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In this paper, we propose two two-level finite element methods based on charge-conservation to solve the 2D/3D stationary inductionless magnetohydrodynamic (MHD) equations in Lipschitz domain. The methods are based on the Picard coupling correction on the fine mesh, one has to solve a nonlinear problem on the coarse mesh, the other one need to solve a linearized problem with Newton iteration on the coarse mesh. In addition, we give the stabilities and error estimates for the proposed methods. Several numerical results are presented to verify the theoretical analysis and show the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

• The polygonal thin plate element is based on SBFEM and discrete Kirchhoff theory. • The polygonal thin plate element can possess the second order completeness. • The plate element is determined by the boundary nodal displacements and rotations. • The computations of the shape functions of SBFEM can be avoided. • The element has good accuracy for some distorted and irregular polygonal meshes. The scaled boundary finite element method (SBFEM) is a powerful method for solving elastostatics problems based on polygonal elements. In this paper, we firstly construct the quadratic polygonal scaled boundary element only depends on the boundary nodal displacements by transforming the additional degrees of freedom derived from the constant body loads to those by the boundary nodes. Further, combining with the discrete Kirchhoff theory, we construct the polygonal scaled boundary thin plate element, which can possess the second order completeness. The element stiffness matrix for the thin plate problem can be transformed by the stiffness matrix for the plane problem directly by avoiding to compute the shape functions of SBFEM. Numerical examples verify that the proposed polygonal scaled boundary thin plate element has good accuracy. [ABSTRACT FROM AUTHOR]

• A simulation of multiple transverse ply cracking in laminates is presented. • Many phase-field variables are used, each one describes the crack of the layer. • Some interesting phenomenons when the cracks appear at different layers of the structure are shown out. The phase-field theory is a well-known mathematical model for solving interface problems, including crack problems in fracture mechanics. In this study, the formula is derived by variational approaches based on the Reissner-Mindlin plate kinematics and the multi-phase-field theory for simulation of the buckling phenomenon in cracked laminates. Phase-field parameters are defined independently in different plies of laminate to capture the crack behavior of each ply. Simulation is carried out to numerically investigate the stiffness reduction and buckling behavior of transverse cracked laminated composite plates. This paper focuses on the consideration of laminated composite plates, which have a crack in each layer. Therefore, this work is more complicated than the case of the plate has one crack throughout the plate thickness. The significant advancement of the phase-field approach for laminated composite plates with complex crack geometries is demonstrated. [ABSTRACT FROM AUTHOR]

Highlights • The paper Steps to increase practical applicability of PragTic software deals with the parallelization of the fatigue solver PragTic at the node level of the finite element model, and its scalability testing. • Because of the intended focus on multiaxial fatigue analysis, the parallelization at the plane level while searching for the critical plane is also discussed. • Impact of such changes is clearly demonstrated. Abstract This paper describes various methods for increasing the computational speed of an existing fatigue solver called PragTic. The paper describes the basic workflow of computational fatigue analysis, including multiaxial fatigue analysis as well. It documents some of the inefficiencies of the original PragTic version, and parallelization and scalability testing of the parallelized PragTic in three different case studies. These studies include a simple model example (hundreds of nodes) and a real world example (millions of nodes). The implemented parallelization techniques are tested using numerical experiments to demonstrate their parallel scalability. As an output of a unique analysis, the number of necessary and feasible evaluated planes in multiaxial analyses is monitored, and the outcome favoring integral methods for the multiaxial fatigue analysis is commented. [ABSTRACT FROM AUTHOR]

In this paper, a new bilinear immersed finite volume element method based on rectangular mesh is presented to solve the elliptic interface problems with non-homogeneous jump conditions and sharp-edged interfaces. This method is capable of dealing with the case when the interface passes through grid points and when the solutions are oscillating. Plenty of numerical experiments show that our method is nearly second-order accuracy for the solution and is first-order accuracy for the gradient of the solution in the L ∞ norm. • The proposed bilinear IFVE method can deal with the case when the solution or its normal derivative is discontinuous. • Numerical results demonstrate that the method is second-order accuracy for the solution in the L ∞ norm. • The new method can solve the crack problem provided that the mesh is fine enough. [ABSTRACT FROM AUTHOR]

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEM gl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEM gl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEM gl closely matches reference fine-scale FEM solutions in the L ∞ norm. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L 2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method. [ABSTRACT FROM AUTHOR]

Simulating multi-body dynamics with both rigid and flexible parts and with frictional contacts is a hard problem. We solve this by expressing the couplings between the bodies as position level constraints. The implicit treatment of the constraint directions gives us improved stability over velocity based methods. Then by employing regularization of nonlinear constraints and a convex minimization formulation, we bridge constraint-based methods to traditional force-based methods. In fact, the former are just a dual variables formulation of the latter. We solve this dual problem using position based dynamics (PBD). We show how PBD is a completely valid modeling technique and we extend it with an accurate contact and Coulomb friction model. We further show for the first time how the same solver can be used to simulate both rigid and deformable solids with two way coupling. For the soft bodies we introduce a novel form of linear finite elements expressed as constraints, that is more accurate than PBD mass-spring systems. More of our results include the energy conserving Newmark integrator and the accelerated Jacobi solver suitable for parallel architectures. Note that this paper is an extended and revised version of the conference paper published in [1]. [ABSTRACT FROM AUTHOR]

Helmholtz resonator is one of the most basic acoustic models in acoustic theoretical research and engineering applications. It is simple and effective to directly apply the theoretical formula for its resonant frequency calculation, but sometimes the calculation error is too large or even wrong. In this paper, the characteristics of Helmholtz resonators are studied based on the finite element numerical analysis. The influence of structural parameters and boundary conditions of the Helmholtz resonator on the resonant frequency is given, several related problems of the theoretical formula are supplemented and the relevant conclusions are obtained. The explanations employed in this paper could be used as a supplement to the theoretical formula for theoretical study and engineering applications of Helmholtz resonators. [ABSTRACT FROM AUTHOR]

This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain. The coupled approach is applied in solving thermomechanical initial–boundary value problem where the heat transport in the domain is non-stationary. In this method, the domain is divided into two subdomains for FEM and MFDM, respectively. Contrary to other coupling techniques, the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level. In the weak form of thermomechanical initial–boundary value problem (variational principle), the appropriate additional coupling integrals are defined a-priori. Subsequently, the FEM and the MFDM approximations, which may differ from each other, are provided to the formulation. It is assumed that there exists a very thin layer of material between the subdomains, which is not spatially discretized. The width of this layer may be considered the coupling parameter and it is the same for both, thermal and mechanical parts. Similar approach is applied to essential boundary conditions (e.g. prescribed temperature and displacements). Consequently, the consistent formulation of the mixed problem for the coupled FEM–MFDM method is derived. The analysis is illustrated with two- and three-dimensional examples of mechanical and thermomechanical problems. [ABSTRACT FROM AUTHOR]

Abstract The paper is aimed at improving computational cost enhanced by a new combination of deep neural network (DNN) and modified symbiotic organisms search (mSOS) algorithm for optimal material distribution of functionally graded (FG) plates. The material distribution is described by control points, in which coordinates of these points are located along the plate thickness using B-spline basis functions. In addition, DNN is used as an analysis tool to supersede finite element analysis (FEA). By using DNN, solutions can directly be predicted by an optimal mapping which is defined by learning relationship between input and output data of a dataset in training process. Each of dataset is randomly created from analysis through iterations by using isogeometric analysis (IGA). The mSOS being a robust metaheuristic algorithm is employed to solve two optimization problems: buckling and free vibration with various volume constraints. Moreover, the power of mSOS is verified by comparing to other algorithms in the open literature. Finally, optimal results in all examples generated by the proposed method are compared to those of a combination of IGA and mSOS to demonstrate its effectiveness and robustness. [ABSTRACT FROM AUTHOR]

• Software is presented for verifying Timoshenko beam finite elements. • A popular fiite element is shown to converge to the wrong answer. • An element modification sufficient for convergence is reviewed. Abstract The classic paper Han et al. Dynamics of transversely vibrating beams using four engineering theories , Journal of Sound and Vibration, 225, 1999, sets up the continuation problem that determines the exact vibrational frequencies and mode shapes of prismatic Timoshenko beams. This article presents a portable implementation of a solver for the problem. In order to reliably compute a complete set of modes, without redundancies, it is crucial to formulate the frequency equation as a continuation problem. The continuation problem is solved for a thickness parameter starting from zero thickness (the Euler beam), and increasing up to the finite thickness of interest. The continuation problem has a bifurcation point, at which a wave number variable passes from real to a purely imaginary value. For finite thickness, the continuation problem for all sufficiently high frequency modes crosses through this bifurcation. Han et al.[7] solve the continuation problem using an unspecified differential equation solver, and dummyTXdummy- do not discuss accuracy. Here a solution of the continuation problem using pseudoarclength continuation is presented. This article presents a description of how to compute this singularity exactly, and develops software that maintains full working precision. Furthermore the details of the computation of the mode shapes from the frequencies are presented for free-free, clamped-clamped and clamped-free boundary conditions. The source code is available under a MIT license from github in the TimoshenkoBeamEigenvalues repository. [ABSTRACT FROM AUTHOR]

In this paper, we develop an online basis enrichment method with the mortar mixed finite element method, using the oversampling technique, to solve for flow problems in highly heterogeneous media. We first compute a coarse grid solution with a certain number of offline basis functions per edge, which are chosen as standard polynomials basis functions. We then iteratively enrich the multiscale solution space with online multiscale basis functions computed by using residuals. The iterative solution converges to the fine scale solution rapidly. We also propose an oversampling online method to achieve faster convergence speed. The oversampling refers to using larger local regions in computing the online multiscale basis functions. We present extensive numerical experiments(including both 2D and 3D) to demonstrate the performance of our methods for both steady state flow, and two-phase flow and transport problems. In particular, for the time dependent two-phase flow and transport problems, we apply the online method to the initial model, without updating basis along the time evolution. Our numerical results demonstrate that by using a few number of online basis functions, one can achieve a fast convergence. [ABSTRACT FROM AUTHOR]

The standard application of the Lehmann–Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes σ ̃ i that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes σ ̃ i are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes σ ̃ i of the same quality. The simplified problem is smaller, it is positive definite, and any H ( div , Ω ) conforming finite elements, such as Raviart–Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions. [ABSTRACT FROM AUTHOR]

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In structural reliability, an important challenge is to reduce the number of calling the performance function, especially a finite element model in engineering problem which usually involves complex computer codes and requires time-consuming computations. To solve this problem, one of the metamodels, Kriging is then introduced as a surrogate for the original model. Kriging presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used as an active learning method. In this paper, a new learning function based on information entropy is proposed. The new learning criterion can help select the next point effectively and add it to the design of experiments to update the metamodel. Then it is applied in a new method constructed in this paper which combines Kriging and Line Sampling to estimate the reliability of structures in a more efficient way. In the end, several examples including non-linearity, high dimensionality and engineering problems are performed to demonstrate the efficiency of the methods with the proposed learning function. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid preconditioning technique for Conjugate Gradient method and discusses its parallel implementation on Graphic Processing Unit (GPU) for solving large sparse linear systems arising from application of interior point methods to conic optimization problems in the context of nonlinear Finite Element Limit Analysis (FELA) for computational Geomechanics. For large 3D problems, the use of direct solvers in general becomes prohibitively expensive due to exponentially growing memory requirements and computational time. Besides, the so-called saddle-point systems resulting from use of optimization framework is not an exemption. On the other hand, although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods, they usually exhibit high number of iterations to reach convergence. In present paper, we show that this problem can be effectively tackled using the proposed hybrid preconditioner along with an elaborate implementation on GPU. Furthermore, numerical results verify the robustness and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law r λ behavior of the displacement field and the corresponding r λ − 1 singularity in the stress field, for 1 2 ≤ λ < 1 . This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width ( P → W ); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it ( P & W ). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O ( h 2 ) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 1 2 < λ < 1 . The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O ( h ) convergence rate. It is also demonstrated that the standard P → W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P & W formulation based on the Hellinger–Reissner variational principle does not have this disadvantage. [ABSTRACT FROM AUTHOR]

The paper presents a frequency-domain h p -adaptive Finite Element (FE) Method for a class of coupled acoustics/elasticity/poroelasticity problems with application to modeling of acoustic logging measurements in complex borehole environments. The paper extends methodology, software, and results presented in Matuszyk et al. (2012). We derive for poroelastic media a mixed weak formulation, which is closely related to the enhanced ( u , p ) formulation developed by Atalla et al. (2001). The formulation is extended to the transversely anisotropic (VTI) case, and makes use of the viscodynamic operator, which enables more accurate calculations for higher frequencies. In this approach, a special split of the differential operator results in boundary and coupling conditions that can be easily implemented within the FE framework. The appropriate PML technique is employed to truncate the computational domain. Accordingly, the original h p -FE code is modified by the implementation of the appropriate scaling of the independent variables to make efficient the automatic adaptation algorithm. Solutions of non-trivial examples involving formations with permeable layers are presented. They positively verify the consistency and accuracy of the presented method. [ABSTRACT FROM AUTHOR]

This paper is aimed to investigate the process design and control method in cold rotary forging of parts with non-rotary upper and lower profiles. Using the analytical and FE simulation methods, three critical technological problems in the cold rotary forging process of this kind of parts are resolved reasonably. The first one is that an accurate design method of the non-rotary upper die is presented based on the geometrical and kinematic relationship between the upper die and upper profile of parts. The second one is that the interference judgment between the upper die and upper profile of parts can be achieved by analytically obtaining the trajectory of any point in the upper die. The third one is that the metal flow and geometrical accuracy of the non-rotary upper profile of parts can be effectively controlled through optimizing the process parameters. On the basis of the above research, the cold rotary forging process of a typical gear part, both the upper and lower profiles of which are non-rotary, is investigated numerically and experimentally. The results indicate that the desirable geometrical accuracy of the non-rotary upper profile of the gear part can be obtained, which verifies that the process design and control method presented in this paper is valid and cold rotary forging can thus be used to manufacture the parts with non-rotary upper and lower profiles. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method for solving two-dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface. We use finite element method coupled with Newton’s method to deal with the jump conditions and to linearize the system. It is easy to implement. The grid used here is body-fitting grids based on the idea of semi-Cartesian grid. Numerical experiments show that this method is nearly second order accurate in the L ∞ norm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples. [ABSTRACT FROM AUTHOR]

The consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties. In this paper, we are interested in the resolution of an exterior Maxwell problem in 3D. As a first step, we focus on the justification of an algorithm described in literature, using an interpretation as a Schwarz method. The study of the convergence indicates that it depends significantly on the thickness of the domain of computation. This analysis suggests the use of the finite element term of Schwarz method as a preconditioner for use of Krylov iterative solvers. An analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach. The consideration of the preconditioner suggested by the Schwarz method leads to a superlinear convergence of the GMRES predicted by the analytical study and verified numerically. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to present the SUPG finite element method for adiabatic flows and to compare the results with those obtained via various computational methods. The incompressibility assumption is often used to solve transient viscous flows. On the other hand, compressible flow analyses are also popular because natural flows include compressibility even if it is a negligible amount. It is well known that incompressibility is a limit state of compressibility. To solve compressible flows, three kinds of governing equations are needed: the conservation of mass, momentum, and energy, in which density, velocity, and internal energy are independent variables. If we assume that there is no heat transfer in or out of a system, the energy equation can be eliminated from the governing equations. Density and velocity can be considered independent variables. These kinds of flows are referred to as adiabatic flows in this paper. The SUPG formulation is one of the most widely used methods in the finite element analyses of fluid flows. In this paper, the SUPG method for adiabatic flows is presented. The polytropic law is introduced for the equation of state. Moreover, the computational results obtained by four models of fluid flows are compared: adiabatic flows, compressible flows, constant acoustic velocity flows, and incompressible flows. Verifications are carried out using a lid-driven cavity flow. Boundary effects are estimated using a wide computational domain. Drag forces are compared with those computed using incompressible flows. The pressure coefficients over a surface of a circular cylinder located in fluid flows computed by the present method show good correspondence with the experimental measurement. On the other hand, significant discrepancies are observed between the pressure coefficients computed assuming constant density and the experimental results. [Copyright &y& Elsevier]

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium. The proposed method is based on the time discontinuous Galerkin Space-Time Finite Element Method (ST/FEM). It solves the seepage problem in the saturated region. The primary unknown in ST/FEM is piezometric pressure. Fluid velocities are derived from the pressure using Darcy's law. Further, an iterative algorithm has been proposed in this paper to implement the proposed method. In each iteration step, the computation domain is updated according to the flow velocity on the phreatic boundary. Subsequently, internal nodes are moved using the mesh moving technique to accommodate the newly updated computation domain. The mesh moving technique, which is discussed in this paper, is based on an elasticity problem. ST/FEM is employed to analyze several unconfined seepage flow problems, and results of steady state solutions are compared with those available in the literature to demonstrate the efficacy of the proposed scheme. • Moving mesh type finite element method is developed for transient and unconfined seepage flow analysis. • The proposed scheme requires 2 to 3 iterations in a time step to achieve convergence. • Elasticity equation based automatic mesh moving technique is employed to move the internal nodes. • Initial computation domain significantly affects the deformation characteristics of mesh. • Proposed iterative scheme can be employed with any type of mesh moving technique. [ABSTRACT FROM AUTHOR]

The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k ≥ 1 . Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment. [ABSTRACT FROM AUTHOR]

We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix–Raviart P 1 -nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian meshes is developed for solving planar elasticity interface problems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover, the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimentally, extensive numerical examples are given to show that the approximation orders in L 2 norm and semi- H 1 norm are optimal under various Lamé parameters settings and different interface geometry configurations. [ABSTRACT FROM AUTHOR]

Solving elliptic equations with interfaces has wide applications in engineering and science. The real world problems are mostly in three dimensions, while an efficient and accurate solver is a challenge. Some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions. Although traditional finite element method using body-fitted grid is well-established, the expensive cost of mesh generation is an issue. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve elliptic interface problems. The special cases when the interface cuts though grid points are handled carefully, rather than perturbing the cutting point away to apply the method for general case. Both Dirichlet and Neumann boundary conditions are considered. Numerical experiments show that this method is approximately second order accurate in the L ∞ norm and L 2 norm for piecewise smooth solutions. The large sparse matrix for our linear system also has nice structure and properties. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear dynamic buckling response of axisymmetric carbon nanotubes reinforced composite sandwich spherical caps subjected to suddenly applied pressure. To solve this problem, a finite element approach is adopted using an axisymmetric shear flexible shell element, free from spurious constraints causing shear and membrane locking especially for the case of thin shells. The geometric nonlinearity is accounted for in the formulation using von Karman’s strain-displacement relations. Newmark’s integration technique along with the modified Newton-Raphson iteration scheme is employed to solve the nonlinear governing equations. The critical dynamic pressure value is considered as the pressure load beyond which the maximum average displacement response shows instant growth in the time history of the shell structure. The obtained results are validated against the available analytical solutions for the case of isotropic spherical caps. A detailed parametric study is carried out to bring out the effects of shell geometry parameter, volume fraction of the CNT, core-to-face sheet thickness, temperature, boundary conditions and different types of pulse on the dynamic snap-through characteristics of spherical shells. [ABSTRACT FROM AUTHOR]

The extended finite element method (XFEM) is widely accepted in academy as the major technique for crack analysis. Starting from 2009, commercial codes started to use this technique for crack analysis, singling the mature and acceptance of the technique in industries. However, the direct extension of the singular tip enrichment of XFEM, the core of the method, to dynamic crack growth simulation has long been a difficulty due to: (a) elevated bad conditioning as crack propagating, (b) extra-dof dynamics and energy inconsistency, and (c) “null” critical time step size and optimal mass lumping at crack tip. Based on an extra-dof-free partition of unity enrichment technique (Tian, 2013), we have improved XFEM through a crack tip enrichment without extra dof (Tian and Wen, 2015). This paper is to answer the question whether the improved XFEM can also be easily extended to dynamic problems. Numerical tests show that the new XFEM is not only straightforward in implementation in dynamic problems, also provides the most accurate dynamic SIF in the benchmark problems and is orders of magnitude faster with an iterative solver. [ABSTRACT FROM AUTHOR]

Mesh updating is a problem commonly encountered in fluid–structure interaction simulations when structures undergo a large displacement. In this paper, a set of mesh updating strategies based on a “pseudo-solid” model is proposed. Generally, a homogeneous analysis is first performed on the pseudo-solid with prescribed boundary displacements. Different element properties and external forces are then introduced based on strain results from the first analysis. A second analysis is performed on this non-homogeneous model with the goal to maintain element aspect ratio while preserving element volumes. Several numerical simulations are conducted using different combinations of strategies proposed in this paper. In general, the proposed strategies significantly enhance the performance of the pseudo-solid mesh updating method. [Copyright &y& Elsevier]

This paper presents and validates a new local to global (L2G) FEM approach that can analyze multiple, interactive fracture processes in 2D solids with improved numerical efficiency and robustness. The method features: 1) forming local problems for individual and interactive cracks; and 2) parallel solving local problems and returning local solutions as part of the trial solution for global iteration. It has been demonstrated analytically (through a simple 1D problem) and numerically (through several benchmarking examples) that, the proposed method can substantially improve the robustness of the global solution process and significantly reduce the costly global iteration for convergence. The demonstrated improvement in numerical efficiency is up to 20 ∼ 40 % for mildly unstable problems. For problems with severely unstable crack initiation and propagation, the improvement can be more significant. This new method is readily applicable to other popular methods such as the extended FEM (X-FEM), Augmented FEM (A-FEM) and Phantom-node method (PNM). [ABSTRACT FROM AUTHOR]

The paper proposes a 3D mixed finite element and tests its performance in elasto-plastic and limit analysis problems. A composite tetrahedron mesh is assumed over the domain. Within each element the displacement field is described by a quadratic interpolation, while the stress field is represented by a piece-wise constant description by introducing a subdivision of the element into four tetrahedral regions. The assumptions for the unknown fields make the element computationally efficient and simple to implement also in existing codes. The limit and elasto-plastic analyses are formulated as a unified mathematical programming problem allowing the use of Interior Point like algorithms. A series of numerical experiments shows that the proposed finite element is locking free and has a good plastic behavior. [ABSTRACT FROM AUTHOR]

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

In this paper we introduce a Nitsche-XFEM method for fluid–structure interaction problems involving a thin-walled elastic structure (Lagrangian formalism) immersed in an incompressible viscous fluid (Eulerian formalism). The fluid domain is discretized with an unstructured mesh not fitted to the solid mid-surface mesh. Weak and strong discontinuities across the interface are allowed for the velocity and pressure, respectively. The fluid–solid coupling is enforced consistently using a variant of Nitsche’s method with cut-elements. Robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization. Several coupling schemes with different degrees of fluid–solid time splitting (implicit, semi-implicit and explicit) are investigated. A series of numerical test in 2D, involving static and moving interfaces, illustrates the performance of the different methods proposed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stable nonconforming mixed finite element method (MFEM) for the constrained optimal control problems (OCPs) governed by Stokes equations, in which the E Q 1 r o t -constant scheme just satisfies the discrete inf–sup condition. The superclose and superconvergence results are obtained. [ABSTRACT FROM AUTHOR]

In this paper we survey all known (including own recent results) properties of the longest-edge n -section algorithms. These algorithms (in classical and recently designed conforming form) are nowadays used in many applications, including finite element simulations, computer graphics, etc. as a reliable tool for controllable mesh generation. In addition, we present a list of open problems arising in and around this topic. [ABSTRACT FROM AUTHOR]

In this paper we analyze and numerically solve a problem related to the optimal location of green zones in metropolitan areas in order to mitigate the urban heat island effect. So, we consider a microscale climate model and analyze the problem within the framework of optimal control theory of partial differential equations. Finally we compute its numerical solution using the finite element method, with the help of the interior point algorithm IPOPT, interfaced with the FreeFem++ software package. [ABSTRACT FROM AUTHOR]

In this paper, an Uzawa -type augmented Lagrangian contact formulation is presented for modeling frictional discontinuities in the framework of the X-FEM technique. The kinematically nonlinear contact problem is resolved based on an active set strategy to fulfill the Kuhn–Tucker inequalities in the normal direction of contact. The Coulomb’s friction rule is employed to address the stick–slip behavior on the contact interface through a return mapping algorithm in conjunction with a symmetrized (nested) augmented Lagrangian approach. A stabilization algorithm is proposed for the robust imposition of the frictional contact constraints within the proposed augmented Lagrangian framework. Several numerical examples are presented to demonstrate various aspects of the proposed computational algorithm in simulation of the straight, curved and wave-shaped discontinuities. [ABSTRACT FROM AUTHOR]

This paper introduces a new recovery-based a posteriori error estimator for the lowest order Nédélec finite element approximation to the H ( curl ) interface problem. The error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results. Under certain assumptions, it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we will construct a discrete de Rham complex with commuting interpolation operators for Fourier finite element methods in axisymmetric domains. To our knowledge, this will be the first construction of a discrete de Rham complex and commuting projectors that can be applied to axisymmetric problems with general data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w : = Δ u (with u representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments supporting our theoretical results. [ABSTRACT FROM AUTHOR]

The possibility to use a Lagrangian frame to solve problems with large time-steps was successfully explored previously by the authors for the solution of homogeneous incompressible fluids and also for solving multi-fluid problems (Idelsohn et al. 2012; 2014; 2013). The strategy used by the authors was named Particle Finite Element Method second generation (PFEM-2). The objective of this paper is to demonstrate in which circumstances the use of a Lagrangian frame with particles is more accurate than a classical Eulerian finite element method, and when large time-steps and/or coarse meshes may be used. [ABSTRACT FROM AUTHOR]

In this paper, goal-oriented error estimation for Timoshenko beams on Pasternak foundation, which involves double shear effect, is performed. The constitutive relation error (CRE) estimation is used in finite element analysis (FEA) to acquire strict bounds on quantities of interest. Due to the coupling of the displacement field and the internal force field in the equilibrium equations of the beam, the prolongation condition for construction of the admissible internal force field, a pillar of the CRE estimation, is not directly applicable. To overcome this difficulty, an auxiliary approximate problem whose stress solution enables the CRE estimation to proceed is introduced. Thereafter, strict bounds of outputs for the beam are obtained by dual analysis to which a significant adjunct is the circumvention of shear locking in low-order finite element analysis. Numerical results are presented to validate the strict bounding properties for quantities of beams on elastic foundation and accurate estimation of displacement quantities that is impervious to shear locking. [ABSTRACT FROM AUTHOR]

The solution of physical problems discretized using the finite element methods using NISQ quantum hardware remains relatively unexplored. Here, we present a unified formulation (FEqa) to solve such problems using quantum annealers. FEqa is a hybrid technique in which the finite element problem is formulated on a classical computer, and the residual is minimized using a quantum annealer. The advantages of FEqa include utilizing a single qubit per degree of freedom, enforcing Dirichlet boundary conditions a priori , reaching arbitrary solution precision, and eliminating the possibility of the annealer generating invalid results. FEqa is scalable on the classical portion of the algorithm due to its Single Program Multiple Data (SPMD) nature and does not rely on ground state solutions from the annealer. The exponentially large number of collocation points used in quantum annealing are investigated for their cosine measures, and new iterative techniques are developed to exploit their properties. The quantum annealer has clear advantages in computational time over simulated annealing, for the example problems presented in this paper solved on the D-Wave machine. The presented work provides a pathway to solving physical problems using quantum annealers. [ABSTRACT FROM AUTHOR]

The paper generalizes the classical C 1 cubic Clough–Tocher spline space over a triangulation to C 1 spaces of any degree higher that three. It shows that the considered spaces can be equipped with a basis consisting of non-negative locally supported functions forming a partition of unity and demonstrates the applicability of the basis in the context of the finite element method. The studied spaces have optimal approximation power and are defined by enforcing additional smoothness inside the triangles of the triangulation where the Clough–Tocher splitting is used. Locally, over each triangle of the triangulation, the splines are expressed in the Bernstein–Bézier form, which enables one to take the full advantage of the geometric properties and computational techniques that come with such a representation. Solving boundary problems with Galerkin discretization is thus relatively straightforward and is illustrated with several examples. [ABSTRACT FROM AUTHOR]

Based on the artificial viscosity approach, a characteristic stabilized finite element method is proposed for approximating solutions to the incompressible Navier–Stokes equations in this paper. The natural combination of characteristic method and subgrid artificial viscosity stabilized finite element method retains the best features of both methods. The stability analysis and error estimates are deduced rigorously. Finally, some numerical experiments are given to demonstrate that this method is highly efficient for the nonstationary Navier–Stokes problems. [ABSTRACT FROM AUTHOR]

Based on the rectangular cell model cut from doubly periodic anisotropic inclusion medium, this paper presents the effective elastic properties of the mentioned problems by an iterative FE–BE coupling method. This method is easy to be numerically implemented and especially suitable for the analysis of anisotropic inclusions embedded in an isotropic matrix, allows a wide range of microgeometries of the composite and determination of the complete set of effective elastic properties. Besides, the adopted method also avoids using the fundamental solutions of anisotropic materials and overcomes the difficulties of solving the inclusions with irregular shapes in the BEM. In calculation, the anisotropic inclusion is discretized into finite elements, whereas the boundary of the rectangular cell and the inclusion-matrix interface are meshed into a series of boundary elements. Some numerical examples are used to validate the applicability and reliability of the present scheme. [ABSTRACT FROM AUTHOR]

We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dual-weighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart–Thomas method is used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size. [ABSTRACT FROM AUTHOR]