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This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT k space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented. • Numerical comparison of the mixed, primal hybrid, and other locally conservatives finite element methods. • Introduction of a twice hybridized primal hybrid method, yielding a positive definite matrix after static condensation. • Comparison of multi-threaded strategies and thread-safety mechanisms. • A numerical procedure to stabilize the linear system resolution for indefinite matrices is described. • A general procedure that enables static condensation and fulfills Neumann's compatibility condition. [ABSTRACT FROM AUTHOR]

This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time-dependent, multi-fidelity problems, and use the trained hybrid models to perform bias correction of the low-fidelity models. The hybrid model uses FE basis functions as a spatial basis and RNNs for the approximation of the time dependencies of the FE basis' degrees of freedom. The training data sets consist of sparse, non-uniformly sampled snapshots of the discrepancy function, pre-computed from trajectory data of low- and high-fidelity dynamic FE models. To account for data sparsity and prevent overfitting, data upsampling and local weighting factors are employed, to instigate a trade-off between physically conforming model behavior and neural network regression. The proposed hybrid modeling methodology is showcased in three highly non-trivial engineering test-cases, all featuring transient FE models, namely, heat diffusion out of a heat sink, eddy-currents in a quadrupole magnet, and sound wave propagation in a cavity. The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models. • A RNN/FE hybrid model is developed to learn model discrepancies of multi-fidelity FE simulations based on trajectory data. • The RNN captures the dynamics of the discrepancy function while spatial discrepancies are resolved by a FE basis. • Upsampling and Gaussian priors are employed to account for sparsity in the discrepancy data and prevent overfitting. • The model is employed as a correction operator for low-fidelity models in three non-trivial test cases. • The bias corrected models are significantly more accurate compared to the original low-fidelity models. [ABSTRACT FROM AUTHOR]

This paper presents application and tuning of the equilibrated residual method (ERM) of a posteriori error estimation for coupled electromechanical problems of direct, inverse and general piezoelectricity. In these three cases, either electric potential is induced by strains or strains appear due to the applied electric potential or both phenomena occur simultaneously. The mentioned ERM is assigned for the assessment of modeling and approximation errors of the numerical finite element solution. Such error values usually serve as indication for adaptive hierarchical modeling and adaptive mesh changes within thin and/or solid piezoelectric members so as to obtain the solution of assumed accuracy. [ABSTRACT FROM AUTHOR]

Determining the water table shape and position in unconfined aquifers is fundamental to many groundwater flow assessment studies. The commonly used industry-standard fixed mesh models, contrary to popular belief, do not provide an accurate description of the phreatic surface. When using such models, the water table position is post-processed from the simulated groundwater heads, leading to an approximation error. This error becomes larger for coarse vertical grids. This paper introduces a novel moving mesh technique to simulate the groundwater table in three-dimensional unconfined aquifers under steady-state or transient conditions. We adopt the face-based mixed-hybrid finite element discretization approach in space, leading to a more accurate approximation of the specific discharge field. The model uses an adaptive unstructured but layered mesh which is iteratively adjusted until its top fits the phreatic surface. The developed algorithm accounts for a linearized form of the kinematic boundary condition prescribed on the moving boundary and also supports usual boundary conditions as well. The model was compared to the existing analytical, fixed mesh, and previously published solutions. The obtained results show that the developed model is superior in terms of its numerical stability, convergence behavior, and accuracy. Furthermore, the simulated phreatic surface is free from a cellwise interpolation error and independent of the vertical grid size as used in fixed mesh methods. We also found that the robustness of the moving mesh method cannot be surpassed by a fixed mesh alternative. The model's efficiency is supported by an almost quadratic rate of convergence of the outer iteration loop. [ABSTRACT FROM AUTHOR]

In [1] , we proved that the inverse of the stiffness matrix of an h -version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by H -matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree p of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified. [ABSTRACT FROM AUTHOR]

The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C 0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators. [ABSTRACT FROM AUTHOR]

Seismic fragility analysis of a mega-frame with vibration control substructure (MFVCS) considering structural uncertainties is computationally expensive. Dual surrogate model (DSM) can be used to improve computational efficiency, whereas the proper selection of design of experiments (DoE) is a difficult work in the DSM-based seismic fragility analysis (DSM-SFA) method. To efficiently assess the seismic fragility with sufficient accuracy, this paper proposes an improved DSM-SFA method based on active learning (AL). In this method, the Kriging model is employed for surrogate modeling to obtain the predicted error of approximation. An AL sampling strategy is presented to update the DoE adaptively, and the refinement of the surrogate models can reduce the error of the probability result computed by the Monte Carlo (MC) simulation. A numerical example was studied to verify the effectiveness and feasibility of the improved procedure. This method was applied to the fragility analysis of an MFVCS and a mega-frame structure (MFS). The finite element models were established using OpenSeesPy and SAP2000 software, respectively, and the correctness of the MFVCS model was verified. The results show that MFVCS is less vulnerable than MFS and has better seismic performance. [ABSTRACT FROM AUTHOR]

We present an improvement of the Multiscale Finite Element Method (MsFEM). MsFEM incorporates fine-scale features of a domain through special oscillatory coarse-scale trial functions and its essential step is an online computation of these special functions defined using local auxiliary problems. Our enhancement makes use of the dynamic condensation that transmits the construction of the oscillatory basis functions exclusively to the coarse mesh skeleton (element interfaces) where we defined special 1D eigenproblems. Consequently, both the approximation error and computational cost are reduced. Moreover, arbitrary coarse mesh element geometries, like polytopal ones, may be used without any additional modifications. We illustrate the accuracy and convergence of the improved method with higher order approximation through selected plane strain examples. [ABSTRACT FROM AUTHOR]

Large deformation analyses of structures are of great importance to the evaluation of structural performance under extreme environmental loads, but currently available methods are time-consuming because of the requirement of factorizing large-scale matrices. The inelasticity-separated finite-element method (IS FEM), which can keep the global stiffness matrix unchanged and uses the Woodbury formula as the solver, was presented recently to provide a highly efficient tool for local material nonlinear analysis. To extend the high efficiency advantage of the IS FEM to large deformation analyses, in which the material nonlinearity may be nonlocal and the geometric nonlinearity should be considered, this paper proposes a novel numerical solution scheme by incorporating the updated Lagrangian (UL) formulation into the IS FEM framework. Within this scheme, a Woodbury approximation method (WAM) is introduced as an efficient solver, in which the changing global stiffness matrix is approximated as a constant matrix within a short time period, and a linear equation related to the Schur complement matrix is solved by the combined approximations (CA) method. To eliminate the additional error induced by the approximation, an adaptive iteration strategy (AIS) is presented, in which the approximation error involved in WAM solution is evaluated based on energy norm concept, and the global stiffness matrix is required to be updated adaptively according to the calculated error. The high efficiency and accuracy of the proposed method are finally demonstrated by the time complexity analysis and numerical examples. [ABSTRACT FROM AUTHOR]

The aim of this work is to show an abstract framework to analyze the numerical approximation by using a finite element method in space and a Backward Euler scheme in time of a family of degenerate parabolic problems. We deduce sufficient conditions to ensure that the fully-discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we show a degenerate parabolic problem which arises from electromagnetic applications and deduce its well-posedness and convergence by using the developed abstract theory, including numerical tests to illustrate the performance of the method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We consider sufficient conditions to prove that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. Furthermore, we show that mixed finite element formulations arising from dynamics fluids (time-dependent Stokes problem) and from electromagnetic applications (eddy current models) , can be analyzed as applications of the developed theory. Finally, we include numerical tests to illustrate the performance of the method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present the application of the Discontinuous Petrov–Galerkin (DPG) methodology for the mixed Multiscale Finite Element Method (MsFEM). The MsFEM upscaling technique relies on incorporating fine-scale features through special, in a sense optimized for approximability, trial functions while the DPG methodology allows for the selection of the optimal test functions to provide stability of the FEM approximation. The special trial functions are computed online by the solution of local boundary value problems. We improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton (element interfaces) only. We have verified by numerical tests that the proposed improvement of MsFEM reduced both the approximation error and computational cost. Moreover, it simplified the algorithm significantly. The key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape. The proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters, using an arbitrary well-posed functional setting with or without the DPG methodology. [ABSTRACT FROM AUTHOR]

We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates' transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

The article examines local approximation errors of finite-difference schemes in electromagnetic analysis. Despite a long history of the subject, several accuracy-related issues have been overlooked and/or remain controversial. For example, conflicting claims have been made in the literature about the order of Yee-like schemes in the vicinity of slanted or curved material interfaces. Two novel practical methods for comparison of local accuracy of difference schemes are proposed: one makes use of Trefftz test matrices, and the other one relies on a new measure of approximation accuracy in scheme-exact Trefftz subspaces. One particular conclusion is that a loss of accuracy for Yee-like schemes at slanted material boundaries is unavoidable. [ABSTRACT FROM AUTHOR]

To address the issue of low accuracy resulting from approximation errors in conventional interval finite element methods, this study presents an ɛ -accelerated bivariate dimension-reduction interval finite element method. The proposed method aims to accurately and efficiently predict the static response of structures with high dimensionality and large uncertainty parameters. We first approximate the structural interval equilibrium equation using a bivariate dimension-reduction method. An explicit expression for the sequence of displacements is derived by approximating the inverse of the structural interval stiffness matrix using the Neumann series. Adjoint-based sensitivity analysis can effectively avoid the response interval expansion caused by parameter coupling. For improving the convergence speed and computational accuracy, the displacement response sequence is accelerated by a vector ɛ acceleration algorithm and an adaptive strategy. Finally, three engineering structures with uncertain-but-bounded parameters are analyzed with the proposed method. Compared with conventional interval analysis methods, the proposed method effectively balances computational accuracy and efficiency, making it more suitable for non-linear and large uncertainty problems with multiple interval parameters. The study shows that ɛ -accelerated bivariate dimension-reduction interval finite element method has a promising application. • An interval finite element method based on bivariate dimension-reduction is proposed. • Explicit sequences and errors for the bounds of displacements were derived. • Dependencies between parameters are addressed by adjoint-based sensitivity analysis. • Convergence speed and accuracy are improved by the adaptive ɛ algorithm. • Structural response intervals are efficiently and accurately predicted. [ABSTRACT FROM AUTHOR]

The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs. [ABSTRACT FROM AUTHOR]

In this article, we consider recovery-based a posteriori error estimators for the Virtual Element Method (VEM) and the Boundary Element Method based Finite Element Method (BEM-based FEM). Both methods are Galerkin methods on polygonal and polyhedral grids for the numerical solution of partial differential equations. Thus, they are highly flexible and particularly efficient in combination with adaptive refinement. Our error estimator computes the distance between the finite element gradient and its post-processed version to obtain an accurate approximation of the true error. The post-processing is realised by local averaging and is therefore easy to implement and fast. Furthermore, we have found points of extraordinary accuracy, so called stress points, on specific regular elements for the BEM-based FEM. We demonstrate that, when sampling these stress points, the recovered gradient becomes superconvergent, which means that it converges at a greater rate than the unprocessed gradient. • Discovery of superconvergent points and gradient recovery for BEM-based FEM. • Fast and efficient construction of recovered gradients for VEM and BEM-based FEM. • Application to a posteriori error estimation leads to very accurate results. [ABSTRACT FROM AUTHOR]

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient. [ABSTRACT FROM AUTHOR]

Aim of the present study is to propose an enhanced method for the domain decomposition (clustering) of the representative volume element (RVE) of composite materials to be used with homogenization techniques, based on the PieceWise Uniform Transformation Field Analysis (PWUTFA). With PWUTFA, both constitutive and evolutive equations for the constituents of the composite material are written in terms of averages in each cluster; moreover, it is not required to solve via FEM the nonlinear micro-mechanical problem, allowing to drastically reduce the number of internal variables. PWUTFA is founded on the idea that it is possible to divide the RVE into large subdomains (clusters) that should group together finite elements having, under any applied loading condition, the most similar values of strain. Accordingly, in this study a multi-objective optimization-based approach is proposed with the aim to simultaneously reduce both the error in the approximation of the strain fields and the number of clusters in which the domain is decomposed. Different clustering solutions, obtained through the proposed optimization approach are analyzed, and the corresponding mechanical responses are compared with the ones obtained by the finite element analysis and by uniform transformation field analysis. [ABSTRACT FROM AUTHOR]

Most of studies on the hierarchical models were restricted to beam- and plate-like structures using the finite element method. In this context, this study intends to present the hierarchical models for cylindrical panels and investigate their characteristics using 2-D natural element method (NEM). The displacement field of cylindrical panel is decomposed into the triple-vectored in-plane functions and the assumed thickness polynomials, and the hierarchical models are defined by sequentially increasing the maximum orders of thickness polynomials from (1,1,0). The triple-vectored in-plane functions are approximated by applying 2-D NEM to the unfolded rectangular plane of curved mid-surface of cylindrical panels, for which the stiffness matrices corresponding to the membrane and transverse strains and stresses are calculated using the selectively reduced integration technique to avoid shear-membrane locking. The hierarchical models are demonstrated and validated from the comparative numerical experiments of bending and free vibration of cylindrical panels. The present hierarchical models provide the central deflections and the natural frequencies which are in good agreement with the reference solutions. In addition, the characteristics of hierarchical models such as the limit property and the modeling and approximation errors are investigated with respect to the model level and the width-thickness ratio and the grid density. It is found that the hierarchical models show the locking-free robust and spectral variation in the central deflection and natural frequencies. [ABSTRACT FROM AUTHOR]

A simple flux reconstruction for finite element solutions of reaction–diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction–diffusion problem. The flux reconstruction is based on simple, independent post-processing operations over patches of elements in conjunction with standard Raviart–Thomas vector fields. We prove that the corresponding local error indicators are locally efficient and robust (up to a nonstandard oscillation term) with respect to any mesh size and any size of the reaction coefficient, including the singularly perturbed limit. • A posteriori error bounds for singularly perturbed reaction–diffusion problems. • Proof of guaranteed upper bound on the error and robust local efficiency. • Fully computable estimator, fast to evaluate, and easily parallelizable. • Non-constant reaction coefficient, mixed boundary conditions, arbitrary dimension. [ABSTRACT FROM AUTHOR]

This paper presents a quasistatic problem of an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. We recall existence and uniqueness results obtained in Sofonea et al. (2017). The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model. [ABSTRACT FROM AUTHOR]

Abstract A mixed generalized multiscale finite element method for linear elasticity based on Hellinger Reissner principle with a strong symmetry enforcement for the stress tensor is introduced. The multiscale approximation space for the stress tensor is built on a coarse grid with carefully designed local problems so that the basis functions are also symmetric. Using eigenfunctions of local spectral problems as basis functions allows the approximation error of the multiscale finite element space to have a rapid spectral decay. Together with a properly chosen approximation space for the displacement, the method is shown to be inf–sup stable and robust as the first Lamé coefficient λ → ∞. Numerical experiments are supplemented to demonstrate fast convergence of the method with respect to local enrichment, and robustness of the method with respect to high contrast heterogeneity of the Poisson ratio. [ABSTRACT FROM AUTHOR]

In this paper, the artificial boundary method for anisotropic problems in semi-infinite strips is investigated. The exact and approximate boundary conditions on a segment artificial boundary are given. Finite element approximations are applied to the problem in a bounded computational domain and error estimates are obtained. Finally, some numerical examples show the effectiveness of this method. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimators for an optimal control problem governed by a convection–reaction–diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to approximate the solutions to the state and adjoint equations. We obtain a fully computable a posteriori error estimator for the optimal control problem. All the constants that appear in the upper bound for the error are fully specified. Therefore, the proposed estimator can be used as a stopping criterion in adaptive algorithms. We also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative. Numerical examples, in two and three dimensions, are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This work provides a new mixed finite element method for the Brinkman problem over arbitrary convex quadrilateral meshes. The velocity is approximated by piecewise polynomial element space which is H ( div ) -nonconforming, and the pressure is approximated by piecewise constant. We give the convergence analysis of our element, and especially show the robustness with respect to the Darcy limit. Moreover, via a discrete de Rham complex, a higher-order approximation error term is obtained for incompressible flow. Numerical examples verify our theoretical findings. [ABSTRACT FROM AUTHOR]

Large-margin classifiers are powerful techniques for classification problems. Although binary large-margin classifiers are heavily studied, multicategory problems are more complicated and challenging. A common approach is to construct k different decision functions for a k -class problem with a sum-to-zero constraint. However, such a constraint can be inefficient. Moreover, many large-margin classifiers can be sensitive to outliers in the training sample. In this article, we use the angle-based classification framework to avoid the explicit sum-to-zero constraint, and we propose two adaptively weighted large-margin classification techniques. Our new methods are Fisher consistent and more robust against outliers under suitable conditions. Numerical experiments further indicate that our methods give competitive and stable performance when compared with existing approaches. [ABSTRACT FROM AUTHOR]

The effect of fiber curvature on the overall material stiffness of fiber-reinforced composites is discussed using the formalism of stiffness contribution tensor for the special case of a continuous sinusoidal fiber. Stiffness contribution tensors of individual sinusoidal fibers with different crimp ratios are presented for the first time. The tensors are calculated numerically using Finite Element Analysis and analytically by representing fibers as equivalent sets of ellipsoids following the procedure available in the literature for approximation of effective composite stiffness. It is demonstrated that the existing procedure results in large approximation errors in several components of the stiffness contribution tensors of the considered fibers. A modification to the procedure is proposed to improve the accuracy of the predictions. Replacement relations that interrelate stiffness contribution tensors of inhomogeneities having the same shape but different material properties are tested for the studied fiber geometries. [ABSTRACT FROM AUTHOR]

In this paper, inverted finite element method is used for solving two-dimensional second order elliptic equations with a Dirichlet boundary condition in an exterior domain. After laying down the method, and after giving an estimate of the error, we detail how its implementation can be accomplished. Numerical results show the high efficiency and the accuracy of the method, especially for equations with infinitely varying coefficients. [ABSTRACT FROM AUTHOR]

This paper aims at providing recommendations for the practical application of relatively novel fatigue assessment approaches. Although the concept of notch stress intensity factor (N-SIF) itself was proposed decades ago to provide the analytical solution of the stress field in way of sharp notches, only recently it has been considered as the basis for fatigue strength assessment of welded joints. N-SIF itself and derived quantities such as the equivalent peak stress or the averaged strain energy density (SED) can be assumed as fatigue governing parameters and applied in everyday engineering practice being supported by powerful computation facilities. The N-SIF approach is still too cumbersome for practical applications; the SED approach seems promising to overcome shortcomings, to be applied flexibly to different joint geometries and possibly to automate the calculation procedure in finite element analysis. Software having different features and different refinement strategies are applied in order to benchmark advantages and disadvantages of possible practical solutions for the approaches based on N-SIF and averaged SED, thus offering various element types and modelling strategies. In general, high computational accuracy has been verified, despite relatively coarse meshes, if the element type, shape and mesh pattern is selected properly. However, the approach based on the averaged SED can be unsuitable for studying the notch effect of varying geometries if the averaging area increases more than the local stress concentration rises, as shown for the simplified geometry of a butt joint. [ABSTRACT FROM AUTHOR]

We provide a simple condition, which is both necessary and sufficient, that guarantees the existence of an enriched Crouzeix-Raviart element. Our main result shows that the latter can be easily expressed in terms of the approximation error in a multivariate generalized trapezoidal type cubature formula. Furthermore, we derive simple explicit formulas for its associated basis functions, and then prove how to use them to characterize all admissible added degrees of freedom, that generate well defined enriched Crouzeix-Raviart elements. We also show that the approximation error using the proposed enriched element can be written as the error of the (non-enriched) Crouzeix-Raviart element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L2 norm, with explicit constants in both two and three dimensions. A complement to this result is also given for any dimension. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H ( curl ) problem with inhomogeneous media and with the right-hand side only in L 2 . The estimator is of the recovery type. Independent with the current approximation to the primary variable (the electric field), an auxiliary variable (the magnetizing field) is recovered in parallel by solving a similar H ( curl ) problem. An alternate way of recovery is presented as well by localizing of the error flux. The estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable. It is proved that the estimator is approximately equal to the true error in the energy norm without the quasi-monotonicity assumption. Finally, we present numerical results for several H ( curl ) interface problems. [ABSTRACT FROM AUTHOR]

In this paper we introduce a posteriori error estimator based on the concept of error in the constitutive relation to verify parametric models computed with a reduced basis approximation. We develop a global error estimator which leads to an upper bound for the exact error and takes into account all the error sources: the error due to the reduced basis approximation as well as the error due to the finite element approximation. We propose an error indicator to measure the quality of the reduced basis approximation and we deduce an error indicator on the finite element approximation. [ABSTRACT FROM AUTHOR]

Projection-based model reduction is a well-established methodology for computing fast frequency sweeps of finite-element (FE) approximations to passive microwave structures. This contribution presents a novel provable error bound for moment-matching reduced-order models of lossless systems. It improves over existing methods by increasing the accuracy of the estimate and by reducing numerical costs. Numerical studies demonstrate the benefits of the suggested approach. [ABSTRACT FROM AUTHOR]

A systematic identification method for non-linear stochastic spatiotemproal (SST) systems described by non-linear stochastic partial differential equations (SPDEs) is investigated in this study based on pointwise observation data. A theoretical framework for a semi-finite element model approximating to an infinite-dimensional system is established, and several fundamental issues are discussed including the approximation error between the underlying infinite-dimensional dynamics and the model to be identified, and its rationality etc. Based on the proposed theoretical framework, a general identification method with irregular observation data is provided. These results not only provide an effective method for the identification of non-linear SST systems using measurement data (both offline and online), but also demonstrate a potential solution for the analysis, design and control of non-linear SST systems from a numerical point of view. [ABSTRACT FROM AUTHOR]

The semiclassical Schrödinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrödinger equation is approximated by these localized multiscale basis functions in space and is solved by the Crank–Nicolson method in time. These multiscale basis functions have compact supports in space, leading to the sparsity of the stiffness matrix, and thus the computational complexity of these two methods is comparable to that of the standard finite element method. The spatial mesh size in the multiscale finite element methods is O (ε) , while the mesh size in the standard finite element method is o (ε) , where ε is the semiclassical parameter. Through a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude. • Two multiscale methods are proposed to solve semiclassical Schrödinger equations with multiscale and time-dependent potentials. • A greedy algorithm to enrich multiscale basis functions and reduce the approximation error. • The computational complexity of our methods is comparable with the standard FEM. • More than two orders of magnitude reduction of the approximation error is achieved in our methods compared to the standard FEM. [ABSTRACT FROM AUTHOR]

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction. • New randomised algebra approach to expedite the computation of the FEM. • Efficient sampling that preserves aspects of leverage score sampling. • Rigorous analysis of the errors incurred in the methodology. • For a moderate approximation error we obtain substantial time savings. [ABSTRACT FROM AUTHOR]

In this work we extend our recently proposed adaptive refinement strategy for h p -finite element approximations of elliptic problems by taking into account an inexact algebraic solver. Namely, on each level of refinement and on each iteration of an (arbitrary) iterative algebraic solver, we compute guaranteed a posteriori error bounds on the algebraic and the total errors in energy norm. The algebraic error is the difference between the inexact discrete solution obtained by an iterative algebraic solver and the (unavailable) exact discrete solution. On the other hand, the total error stands for the difference between the inexact discrete solution and the (unavailable) exact solution of the partial differential equation. For the algebraic error upper bound, we crucially exploit the whole nested hierarchy of h p -finite element spaces created by the adaptive algorithm, whereas the remaining parts of the total error upper and lower bounds are computed using the finest space only. These error bounds allow us to formulate adaptive stopping criteria for the algebraic solver ensuring that the algebraic error does not significantly contribute to the total error. Next, we use the total error bound to mark mesh vertices for refinement via Dörfler's bulk-chasing criterion. On patches associated with marked vertices only, we solve two separate primal finite element problems with homogeneous Dirichlet (Neumann) boundary conditions, which serve to decide between h -, p -, or h p -refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the total errors of the inexact approximations between successive refinements (the error reduction factor), when the stopping criteria are satisfied. Finally, in a series of numerical experiments, we investigate the practicality of the proposed adaptive solver, the accuracy of our bound on the reduction factor, and show that exponential convergence rates are achieved even in the presence of an inexact algebraic solver. • Adaptive h p -refinement strategy taking into account an inexact algebraic solver. • Computable guaranteed bound on the ratio of errors between successive refinements. • Adaptive stopping criteria for the algebraic solver on each level of refinement. • Numerics: exponential convergence rates in presence of an inexact algebraic solver. [ABSTRACT FROM AUTHOR]

A novel numerical method, which uses linear edge elements in eddy current region and nodal finite elements in non eddy current regions, is presented for solving 3D eddy current problems. The mathematical models are divided into two parts according to their properties. The theory of linear edge elements is developed. The test results show the validity of the algorithm. [ABSTRACT FROM PUBLISHER]