Covering sharing trees: a compact data structure for parameterized verification

The control state reachability problem is decidable for well-structured infinite-state systems like (Lossy) Petri Nets, Vector Addition Systems, and broadcast protocols. An abstract algorithm that solves the problem is the backward reachability algorithm of [1, 21 ]. The algorithm computes the closure of the predecessor operator with respect to a given upward-closed set of target states. When applied to this class of verification problems, symbolic model checkers based on constraints like [7, 26 ] suffer from the state explosion problem.In order to tackle this problem, in [13] we introduced a new data structure, called covering sharing trees, to represent in a compact way collections of infinite sets of system configurations. In this paper, we will study the theoretical complexity of the operations over covering sharing trees needed in symbolic model checking. We will also discuss several optimizations that can be used when dealing with Petri Nets. Among them, in [14] we introduced a new heuristic rule based on structural properties of Petri Nets that can be used to efficiently prune the search during symbolic backward exploration. The combination of these techniques allowed us to turn the abstract algorithm of [1, 21 ] into a practical method. We have evaluated the method on several finite-state and infinite-state examples taken from the literature [2, 18 , 20 , 30 ]. In this paper, we will compare the results we obtained in our experiments with those obtained using other finite and infinite-state verification tools.     

Benchmark study of numerical methods for reliability-based design optimization

The reliability-based design optimization (RBDO) seeks for the best compromise between cost and safety, by considering system uncertainties. In order to overcome computational difficulties, many formulations have been recently developed, leading to confusion about what method should be selected for a given application, due to the lack of full-scale comparative studies. In this context, the present paper aims at giving an overview of various RBDO approaches which are tested on a benchmark constituted of four examples using mathematical and finite element models, with different levels of difficulties. The study is focused on the three main approaches, namely the two-level approach, the single loop approach and the decoupled approach; for each category, two RBDO formulations are discussed, implemented and tested for numerical examples. The benchmark study allows us to give comprehensive overview of various approaches, to give clear ideas about their capabilities and limitations, and to draw useful conclusions regarding robustness and numerical performance.     

Adaptive full domain covering meshes for parallel finite element computations

Summary  In this work, we present a new parallelization concept for adaptive finite element methods. Compared to classical domain decomposition approaches, the concept of adaptive full domain covering meshes reduces the parallel communication overhead. Furthermore, it provides an easy way to transform sequential codes into parallel software by changing only a few lines of source code.      

Implementation of the Arlequin method into ABAQUS: Basic formulations and applications

This article is concerned with implementing the Arlequin method into the commercial finite element software ABAQUS and demonstrating its versatility in various typical applications. According to the basic idea of the Arlequin method, the particular formulations for the ABAQUS user element are presented. A unified method for calculating coupling matrices is proposed in an obvious and straightforward way. The user element subroutines for coupling two relatively independent models with different or same dimensions are developed. Four numerical examples are considered: One is to demonstrate the coupling of elements with the same element type but different grid density by the Arlequin method, and the other three examples involve different types of elements. The numerical results justify the correctness of the codes developed for the user defined elements as well as the feasibility of the Arlequin method which is successfully integrated into ABAQUS.     

A Qualitative Approach to Syllogistic Reasoning

In this paper we present a new approach to a symbolic treatment of quantified statements having the following form Q A's are B's, knowing that A and B are labels denoting sets, and Q is a linguistic quantifier interpreted as a proportion evaluated in a qualitative way. Our model can be viewed as a symbolic generalization of statistical conditional probability notions as well as a symbolic generalization of the classical probabilistic operators. Our approach is founded on a symbolic finite M-valued logic in which the graduation scale of M symbolic quantifiers is translated in terms of truth degrees. Moreover, we propose symbolic inference rules allowing us to manage quantified statements.     

The inverse electromagnetic shaping problem

The inverse problem concerning electromagnetic casting of molten metals consists of looking for an electric current density distribution such that the induced electromagnetic field makes a given mass of liquid metal acquire a predefined shape. This problem is formulated here as an optimization problem where the positions of a finite set of inductors are the design variables. Two different formulations for this optimization problem for the two-dimensional case are proposed. The first one minimizes the difference between the target and the equilibrium shapes while the second approach minimizes the L ² norm of a fictitious surface pressure that makes the target shape to be in mechanical equilibrium. The optimization problems are solved using Feasible Arc Interior Point Algorithm, a line search interior-point algorithm for nonlinear optimization. Some examples are presented to show the effectiveness of the proposed approaches.     

The analysis of solids without mesh generation using trimmed patch boundary elements

The tools available to the mechanical engineer—for example, geometric modeling and computer-aided analysis—are individually quite powerful, but they are based on different geometric representations. Hence, they do not always work well together. In this paper an analysis method is presented that operates directly on the geometric modeling representation. Therefore, the time-consuming and error-prone procedure of generating a mesh is skipped. The method is based on boundary integral equations, but unlike previously published methods, the boundary elements aren-sided trimmed patches, the same patches that are used by modern geometric modelers to represent complex solids. The method is made practical by defining shape functions over the trimmed patches in such a way that the number of degrees of freedom can be controlled. This is done by using a concept called virtual nodes. The paper begins by deriving the trimmed patch boundary element. Then its properties are discussed in comparison with existing boundary element and finite element methods, and several examples are given.     

The Approximation and Computation of a Basis of the Trace Space <Emphasis Type="Italic">H</Emphasis><Superscript>1/2</Superscript>

We present a method for construction of an approximate basis of the trace space H ^1/2 based on a combination of the Steklov spectral method and a finite element approximation. Specifically, we approximate the Steklov eigenfunctions with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. We provide a reformulation of the discrete Steklov eigenproblem as a generalized eigenproblem that we solve by the implicitly restarted Arnoldi method of ARPACK. We include examples highlighting the computational properties of the proposed method for the solution of elliptic problems on bounded domains using both a conforming bilinear finite element and a non-conforming harmonic finite element. In addition, we document the efficiency of the proposed method by solving a Dirichlet problem for the Laplace equation on a densely perforated domain.     

Computational robot dynamics: Foundations and applications

In 1984, the authors unveiled the computer program Algebraic Robot Modeler (ARM) for the symbolic generation of complete closed-form dynamic robot models. In this paper, we introduce computational robot dynamics as the synthesis of classical mechanics and computer software for the symbolic and numeric modeling of robotic mechanisms, and branch-out in three directions. First, we outline the foundations of computational robot dynamics. From its inception (in 1973), we review chronologically the contributions of prominent roboticists, tracing the parallel development of robot dynamics formulations and computational robot dynamics. We then highlight our research activities, the current capabilities of ARM, and our plans for the continuing development and application of ARM and computational robot dynamics. Finally, we focus on practical applications of computational robot dynamics. We apply ARM to produce examples illustrating the comparative computational requirements of robot dynamics formulations for symbolic processing and customized algorithms for numeric processing.     

T-elements: a finite element approach with advantages of boundary solution methods

Since their introduction in 1977, the so-called T-elements have considerably evolved and have now become a highly efficient and well established tool for the solution of complex boundary value problems. This class of finite elements, associated with the Trefftz method, is based on enforcement of interelement continuity and boundary conditions on assumed displacement fields chosen so as to a priori satisfy the governing differential equations of the problem. Several alternative T-element formulations are available which yield for a particular subdomain the customary force-displacement relationship with a symmetric positive definite stiffness matrix which makes it possible for such elements to be implemented into the standard finite element (FE) codes.

Owing to their nature, the T-elements may either be considered as a new FE model or as a non-conventional symmetric substructure-oriented form of the boundary element method (BEM). From the point of view of the latter, the outstanding features of the T-element approaches are the use of T-complete sets of non-singular solutions (rather than the singular Kelvin's type fundamental solutions) and the replacement of the customary integral equations form by a symmetric variational formulation.

This paper reviews and critically assesses the most important T-element formulations developed over the past years. It shows that such elements not only cumulate the advantages and discard the drawbacks of the conventional finite element and boundary element methods, but also offer additional advantages not available in the standard form of these methods.     

An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations

For the mathematically sound, cost effective, flexible and automatic computation of structural mechanical problems with error tolerances, adaptive finite element meshes (h-adaptivity) and elements with different Ansatz order (p-adaptivity) and dimension (d-adaptivity) are desirable. Furthermore, because of the numerical effort, the use of parallel computers is adequate. Object-oriented data structures and algorithms are presented which support these adaptive formulations. In this paper, we describe a refinement algorithm which adapts hexahedral meshes in a node regular way, i.e. without hanging nodes. Moreover, classes implementing the mathematical operators and structures arising in the finite element formulation are introduced within the object oriented concept. They offer the means to implement FE formulations in a way, very similar to the mathematical notation. For these purposes, an object-oriented language is strongly required in order to get a general and simple program structure, even for highly complex tasks with h-, p- and d-adaptivity and distributed data.     

Finite element analysis of lateral buckling for beam structures

The application of finite element analysis to lateral buckling problems, locating the critical points and tracing the postbifurcation path, is treated on the basis of a geometrically nonlinear formulation for a beam with small elastic strain but with possibly large rotations. The existing finite element formulations for thin beams are examined in the aspect of application to bifurcation problems, such as lateral buckling, and the choice of an appropriate rotation parameter for representing incremental or variational rotations in finite element formulations is discussed in relation to locating bifurcation points. This is illustrated through several numerical examples and followed by appropriate discussion.     

Measuring convergence of mixed finite element discretizations: an application to shell structures

We consider the problem of assessing the convergence of mixed-formulated finite elements. When displacement-based formulations are considered, convergence measures of finite element solutions to the exact solution of the mathematical problem are well known. However when mixed formulations are considered, there is no well-established method to measure the convergence of the finite element solution. We first review a number of approaches that have been employed and discuss their limitations. After having stated the properties that an ideal error measure would possess, we introduce a new physics-based procedure. The new proposed error measure can be used for many different types of mixed formulations and physical problems. We illustrate its use in an assessment of the performance of the MITC family of shell elements.     

High order finite elements for shells

In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.     

Accurate evaluation of support reactions in finite element plate-bending analysis

Two simple approaches are presented which allow the distribution of support reactions to be predicted with as high degree of accuracy as the displacements. In the first approach the plate element assembly is completed with special one-dimensional elastic support elements. If their Winkler coefficient is suitably tuned, an accurate prediction of reactions is obtained as a part of the finite element analysis without unduly affecting the displacements and moments of the plate. In the second approach, a standard finite element calculation (without elastic support elements) is performed first and the distribution of reactions is then evaluated based on the known nodal forces at boundary nodes of the plate.

The two approaches are indiscriminately applicable with Kirchhoff and Reissner-Mindlin plate bending elements. Their practical efficiency is illustrated by numerical examples.     

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the finite element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture.The purpose of this paper is to develop this formulation for Kirchhoff–Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of the shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method.     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

On using deterministic FEA software to solve problems in stochastic structural mechanics

Over the last three decades there has been an outstanding growth in the development of deterministic finite element codes with extensive analysis capabilities. Extension of such deterministic codes to solve problems in stochastic mechanics is of much interest to the academic research community and industry. In this paper we discuss some of the issues involved in integrating fully grown third-party deterministic finite element codes with stochastic projection schemes. The objective of this study is to lay the foundation for development of an easy-to-use general-purpose stochastic finite element software for carrying out probabilistic analysis of large-scale engineering systems. We present a brief introduction to stochastic reduced basis projection schemes and the steps involved in coupling them with a typical deterministic finite element software. We demonstrate with the help of a number of case studies how a coupled framework can be used for solving problems in probabilistic mechanics.