Showing papers on "Smoothed finite element method published in 2007"

A Smoothed Finite Element Method for Mechanics Problems

Abstract: In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180° and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.

Theoretical aspects of the smoothed finite element method (SFEM)

Abstract: This paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell-wise strain smoothing operation into standard compatible finite element method (FEM) The weak form of SFEM can be derived from the Hu–Washizu three-field variational principle For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC → ∞) It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution The properties of SFEM are confirmed by numerical examples Copyright © 2006 John Wiley & Sons, Ltd

Finite cell method

Abstract: A simple yet effective modification to the standard finite element method is presented in this paper. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. If this extension is smooth, the extended solution can be well approximated by high order polynomials. This way, the finite element mesh can be replaced by structured or unstructured cells embedding the domain where classical h- or p-Ansatz functions are defined. An adequate scheme for numerical integration has to be used to differentiate between inside and outside the physical domain, very similar to strategies used in the level set method. In contrast to earlier works, e.g., the extended or the generalized finite element method, no special interpolation function is introduced for enrichment purposes. Nevertheless, when using p-extension, the method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities. The formulation in this paper is applied to linear elasticity problems and examined for 2D cases, although the concepts are generally valid.

Numerical Treatment of Partial Differential Equations

Abstract: Contents Notation 1 Basics 1.1 Classification and Correctness 1.2 Fourier's Method, Integral Transforms 1.3 Maximum Principle, Fundamental Solution 2 Finite Difference Methods 2.1 Basic Concepts 2.2 Illustrative Examples 2.3 Transportation Problems and Conservation Laws 2.4 Elliptic Boundary Value Problems 2.5 Finite Volume Methods as Finite Difference Schemes 2.6 Parabolic Initial-Boundary Value Problems 2.7 Second-Order Hyperbolic Problems 3 Weak Solutions 3.1 Introduction 3.2 Adapted Function Spaces 3.3 VariationalEquationsand conformingApproximation 3.4 WeakeningV-ellipticity 3.5 NonlinearProblems 4 The Finite Element Method 4.1 A First Example 4.2 Finite-Element-Spaces 4.3 Practical Aspects of the Finite Element Method 4.4 Convergence of Conforming Methods 4.5 NonconformingFiniteElementMethods 4.6 Mixed Finite Elements 4.7 Error Estimators and adaptive FEM 4.8 The Discontinuous Galerkin Method 4.9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems 5.1 Parabolic Problems 5.2 Second-Order Hyperbolic Problems 6 Singularly Perturbed Boundary Value Problems 6.1 Two-Point Boundary Value Problems 6.2 Parabolic Problems, One-dimensional in Space 6.3 Convection-Diffusion Problems in Several Dimensions 7 Variational Inequalities, Optimal Control 7.1 Analytic Properties 7.2 Discretization of Variational Inequalities 7.3 Penalty Methods 7.4 Optimal Control of PDEs 8 Numerical Methods for Discretized Problems 8.1 Some Particular Properties of the Problems 8.2 Direct Methods 8.3 Classical Iterative Methods 8.4 The Conjugate Gradient Method 8.5Multigrid Methods 8.6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs Bibliography: Original Papers Index

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

Abstract: In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.

The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion

Abstract: Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth’s structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finite-difference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finite-difference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finitedifference (FD), finite-element (FE), and hybrid FD-FE methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. We present alternative formulations of equation of motion for a smooth elastic continuum. We then develop alternative formulations for a canonical problem with a welded material interface and free surface. We continue with a model of an earthquake source. We complete the general theoretical introduction by a chapter on the constitutive laws for elastic and viscoelastic media, and brief review of strong formulations of the equation of motion. What follows is a block of chapters on the finite-difference and finite-element methods. We develop FD targets for the free surface and welded material interface. We then present various FD schemes for a smooth continuum, free surface, and welded interface. We focus on the staggered-grid and mainly optimally-accurate FD schemes. We also present alternative formulations of the FE method. We include the FD and FE implementations of the traction-at-split-nodes method for simulation of dynamic rupture propagation. The FD modeling is applied to the model of the deep sedimentary Grenoble basin, France. The FD and FE methods are combined in the hybrid FD-FE method. The hybrid method is then applied to two earthquake scenarios for the Grenoble basin. Except chapters 1, 3, 5, and 12, all chapters include new, previously unpublished material and results.

A finite element method for animating large viscoplastic flow

Abstract: We present an extension to Lagrangian finite element methods to allow for large plastic deformations of solid materials. These behaviors are seen in such everyday materials as shampoo, dough, and clay as well as in fantastic gooey and blobby creatures in special effects scenes. To account for plastic deformation, we explicitly update the linear basis functions defined over the finite elements during each simulation step. When these updates cause the basis functions to become ill-conditioned, we remesh the simulation domain to produce a new high-quality finite-element mesh, taking care to preserve the original boundary. We also introduce an enhanced plasticity model that preserves volume and includes creep and work hardening/softening. We demonstrate our approach with simulations of synthetic objects that squish, dent, and flow. To validate our methods, we compare simulation results to videos of real materials.

Automating the Finite Element Method

Abstract: The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations (realizations), which are often specialized and can handle only a small set of variational problems and finite elements (but are typically parametrized over the choice of mesh).

Object-oriented finite element analysis of thermo-hydro-mechanical (THM) problems in porous media

Abstract: The design, implementation and application of a concept for object-oriented in finite element analysis of multi-field problems is presented in this paper. The basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations (PDEs). In principle, similar types of PDEs for diverse physical problems differ only in material coefficients. Local element matrices and vectors arising from the finite element discretization of the PDEs are categorized into several types, regardless of which physical problem they belong to (i.e. fluid flow, mass and heat transport or deformation processes). Element (ELE) objects are introduced to carry out the local assembly of the algebraic equations. The object-orientation includes a strict encapsulation of geometrical (GEO), topological (MSH), process-related (FEM) data and methods of element objects. Geometric entities of an element such as nodes, edges, faces and neighbours are abstracted into corresponding geometric element objects (ELE–GEO). The relationships among these geometric entities form the topology of element meshes (ELE–MSH). Finite element objects (ELE–FEM) are presented for the local element calculations, in which each classification type of the matrices and vectors is computed by a unique function. These element functions are able to deal with different element types (lines, triangles, quadrilaterals, tetrahedra, prisms, hexahedra) by automatically choosing the related element interpolation functions. For each process of a multi-field problem, only a single instance of the finite element object is required. The element objects provide a flexible coding environment for multi-field problems with different element types. Here, the C++ implementations of the objects are given and described in detail. The efficiency of the new element objects is demonstrated by several test cases dealing with thermo-hydro-mechanical (THM) coupled problems for geotechnical applications. Copyright © 2006 John Wiley & Sons, Ltd.

Computational aspects of the stochastic finite element method

Abstract: We present an overview of the stochastic finite element method with an emphasis on the computational tasks involved in its implementation.

Direct simulation of fluid-solid mechanics in porous media using the discrete element and lattice-Boltzmann methods

Abstract: [1] A detailed understanding of the coupling between fluid and solid mechanics is important for understanding many processes in Earth sciences. Numerical models are a popular means for exploring these processes, but most models do not adequately handle all aspects of this coupling. This paper presents the application of a micromechanically based fluid-solid coupling scheme, lattice-Boltzmann discrete element method (LBDEM), for porous media simulation. The LBDEM approach couples the lattice-Boltzmann method for fluid mechanics and a discrete element method for solid mechanics. At the heart of this coupling is a previously developed boundary condition that has never been applied to coupled fluid-solid mechanics in porous media. Quantitative comparisons of model results to a one-dimensional analytical solution for fluid flow in a slightly deformable medium indicate a good match to the predicted continuum-scale fluid diffusion-like profile. Coupling of the numerical formulation is demonstrated through simulation of porous medium consolidation with the model capturing poroelastic behavior, such as the coupling between applied stress and fluid pressure rise. Finally, the LBDEM model is used to simulate the genesis and propagation of natural hydraulic fractures. The model provides insight into the relationship between fluid flow and propagation of fractures in strongly coupled systems. The LBDEM model captures the dominant dynamics of fluid-solid micromechanics of hydraulic fracturing and classes of problems that involve strongly coupled fluid-solid behavior.

Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method

Abstract: In this work we discuss the finite element model using the embedded discontinuity of the strain and displacement field, for dealing with a problem of localized failure in heterogeneous materials by using a structured finite element mesh. On the chosen 1D model problem we develop all the pertinent details of such a finite element approximation. We demonstrate the presented model capabilities for representing not only failure states typical of a slender structure, with crack-induced failure in an elastic structure, but also the failure state of a massive structure, with combined diffuse (process zone) and localized cracking. A robust operator split solution procedure is developed for the present model taking into account the subtle difference between the types of discontinuities, where the strain discontinuity iteration is handled within global loop for computing the nodal displacement, while the displacement discontinuity iteration is carried out within a local, element-wise computation, carried out in parallel with the Gauss-point computations of the plastic strains and hardening variables. The robust performance of the proposed solution procedure is illustrated by a couple of numerical examples. Concluding remarks are stated regarding the class of problems where embedded discontinuity finite element method (ED-FEM) can be used as a favorite choice with respect to extended FEM (X-FEM).

Particle Finite Element Methods in Solid Mechanics Problems

Abstract: The paper examines the possibilities of extending the Particle finite element methods (PFEM), which have been successfully applied in fluid mechanics, to solid mechanics problems. After a review of the fundamentals of the method, their specific features in solid mechanics are presented. A methodology to face contact problems, the anticipating contact interface mesh, is presented on the basis of a penalty-like constitutive models for imposing the contact and friction conditions. Finally, the PFEM is applied to same representative solid mechanics problems to display the capabilities of the method and some final conclusions are obtained.

Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity

Abstract: Thermoelastic interactions in an infinite homogeneous elastic medium with a spherical or cylindrical cavity are studied. The cavity surface is subjected to a ramp-type heating of its internal boundary which is assumed to be traction free. A general finite element model is proposed to analyze transient phenomena in thermoelastic solids. Lord–Shulman and Green–Lindsay for the generalized thermoelasticity model are selected for that purpose since it allows for “second sound” effects and reduces to the classical model by appropriate choice of the parameters. The problem has been solved numerically using a finite element method (FEM). Numerical results for the temperature distribution, displacement, radial stress and hoop stress are represented graphically. A comparison is made with the results predicted by the three theories.

A Global-Local Approach for the Construction of Enrichment Functions for the Generalized FEM and Its Application to Three-Dimensional Cracks

Abstract: Existing generalized or extended finite element methods for modeling cracks in three-dimensions require the use of a sufficiently refined mesh around the crack front. This offsets some of the advantages of these methods specially in the case of propagating three-dimensional cracks. In this paper, a strategy to overcome this limitation is investigated. The approach involves the development of enrichment functions that are computed using a new global-local approach. This strategy allows the use of a fixed global mesh around the crack front and is specially appealing for non-linear or time dependent problems since it avoids mapping of solutions between meshes. The resulting technique enjoys the same flexibility of the so-called meshfree methods for this class of problem while being more computationally efficient.

Adaptive Finite Element Methods for PDE-Constrained Optimal Control Problems

Abstract: We present a systematic approach to error control and mesh adaptation in the numerical solution of optimal control problems governed by partial differential equations. By the Lagrangian formalism the optimization problem is reformulated as a saddle-point boundary value problem which is discretized by a finite element Galerkin method. The accuracy of the discretization is controlled by residual-based a posteriori error estimates. The main features of this method are illustrated by examples from optimal control of heat transfer, fluid flow and parameter estimation. The contents of this article is as follows: Preliminary thoughts A general framework for a posteriori error estimation Solution process and mesh adaptation Examples of optimal control problems Conclusion and outlook References

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Abstract: Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.

X-SFEM, a computational technique based on X-FEM to deal with random shapes

Abstract: We propose a new method to deal with random geometries. It is an extension to the stochastic context of the eXtended Finite Element Method. This method lies on two majors points: the implicit description of geometry by the level set technique and the use of the partition of unity method for the enrichment of the finite element approximation space. This new technique leads by a direct calculus on a fixed finite element mesh to a solution which is explicit in terms of the basic random variables describing the geometry. We present here the basis of this approach and several examples to illustrate its performances.

A multiscale mortar mixed finite element method for slightly compressible flows in porous media

Abstract: We consider multiscale mortar mixed finite element discretizations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. Optimal fine scale convergence is obtained by an appropriate choice of mortar grid and polynomial degree of approximation. Parallel numerical simulations on some multiscale benchmark problems are given to show the efficiency and effectiveness of the method.

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

Abstract: This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream-Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path-dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.

Discrete Element Modeling

Abstract: The Distinct Element Method (also frequently referred to as the Discrete Element Method) (DEM) is a Lagrangian numerical technique where the computational domain consists of discrete solid elements which interact via compliant contacts. This can be contrasted with Finite Element Methods where the computational domain is assumed to represent a continuum (although many modern implementations of the FEM can accommodate some Distinct Element capabilities). Often the terms Discrete Element Method and Distinct Element Method are used interchangeably in the literature, although Cundall and Hart (1992) suggested that Discrete Element Methods should be a more inclusive term covering Distinct Element Methods, Displacement Discontinuity Analysis and Modal Methods. In this work, DEM specifically refers to the Distinct Element Method, where the discrete elements interact via compliant contacts, in contrast with Displacement Discontinuity Analysis where the contacts are rigid and all compliance is taken up by the adjacent intact material.

Numerical Methods in Micromagnetics (Finite Element Method)

Abstract: Micromagnetic simulations provide guidelines for the development of composite magnetic materials and magnetic devices. The granular structure of a material and the magnetic interactions between the different magnetic parts of a device can be easily taken into account by the finite element method and the boundary element method, respectively. The finite element equations are derived for the calculation of the magnetostatic field. Similarly, it is shown how the total energy of a magnetic system can be computed by matrix-vector operations. Efficient methods for calculating equilibrium magnetic states and the simulation of the dynamic response of a magnetic material to a time varying external field are discussed. Numerical schemes for the simulation of magnetization dynamics in systems involving moving parts are introduced. The numerical methods are demonstrated by showing results of perpendicular magnetic recording simulation on composite media. Keywords: micromagnetics; finite element method; magnetic recording