A quasi‐implicit characteristic–based penalty finite‐element method for incompressible laminar viscous flows

In this paper, a novel characteristic–based penalty (CBP) scheme for the finite‐element method (FEM) is proposed to solve 2‐dimensional incompressible laminar flow. This new CBP scheme employs the characteristic‐Galerkin method to stabilize the convective oscillation. To mitigate the incompressible constraint, the selective reduced integration (SRI) and the recently proposed selective node–based smoothed FEM (SNS‐FEM) are used for the 4‐node quadrilateral element (CBP‐Q4SRI) and the 3‐node triangular element (CBP‐T3SNS), respectively. Meanwhile, the reduced integration (RI) for Q4 element (CBP‐Q4RI) and NS‐FEM for T3 element (CBP‐T3NS) with CBP scheme are also investigated. The quasi‐implicit CBP scheme is applied to allow a large time step for sufficient large penalty parameters. Due to the absences of pressure degree of freedoms, the quasi‐implicit CBP‐FEM has higher efficiency than quasi‐implicit CBS‐FEM. In this paper, the CBP‐Q4SRI has been verified and validated with high accuracy, stability, and fast convergence. Unexpectedly, CBP‐Q4RI is of no instability, high accuracy, and even slightly faster convergence than CBP‐Q4SRI. For unstructured T3 elements, CBP‐T3SNS also shows high accuracy and good convergence but with pressure oscillation using a large penalty parameter; CBP‐T3NS produces oscillated wrong velocity and pressure results. In addition, the applicable ranges of penalty parameter for different proposed methods have been investigated.     

Computation of limit and shakedown loads using a node‐based smoothed finite element method

This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node‐based smoothed FEM in combination with a primal–dual algorithm. An associated primal–dual form based on the von Mises yield criterion is adopted. The primal‐dual algorithm together with a Newton‐like iteration are then used to solve this associated primal–dual form to determine simultaneously both approximate upper and quasi‐lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of photonic crystals using the hybrid finite‐element/finite‐difference time domain technique based on the discontinuous Galerkin method

Two‐dimensional photonic crystal structures are analyzed by a recently developed hybrid technique combining the finite‐element time‐domain (FETD) method and the finite‐difference time‐domain (FDTD) method. This hybrid FETD/FDTD method uses the discontinuous Galerkin method as framework for domain decomposition. To the best of our knowledge, this is the first hybrid FETD/FDTD method that allows non‐conformal meshes between different FETD and FDTD subdomains. It is also highly parallelizable. These properties are very suitable for the computation of periodic structures with curved surfaces. Numerical examples for the computation of the scattering parameters of two‐dimensional photonic bandgap structures are presented as applications of the hybrid FETD/FDTD method. Numerical results demonstrate the efficiency and accuracy of the proposed hybrid method. Copyright © 2012 John Wiley & Sons, Ltd.     

Generation of large finite‐element matrices on multiple graphics processors

This paper presents techniques for generating very large finite‐element matrices on a multicore workstation equipped with several graphics processing units (GPUs). To overcome the low memory size limitation of the GPUs, and at the same time to accelerate the generation process, we propose to generate the large sparse linear systems arising in finite‐element analysis in an iterative manner on several GPUs and to use the graphics accelerators concurrently with CPUs performing collection and addition of the matrix fragments using a fast multithreaded procedure. The scheduling of the threads is organized in such a way that the CPU operations do not affect the performance of the process, and the GPUs are idle only when data are being transferred from GPU to CPU. This approach is verified on two workstations: the first consists of two 6‐core Intel Xeon X5690 processors with two Fermi GPUs: each GPU is a GeForce GTX 590 with two graphics processors and 1.5 GB of fast RAM; the second workstation is equipped with two Tesla C2075 boards carrying 6 GB of RAM each and two 12‐core Opteron 6174s. For the latter setup, we demonstrate the fast generation of sparse finite‐element matrices as large as 10 million unknowns, with over 1 billion nonzero entries. Comparing with the single‐threaded and multithreaded CPU implementations, the GPU‐based version of the algorithm based on the ideas presented in this paper reduces the finite‐element matrix‐generation time in double precision by factors of 100 and 30, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Development of a 4‐node hybrid stress tetrahedral element using a node‐based smoothed finite element method

In this paper, a new 4‐node hybrid stress element is proposed using a node‐based smoothing technique of tetrahedral mesh. The conditions for hybrid stress field required are summarized, and the field should be continuous for better performance of a constant‐strain tetrahedral element. Nodal stress is approximated by the node‐based smoothing technique, and the stress field is interpolated with standard shape functions. This stress field is linear within each element and continuous across elements. The stress field is expressed by nodal displacements and no additional variables. The element stiffness matrix is calculated using the Hellinger‐Reissner functional, which guarantees the strain field from displacement field to be equal to that from the stress field in a weak sense. The performance of the proposed element is verified by through several numerical examples.     

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

We propose a novel finite‐element method for polygonal meshes. The resulting scheme is hp‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.     

Finite element analyses of wave propagations due to a high‐speed train across bridges

The finite element mesh of the unbounded wave problem often contains a large number of degrees of freedom. Moreover, the wave reflection along the mesh boundary cannot be avoided if only the traditional time‐domain finite element method (FEM) is used. Thus, the main purpose of this study is to overcome those drawbacks and discuss how to use the three‐dimensional FEM to simulate the soil vibration due to a moving high‐speed train across bridges. A number of finite element analyses (FEA) including the isolation schemes of in‐filled and open trenches were performed. The appropriate mesh dimension in the direction of the railroad is suggested and validated. Moreover, the 3‐node wheel element for simulating the moving train and the least‐squares method for finding foundation properties are proposed. Since the finite element mesh contains a large number of degrees of freedom, the conjugated gradient method with the SSOR preconditioning scheme is recommended. In conclusion, this study indicates that the difficulty of the FEA for unbounded problems can be overcome efficiently even by using a personal computer. Copyright © 2002 John Wiley & Sons, Ltd.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Large‐scale parallel finite‐element analysis using the internet: a performance study

This paper describes a parallel finite‐element system implemented using the domain decomposition method on a cluster of remote computers connected via the Internet. This technique is also readily applicable to a grid computing environment. A three‐dimensional finite‐element elastic analysis involving more than one million degrees of freedom was solved using this system, and a good approximate solution was obtained with high parallel efficiency of over 90% using remote computers located in three different countries. Copyright © 2005 John Wiley & Sons, Ltd.     

Rotation‐free triangular shell element using node‐based smoothed finite element method

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.     

Mesh‐independent p‐orthotropic enrichment using the generalized finite element method

This paper is aimed at presenting a simple yet effective procedure to implement a mesh‐independent p‐orthotropic enrichment in the generalized finite element method. The procedure is based on the observation that shape functions used in the GFEM can be constructed from polynomials defined in any co‐ordinate system regardless of the underlying mesh or type of element used. Numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p‐orthotropic enrichment. Copyright © 2002 John Wiley & Sons, Ltd.     

Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)

The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge‐based smoothed finite element method (ES‐FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the ‘overly stiff’ feature of the discrete model. In such an ‘overly stiff’ FEM model, the wave propagates with an artificially higher ‘numerical’ speed, and hence the numerical wave‐number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge‐based gradient smoothing operations, the ES‐FEM model, however, behaves much softer than the standard FEM model, leading to the so‐called very ‘close‐to‐exact’ stiffness. Therefore the ES‐FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES‐FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model. Copyright © 2010 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

A quadtree data structure for the coupled finite‐element/element‐free Galerkin method

In the present paper, we derive an efficient data structure for the organization of the nodes in the coupled finite element/element‐free Galerkin method. With respect to its implementation, we compare various approaches of recursive spatial discretizations that facilitate most flexible handling of the nodes. The goal of the paper is to refine the implementation issues of the data structure which is fundamental to the element‐free Galerkin method and thus to speed‐up this otherwise computationally rather expensive meshfree method. Copyright © 2001 John Wiley & Sons, Ltd.     

On the computation of steady‐state compressible flows using a discontinuous Galerkin method

Computation of compressible steady‐state flows using a high‐order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics‐based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low‐storage p‐multigrid method is used for solving the governing compressible Euler equations to obtain steady‐state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady‐state flows. Copyright © 2007 John Wiley & Sons, Ltd.     

A shape‐free 8‐node plane element unsymmetric analytical trial function method

The unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress‐function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth‐order completeness in Cartesian coordinates. The resulting element model, denoted by US‐ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US‐ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.