Showing papers on "Smoothed finite element method published in 2003"
Book•
19 Jun 2003
TL;DR: In this paper, a survey of finite element methods for approximating the time harmonic Maxwell equations is presented, and error estimates for problems with spatially varying coefficients are compared for three DG families: interior penalty type, hybridizable DG, and Trefftz type methods.
Abstract: We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.
1,453 citations
Book•
17 Nov 2003TL;DR: In this paper, the authors present a theoretical analysis of the Spectral Element Method and Spectral Analysis of Signals (SAM) in terms of the following: 1.1 Fourier Series. 2.2 Discrete Fourier Transform and FFT. 3.3 Aliasing. 4.4 Reduction from the Finite Models.
Abstract: Preface. Part One Introduction to the Spectral Element Method and Spectral Analysis of Signals. 1 Introduction. 1.1 Theoretical Background. 1.2 Historical Background. 2 Spectral Analysis of Signals. 2.1 Fourier Series. 2.2 Discrete Fourier Transform and the FFT. 2.3 Aliasing. 2.4 Leakage. 2.5 Picket-Fence Effect. 2.6 Zero Padding. 2.7 Gibbs Phenomenon. 2.8 General Procedure of DFT Processing. 2.9 DFTs of Typical Functions. Part Two Theory of Spectral Element Method. 3 Methods of Spectral Element Formulation. 3.1 Force-Displacement Relation Method. 3.2 Variational Method. 3.3 State-Vector Equation Method. 3.4 Reduction from the Finite Models. 4 Spectral Element Analysis Method. 4.1 Formulation of Spectral Element Equation. 4.2 Assembly and the Imposition of Boundary Conditions. 4.3 Eigenvalue Problem and Eigensolutions. 4.4 Dynamic Responses with Null Initial Conditions. 4.5 Dynamic Responses with Arbitrary Initial Conditions. 4.6 Dynamic Responses of Nonlinear Systems. Part Three Applications of Spectral Element Method. 5 Dynamics of Beams and Plates. 5.1 Beams. 5.2 Levy-Type Plates. 6 Flow-Induced Vibrations of Pipelines. 6.1 Theory of Pipe Dynamics. 6.2 Pipelines Conveying Internal Steady Fluid. 6.3 Pipelines Conveying Internal Unsteady Fluid. Appendix 6.A: Finite Element Matrices: Steady Fluid. Appendix 6.B: Finite Element Matrices: Unsteady Fluid. 7 Dynamics of Axially Moving Structures. 7.1 Axially Moving String. 7.2 Axially Moving Bernoulli-Euler Beam. 7.3 Axially Moving Timoshenko Beam. 7.4 Axially Moving Thin Plates. Appendix 7.A: Finite Element Matrices for Axially Moving String. Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli-Euler Beam. Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam. Appendix 7.D: Finite Element Matrices for Axially Moving Plate. 8 Dynamics of Rotor Systems. 8.1 Governing Equations. 8.2 Spectral Element Modeling. 8.3 Finite Element Model. 8.4 Numerical Examples. Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration. 9 Dynamics of Multi-Layered Structures. 9.1 Elastic-Elastic Two-Layer Beams. 9.2 Elastic-Viscoelastic-elastic-Three-Layer (PCLD) Beams. Appendix 9.A: Finite Element Matrices for the Elastic-Elastic Two-Layer Beam. Appendix 9.B: Finite Element Matrices for the Elastic-VEM-Elastic Three-Layer Beam. 10 Dynamics of Smart Structures. 10.1 Elastic-Piezoelectric Two-Layer Beams. 10.2 Elastic-Viscoelastic-Piezoelctric Three-Layer (ACLD) Beams. 11 Dynamics of Composite Laminated Structures. 11.1 Theory of Composite Mechanics. 11.2 Equations of Motion for Composite Laminated Beams. 11.3 Dynamics of Axial-Bending-Shear Coupled Composite Beams. 11.4 Dynamics of Bending-Torsion-Shear Coupled Composite Beams. Appendix 11.A: Finite Element Matrices for Axial-Bending-Shear Coupled Composite Beams. Appendix 11.B: Finite Element Matrices for Bending-Torsion-Shear Coupled Composite Beams. 12 Dynamics of Periodic Lattice Structures. 12.1 Continuum Modeling Method. 12.2 Spectral Transfer Matrix Method. 13 Biomechanics: Blood Flow Analysis. 13.1 Governing Equations. 13.2 Spectral Element Modeling: I. Finite Element. 13.3 Spectral Element Modeling: II. Semi-Infinite Element. 13.4 Assembly of Spectral Elements. 13.5 Finite Element Model. 13.6 Numerical Examples. Appendix 13.A: Finite Element Model for the 1-D Blood Flow. 14 Identification of Structural Boundaries and Joints. 14.1 Identification of Non-Ideal Boundary Conditions. 14.2 Identification of Joints. 15 Identification of Structural Damage. 15.1 Spectral Element Modeling of a Damaged Structure. 15.2 Theory of Damage Identification. 15.3 Domain-Reduction Method. 16 Other Applications. 16.1 SEM-FEM Hybrid Method. 16.2 Identification of Impact Forces. 16.3 Other Applications. References. Index.
430 citations
TL;DR: In this article, the authors present an extended finite element method (X-FEM) for modeling strong (displacement) and weak (strain) discontinuities within a standard finite element framework.
401 citations
01 Jan 2003
289 citations
TL;DR: In this paper, a finite element method for linear elastic fracture mechanics using enriched quadratic interpolations is presented, which is enriched with the asymptotic near tip displacement solutions and the Heaviside function so that the finite element approximation is capable of resolving the singular stress field at the crack tip as well as the jump in the displacement field across the crack face.
Abstract: A finite element method for linear elastic fracture mechanics using enriched quadratic interpolations is presented. The quadratic finite elements are enriched with the asymptotic near tip displacement solutions and the Heaviside function so that the finite element approximation is capable of resolving the singular stress field at the crack tip as well as the jump in the displacement field across the crack face without any significant mesh refinement. The geometry of the crack is represented by a level set function which is interpolated on the same quadratic finite element discretization. Due to the higher-order approximation for the crack description we are able to represent a crack with curvature. The method is verified on several examples and comparisons are made to similar formulations using linear interpolants.
233 citations
TL;DR: The partition of unity property of nodal shape functions has been exploited to introduce a higher-order displacement discontinuity in a standard finite element model, able to reproduce the typical cusp-like shape of the process zone at the tip of a cohesive crack.
Abstract: A methodology for the simulation of quasi-static cohesive crack propagation in quasi-brittle materials is presented. In the framework of the recently proposed extended finite element method, the partition of unity property of nodal shape functions has been exploited to introduce a higher-order displacement discontinuity in a standard finite element model. In this way, a cubic displacement discontinuity, able to reproduce the typical cusp-like shape of the process zone at the tip of a cohesive crack, is allowed to propagate without any need to modify the background finite element mesh. The effectiveness of the proposed method has been assessed by simulating mode-I and mixed-mode experimental tests. Copyright © 2003 John Wiley & Sons, Ltd.
210 citations
TL;DR: In this article, the authors present several techniques for modeling cracking within the finite element (FE) framework which use meshes independent of the crack configuration and thus avoid remeshing, combining the traditional FE method with the partition of unity method for modelling individual cracks.
196 citations
TL;DR: In this paper, a finite element method for axisymmetric two-phase flow problems is presented, which uses an enriched finite element formulation, in which the interface can move arbitrarily through the mesh without remeshing.
Abstract: A finite element method for axisymmetric two-phase flow problems is presented. The method uses an enriched finite element formulation, in which the interface can move arbitrarily through the mesh without remeshing. The enrichment is implemented by the extended finite element method (X-FEM) which models the discontinuity in the velocity gradient at the interface by a local partition of unity. It provides an accurate representation of the velocity field at interfaces on an Eulerian grid that is not conformal to the weak discontinuity. The interface is represented by a level set which is also used in the construction of the element enrichment. Surface tension effects are considered and the interface curvature is computed from the level set field. The method is demonstrated by several examples. Copyright © 2003 John Wiley & Sons, Ltd.
128 citations
TL;DR: In this article, a 3D low-order element, identified as HIS, is proposed for analysis of both linear and non-linear problems including finite strain plasticity, which is shown to be suitable in the analysis of linear problems and general nonlinear problems.
Abstract: The now classical enhanced strain technique, employed with success for more than 10 years in solid, both 2D and 3D and shell finite elements, is here explored in a versatile 3D low-order element which is identified as HIS. The quest for accurate results in a wide range of problems, from solid analysis including near-incompressibility to the analysis of locking-prone beam and shell bending problems leads to a general 3D element. This element, put here to test in various contexts, is found to be suitable in the analysis of both linear problems and general non-linear problems including finite strain plasticity. The formulation is based on the enrichment of the deformation gradient and approximations to the shape function material derivatives. Both the equilibrium equations and their variation are completely exposed and deduced, from which internal forces and consistent tangent stiffness follow. A stabilizing term is included, in a simple and natural form. Two sets of examples are detailed: the accuracy tests in the linear elastic regime and several finite strain tests. Some examples involve finite strain plasticity. In both sets the element behaves very well, as is illustrated in numerous examples. Copyright © 2003 John Wiley & Sons, Ltd.
104 citations
TL;DR: In this paper, a finite element approach for the analysis of the wave propagation in an infinitely long plate is presented, where a nonreflecting boundary condition such that there will be no spurious reflections generated by the finite boundary of the finite element model is introduced.
102 citations
TL;DR: In this article, a new finite element is presented for the simulation of delamination growth in thin-layered composite structures, based on a solid-like shell element: a volume element that can be used for very thin applications due to a higher-order displacement field in the thickness direction.
Abstract: In this contribution a new finite element is presented for the simulation of delamination growth in thin-layered composite structures. The element is based on a solid-like shell element: a volume element that can be used for very thin applications due to a higher-order displacement field in the thickness direction. The delamination crack can occur at arbitrary locations and is incorporated in the element as a jump in the displacement field by using the partition of unity property of finite element shape functions. The kinematics of the element as well as the finite element formulation are described. The performance of the element is demonstrated by means of two examples. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: A new mixed finite element method for the diffusion equations on general polygonal and polyhedral meshes is presented and Numerical results for the Poisson equation on distorted prismatic meshes are given.
Abstract: A new mixed finite element method for the diffusion equations on general polygonal and polyhedral meshes is presented. The basis vector functions in macrocells are designed by solving the local mixed finite element problems with the lowest order Raviart-Thomas elements. Numerical results for the Poisson equation on distorted prismatic meshes are given.
TL;DR: In this article, the p-version finite element method is applied to finite strain problems and the behavior of high order finite elements is studied for an isotropic hyperelastic material in the case of near incompressibility.
TL;DR: In this article, the divergence theorem is applied once and twice, respectively, for polygonal and polyhedral integration domains, to construct integrals on boundary wireframes, and the sum of Gaussian quadrature values on linear segments of the wireframe yields the final result of numerical integration.
Abstract: Engineering mechanics formulations of aerospace industry problems overwhelmingly rely upon spatial averaging techniques. Crucial applications in the area of dynamic response analysis and stochastic estimation of material degradation can be cited as important cases. Integration procedures on finite domains underlie physically acceptable averaging processes. Unlike one-dimensional cases, integrals within arbitrary areas and volumes cannot be approximated by a Gaussian form of numerical quadrature. Here the divergence theorem is applied once and twice, respectively, for polygonal and polyhedral integration domains, to construct integrals on boundary wireframes. The sum of Gaussian quadrature values on linear segments of the wireframe yields the final result of numerical integration on a finite element.
TL;DR: A duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems and criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p.
Abstract: In this paper a duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems. Duality arguments combined with Galerkin orthogonality yield representations of the error in arbitrary quantities of interest. From these error estimates, criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p. The effectivity of this procedure is confirmed by numerical tests.
TL;DR: This paper develops a sub-domain inverse finite element method for characterizing the material properties of inflated hyperelastic membranes, including soft tissues, which can assume homogeneity of the material behavior as well as of the local stress and strain fields.
Abstract: Quantification of the mechanical behavior of hyperelastic membranes in their service configuration, particularly biological tissues, is often challenging because of the complicated geometry, material heterogeneity, and nonlinear behavior under finite strains. Parameter estimation thus requires sophisticated techniques like the inverse finite element method. These techniques can also become difficult to apply, however, if the domain and boundary conditions are complex (e.g. a non-axisymmetric aneurysm). Quantification can alternatively be achieved by applying the inverse finite element method over sub-domains rather than the entire domain. The advantage of this technique, which is consistent with standard experimental practice, is that one can assume homogeneity of the material behavior as well as of the local stress and strain fields. In this paper, we develop a sub-domain inverse finite element method for characterizing the material properties of inflated hyperelastic membranes, including soft tissues. We illustrate the performance of this method for three different classes of materials: neo-Hookean, Mooney Rivlin, and Fung-exponential.
TL;DR: In this article, the authors consider the problem of assessing the convergence of mixed-formulated finite elements and propose a new physics-based procedure to evaluate the performance of shell elements.
TL;DR: This paper presents a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two.
Abstract: Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM2 three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.
01 Jan 2003
TL;DR: In this article, a spectral element method was developed for solving the two-dimensional Helmholtz's equation, which is the equation governing time-harmonic acoustic waves, and the spectral element led to fewer grid points per wavelength and less computational cost, for the same accuracy.
Book•
01 Jan 2003
TL;DR: The Finite Element Method for Stochastic Structures (FEM) as mentioned in this paper is based on the exact inverse of stiffness matrix (INF) matrix, which is used to measure stiffness.
Abstract: 1. Fundamentals of Finite Element Method 2. Finite Element Method for Stochastic Structures - A Review and Improvement 3. Finite Element Method for Stochastic Structures Based on Exact Inverse of Stiffness Matrix 4. FEM Based on Direct Exact Inverse of Stiffness Matrix 5. Variational Principles-Based FEM for Stochastic Beams 6. Element-Level Flexibility-Based Finite Element Method for Stochastic Structures 7. A Comparison of Stochastic and Interval Finite Elements Biblography Appendices
TL;DR: A list of programs developed in recent studies and example problems of the finite element method application to thin-walled structures are given in this article, along with a discussion of the analysis types, material models, elements and initial conditions that are taken into account in the finite-element analysis of thin-wall structures.
Abstract: Traditionally numerous physical tests have been required to develop and verify newly proposed design procedures. The availability of powerful computers and software makes the finite element method an essential tool in such research. Analysis types, material models, elements and initial conditions that are taken into account in the finite element analysis of thin-walled structures are discussed. A list of programs developed in recent studies and example problems of the finite element method application to thin-walled structures are given.
TL;DR: In this article, the authors describe an implementation of the infinite element for three-dimensional, time harmonic Maxwell's equations, proposed in Reference [15] (Demkowicz and Pal, Computer Methods in Applied Mechanics and Engineering 1998; 164: 77−94).
Abstract: This paper is a continuation of Reference [26] (Cecot, Demkowicz and Rachowicz, Computer Methods in Applied Mechanics and Engineering 2000; 188: 625–643) and describes an implementation of the infinite element for three-dimensional, time harmonic Maxwell's equations, proposed in Reference [15] (Demkowicz and Pal, Computer Methods in Applied Mechanics and Engineering 1998; 164: 77–94). The element is compatible with the hp finite element discretizations for Maxwell's equations in bounded domains reported in References [16–18] (Computer Methods in Applied Mechanics and Engineering 1998; 152: 103–124, 1999; 169: 331–344, 2000; 187: 307–337). Copyright © 2003 John Wiley & Sons, Ltd.
01 Jan 2003
TL;DR: A modification of the immersed boundary method based on a finite element approximation of the fluid and two-dimensional numerical examples which confirm the good behavior of the computed solutions are presented.
Abstract: In this paper we consider a modification of the immersed boundary method based on a finite element approximation of the fluid. Besides some theoretical considerations, we present two-dimensional numerical examples which confirm the good behavior of the computed solutions.
TL;DR: In this article, the authors focus on possible sources of error in linear and non-linear finite element solutions, and give suggestions how to check and prevent these errors, and how to prevent the errors.
Abstract: The application of the finite element method to thin-walled structures often requires non-linear analysis. Whereas in linear finite element analyses errors are easily made, this is even more so in the non-linear analyses. This paper focuses on possible sources of error in linear and non-linear finite element solutions, and gives suggestions how to check and prevent these errors.
01 Jan 2003
TL;DR: In this paper, the problem domain is first discretized into small elements, while using the FEM to solve mechanics problems governed by a set of partial differential equations, and the equations obtained for each element are then assembled together with adjoining elements to form the global finite element (FE) equation for the whole problem domain.
Abstract: This chapter focuses on the fundamentals of finite element method (FEM). The problem domain is first discretized into small elements, while using the FEM to solve mechanics problems governed by a set of partial differential equations. In each of these elements, the profile of the displacements is assumed in simple forms to obtain element equations. The equations obtained for each element are then assembled together with adjoining elements to form the global finite element (FE) equation for the whole problem domain. Thus, the equations created for the global problem domain can be solved easily for the entire displacement field.
TL;DR: In this paper, a finite element based Inverse analysis technique was developed to determine the flow stress and friction at the tool/workplace interface simultaneously from one set of material tests, aimed at minimizing the error between experimental data and predictions made by rigid-plastic finite element simulations.
01 Jan 2003
TL;DR: In this paper, numerical modeling techniques for hysteresis properties in magnetic materials are discussed, and the authors distinguish between rate-independent and rate dependent material behaviour, which can be included into finite element formulations.
Abstract: This paper deals with numerical modelling techniques for hysteresis properties in magnetic materials. Physical and phenomenological models are discussed. First some basic hysteresis properties are described in order to proceed gradually to more sophisticated models, which can be included into finite element formulations. Scalar and vector hysteresis models are considered. We distinguish between rate-independent and rate dependent material behaviour.
TL;DR: Optimal-order error estimates in some H1-equivalence norms are established for the proposed discontinuous finite element methods and an optimal- order error estimate is derived in the L2 norm for the symmetric formulation.
Abstract: A new finite element method is proposed and analysed for second order elliptic equations using discontinuous piecewise polynomials on a finite element partition consisting of general polygons. The new method is based on a stabilization of the well-known primal hybrid formulation by using some least-squares forms imposed on the boundary of each element. Two finite element schemes are presented. The first one is a non-symmetric formulation and is absolutely stable in the sense that no parameter selection is necessary for the scheme to converge. The second one is a symmetric formulation, but is conditionally stable in that a parameter has to be selected in order to have an optimal order of convergence.
Optimal-order error estimates in some H1-equivalence norms are established for the proposed discontinuous finite element methods. For the symmetric formulation, an optimal-order error estimate is also derived in the L2 norm. The new method features a finite element partition consisting of general polygons as opposed to triangles or quadrilaterals in the standard finite element Galerkin method. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: Novel high-accuracy computational techniques for solid mechanics problems are presented, including fourth-order and arbitrary-order finite difference methods based on Pade-type differencing formulas and a meshless method which uses radial basis functions in a "finite difference" mode.
Abstract: Novel high-accuracy computational techniques for solid mechanics problems are presented. They include fourth-order and arbitrary-order finite difference methods based on Pade-type differencing formulas and a meshless method which uses radial basis functions in a "finite difference" mode. Some results illustrating high performance of the suggested numerical methods are displayed.