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Element Space (element + space)

Distribution by Scientific Domains
Mathematics and Statistics76%
Engineering23%

Kinds of Element Space

  • finite element space
    		•

    Selected Abstracts

    A discontinuous Galerkin method for elliptic interface problems with application to electroporation

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2009
    Grégory Guyomarc'h
    Abstract We solve elliptic interface problems using a discontinuous Galerkin (DG) method, for which discontinuities in the solution and in its normal derivatives are prescribed on an interface inside the domain. Standard ways to solve interface problems with finite element methods consist in enforcing the prescribed discontinuity of the solution in the finite element space. Here, we show that the DG method provides a natural framework to enforce both discontinuities weakly in the DG formulation, provided the triangulation of the domain is fitted to the interface. The resulting discretization leads to a symmetric system that can be efficiently solved with standard algorithms. The method is shown to be optimally convergent in the L2 -norm. We apply our method to the numerical study of electroporation, a widely used medical technique with applications to gene therapy and cancer treatment. Mathematical models of electroporation involve elliptic problems with dynamic interface conditions. We discretize such problems into a sequence of elliptic interface problems that can be solved by our method. We obtain numerical results that agree with known exact solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Reducing dimensionality in topology optimization using adaptive design variable fields

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2010
    James K. Guest
    Abstract Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    c-Type method of unified CAMG and FEA.

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003
    2D non-linear, 3D linear, Part 1: Beam, arch mega-elements
    Abstract Computer-aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce ,mega-sized', arbitrary-shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis (FEA) and reduced engineering manpower requirement for final quality assurance. A method, denoted as c-type, has been proposed by constructively defining a finite element space whereby the above hurdles may be overcome with a minimal number of hyper-sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B-splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter-element continuity on demand unifies modelling and finite element method. The c-type method may introduce additional control, namely, an inter-element continuity condition to the existing h-type and p-type methods. Adaptation of the c-type method to existing commercial and general-purpose computer programs based on a conventional displacement-based finite element method is straightforward. The c-type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non-linear formulation for the two-dimensional curved beams/arches is presented. Several beam problems ranging from truly three-dimensional tortuous linear curved beams to geometrically extremely non-linear two-dimensional arches are solved to establish numerical efficiency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non-linearity and to treat general material non-linearity. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    On the quadrilateral Q2,P1 element for the Stokes problem

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2002
    Daniele Boffi
    Abstract The Q2 , P1 approximation is one of the most popular Stokes elements. Two possible choices are given for the definition of the pressure space: one can either use a global pressure approximation (that is on each quadrilateral the finite element space is spanned by 1 and by the global co-ordinates x and y) or a local approach (consisting in generating the local space by means of the constants and the local curvilinear co-ordinates on each quadrilateral , and ,). The former choice is known to provide optimal error estimates on general meshes. This has been shown, as it is standard, by proving a discrete inf,sup condition. In the present paper we check that the latter approach satisfies the inf,sup condition as well. However, recent results on quadrilateral finite elements bring to light a lack in the approximation properties for the space coming out from the local pressure approach. Numerical results actually show that the second choice (local or mapped pressure approximation) is suboptimally convergent. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Numerical analysis of a non-singular boundary integral method: Part II: The general case

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2002
    P. Dreyfuss
    In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non-singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single-layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any ,,, domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Efficient algorithms for multiscale modeling in porous media

    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010
    Mary F. Wheeler
    Abstract We describe multiscale mortar mixed finite element discretizations for second-order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux 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. We discuss the construction of multiscale mortar basis and extend this concept to nonlinear interface operators. We present a multiscale preconditioning strategy to minimize the computational cost associated with construction of the multiscale mortar basis. We also discuss the use of appropriate quadrature rules and approximation spaces to reduce the saddle point system to a cell-centered pressure scheme. In particular, we focus on multiscale mortar multipoint flux approximation method for general hexahedral grids and full tensor permeabilities. Numerical results are presented to verify the accuracy and efficiency of these approaches. Copyright © 2010 John Wiley & Sons, Ltd. [source]

    A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity

    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2009
    Yingxiong Xiao
    Abstract In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    A robust multilevel approach for minimizing H(div)-dominated functionals in an H1 -conforming finite element space

    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2004
    Travis M. Austin
    Abstract The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss,Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator I-grad div by a negative , ,. For ,near to one, this operator is elliptic, but as ,approaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part. Previous research on the non-elliptic part has revealed that discretizing I-grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high-frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any ,is bounded by the convergence for ,equal to zero. Then we prove that convergence of this multigrid algorithm for the case of , equal to zero is independent of mesh size. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Approximation capability of a bilinear immersed finite element space

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008
    Xiaoming He
    Abstract This article discusses a bilinear immersed finite element (IFE) space for solving second-order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source]

    An algebraic multigrid method for finite element discretizations with edge elements

    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002
    S. Reitzinger
    Abstract This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,,). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ,discrete' gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Unified finite element discretizations of coupled Darcy,Stokes flow

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009
    Trygve Karper
    Abstract In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor,Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions are straightforward. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

    Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2007
    Mary F. Wheeler
    Abstract We present a finite element formulation for coupled flow and geomechanics. We use mixed finite element spaces to approximate pressure and continuous Galerkin methods for displacements. In solving the coupled system, pressure and displacements can be solved either simultaneously in a fully coupled scheme or sequentially in a loosely coupled scheme. In this paper we formulate an iterative method where pressure and displacement solutions are staggered during a time step until a convergence tolerance is satisfied. A priori convergence results for the iterative coupling are also presented, along with a summary of the convergence results for the fully coupled scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 785,797, 2007 [source]

    A preconditioner for generalized saddle point problems: Application to 3D stationary Navier-Stokes equations

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
    C. Calgaro
    Abstract In this article we consider the stationary Navier-Stokes system discretized by finite element methods which do not satisfy the inf-sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen-type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices ("generalized saddle point problems"). We show that if the underlying finite element spaces satisfy a generalized inf-sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1-P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier-Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source]

    Discrete compactness property for quadrilateral finite element spaces,

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005
    Francesca Gardini
    Abstract The main purpose of the present article is to prove the discrete compactness property for Arnold-Boffi-Falk spaces of any order. Results of numerical experiments confirming the theory are also reported. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source]

    Finite element analysis and application for a nonlinear diffusion model in image denoising

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2002
    Jichun Li
    Abstract The stability analysis and error estimates are presented for a nonlinear diffusion model, which appears in image denoising and solved by a fully discrete time Galerkin method with kth (k , 1) order conforming finite element spaces. Numerical experiments are provided with denoising several grayscale noisy images by our Galerkin method on bilinear finite elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 649,662, 2002; DOI 10.1002/num.10017 [source]