetic (EM) forward problems. These approaches, when combined with the vector finite-element (FE) method and with refinement near transmitters and receivers, can give accurate solutions and can easily handle realistic models with complex geometry and topography. We have applied this approach to 3D forward modeling for fixed- and moving-loop configurations. MUMPS, a direct solver, was used to solve the linear system of equations generated by FE analysis. A direct solver is particularly suited to the moving-loop configuration for which the right side is different for every transmitter loop, but for which the coefficient matrix is unchanged. Therefore, the coefficient matrix need only be factorized once, and then the system can be solved efficiently for all different right sides. We compared our results with several typical scenarios from the literature: a conductive brick in a homogeneous half-space and a complex conductor at a vertical contact both for fixed-loop configurations, and a homogeneous half-space for a moving-loop configuration. We also evaluated results for the massive sulfide ore deposit of the Ovoid Zone at Voisey’s Bay, Labrador, Canada, for which we considered fixed- and moving-loop configurations. This model also provides an illustration of the complex vortex current systems that are generated by time-domain EM methods within highly conductive ore bodies in a resistive host.

Buckling of the rectangular thin plates is a class of problems of fundamental importance in mechanical engineering. Although various theoretical and numerical approaches have been developed, benchmark analytic solutions are still rare due to the mathematical difficulty in solving the complex boundary value problems of the governing high-order partial differential equation. Actually, most available solutions can be categorized as either “accurate” for the plates with two opposite edges simply supported or “approximate” for those without two opposite edges simply supported. In this paper, we present the first work on the symplectic superposition method-based analytic buckling solutions of the rectangular thin plates. A Hamiltonian system-based variational principle via the Lagrangian multiplier method is proposed to formulate the thin plate buckling in the symplectic space. Then the governing equation is analytically solved for some fundamental subproblems which are superposed to yield the final solutions of the original problems. For each problem, a set of equations are produced with respect to the expansion coefficients of the quantities imposed on the plate edges. The existence of the nontrivial solutions of the equations sets the requirement that the determinant of the coefficient matrix be zero, which leads to a transcendental equation with respect to the buckling loads. The buckling mode shapes are obtained by substituting the nontrivial coefficient solutions into the mode shape solutions of the subproblems, followed by superposition. Four types of buckling problems are studied for the plates with combinations of clamped and simply supported edges, without two opposite edges simply supported. The developed method as well as the accurate analytic results is well validated by the finite element method and very few analytic results from the literature.

Purpose The purpose of this paper is to introduce a new method called simultaneous elimination and back-substitution method (SEBSM) to solve a system of linear equations as a new finite element method (FEM) solver. Design/methodology/approach In this paper, a new technique assembling the global stiffness matrix will be proposed and meanwhile the direct method SEBSM will be applied to solve the equations formed in FEM. Findings The SEBSM solver for FEM with the present assembling technique has distinct advantages in both computational time and memory space occupation over the conventional methods, such as the Gauss elimination and LU decomposition methods. Originality/value The developed solver requires less memory space no matter the coefficient matrix is a typical sparse matrix or not, and it is applicable to both symmetric and unsymmetrical linear systems of equations. The processes of assembling matrix and dealing with constraints are straightforward, so it is convenient for coding. Compared to the previous solvers, the proposed solver has favorable universality and good performances.

In electric machine drive systems, hardware-in-theloop (HIL) emulation provides accurate testing of actual control system prototypes and protection devices interfaced with the electric machine model on a real-time simulator in a non-destructive environment particularly when faults are studied. A compromise between the model accuracy and computational burden makes the magnetic equivalent circuit (MEC) model ideal for real-time simulation of electric machines. However, satisfying the timing constraints of real-time simulation to accommodate internal machine faults is still challenging due to the nonlinearity and rotation of electric machines. In this paper, the transmission line modeling (TLM) method is utilized to keep the MEC coefficient matrix unchanged during nonlinear iterations. Afterward, for the first time, the entire potential of the TLM method for pre-calculation is exploited by proposing an efficient matrix re-ordering combined with the left-looking Gilbert–Peierls algorithm to minimize the computational burden of the sparse MEC matrix LU decomposition required in each time-step due to rotation. Furthermore, the massive hardware architecture of the field programmable gate array is used as the platform for implementation to fully exploit parallelism. With the proposed MEC-based real-time TLM method, the minimum timestep as low as 500 microsecondscan be achieved and the results validation with two-dimensional finite element model (FEM) of the commercial Jmag-Designer software shows the accuracy and efficiency of the proposed methodology

The problems of Acoustic-Structural Interaction (ASI) are commonly encountered in the simulation of the thin-walled structures immersed in the fluid. This interaction can affect the acoustic properties of the fluid and the dynamic characteristics of the structure. In this paper, a Non-Uniform Rational B-Splines (NURBS)-based Isogeometric Finite Element (FE) - Boundary Element (BE) symmetric coupling method is developed to study the ASI problems and the free vibration of the elastic structures submerged in the fluid. The geometry of the thin-walled structures is exactly modelled and discretized by utilizing the Isogeometric Reissner-Mindlin shell elements. This shell theory simplifies the continuity enforcement between NURBS patch boundaries. The Isogeometric Boundary Element Method (IGABEM) is applied to describe the acoustic field. In order to impose coupling constraints, a transformation equation of unknown variables on the interface between sound field and structure field is established, that is, sound pressure versus external force and particle velocity versus displacement. Different from the non-symmetric matrix obtained by the traditional direct BEM, a new variational formulation is introduced to obtain the symmetric coupling coefficient matrix, which maks this coupling approach suitable for a broad class of solvers and improves the robustness of the computation. The reliability and stability of the symmetric coupling method are verified by numerical examples considering acoustic and structural loadings. It is concluded that the symmetric coupling method possesses high accuracy, and can circumvent the limitation of traditional coupling method in solving open boundary problems.

The main purpose of this paper is to develop the two-step implicit-explicit(IMEX) time discretization coupled with finite element methods for solving delayed predator-prey competition reaction-diffusion system. Finite element methods are used to discretize the space variables, both IMEX two-step one-leg methods and IMEX linear two-step methods are considered in time discretization, where the nonlinear reaction part is discretized explicitly and the diffusion part is discretized implicitly. The L 2 -norm error estimates with handling the breaking points are derived. These methods only require solving two independent linear systems with the same coefficient matrix at different time, and avoid solving large-scale systems of nonlinear algebraic equations, thus the resulting schemes are efficient. Some numerical experiments are presented to confirm the theoretical results.

In this paper two numerical procedures are presented for solving a class of Turing system. Firstly, we obtain a time discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and radial point interpolation and also obtain their shape functions. We use the element free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element free Galerkin method test and trial functions are moving least squares (MLS) approximation shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. The element free Galerkin method is a global method while finite elements method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. The coefficient matrix of the element free Galerkin is symmetric. Also, using numerical algorithms, we can conclude that the eigenvalues of the coefficient matrix are positive. Thus, for solving the obtained linear system of equations from the discretization, we use the conjugant gradient method. To keep away from solving a nonlinear algebraic system of equations and obtaining the acceptable numerical results, we use a predictor–corrector algorithm. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

In this paper, we present a finite element implementation of fluctuating hydrodynamics with a moving boundary fitted mesh for treating the suspended particles. The thermal fluctuations are incorporated into the continuum equations using the Landau and Lifshitz approach [1]. The proposed implementation fulfills the fluctuation-dissipation theorem exactly at the discrete level. Since we restrict the equations to the creeping flow case, this takes the form of a relation between the diffusion coefficient matrix and friction matrix both at the particle and nodal level of the finite elements. Brownian motion of arbitrarily shaped particles in complex confinements can be considered within the present formulation. A multi-step time integration scheme is developed to correctly capture the drift term required in the stochastic differential equation (SDE) describing the evolution of the positions of the particles.The proposed approach is validated by simulating the Brownian motion of a sphere between two parallel plates and the motion of a spherical particle in a cylindrical cavity. The time integration algorithm and the fluctuating hydrodynamics implementation are then applied to study the diffusion and the equilibrium probability distribution of a confined circle under an external harmonic potential.

This work shows when using the boundary element method to solve 3D acoustic scattering problems from periodic structures, the coefficient matrix can be represented as a block Toeplitz matrix. By exploiting the Toeplitz structure, the computational time and storage requirements to construct the coefficient matrix are significantly reduced. To solve the linear system of equations, the original matrix is embedded into a larger and more structured matrix called the block circulant matrix. Discrete Fourier transform is then employed in an iterative algorithm to solve the block Toeplitz system. To demonstrate the effectiveness of the formulation for periodic acoustic problems, two exterior acoustic case studies are considered. The first case study examines a continuous structure to predict the noise generated by a sharp-edged flat plate under quadrupole excitation. Directivity plots obtained using the periodic boundary element method technique are compared with numerical results obtained using a conventional boundary element model. The second case study examines a discrete periodic structure to predict the acoustic performance of a sonic crystal noise barrier. Results for the barrier insertion loss are compared with both finite element results and available data in the literature.

The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.

The free flexural vibration of thin rectangular plates is revisited. A new, quasi-exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam-like expressions. Using the proposed quasi-exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB®, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS® and in-house conventional FEM programs using both 12- and 16-DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi-exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two-dimensional structural components during the preliminary design stages, before a detailed design begins.

This chapter briefly reviews two important mathematical topics in the scope of the finite element context. To solve a finite element problem means finally to solve a system of equations. This is, in the simplest case, a linear system of equations and this chapter introduces two simple solution strategies, i.e. the Gaussian elimination and the inversion of the coefficient matrix. The second topic covers the analytical and numerical integration which is needed, for example, to evaluate the elemental stiffness matrix and the column matrix of equivalent nodal loads.

The dynamic stability of dynamical systems with time-periodic stiffness is addressed here. Cracked rotor systems with time-periodic stiffness are well-known examples of such systems. Time-varying area moments of inertia at the cracked element cross-section of a cracked rotor have been used to formulate the time-periodic finite element stiffness matrix. The semi-infinite coefficient matrix obtained by applying the harmonic balance (HB) solution to the finite element (FE) equations of motion is employed here to study the dynamic stability of the system. Consequently, the sign of the determinant of a scaled version of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics in the HB solution is found to be sufficient for identifying the major unstable zones of the system in the parameter plane. Specifically, it is found that the negative determinant always corresponds to unstable zones in all of the systems considered. This approach is applied to a parametrically excited Mathieu’s equation, a two degree-of-freedom linear time-periodic dynamical system, a cracked Jeffcott rotor and a finite element model of the cracked rotor system. Compared to the corresponding results obtained by Floquet’s theory, the sign of the determinant of the scaled sub-matrix is found to be an efficient tool for identifying the major unstable zones of the linear time-periodic parametrically excited systems, especially large-scale FE systems. Moreover, it is found that the unstable zones for a FE cracked rotor with an open transverse crack model only appear at the backward whirl. The theoretical and experimental results have been found to agree well for verifying that the open crack model excites the backward whirl amplitudes at the critical backward whirling rotational speeds.