Gradient Recovery for Singularly Perturbed Boundary Value Problems I: One-Dimensional Convection-Diffusion

We consider a Galerkin finite element method that uses piecewise linears on a class of Shishkin-type meshes for a model singularly perturbed convection-diffusion problem. We pursue two approaches in constructing superconvergent approximations of the gradient. The first approach uses superconvergence points for the derivative, while the second one combines the consistency of a recovery operator with the superconvergence property of an interpolant. Numerical experiments support our theoretical results. Received November 12, 1999; revised September 9, 2000     

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.     

A Finite Element Recovery Approach to Eigenvalue Approximations with Applications to Electronic Structure Calculations

In this paper, we propose and analyze a recovery approach for trilinear finite element approximations on locally-refined hexahedral meshes for a class of elliptic eigenvalue problems. In the approach a local high-order interpolation recovery is followed by some gradient averaging based defect correction scheme. It is proved theoretically and shown numerically that our recovery approach can produce highly accurate eigenpair approximations. And we observe from our numerical experiments that the recovered eigenvalue approximation from the gradient averaging based defect correction approximates the exact eigenvalue from below. Furthermore, this approach has been applied to electronic structure calculations to improve the total energy approximations with small extra overheads.     

SSD: Smooth Signed Distance Surface Reconstruction

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state‐of‐the‐art algorithms. An efficient implementation based on a primal‐graph octree‐based hybrid finite element‐finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack‐free adaptive manifold polygon meshes.     

Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery

In this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented.     

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.     

Application of the Petrov-Galerkin method to chemical-flooding reservoir simulation in one dimension

A one-dimensional, chemical-flooding simulator is described. This program models the effect that polymers, surfactants and salts have on the enhanced recovery of oil. The flow equations are convection dominated and often shock-like profiles develop which traverse the domain. The simulator incorporates, with a few modifications, most of the physical and chemical package which is set forth by Pope and Nelson. The spacial discretization is performed via the Petrov-Galerkin, finite element method. This technique is mainly derived from the work of Hughes and Brooks and is modified for the case of chemical flooding. We compare this finite element method to the upstream, finite difference method and to Galerkin's method. Generally speaking, the Petrov-Galerkin results are significantly less sensitive to the number of grid points than are those of the upstream method. Its stability properties are nearly the same as those of the upstream techniques, which in turn, are superior to those of Galerkin's method.     

A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

In this paper, we derive a posteriori error estimates of recovery type, and present the superconvergence analysis for the finite element approximation of distributed convex optimal control problems. We provide a posteriori error estimates of recovery type for both the control and the state approximation, which are generally equivalent. Under some stronger assumptions, they are further shown to be asymptotically exact. Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as a reliability bound for the control problems. Numerical results demonstrating our theoretical results are also presented in this paper.     

基于COMSOL的磁共振成像双平面梯度线圈的仿真研究

目的：采用COMSOL有限元软件对磁共振成像横向双平面梯度线圈进行仿真分析,为高性能梯度线圈的设计及制作提供技术支持。方法：首先采用改进的目标场方法设计得到梯度线圈绕线的点数据,然后利用AUTOCAD建立三维模型,最后将模型导入COMSOL中,进行电-磁多场耦合模型仿真和结果分析。并且提出结合3D打印技术为复杂梯度线圈的制作提供技术支持。结果：根据设计的不同参数建立不同的梯度线圈模型进行仿真比较,所设计梯度线圈的梯度磁场可以满足非线性度小于5%的应用要求。结论：通过横向双平面梯度线圈三维模型的有限元仿真,可以为梯度线圈的设计和制作提供一定的参考,对优化设计和制作性能更优的梯度线圈具有重要意义。     

Curvature-dependent triangulation of implicit surfaces

Implicit surfaces appear in many applications, including medical imaging, molecular modeling, computer aided design, computer graphics and finite element analysis. Despite their many advantages, implicit surfaces are difficult to render efficiently. Today's real-time graphics systems are heavily optimized for rendering triangles, so an implicit surface should be converted to a mesh of triangles before rendering. Our algorithm polyonalizes an implicit surface. The algorithm generates a mesh of close-to-equilateral triangles with sizes dependent on the local surface curvature. We assume that the implicit surface is connected and G¹ is smooth (that is, the tangent plane varies continuously over the surface). The algorithm requires an evaluator for the implicit function defined at all points in space, an evaluator for the function gradient defined at points near the surface, and a bounding box around the surface. The output of the algorithm is good for applications requiring a well-behaved triangulation, such as rendering systems and finite element partial differential equation (PDE) solvers     

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

The paper presents postprocessing techniques based on locally improved finite element (FE) solutions of the basic field variables. This opens up the possibility to control both “strain energy” terms and “kinetic energy” terms in the governing equations. The proposed postprocessing technique on field variables is essentially a least square fit of the prime variables (displacements) at superconvergent points. Its performance is compared with other well-known techniques, showing a good performance. A h-adaptive FE strategy for acoustic problems is presented where, for adaptive mesh generation and remeshing the commercial software package i-deas has been applied and for the FE analysis the commercial software package sysnoise. The paper also presents an adaptive h-version FE approach to control the discretisation error in free vibration analysis. The postprocessing technique used here is a mix of local and global updating methods. Rapid convergence of the preconditioned conjugate gradient method is enhanced by choosing the initial trial eigenmodes as the superconvergent patch recovery technique for displacements improved FE eigenmodes. Numerical examples show nice properties of the final local and global updated solution as a basis for an error estimator and the error indicator in an adaptive process.     

Deflection analysis of expanded open-web steel beams

The finite element method is very appropriate for calculating stresses at isolated areas of expanded open-web steel beams. For deflection analysis, however, the entire beam, or one-half the beam if the system is symmetrical, must be included in the idealization and therefore it is not practical to obtain deflections by a direct application of the finite element method. In this paper, the authors present a method of deflection analysis which treats the castellated beam as an assemblage of typical segments and utilizes the finite element method to form the stiffness matrix for the typical segment. The beam deflections at the panel points are then computed by the conventional stiffness method. Two different idealizations were tried for the finite element analysis of a typical segment. These idealizations resulted in a 13 × 13 and a 7 × 7 stiffness matrix. Deflection values calculated by using the 7 × 7 stiffness matrix showed close agreement with those obtained experimentally.     

Domain decomposition method for elliptic mixed boundary value problems

A method for solving the elliptic second order mixed boundary value problem is discussed. The finite element problem is divided into subproblems, associated with subregions into which the region has been partitioned, and an auxiliary problem connected with intersect curves. The subproblems are solved directly, while the auxiliary problem is handled by a conjugate gradient method. The rate of the convergence of the cg-method is discussed also for the cases when the Neumann and Dirichlet boundary conditions change at points belonging to the intersecting curves. Results from numerical experiments are also reported.     

Fatigue analysis by local stress concept based on finite element results

The assessment of finite element results for dynamically loaded components requires local S/N-curves, which depend, under further effects, on the type of loading and the stress flow within the component. These influences are described with help of the stress gradient, which can easily be determined with finite element calculations. A model for the calculation of S/N-curves is presented, which takes into account the stress gradient to define the local stress limit, the number of cycles at the fatigue limit and the slope.     

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.     

Error estimation and adaptive procedures based on superconvergent patch recovery (SPR) techniques

Summary  The paper discusses error estimation and adaptive finite element procedures for elasto-static and dynamic problems based on superconvergent patch recovery (SPR) techniques. The SPR is a postprocessing procedure to obtain improved finite element solutions by the least squares fitting of superconvergent stresses at certain sampling points in local patches. An enhancement of the original SPR by accounting for the equilibirum equations and boundary conditions is proposed. This enhancement improves the quality of postprocessed solutions considerably and thus provides an even more effective error estimate. The patch configuration of SPR can be either the union of elements surrounding a vertex node, thenode patch, or, the union of elements surrounding an element, theelement patch. It is shown that these two choices give normally comparable quality of postprocessed solutions. The paper is also concerned with the application of SPR techniques to a wide range of problems. The plate bending problem posted in mixed form where force and displacement variables are simultaneously used as unknowns is considered. For eigenvalue problems, a procedure of improving eigenpairs and error estimation of the eigenfrequency is presented. A postprocessed type of error estimate and an adaptive procedure for the semidiscrete finite element method are discussed. It is shown that the procedure is able to update the spatial mesh and the time step size so that both spatial and time discretization errors are controlled within specified tolerances. A discontinuous Galerkin method for solving structural dynamics is also presented.     

A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

Efficient solution of lattice equations by the recovery method Part I: Scalar elliptic problems

In this paper, an efficient solver for high dimensional lattice equations will be introduced. We will present a new concept, the recovery method, to define a bilinear form on the continuous level which has equivalent energy as the original lattice equation. The finite element discretisation of the continuous bilinear form will lead to a stiffness matrix which serves as an quasi-optimal preconditioner for the lattice equations. Since a large variety of efficient solvers are available for linear finite element problems the new recovery method allows to apply these solvers for unstructured lattice problems.     

Identification and elimination of parasitic shear in a laminated composite beam finite element

A displacement-based finite element for the analysis of laminated composite beams is formulated using strain gradient notation. The definition of the beam’s longitudinal displacement possesses only the independent term (axial displacement) and a term which is linear in the thickness coordinate z. Thus, the finite element is first-order shear deformable. As strain gradient notation is physically interpretable, the contents of the coefficients of the polynomial expansions are identified a priori. Thus, the modeling capabilities as well as modeling deficiencies of the element are identified during the formulation procedure. A single parasitic shear term (spurious) is found to be present in the transverse shear strain expression of the element, which is responsible for locking. This parasitic shear term is also found to be the cause of a qualitative error existing in the representation of transverse shear strain along the length of a typical beam model. As the spurious term has been clearly identified, it can easily be removed to correct the element. The effectiveness of the procedure is shown through numerical analyses performed using the element containing the spurious term and then corrected for it. The beam model is validated by comparing numerical solutions with analytical solutions provided by the minimization of the total potential energy for a given laminated composite beam.