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In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order 2α, 1/2 < α; ≤ 1, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the nth eigenvalue, the correction technique reduces the error order from O(n4h²) to O(n²h²). [ABSTRACT FROM AUTHOR]

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order m = 1 Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in $[{0,2}]\cup \{\infty \}$) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 1 as well as (lower and upper) bounds for the case when the same parameter is smaller than 1. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than 1, consistent with the traditional case where the Rényi parameter is equal to 1, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 1 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d. random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence. This tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to present several methods on how to deal with yield surface discontinuities. The explicit formulations, first presented by Koiter (1953), result in multisingular constitutive matrices which can cause numerical problems in elasto-plastic finite element calculations. These problems, however, are not documented in previous literature. In this paper an amendment to the Koiter formulation of the constitutive matrices for stress points located on discontinuities is proposed. Design/methodology/approach – First, a review of existing methods of handling yield surface discontinuities is given. Examples of the numerical problems of the methods are presented. Next, an augmentation of the existing methods is proposed and its robustness is demonstrated through footing bearing capacity calculations that are usually considered “hard”. Findings – Previous studies documented in the literature all present “easy” calculation examples, e.g. low friction angles and few elements. The amendments presented in this paper result in robust elasto-plastic computations, making the solution of “hard” problems possible without introducing approximations in the yield surfaces. Examples of “hard” problems are highly frictional soils and/or three-dimensional geometries. Originality/value – The proposed method makes finite element calculations using yield criteria with corners and apices, e.g. Mohr-Coulomb and Hoek-Brown, much more robust and stable. [ABSTRACT FROM AUTHOR]

This paper proposes an improved extended finite element method for modeling electromagnetic devices with multiple nearby geometrical interfaces. In regions near these interfaces, the magnetic vector potential approximation is enriched by incorporating multiple derivative discontinuous fields based on the partition of unity method such that the interfaces can be represented independent of the mesh. The support of a node or an element can be cut by several interfaces. This method results in the high accuracy in the approximation field and derivative field. Numerical examples applied to the iron core in 1-D eddy current field involving level set-based parts, error analysis, and electromagnetic field computations are provided to demonstrate the utility of the proposed approach. [ABSTRACT FROM PUBLISHER]

This paper studies level set topology optimization of structures predicting the structural response by the eXtended Finite Element Method (XFEM). In contrast to Ersatz material approaches, the XFEM represents the geometry in the mechanical model by crisp boundaries. The traditional XFEM approach augments the approximation of the state variable fields with a fixed set of enrichment functions. For complex material layouts with small geometric features, this strategy may result in interpolation errors and non-physical coupling between disconnected material domains. These defects can lead to numerical instabilities in the optimized material layout, similar to checker-board patterns found in density methods. In this paper, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach. This XFEM formulation is embedded into a level set topology optimization framework and studied with 'material-void' and 'material-material' design problems, optimizing the compliance via a mathematical programming method. The numerical results suggest that the generalized formulation of the XFEM resolves numerical instabilities, but regularization techniques are still required to control the optimized geometry. It is observed that constraining the perimeter effectively eliminates the emergence of small geometric features. In contrast, smoothing the level set field does not provide a reliable geometry control but mainly improves the convergence rate of the optimization process. [ABSTRACT FROM AUTHOR]

We consider the scattering of time-harmonic electromagnetic waves from a chiral medium. It is known for the Drude–Born–Fedorov model that the forward scattering problem can be described by an integro-differential equation. In this paper we study a Galerkin finite element approximation for this integro-differential equation. Our Galerkin scheme, which relies on a suitable periodization of the integral equation, enables the use of the fast Fourier transform and a simple numerical implementation. We establish a quasi-optimal convergence analysis for the Galerkin method. Explicit formulas for the discrete scheme are also provided. [ABSTRACT FROM AUTHOR]

This paper is the second part of a work which aims at proposing a global approach for the numerical modeling of parasitic capacitances in high-frequency multiwinding transformers. In this part, we focus on the numerical identification using the finite-element method (FEM) of the capacitive circuits which have been introduced in a companion paper in the same issue. An original procedure based on the FEM in 2-D electrostatics is first presented. The validity of the 2-D approximation is thereafter discussed, and a new method which consists in solving two successive 3-D models (an electrokinetic problem in the windings and an electrostatic one in the dielectrics) is then exposed. The two methods are illustrated and experimentally validated on a three-winding transformer. [ABSTRACT FROM AUTHOR]

In this paper we study the a priori error estimates for the finite element approximations of parabolic equations with measure data, especially we consider problems with separate measure data in time and space, respectively. The solutions of these kinds of problems exhibit low regularities due to the existence of measure data, this introduces some difficulties in both theoretical and numerical analysis. For both cases we use standard piecewise linear and continuous finite elements for the space discretization and derive the a priori error estimates for the semi-discretization problems, while the backward Euler method is then used for time discretization and a priori error estimates for the fully discrete problems are also derived. Numerical results are provided at the end of the paper to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

A concave finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is presented for scattering by a large body with nonuniform deep cavities. Different from the conventional FE-BI-MLFMA, the boundary integral equation in this concave FE-BI-MLFMA is established on a concave surface to reduce the region of the finite element method (FEM), which can significantly reduce the dispersion error from the FEM and improve the efficiency of FE-BI-MLFMA especially for nonuniform cavities. To eliminate the problem of slow convergence caused by concave surface, an efficient preconditioner based on the sparse approximate inverse (SAI) is constructed in this paper. Numerical performance of the constructed preconditioner based on the SAI is investigated in detail. Numerical experiments demonstrate the accuracy and efficiency of this SAI preconditioned concave FE-BI-MLFMA for nonuniform deep and large cavites. This SAI preconditioned concave FE-BI-MLFMA is parallelized to further improve its capability in this paper. An extremely big and deep nonuniform cavity has been calculated, demonstrating the great capability of this parallel concave FE-BI-MLFMA. [ABSTRACT FROM PUBLISHER]

In this paper we study an accuracy-increasing postprocessing technique for full discretizations of nonlinear convection–diffusion problems. The spatial discretization is based on finite elements. Once the fully discrete standard Galerkin approximation is computed at any fixed time, a discrete stationary convection–diffusion problem with data based on the Galerkin approximation is solved. The semidiscrete in space case was first considered in de Frutos, García-Archilla and Novo (2009, IMA J. Numer. Anal., 30, 2010, 1137–1158). A posteriori error estimates, based on this postprocessing technique, are obtained both for the semidiscrete in space and for the fully discrete cases. We prove that the semidiscrete error estimator proposed in this paper is efficient and asymptotically exact. In the fully discrete case the estimates have the property of giving a measure of the spatial errors that is independent of temporal errors. Some numerical experiments are provided. [ABSTRACT FROM PUBLISHER]

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted L2 projection onto H(div) conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient and systematic algorithm for the solution of electromagnetic scattering and radiation problems over a wide frequency interval. The algorithm is based on the evaluation of the Padé approximant of the solution vector, constructed at a minimum number of expansion frequencies within the interval of interest. The bisection algorithm introduced in this paper is robust. It keeps the approximation error bounded by a predefined constant over the entire frequency band. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 20: 376–380, 1999. [ABSTRACT FROM AUTHOR]

Highlights • A finite-buffer vacation queue with batch Markovian arrival process has been analyzed. • The server is subjected to serve under gated-limited service discipline. • Proposed analysis is based on the successive substitution and the supplementary variable method. • The results have been matched with the corresponding infinite-buffer queue. • Numerical results are presented for different service- and vacation-time distributions. Abstract This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process (BMAP) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations. The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms. [ABSTRACT FROM AUTHOR]

This paper aims to investigate an efficient numerical scheme for the optimal control of fractional-order dynamic systems. By using the Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations, the fractional optimal control problem under consideration is converted into a linear programming problem. Then, the internal model principle is employed in order to extend the new scheme for the fractional dynamic systems affected by the external persistent disturbances. Numerical examples and comparative results verify the validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we develop an equivalent approximation approach to obtain the Hamilton‐Jacobi‐Bellman (HJB) equations of intertemporal decision problems under uncertainty. An implicit discount function is adopted in the derivation of the rate of change of value function for the HJB equations to provide the broadest coverage of time preferences. Meanwhile, we provide an illustrative example for application by revisiting the intertemporal consumption and portfolio decisions problem. Furthermore, we verify the validity of our approach in robustness tests by incorporating exponential, nonconstant, and stochastic hyperbolic discounting functions, respectively. The HJB equations and their resulting decision rules in the finite and infinite planning horizons have confirmed that our approach provides an efficient and reliable alternative to acquire the HJB equations in the intertemporal economic issues under uncertainty. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to propose a new three-node triangular element in the framework of the numerical manifold method (NMM), which is designated by Trig3-MLScns.Design/methodology/approach The formulation uses the improved parametric shape functions of classical triangular elements (Trig3-0) to construct the partition of unity (PU) and the moving least square (MLS) interpolation method to construct the local approximation function.Findings Compared with the classical three-node element (Trig3-0), the Trig3-MLScns element has a higher order of approximations, much better accuracy and continuous nodal stress. Moreover, the linear dependence problem associated with many PU-based methods with high-order approximations is eliminated in the present element. A number of numerical examples indicate the high accuracy and robustness of the Trig3-MLScns element.Originality/value The proposed element inherits the individual merits of the NMM and the MLS. [ABSTRACT FROM AUTHOR]

This paper presents an extension of high-order moment-matching model order reduction to frequency-dependent impedance boundary conditions. Such boundary conditions are essential for accurately and efficiently modeling conductors at high frequency, where skin effect is significant. These boundary conditions involve complicated transcendental functions, whereas previous MOR methods assume polynomial dependence. Automatic differentiation is used to easily and accurately calculate the higher derivatives of such functions. Substantial improvement is shown, not only compared to discrete frequency sweeps, but also compared to non-moment-matching S matrix interpolation. [ABSTRACT FROM AUTHOR]

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree k, it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k-1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k-1 are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of H(div)-conformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis, and demonstrate the performance of the method on numerical examples. [ABSTRACT FROM AUTHOR]

SUMMARY Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so-called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well-sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto-poro-acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the classical error estimates of nonconforming Crouzeix-Raviart type finite elements for the Stokes equations. By introducing some quasi-interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high-order term, especially which can be of arbitrary order provided that f in the right-hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L2 norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L2 norms under the regularity assumption ( u, p) ∈ H1 + s(Ω) × H s(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity [ABSTRACT FROM AUTHOR]

In this paper, a meshless local maximum-entropy finite element method (LME-FEM) is proposed to solve 1D Poisson equation and steady state convection-diffusion problems at various Peclet numbers in both 1D and 2D. By using local maximum-entropy (LME) approximation scheme to construct the element shape functions in the formulation of finite element method (FEM), additional nodes can be introduced within element without any mesh refinement to increase the accuracy of numerical approximation of unknown function, which procedure is similar to conventional p-refinement but without increasing the element connectivity to avoid the high conditioning matrix. The resulted LME-FEM preserves several significant characteristics of conventional FEM such as Kronecker-delta property on element vertices, partition of unity of shape function and exact reproduction of constant and linear functions. Furthermore, according to the essential properties of LME approximation scheme, nodes can be introduced in an arbitrary way and the $$C^0$$ continuity of the shape function along element edge is kept at the same time. No transition element is needed to connect elements of different orders. The property of arbitrary local refinement makes LME-FEM be a numerical method that can adaptively solve the numerical solutions of various problems where troublesome local mesh refinement is in general necessary to obtain reasonable solutions. Several numerical examples with dramatically varying solutions are presented to test the capability of the current method. The numerical results show that LME-FEM can obtain much better and stable solutions than conventional FEM with linear element. [ABSTRACT FROM AUTHOR]

In this paper we consider the finite element approximation of the Stokes eigenvalue problems based on projection method, and derive some superconvergence results and the related recovery type a posteriori error estimators. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares strategy. The results are based on some regularity assumptions for the Stokes equations, and are applicable to the finite element approximations of the Stokes eigenvalue problems with general quasi-regular partitions. Numerical results are presented to verify the superconvergence results and the efficiency of the recovery type a posteriori error estimators. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Kármán equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables. Numerical results that justify the theoretical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative approach to a priori $$L^\infty $$ -error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical analysis for three-dimensional Bean's critical-state model in type-II superconductivity. We derive hyperbolic mixed variational inequalities of the second kind for the evolution Maxwell equations with Bean's constitutive law between the electric Field and the current density. On the basis of the variational inequality in the magnetic induction formulation, a semidiscrete Ritz-Galerkin approximation problem is rigorously analyzed, and a strong convergence result is proven. Thereafter, we propose a concrete realization of the Ritz-Galerkin approximation through a mixed Finite element method based on edge elements of Nédélec's First family, Raviart-Thomas face elements, divergence-free Raviart-Thomas face elements, and piecewise constant elements. As a Final result, we prove error estimates for the proposed mixed Finite element method. [ABSTRACT FROM AUTHOR]

Multisecant quasi-Newton methods have been shown to be particularly suited to solve nonlinear fixed-point equations that arise from partitioned multiphysics simulations where the exact Jacobian is inaccessible. In all these methods, the underdetermined multisecant equation for the approximate (inverse) Jacobian is enhanced by a norm minimization condition. The standard choice is the minimization of the Frobenius norm of the approximate inverse Jacobian. In this setting, it is well known that transient fluid-structure simulations typically require the use of secant information also from previous time steps to achieve a small enough number of iterations per implicit time step. The number of these time steps highly depends on the application, the physical parameters, the used solvers, and the mesh resolution. Using too few leads to a relatively high number of iterations, while using too many leads not only to a computational overhead but also to an increase in the number of iterations as well. Determining the optimal number requires a costly trial-and-error process. In this paper, we present results for two different approaches to overcome this issue: The first approach is to use a modified method (presented in [F. Lindner, M. Mehl, K. Scheufele, and B. Uekermann, "A Comparison of Various Quasi-Newton Schemes for Partitioned Fluid-Structure Interaction," in Proceedings of ECCOMAS Coupled Problems, Venice, 2015, pp. 1-12]) that minimizes the Frobenius norm of the difference between the current approximate (inverse) Jacobian and that of the previous time step. Thus, previous time step information is taken into account in an implicit and automatized way without magic parameters. The second approach is to use an unrestricted number of previous time steps in combination with a suitable filtering algorithm automatically removing secant information that is outdated (thus, slowing down convergence) or contradicting newer information (deteriorating the condition of the multisecant equation system). We present a novel algorithm to realize the first idea with linear complexity in the number of coupling surface unknowns (note that already storing the approximate inverse Jacobian would induce quadratic complexity) and the efficient parallel implementation for both approaches. This results in highly efficient, parallelizable, and robust iterative solvers applicable for surface coupling in many types of multiphysics simulations using black-box solver software. In addition, our numerical results for the fluid-structure benchmark (FSI3) from Turek and Hron ["Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow," in Fluid-Structure Interaction, Springer, Berlin, 2006, pp. 371-385] and for a flow through a flexible tube with a large variety of parameter settings prove the robustness and numerical efficiency of the first approach in particular. The second approach can be shown to be highly sensitive to the choice of the filtering method for the secant information and not always robust for the filters we currently use. [ABSTRACT FROM AUTHOR]

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q, EQ and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the unilateral contact model in linear elastostatics, the so-called Signorini problem. (Our results can also be applied to the scalar Signorini problem.) A standard continuous linear finite element approximation is first chosen to approach the two-dimensional problem. We develop a new error analysis in the H¹-norm using estimates on Poincaré constants with respect to the size of the areas of the noncontact sets. In particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact. This approach allows us to establish better error bounds under sole Hτ assumptions on the solution: if 3/2 < τ < 2 we improve the existing rate by a factor h(τ-3/2)² and if τ = 2 the existing rate (h¾) is improved by a new rate of h √"ln(h)|. Using the same finite element spaces as previously we then consider another discrete approximation of the (nonlinear) contact condition in which the same kind of analysis leads to the same convergence rates as for the first approximation. [ABSTRACT FROM AUTHOR]

A simple analytical approximation is proposed in this paper to calculate the crack tip opening displacement under general random variable amplitude loadings. This approximation is based on a modified Dugdale model for cyclic loadings. The discussion is first given under constant amplitude loading and is extended to several simple cases under variable amplitude loadings. Following this, a general algorithm is proposed under general random variable loadings. Numerical examples are verified with finite element simulations. Following this, hardening effect is included by including a hardening correction function. The proposed analytical approximation is very efficient compared to the direct finite element simulation. The solution can be used for detailed fatigue crack growth analysis under random variable amplitude loadings. [ABSTRACT FROM AUTHOR]

This paper studies the finite element (FE) approximation to a second-type variational inequality. The supe rclose and superconvergence results are obtained for conforming bilinear FE and nonconforming EQrot FE schemes under a reasonable regularity of the exact solution u ∊ H5/2(O), which seem to be never discovered in the previous literature. The optimal L²-norm error estimate is also derived for EQrot FE. At last, some numerical results are provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O( h). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU. [ABSTRACT FROM AUTHOR]

Purpose - The purpose of this paper is to achieve numerical simulation of discontinuous rock masses. Design/methodology/approach - The extended finite element method (XFEM) was used. Discontinuities (such as joints, faults, and material interfaces) are contained in the elements, thus the mesh can be generated without taking into account the existence of discontinuities. When one element contains no discontinuity, the displacement function is degenerated into that of the conventional finite element. For the element containing discontinuities, the standard displacement-based approximation is enriched by incorporating level-set-based enrichment functions that model the discontinuities, and an element subdivision procedure is used to integrate the domain of the element. Findings - Mesh generation can be simplified considerably and high-quality meshes can be obtained. A solution with good precision can also be achieved. It is concluded that the XFEM technique is especially suitable in simulating discontinuous rock masses problems. Research limitations/implications - Crack initiation and propagation should be considered in further studies. Practical implications - The paper presents a very useful numerical method for a geotechnical engineering problem that has the ability to simulate the failure process of discontinuous rock masses. The method is expected to be used widely in the deformation and stability analysis of complicated rock masses. Originality/value - The paper provides a new numerical method for discontinuous rock masses that is very convenient for pre-processing. [ABSTRACT FROM AUTHOR]

In an earlier paper the last two authors studied spatially semidiscrete piecewise linear finite element approximations of the heat equation and showed that, in the case of the standard Galerkin method, the solution operator of the initial-value problem is neither positive nor contractive in the maximum-norm for small time, hut that for the lumped mass method these properties hold, if the triangulations are essentially of Delaunay type. In this paper we continue the study by considering fully discrete analogues obtained by discretization also in time. The above properties then carry over to the backward Euler time stepping method, but for other methods the results are more restrictive. We discuss in particular the 9-method and the (0, 2) Padé approximation in one space dimension. [ABSTRACT FROM AUTHOR]

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier-Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well. [ABSTRACT FROM AUTHOR]

The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm based on the Mumford-Shah model for simultaneously detecting the edge features of two images and jointly estimating a consistent set of transformations to match them. Compared to the current asymmetric methods in the literature, this fully symmetric method allows one to determine one-to-one correspondences between the edge features of two images. The entire variational model is realized in a multiscale framework of the finite element approximation. The optimization process is guided by an estimation minimization-type algorithm and an adaptive generalized gradient flow to guarantee a fast and smooth relaxation. The algorithm is tested on Ti and T2 magnetic resonance image data to study the parameter setting. We also present promising results of four applications of the proposed algorithm: interobject monomodal registration, retinal image registration, matching digital photographs of neurosurgery with its volume data, and motion estimation for frame interpolation. [ABSTRACT FROM AUTHOR]

This paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the Navier-Stokes equations and the Cahn-Hilliard equation, which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids. In the model the two sets of equations are coupled through an extra phase induced stress term in the Navier-Stokes equations and a fluid induced transport term in the Cahn-Hilliard equation. Fully discrete mixed finite element methods are proposed for approximating the coupled system, it is shown that the proposed numerical methods satisfy a mass conservation law, and a discrete energy law which is analogous to the basic energy law for the phase field model. The convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law. As a by-product, the convergence result also provides a constructive proof of the existence of weak solutions to the Navier-Stokes-Cahn-Hilliard phase field model. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach. [ABSTRACT FROM AUTHOR]

This paper introduces the meshless local radial point interpolation method to model eddy-current problems for the first time. The formulation is described and results are compared with those from the ordinary finite-element simulation and the analytical solution. [ABSTRACT FROM AUTHOR]

This paper presents an enhancement of the finite element method (FEM) by adopting the moving Kriging (MK) interpolation as a substitute for the traditional hat functions. The MK shape functions can be referred as element-free because their construction is not tied to the element geometry. Kriging interpolation is a geostatistical technique for spatial interpolation. The basic idea of Kriging is that any unknown point can be interpolated from known scatter points in a specific domain. Using the moving Kriging interpolation, shape functions can be generated over any finite set of nodes. This leads to an idea to extend the influence of a node beyond the layer of surrounding elements to enhance the global smoothness of the field variable and its derivatives. The present paper thus proposes a concept of layered domain of influence. Hence, characteristic arrays of an element, such as the element stiffness, have contributions from all visible nodes that include a set of satellite nodes unattached to the element. The validation of the method was confirmed through numerical tests of one and two-dimensional problems. The results show remarkable accuracy and global smoothness. Existing general-purposed FE programs can be easily modified to accommodate this new element concept; thus, the method has a higher chance to be accepted in practice. [ABSTRACT FROM AUTHOR]

In this paper, we provide three direct procedures to extrapolate the early-time and the low-frequency response of a causal signal simultaneously in the time-and frequency domain. Compared with the extrapolation by orthonormal basis functions, direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions. We show that the extrapolation introduced by Adve and Sarkar is equivalent to a Neumann-series solution of an integral equation of the second kind. It is further shown that this iterative Neumann expansion is an error-reducing method. We propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme. The convergence of this scheme is also demonstrated. We provide the matrix equations and show the equivalence to the integral equations, and demonstrate that the method of singular value decomposition (SVD) of solving the matrix equation provides accurate and stable results. Finally, a number of illustrative numerical examples are presented and the performances of the three direct methods are compared. [ABSTRACT FROM AUTHOR]

This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used. We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical approximation of the bending of a plate modeled by Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is based on the family of elements called MITC (mixed interpolation of tensorial components). We consider two lowest-order methods of this family on quadrilateral meshes. Under mild assumptions we obtain optimal H¹ and L² error estimates for both methods. These estimates are valid with constants independent of the plate thickness. We also obtain error estimates for the approximation of the plate vibration problem. Finally, we report some numerical experiments showing the very good behavior of the methods, even in some cases not covered by our theory. [ABSTRACT FROM AUTHOR]

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform 'virtual' (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems' scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented. [ABSTRACT FROM AUTHOR]

This paper proposes a method of two dimenslonal analysis for the thickness-flexural vibration, where the one-dimensional finite-element method (FEM) is applied to the planar equation. This approach aims at application to the analysis of the flexural spurious vibrations of strip rectangular AT-cut plate often used in practice. The feature of the proposed method is that the two-dimensional coupled vibration can be analyzed by applying twice the one-dimensional FEM. in other words, the one-dimensional FEM is applied to the eigenmode guided by the X-edge, as well as to the resonant expression derived from the boundary condition on the Z'-edge. As a result, the considered matrix is reduced greatly in size compared to the direct analysis by the two-dimensional FEM. The result of calculation for the change of the resonant frequency due to the change of the Z'-edge ratio is compared to the result of experiment, and the usefulness of the proposed method is demonstrated. It is shown also that most of the spurious vibrations depending on the Z'-edge ratio are flexural vibrations, which is a new finding. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for solving systems of partial differential equations describing flow in porous media with an embedded and inclined conduit pipe. This work considers a coupled continuum pipe-flow/Darcy model. The numerical schemes presented are based on combinations of the quasi-Wilson element on anisotropic mesh and the conforming finite element on regular mesh. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in bothL2andH1norms are obtained independent of the regularity condition on the mesh. Numerical examples show the accuracy and efficiency of the proposed scheme. [ABSTRACT FROM PUBLISHER]

This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation-consolidation processes. The model focuses on the steady-state regime of a solid-liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection - nonlinear diffusion equation describing the transport of the solids volume fraction. The variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart-Thomas approximations of order for the stress and piecewise continuous polynomials of order for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed-primal finite element method, and the properties of the model. [ABSTRACT FROM AUTHOR]

In this paper, we propose an a posteriori error estimator for the numerical approximation of a stochastic magnetostatic problem, whose solution depends on the spatial variable but also on a stochastic one. The spatial discretization is performed with finite elements and the stochastic one with a polynomial chaos expansion. As a consequence, the numerical error results from these two levels of discretization. In this paper, we propose an error estimator that takes into account these two sources of error, and which is evaluated from the residuals. [ABSTRACT FROM AUTHOR]