Goal-oriented error estimation and adaptivity for the finite element method

In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

A new approach to automatic and a priori mesh adaptation around circular holes for finite element analysis

Through our research on the integration of finite element analysis in the design and manufacturing process with CAD, we have proposed the concept of mesh pre-optimization. This concept consists in converting shape and analysis information in a size map (a mesh sizing function) with respect to various adaptation criteria (refining the mesh around geometric form features, minimizing the geometric discretization error, boundary conditions, etc.). This size map then represents a constraint that has to be respected by automatic mesh generation procedures. This paper introduces a new approach to automatic mesh adaptation around circular holes. This tool aims at optimizing, before any FEA, the mesh of a CAD model around circular holes. This approach, referred to as “a priori” mesh adaptation, should not be regarded as an alternative to adaptive a posteriori mesh refinement but as an efficient way to obtain reasonably accurate FEA results before a posteriori adaptation, which is particularly interesting when evaluating design scenarios. The approach is based on performing many offline FEA analyses on a reference case and deriving, from results and error distributions obtained, a relationship between mesh size and FEA error. This relationship can then be extended to target user specified FEA accuracy objectives in a priori mesh adaptation for any distribution of circular holes. The approach being purely heuristic, fulfilling FEA accuracy objectives, in all cases, cannot be theoretically guaranteed. However, results obtained using varying hole diameters and distributions in 2D show that this heuristic approach is reliable and useful. Preliminary results also show that extension of the method can be foreseen towards a priori mesh adaptation in 3D and mesh adaptation around other types of 2D features.     

Adaptive finite element procedures in elasticity and plasticity

The paper discusses error estimation and h-adaptive finite element procedures for elasticity and plasticity problems. For the spatial discretization error, an enhanced Superconvergent Patch Recovery (SPR) technique which improves the error estimation by including fulfillment of equilibrium and boundary conditions in the smoothing procedure is discussed. It is known that an accurate error estimation on an early stage of analysis results in a more rapid and optimal adaptive process. It is shown that node patches and element patches give similar quality of the postprocessed solution. For dynamic problems, 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 time-discontinuous Galerkin method for solving the second-order ordinary differential equations in structural dynamics is also presented. Many advantages of the new approach such as high order accuracy, possibility to filter effects of spurious modes and convenience to apply adaptive analysis are observed. For plasticity problems, some recent work that improved plastic strains and plastic localization is discussed.     

Error estimates for conforming finite element solutions

A method for determining realistic error estimates for conforming finite element solutions is presented. The method requires solution of the problem by at least two, and preferably three mesh schemes that yield monotonic solution covergence. This in turn will automatically yield one solution bound, upper or lower. The paper describes a simple and practical scheme for obtaining the other bound by utilizing the solutions from the multiple mesh schemes. These bounds bracket the exact solution within relatively narrow limits and provide the basis of the error estimate. The solution quantities considered are the system energy quantities; and for eigenvalue problems these correspond to the eigenvalues themselves. As in convergence proofs, it is expected that the displacement and stress quantities will follow the behavior of the energy quantities. The proposed bounding method is applicable to eigenvalue and static problems devoid of stress singularities, and considers only the discretization error of conforming finite element models. The validity of the proposed bounding method has not been proved mathematically; however, extensive numerical applications of the method indicate its workability in every case tested. Results of some applications are included in this article.     

Adaptive Finite Element Methods for the Identification of Elastic Constants

In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.     

A space–time adaptation scheme for unsteady shallow water problems

We provide a space–time adaptation procedure for the approximation of the Shallow Water Equations (SWE). This approach relies on a recovery based estimator for the global discretization error, where the space and time contributions are kept separate. In particular we propose an ad hoc procedure for the recovery of the time derivative of the numerical solution and then we employ this reconstruction to define the error estimator in time. Concerning the space adaptation, we move from an anisotropic error estimator able to automatically identify the density, the shape and the orientation of the elements of the computational mesh. The proposed global error estimator turns out to share the good properties of each recovery based error estimator. The whole adaptive procedure is then combined with a suitable stabilized finite element SW solver. Finally the reliability of the coupled solution–adaptation procedure is successfully assessed on two unsteady test cases of interest for hydraulics applications.     

A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.     

三维实体仿真建模的网格自动生成方法

有限元网格模型的生成与几何拓扑特征和力学特性有直接关系。建立网格模型时，为了更真实地反映原几何形体的特征，在小特征尺寸或曲率较大等局部区域网格应加密剖分；为提高有限元分析精度和效率，在待分析的开口、裂纹、几何突变、外载、约束等具有应力集中力学特性的局部区域，网格应加密剖分。为此，该文提出了基于几何特征和物理特性相结合的网格自动生成方法。该方法既能有效地描述几何形体，又能实现应力集中区域的网格局部加密及粗细网格的均匀过渡。实例表明本方法实用性强、效果良好。     

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.     

Automatic mesh generation for finite element analysis

The finite element analysis in engineering applications comprises three phases: domain discretization, equation solving and error analysis. The domain discretization or mesh generation is the pre-processing phase which plays an important role in the achievement of accurate solutions. In this paper, the improvement of one particularly promising technique for generating two-dimensional meshes is presented. Our technique shows advantages and efficiency over some currently available mesh generators.     

A finite element alternative to infinite elements

In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.     

Adaptive finite element methods for shape optimization of linearly elastic structures

Grid adaptive methods combined with means for automatic remeshing are applied to problems in shape optimal design of linearly elastic structures. The quantitative effect of element distortion near the design boundaries is identified in terms of interpolation error associated with the finite element discretization. The grid adaptation is itself formulated as a structural optimization problem, with an objective function that reflects the discretization error. A ‘necessary condition’ from this formulation provides the basis for a computational procedure to predict the modified grid.To avoid the sometimes drastic distortion of the FEM grid that might otherwise occur in conjunction with design change, remeshing must be performed at intermediate stages of the overall solution process. In order to produce results for the optimal shape design without interruption in this process, the computer program combines numerical grid generation and automatic remeshing with the grid adaptation and design change. Results for several shape design problems obtained with the use of grid adaptation are compared to computational results predicted from a fixed grid. Both ‘r-’ and ‘h-adaptation’ are tested.     

On boundary-hybrid finite element methods for the Laplace equation

About two decades ago, I. Babu ka, J.T. Oden and J.K. Lee introduced finite element methods that calculate the normal derivative of the solution along the mesh interfaces and recover the solution via local Neumann problems. These methods for the treatment of the homogeneous Laplace equation were called ‘boundary-hybrid methods’. The approach was revisited in [12] for general symmetric and positive definite elliptic equations with homogeneous boundary conditions. The new approximation is nonconforming and lends itself well for an a posteriori error estimator for conforming finite element approximations. Numerical tests presented in [12] corroborated that the error estimates are accurate and cheap for conforming approximations. This paper provides the iterative solution methods and Galerkin discretization methods on which the numerical approximations in [12] were based.     

Image Processing to Automate Mesh Generation

Finite element modeling is now a standard approach used in the industry to minimize costly trials and help reduce the overall lead time in product and process design. However, building surface/solid models defining the product shape and generating finite element meshes for analyses still require a significant amount of the engineers’ time. In this paper, we present a new method for automatically generating a finite element mesh directly from bitmap images obtained from an artist’s concept of a label for an embossed aluminum beverage can, and demonstrate its application towards the building of tooling mesh models used in the modeling of the embossing process. This approach completely eliminates the need for creating a surface/solid model, thus resulting in a dramatic reduction in the time required for process design.     

Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method

A new approach to nonlinear state estimation and object tracking from indirect observations of a continuous time process is examined. Stochastic differential equations (SDEs) are employed to model the dynamics of the unobservable state. Tracking problems in the plane subject to boundaries on the state-space do not in general provide analytical solutions. A widely used numerical approach is the sequential Monte Carlo (SMC) method which relies on stochastic simulations to approximate state densities. For off-line analysis, however, accurate smoothed state density and parameter estimation can become complicated using SMC because Monte Carlo randomness is introduced. The finite element (FE) method solves the Kolmogorov equations of the SDE numerically on a triangular unstructured mesh for which boundary conditions to the state-space are simple to incorporate. The FE approach to nonlinear state estimation is suited for off-line data analysis because the computed smoothed state densities, maximum a posteriori parameter estimates and state sequence are deterministic conditional on the finite element mesh and the observations. The proposed method is conceptually similar to existing point-mass filtering methods, but is computationally more advanced and generally applicable. The performance of the FE estimators in relation to SMC and to the resolution of the spatial discretization is examined empirically through simulation. A real-data case study involving fish tracking is also analysed.     

Isogeometric analysis in option pricing

ABSTRACT

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.     

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

A new scheme for implementing a reduced order model for complex mesh-based numerical models (e.g. finite element unstructured mesh models), is presented. The matrix and source term vector of the full model are projected onto the reduced bases. The proper orthogonal decomposition (POD) is used to form the reduced bases. The reduced order modeling code is simple to implement even with complex governing equations, discretization methods and nonlinear parameterizations. Importantly, the model order reduction code is independent of the implementation details of the full model code. For nonlinear problems, a perturbation approach is used to help accelerate the matrix equation assembly process based on the assumption that the discretized system of equations has a polynomial representation and can thus be created by a summation of pre-formed matrices.In this paper, by applying the new approach, the POD reduced order model is implemented on an unstructured mesh finite element fluid flow model, and is applied to 3D flows. The error between the full order finite element solution and the reduced order model POD solution is estimated. The feasibility and accuracy of the reduced order model applied to 3D fluid flows are demonstrated.     

A Posteriori Estimates for a Natural Neumann–Neumann Domain Decomposition Algorithm on a Unilateral Contact Problem

In this paper we present an error estimator for unilateral contact problems solved by a Neumann–Neumann Domain Decomposition algorithm. This error estimator takes into account both the spatial error due to the finite element discretization and the algebraic error due to the domain decomposition algorithm. To differentiate specifically the contribution of these two error sources to the global error, two quantities are introduced: a discretization error indicator and an algebraic error indicator. The effectivity indices and the convergence of both the global error estimator and the error indicators are shown on several examples.