An Adaptive SDG Method for the Stokes System

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.     

A general purpose adaptivity driver for FE software

This paper describes a tool that serves as an automatic mesh adaptivity driver program for general purpose finite element (FE) software packages. Many commercially available FE programs lack a feature to control the numerical solution's accuracy properly. Our tool implements a mesh adaptive method that, in conjunction with separate finite element software, allows one to fully automatically improve the quality of the numerical solution up to a user specified accuracy. We demonstrate the use of the package with selected computational examples performed with a commercial FE package, ANSYS, and with our FE program FEMEngine. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

An efficient truss structure optimization framework based on CAD/CAE integration and sequential radial basis function metamodel

In order to improve the performance and efficiency of truss structure optimization, this paper presents a general framework that embeds and seamlessly integrates commercial CAD and CAE software through common programming languages and application programming interface (API). Along with the automatic CAD/CAE integration, an adaptive metamodel-based optimization called sequential radial basis function (SRBF) is applied to truss structure optimization involving sizing, geometry and topology variables. SRBF distinguishingly features two-loops searching strategy, the “inner loop” and the “outer loop”. The “inner loop” aims to search a feasible point through updating the factors of the augmented Lagrangian function. With the improved significant sampling space (ISSS) method, the “outer loop” sequentially generates new additional samples to update the RBF model. The continuous relaxation method is developed to deal with the mixed-discrete variables during the truss structure optimization. Applied to practical truss structure optimization problems from small scale to large scale, the proposed framework demonstrates feasibility of the CAD/CAE integration system during the structure modeling and analysis, and facilitates the truss structure optimization process. The comparison results between the SRBF and other approaches show that SRBF improves merit of searching global optimum and reduces the computation cost.     

Numerical simulation of the seismic behavior of building structures equipped with friction energy dissipators

This paper presents a new algorithm to simulate the seismic response of N-story building frames incorporating friction energy dissipators; a device per floor is considered. The frames with the dissipators are described by 2D lumped masses models with two degrees of freedom per floor, namely the horizontal displacements of the main structure and of the dissipators. The proposed algorithm consists of a modification of the linear acceleration method; the main innovation consists of checking at each calculation instant the sliding or sticking condition at each floor, hence, the number of “active” degrees of freedom changes continuously, ranging in between N (there is sticking condition at every dissipator) and 2N (there is sliding condition at every dissipator). Some results given by this algorithm are compared to experimental results from ad-hoc testing and to numerical results obtained with the ADINA software package. In both cases, agreement is satisfactory while the proposed method is more computationally efficient.     

An integrated approach to structural shape optimization of linearly elastic structures. Part II: Shape definition and adaptivity

In this paper we initially study how the number of design variables used affects the final optimum shape of the structure when employing two different types of curves to describe the boundary of the structure, i.e. quadratic Bezier and cubic B-spline curves. The advantage of using better shape definition is highlighted with several examples. An adaptive mesh refinement (AMR) procedure using six-node triangular elements is adopted in the structural shape optimization process. The procedure makes use of an h-version adaptive refinement technique based on error estimates determined from either best-guess stress values or residual terms in the governing equation. An example is presented to illustrate the performance of these error estimators with respect to their convergence, accuracy and cost of computation. Different strategies for the inclusion of AMR procedures in the shape optimization process are also proposed. Anomalies in predicting the optimum shape due to discretization errors are demonstrated using several examples.     

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h^p+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h^2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h^2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.     

Methods for interfacing analysis software to optimization software

Three methods are presented for interfacing analysis software to optimization software to create design software. These methods are referred to as the “conventional interface”, the “pro-gramming-free interface”, and the “generalized interface”. The latter two methods introduce new ideas which are attractive from the user's standpoint. The programming-free interface simplifies the interface process by eliminating the necessity for tlie user to modify the analysis source code. The generalized interface allows one to create a general-purpose design package from a general-purpose analysis package. Support for the methods has been implemented in a software package named OPTDES.BYU. Use of the methods with this package is illustrated with a simple example.     

Recent developments in damage detection based on system identification methods

The output error and equation error methods of system identification are compared for their effectiveness in assessing damage in structural systems. Damage is modeled on an element-by-element basis as changes in sectional properties, which then contribute to variations in the terms of the structural stiffness matrix. Both static displacements and eigenmodes of the structure are used in the damage detection process. A rational basis for the proper selection of eigenmodes and loading conditions for the identification process is also presented. Characteristics of the unconstrained optimization design space for the two approaches are discussed in context of their ability to yield the location and extent of damage.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

A Measurement and Structural Model for Usability Evaluation of Shared Workspace Groupware

This study presents the development process of a set of questionnaire items to establish a measurement model for the usability of shared workspace groupware systems, which is suggested as a usability scale called SWUS, the Shared Workspace Usability Scale. Manifest variables and latent variables are based on the various dimensions of teamwork collated through the literature. A structural model was built on the measurement model. Models were evaluated through PLS-SEM methods. Data acquired on candidate questionnaire items from 398 international respondents who are users of five different online collaborative word processors was used for the model analysis. Of 37 candidate manifest variables, 22 were retained, which were measuring seven latent constructs: “3C Mechanisms,” “Grounding,” “Team Integration,” “Communication,” “Shared Access,” “Awareness,” and “Usability.” The data provided empirical evidence for the structural model based on these latent variables. The responses of the participants were not sensitive to differences between users in terms of gender and native language but showed sensitivity to age, experience with the evaluated software, and different shared workspace groupware evaluated in the study. Our structural model attempts to integrate several frameworks and models of usability for CSCW environments and provides empirical evidence for its reliability and validity based on subjective responses from users of shared workspace groupware.     

The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h ^p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h ^p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.     

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.     

An efficient technique for finding the desired global optimum of robotic joint displacement

For an industrial robot on a daily operation basis such as pick and place, it is desired to minimize the robotic joint displacements when moving the robot from one location to another. The objective of the optimization here is to simultaneously minimize a robot end effector's positional error and the robotic joint displacements. By modifying the searching algorithm in the existing complex optimization method, this article presents a technique for finding the desired global optimum solution more efficiently. To compare the optimum searching capability between the proposed and existing searching algorithms, a modified Himmelblau's function is used as an objective function. The presented technique is then applied to a spatial three-link robot manipulator for global minimization of the joint displacements. © 1992 John Wiley & Sons, Inc.     

Discontinuous Galerkin Method with Staggered Hybridization for a Class of Nonlinear Stokes Equations

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.     

Toward hp-adaptive solution of 3D electromagnetic scattering from cavities

An effective hp-adaptive finite-element (FE) approach is presented for a reliable and accurate solution of 3D electromagnetic scattering problems. The far field is approximated with the infinite-element method. This allows one to reduce the external domain (discretised with finite elements) to a minimum preserving the possibility of arbitrary reduction of the error as the method does not introduce modelling error. The work is focused on scattering from cavity backed apertures recessed in a ground plane. Near optimal discretisations that can effectively resolve local rapid variations in the scattered field can be obtained adaptively by local mesh refinements (so called h-type refinements) blended with graded polynomial enrichments (p-enrichments). The discretisation error can be controlled by a self-adaptive process, which is driven by a posteriori error estimates in terms of the energy norm or in a quantity of interest. The radar cross section (RCS) is an example of the latter. h- and p-adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for generic cavities are given and compared to previously published results.     

Parallel adaptive mesh generation

Unstructured meshes have proved to be a powerful tool for adaptive remeshing of finite element idealizations. This paper presents a transputer-based parallel algorithm for two dimensional unstructured mesh generation. A conventional mesh generation algorithm for unstructured meshes is reviewed by the authors, and some program modules of sequential C source code are given. The concept of adaptivity in the finite element method is discussed to establish the connection between unstructured mesh generation and adaptive remeshing.After these primary concepts of unstructured mesh generation and adaptivity have been presented, the scope of the paper is widened to include parallel processing for un-structured mesh generation. The hardware and software used is described and the parallel algorithms are discussed. The Parallel C environment for processor farming is described with reference to the mesh generation problem. The existence of inherent parallelism within the sequential algorithm is identified and a parallel scheme for unstructured mesh generation is formulated. The key parts of the source code for the parallel mesh generation algorithm are given and discussed. Numerical examples giving run times and the consequent “speed-ups” for the parallel code when executed on various numbers of transputers are given. Comparisons between sequential and parallel codes are also given. The “speed-ups” achieved when compared with the sequential code are significant. The “speed-ups” achieved when networking further transputers is not always sustained. It is demonstrated that the consequent “speed-up” depends on parameters relating to the size of the problem.     

Adaptive element-free Galerkin method applied to the limit analysis of plates

The implementation of an h-adaptive element-free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.     

Dynamic surface control approach to adaptive robust control of nonlinear systems in semi-strict feedback form

This article considers the adaptive robust control of a class of single-input-single-output nonlinear systems in semi-strict feedback form using radial basis function (RBF) networks. It is well known that the standard backstepping design may suffer from “explosion of terms”. To overcome this problem, the recently developed dynamic surface control technique which employs a first-order low-pass filter at each step of the backstepping design procedure is generalized to the nonlinear system under study. Our attention is paid to achieve guaranteed transient performance of the adaptive controller. At each step of design, a feedback controller strengthened by nonlinear damping terms to counteract nonlinear uncertainties is designed to guarantee input-to-state practical stability of the corresponding subsystem, and then parameter adaptations are introduced to reduce the ultimate error bound. Furthermore, for the output trajectory tracking problem, it is recommended to adopt the partial adaptation policy to reduce the computational burden due to “curse of dimension” of the RBF networks. Finally, numerical examples are included to verify the results of theoretical analysis.