Showing papers on "Smoothed finite element method published in 2016"
01 Jan 2016
TL;DR: The analysis of the finite element method is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
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484 citations
TL;DR: In this article, a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation is introduced. But this method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.
Abstract: This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k?1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.
234 citations
Book•
18 Jul 2016TL;DR: In this paper, the authors cover finite element methods for several typical eigenvalues that arise from science and engineering, and present new methods, such as the discontinuous Galerkin method, and new problems such as transmission eigenvalue problem.
Abstract: This book covers finite element methods for several typical eigenvalues that arise from science and engineering. Both theory and implementation are covered in depth at the graduate level. The background for typical eigenvalue problems is included along with functional analysis tools, finite element discretization methods, convergence analysis, techniques for matrix evaluation problems, and computer implementation. The book also presents new methods, such as the discontinuous Galerkin method, and new problems, such as the transmission eigenvalue problem.
102 citations
TL;DR: In this article, a hybrid smoothed finite element method (HS-FEM) using triangular elements is presented for the two-dimensional underwater acoustic scattering problems, which can provide a close-to-exact stiffness of the continuous system, thus the numerical dispersion error can be significantly decreased.
77 citations
TL;DR: In this paper, a stable node-based smoothed finite element method (SNS-FEM) is formulated for steady and transient heat transfer problems using linear triangular and tetrahedron element.
65 citations
TL;DR: In this paper, a mesh-free Galerkin method based on stabilized conforming nodal integration (SCNI) is proposed to solve in-plane mixed-mode fracture problems.
Abstract: Two-dimensional (2D) in-plane mixed-mode fracture mechanics problems are analyzed employing an efficient meshfree Galerkin method based on stabilized conforming nodal integration (SCNI). In this setting, the reproducing kernel function as meshfree interpolant is taken, while employing the SCNI for numerical integration of stiffness matrix in the Galerkin formulation. The strain components are smoothed and stabilized employing Gauss divergence theorem. The path-independent integral (J-integral) is solved based on the nodal integration by summing the smoothed physical quantities and the segments of the contour integrals. In addition, mixed-mode stress intensity factors (SIFs) are extracted from the J-integral by decomposing the displacement and stress fields into symmetric and antisymmetric parts. The advantages and features of the present formulation and discretization in evaluation of the J-integral of in-plane 2D fracture problems are demonstrated through several representative numerical examples. The mixed-mode SIFs are evaluated and compared with reference solutions. The obtained results reveal high accuracy and good performance of the proposed meshfree method in the analysis of 2D fracture problems.
62 citations
TL;DR: In this article, a stable nodal integration method with strain gradient (SNIM-SG) is proposed for dynamic problems using linear triangular and tetrahedron element, except for considering the smoothed strain into the calculation of potential energy functional as NS-FEM.
Abstract: A stable nodal integration method with strain gradient (SNIM-SG) for curing the temporal instability of node-based smoothed finite element method (NS-FEM) is proposed for dynamic problems using linear triangular and tetrahedron element. In each smoothing domain, except for considering the smoothed strain into the calculation of potential energy functional as NS-FEM, a term related to strain gradient is taken into account as a stabilization term. The proposed SNIM-SG can achieve appropriate system stiffness in strain energy between FEM and NS-FEM solutions and obtains quite favorable results in elastic and dynamic analysis. The accuracy and stability of SNIM-SG solution are studied through detailed analyzes of benchmark cases and practical engineering problems. In elastic-static analysis, it is found that SNIM-SG can provide higher accuracy in displacement field than the reference approaches do. In free vibration analysis, the spurious non-zero energy modes can be eliminated effectively owing to the fact that SNIM-SG solution strengths the original relatively soft NS-FEM, and SNIM-SG is confirmed to obtain fairly accurate natural frequency values in various examples. All in all, SNIM-SG cures the flaws of NS-FEM and enhances the dominant of nodal integration. Thus, the efficacy of the presented formulation in solving solid mechanics problems is well represented and clarified.
61 citations
Book•
29 Jul 2016TL;DR: This book provides a complete, clear, and unified treatment of the static aspects of nonlinear solid mechanics and the associated finite element techniques together, in the first of two books in this series.
Abstract: Designing engineering components that make optimal use of materials requires consideration of the nonlinear static and dynamic characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, which requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both the nonlinear solid mechanics and the associated finite element techniques together, the authors provide, in the first of two books in this series, a complete, clear, and unified treatment of the static aspects of nonlinear solid mechanics. Alongside a range of worked examples and exercises are user instructions, program descriptions, and examples for the FLagSHyP MATLAB computer implementation, for which the source code is available online. While this book is designed to complement postgraduate courses, it is also relevant to those in industry requiring an appreciation of the way their computer simulation programs work.
58 citations
TL;DR: In this paper, a novel numerical optimization procedure with mixed integer and continuous design variables for optimal design of laminated composite plates subjected to buckling loads is proposed, where the objective function is to maximize the buckling load factor.
TL;DR: A cut finite element method for a second order elliptic coupled bulk-surface model problem is developed and a priori estimates for the energy and $$L^2$$L2 norms of the error are proved.
Abstract: We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and $$L^2$$L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.
TL;DR: A Galerkin finite element method (GFEM) is presented for the numerical simulation of the fractional cable equation (FCE) involving two integro-differential operators and it is proved that the numerical solution converges to the exact solution with order O(τ+hl+1) for the lth-order finiteelement method.
Abstract: The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(?+hl+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(?2+hl+1).
TL;DR: In this paper, the embedding of constitutive relations for magneto-active polymers (MAP) into finite element simulations is discussed, and a finite element model is established based on the usual steps of weak form representation, discretization and consistent linearization.
Abstract: This contribution is concerned with the embedding of constitutive relations for magneto-active polymers (MAP) into finite element simulations. To this end, a recently suggested, calibrated, and validated material model for magneto-mechanically coupled and rate-dependent MAP response is briefly summarized in its continuous and algorithmic settings. Moreover, the strongly coupled field equations of finite deformation magneto-mechanics are reviewed. For the purpose of numerical simulation, a finite element model is then established based on the usual steps of weak form representation, discretization and consistent linearization. Two verifying inhomogeneous numerical examples are presented in which a classical 'plate with a hole' geometry is equipped with MAP properties and subjected to different types of time-varying mechanical and magnetic loading.
TL;DR: It is found that the SNS-FEM applied in the stochastic problem can improve the accuracy of static and dynamic results, largely decrease the time cost, and lower the requirement of mesh.
Abstract: The traditional stochastic finite element method based on the finite element method fails to give the fine solution in precise determination of reliable problems when the computer power consumption is limited. To cure this fatal defect, the generalized n th order stochastic perturbation technique based on a stable node-based smoothed finite element method (GS_SNS-FEM) is presented. The framework intends to essentially improve the accuracy, lower the mesh limitation and occupy much less computational consumption for stochastic problems, especially when its second order realization is ineffective for large variations of input random fields. Besides, the n th orders expansion makes it possible to get the prefect accuracy for expected values and variances. Numerical examples including the static and dynamic problems are completed and compared with the solution of Monte Carlo simulation. It is found that the SNS-FEM applied in the stochastic problem can improve the accuracy of static and dynamic results, largely decrease the time cost, and lower the requirement of mesh.
TL;DR: In this paper, a stable node-based smoothed finite element method (SNS-FEM) is presented that cures the "overly soft" property of the original node based FEM method for the analysis of underwater acoustic scattering problems.
Abstract: A stable node-based smoothed finite element method (SNS-FEM) is presented that cures the “overly-soft” property of the original node-based smoothed finite element method for the analysis of underwater acoustic scattering problems. In the SNS-FEM model, the node-based smoothed gradient field is enhanced by additional stabilization term related to the gradient variance items. It is demonstrated that SNS-FEM provides an ideal stiffness of the continuous system and improves the performance of the NS-FEM and FEM. In order to handle the acoustic scattering problems in unbounded domain, the well known Dirichlet-to-Neumann (DtN) boundary condition is combined with the present SNS-FEM to give a SNS-FEM-DtN model for exterior acoustic problems. Several numerical examples are investigated and the results show that the SNS-FEM-DtN model can achieve more accurate solutions compared to the NS-FEM and FEM.
TL;DR: In this paper, the 3D mass-redistributed finite element method (MR-FEM) was further developed to solve more complicated structural-acoustic interaction problems, and the smoothed Galerkin weak form was adopted to formulate the discretized equations for the structure, and MR-FEMS was applied in acoustic domain.
Abstract: A 2D mass-redistributed finite element method (MR-FEM) for pure acoustic problems was recently proposed to reduce the dispersion error. In this paper, the 3D MR-FEM is further developed to solve more complicated structural–acoustic interaction problems. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and MR-FEM is applied in acoustic domain. The global equations of structural–acoustic interaction problems are then established by coupling the MR-FEM for the acoustic domain and the edge-based smoothed finite element method for the structure. The perfect balance between the mass matrix and stiffness matrix is able to improve the accuracy of the acoustic domain significantly. The gradient smoothing technique used in the structural domain can provide a proper softening effect to the “overly-stiff” FEM model. A number of numerical examples have demonstrated the effectiveness of the mass-redistributed method with smoothed strain.
TL;DR: In this article, a selective edge-based smoothed finite element method (sES-FEM) of kinematic theorem for predicting the plastic limit loads in structures is presented.
Abstract: We present a selective edge-based smoothed finite element method (sES-FEM) of kinematic theorem for predicting the plastic limit loads in structures. The basic idea in this method is to use two levels of mesh repartitioning for the finite element limit analysis. The master level begins with an adaptive primal-mesh strategy guided by a dissipation-based indicator. The slave level consists of further subdividing each triangle into three sub-triangles and creating a dual mesh through a careful selection of a pair of two sub-triangles shared by the corresponding common element edge. By applying a strain smoothing projection operator to the strain rates on the dual mesh, the flow rule constraint is enforced over the edge-based strain smoothing domains, and likewise everywhere in the problem domain. This numerical procedure is performed for a cohesive-frictional material. This numerical procedure is also performed necessarily to avoid the volumetric locking problem for a purely cohesive material. The optimization formulation of limit analysis is next presented by the form of a second-order cone programming (SOCP) for the purpose of exploiting the efficiency of interior–point solvers. The present method uses linear triangular elements and handles a low number of optimization variables. This leads to a convenient way to design and solve the large-scale optimization problems effectively. Several numerical examples are given to demonstrate the simplicity and effectiveness of the proposed method.
TL;DR: The numerical results have clearly demonstrated that the modified ES-FEM is very effective to minimize the dispersion errors in the simulation of band gap of acoustic metamaterials.
Abstract: A novel modified edge-based smoothed finite element method (modified ES-FEM) is developed to compute the band gap of acoustic metamaterials. The stiffness in the modified ES-FEM is created by the edge-based smoothed finite element method (ES-FEM) which is aimed at softening the overly stiffness of standard finite element method (FEM). On the other hand, the mass matrix is constructed by mass-redistributed method to tune the balance between the smoothed stiffness and mass matrix. The present modified ES-FEM adopts linear triangular elements generated automatically, which enables automation in computation and saving computational cost in mesh generation. Two numerical examples are presented to verify the computational efficiency of the modified ES-FEM. The numerical results have clearly demonstrated that the modified ES-FEM is very effective to minimize the dispersion errors in the simulation of band gap of acoustic metamaterials.
TL;DR: A group of cell-based smoothed point interpolation methods based on the generalised gradient smoothing technique are proposed for the numerical modelling of saturated porous media through comparison of the numerical results with those obtained using analytical/semi-analytical solutions, point interpolations methods, and standard finite element methods.
TL;DR: The present method, named edge/node-based S-FEM (ENS-F EM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-Fems and NS-Fem to avoid shear-locking phenomenon in Reissner–Mindlin flat shell elements.
Abstract: In this paper, a combined scheme of edge-based smoothed finite element method (ES-FEM) and node-based smoothed finite element method (NS-FEM) for triangular Reissner---Mindlin flat shells is developed to improve the accuracy of numerical results. The present method, named edge/node-based S-FEM (ENS-FEM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-FEM and NS-FEM. A discrete shear gap technique is incorporated into ENS-FEM to avoid shear-locking phenomenon in Reissner---Mindlin flat shell elements. For all practical purpose, we propose an average combination (aENS-FEM) of ES-FEM and NS-FEM for shell structural problems. We compare numerical results obtained using aENS-FEM with other existing methods in the literature to show the effectiveness of the present method.
TL;DR: In this article, a two-grid algorithm based on the Newton iteration method was proposed to solve the full discrete scheme problems in a coarse grid and a fine grid using Newton iteration once, and it was shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2.
Abstract: The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.
TL;DR: In this paper, a new shell finite element is introduced, specifically proposed for constrained shell finite elements analysis, which is derived from the finite strips used in the semi-analytical finite strip method.
Abstract: In this paper a novel shell finite element is introduced, specifically proposed for constrained shell finite element analysis. The proposed element is derived from the finite strips used in the semi-analytical finite strip method. The new finite element shares the most fundamental feature of the finite strips, namely: transverse and longitudinal directions are distinguished. Moreover, the new element keeps the transverse interpolation functions of finite strips, however, the longitudinal interpolation functions are changed from trigonometric functions (or function series) to classic polynomials. It is found that the proper selection of the polynomial longitudinal interpolation functions makes it possible to perform modal decomposition similarly as in the constrained finite strip method (cFSM). This requires an unusual combination of otherwise well-known shape functions. If the so-constructed shell finite elements are used to model a thin-walled member, (hence, with using discretization in both the transverse and the longitudinal directions,) modal decomposition can be done essentially identically as in cFSM, whilst the practical applicability of the method is significantly extended (e.g., various restraints, holes, certain cross-section changes can easily be handled). In this paper the focus is on the derivation of the novel shell finite element. Constraining capability is illustrated by some basic examples. Practical application of the novel element will be presented in subsequent papers.
TL;DR: In this article, a mixed finite element method for the Navier-Stokes equations is introduced in which the stress is a primary variable and the variational formulation retains the mathematical structure of the NST equations and the classical theory extends naturally to this setting.
Abstract: A mixed finite element method for the Navier-Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier-Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf-sup conditions are developed.
TL;DR: In this article, a singular edge-based smoothed finite element method (ES-FEM) was proposed for solving two-dimensional thermoelastic crack problems, where the physical domain is first discretized using linear triangular elements which can be generated easily for complicated geometries, and then the smoothing domains are constructed based on edges of these elements.
TL;DR: Efficiency of enriched finite element methods in attaining accuracy results is observed, as well as the elimination of shear locking in higher level of enrichment.
TL;DR: In this paper, the hybrid smoothed finite element method (HS-FEM) using linear triangular elements is presented to analyze two dimensional radiation problems, which is designed to establish the area-weighted strain field that contains contributions from both the standard FEM and the node-based smoothed FEM.
TL;DR: In this paper, a generalized smoothing technique based on beta finite element method (βFEM) was proposed to improve the performance of standard FEM and the existing smoothed finite element methods in solid mechanics.
Abstract: This paper presents a generalized smoothing techniques based beta finite element method (βFEM) to improve the performance of standard FEM and the existing smoothed finite element methods (S-FEM) in solid mechanics. As we know, the edge-based (for 2D) or face-based (for 3D) strain smoothing techniques can bring much more accurate solutions than standard FEM, and offer lower bounds for force driven problems. The node-based smoothing technique with “overly-soft” feature, on the other hand has a unique property of producing upper bound solutions. This work proposes a novel generalized S-FEM with the smoothing domains generated based on both edges/faces and nodes. An adjustable parameter β is introduced to control the ratio of the area of edge/face-based and node-based smoothing domains. It is found that nearly exact solutions in strain energy can be obtained by tuning the parameter, making use of the important property that the exact solution is bonded by the solutions of NS-FEM and ES/FS-FEM. Standard patch tests are likewise satisfied. A number of numerical examples (static, dynamic, linear and nonlinear) have shown that the present βFEM method is found to be ultra-accurate, insensitive to mesh quality, temporal stable, capable of modeling complex geometry, immune from volumetric locking, etc.