scholarly journals A New Finite Element Method for Darcy-Stokes-Brinkman Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Bishnu P. Lamichhane
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Uniform Convergence ◽  
Finite Element Approximation ◽  
Orthogonal Basis ◽  
Discrete Space ◽  
Element Approximation ◽  
Brinkman Equations ◽  
Primal And Dual ◽  
Element Method

We present a new finite element method for Darcy-Stokes-Brinkman equations using primal and dual meshes for the velocity and the pressure, respectively. Using an orthogonal basis for the discrete space for the pressure, we use an efficiently computable stabilization to obtain a uniform convergence of the finite element approximation for both limiting cases.

Analysis of Multi-Crack Growth in Asphalt Pavement Based on Extended Finite Element Method

2012 ◽  
Vol 588-589 ◽  
pp. 1926-1929
Author(s):  
Yu Zhou Sima ◽  
Fu Zhou Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Crack Growth ◽  
Extended Finite Element Method ◽  
Asphalt Pavement ◽  
Finite Element Approximation ◽  
Propagation Path ◽  
Element Approximation ◽  
Extended Finite Element ◽  
Element Method

An extended finite element method (XFEM) for multiple crack growth in asphalt pavement is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite element with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Finally, the propagation path of the cracks in asphalt pavement under different load conditions is presented.

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

2019 ◽  
Vol 53 (5) ◽  
pp. 1645-1665
Author(s):  
Guanglian Li ◽  
Yifeng Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cathodic Protection ◽  
Finite Element Approximation ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Energy Error ◽  
Element Method

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

Robust Approximation of Singularly Perturbed Delay Differential Equations by the hp Finite Element Method

2013 ◽  
Vol 13 (1) ◽  
pp. 21-37 ◽  
Author(s):  
Serge Nicaise ◽  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Interface Layer ◽  
Singularly Perturbed ◽  
Element Approximation ◽  
Smooth Part ◽  
Delay Differential ◽  
Element Method

Abstract. We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

Construction of the Beam Element Based Second Generation Wavelet

2011 ◽  
Vol 80-81 ◽  
pp. 532-535
Author(s):  
Yu Min He ◽  
Xi Chen ◽  
Xiao Long Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Second Generation ◽  
Finite Element Approximation ◽  
Beam Element ◽  
Adaptive Finite Element Method ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Second Generation Wavelet ◽  
Element Method

Second generation wavelet (SGW) provides diversity and agility for constructing wavelet besides the multiresolution property. By introducing SGW into finite element method, a series sequence of finite element approximation spaces which are nested and hierarchically expanded can be constructed. The method has high calculation speed and precision and is suited for constructing adaptive algorithm. In this paper, the beam element based SEW is constructed, which set up a basis for the adaptive finite element method based on multiresolution analysis.

Crack Extension Calculation under Tensile and Shear Load by XFEM

2014 ◽  
Vol 580-583 ◽  
pp. 3046-3050
Author(s):  
Miao Yu ◽  
Zhi Hong Dai ◽  
Gui Juan Hu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Special Functions ◽  
Partition Of Unity ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Extended Finite Element ◽  
Propagation Behavior ◽  
Shear Loads ◽  
Element Method

The extended finite element method (XFEM) is a numerical method for modeling discontinuity such as cracks, holes, inclusions etc within a standard finite element framework. In the XFEM, special functions which can reflect the problem’s solution characteristics are added to the finite element approximation using the framework of partition of unity. Compared with the standard finite element method, it obtained more accuracy without remeshing. In this paper, we studied crack propagation behavior under different proportions of tension and shear loads by XFEM.

3D magnetotelluric modeling using the T‐Ω finite‐element method

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 108-119 ◽  
Author(s):  
Yuji Mitsuhata ◽  
Toshihiro Uchida
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Scalar Potential ◽  
Finite Element Approximation ◽  
Conservation Equation ◽  
Gauge Condition ◽  
Magnetic Flux Density ◽  
Element Approximation ◽  
Unknown Variable ◽  
Element Method

We present a finite‐element algorithm for computing MT responses for 3D conductivity structures. The governing differential equations in the finite‐element method are derived from the T–Ω Helmholtz decomposition of the magnetic field H in Maxwell's equations, in which T is the electric vector potential and Ω is the magnetic scalar potential. The Coulomb gauge condition on T necessary to obtain a unique solution for T is incorporated into the magnetic flux density conservation equation. This decomposition has two important benefits. First, the only unknown variable in the air is the scalar value of Ω. Second, the curl–curl equation describing T is only defined in the earth. By comparison, the system of curl–curl equations for H and the electric field E are singular in the air, where the conductivity σ is zero. Although the use of a small but nonzero value of σ in the air and application of a divergence correction are usually necessary in the E or H formulation, the T–Ω method avoids this necessity. In the finite‐element approximation, T and Ω are represented by the edge‐element and nodal‐element interpolation functions within each brick element, respectively. The validity of this modeling approach is investigated and confirmed by comparing modeling results with those of other numerical techniques for two 3D models.

scholarly journals Finite element approximation of the Isaacs equation

2019 ◽  
Vol 53 (2) ◽  
pp. 351-374
Author(s):  
Abner J. Salgado ◽  
Wujun Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mesh Size ◽  
Finite Element Approximation ◽  
Rates Of Convergence ◽  
Discrete Maximum Principle ◽  
Element Approximation ◽  
Differential Approximation ◽  
Isaacs Equation ◽  
Element Method

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε, h → 0, and ε ≳ (h|log h|)1/2. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.

scholarly journals ABOUT SOLUTION OF MULTIPOINT BOUNDARY PROBLEMS OF TWO-DIMENSIONAL STRUCTURAL ANALYSIS WITH THE USE OF COMBINED APPLICATION OF FINITE ELEMENT METHOD AND DISCRETE-CONTINUAL FINITE ELEMENT METHOD PART 2: SPECIAL ASPECTS OF FINITE ELEMENT APPROXIMATION

2017 ◽  
Vol 13 (4) ◽  
pp. 14-36
Author(s):  
Pavel A. Akimov ◽  
Alexander M. Belostotsky ◽  
Taymuraz B. Kaytukov ◽  
Oleg A. Negrozov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Problem ◽  
Finite Element Approximation ◽  
Element Model ◽  
Discrete Models ◽  
Element Approximation ◽  
Multipoint Boundary ◽  
Joint Application ◽  
Element Method

As is well known, the formulation of a multipoint boundary problem involves three main components: a description of the domain occupied by the structure and the corresponding subdomains; description of the conditions inside the domain and inside the corresponding subdomains, the description of the conditions on the boundary of the domain, conditions on the boundaries between subdomains. This paper is a continuation of another work published earlier, in which the formulation and general principles of the approximation of the multipoint boundary problem of a static analysis of deep beam on the basis of the joint application of the finite element method and the discrete-continual finite element method were considered. It should be noted that the approximation within the fragments of a domain that have regular physical-geometric parameters along one of the directions is expedient to be carried out on the basis of the discrete-continual finite element method (DCFEM), and for the approximation of all other fragments it is necessary to use the standard finite element method (FEM). In the present publication, the formulas for the computing of displacements partial derivatives of displacements, strains and stresses within the finite element model (both within the finite element and the corresponding nodal values (with the use of averaging)) are presented. Boundary conditions between subdomains (respectively, discrete models and discrete-continual models and typical conditions such as “hinged support”, “free edge”, “perfect contact” (twelve basic (basic) variants are available)) are under consideration as well. Governing formulas for computing of elements of the corresponding matrices of coefficients and vectors of the right-hand sides are given for each variant. All formulas are fully adapted for algorithmic implementation.

scholarly journals NUMERICAL SOLUTION OF THE PROBLEM FOR POISSON’S EQUATION WITH THE USE OF DAUBECHIES WAVELET DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (4) ◽  
pp. 123-133
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov ◽  
Mojtaba Aslami
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Finite Element Approximation ◽  
Poisson’S Equation ◽  
Element Approximation ◽  
Daubechies Wavelet ◽  
Numerical Implementation ◽  
Poisson's Equation ◽  
Element Method

Numerical solution of the problem for Poisson’s equation with the use of Daubechies wavelet discrete continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The operational initial continual and discrete-continual formulations of the problem are given, several aspects of finite element approximation are considered. Some information about the numerical implementation and an example of analysis are presented.

scholarly journals Second-order time discretization with finite-element method for partial integro-differential equations with a weakly singular kernel

1999 ◽  
Vol 40 (4) ◽  
pp. 513-524
Author(s):  
Chang Ho Kim ◽  
U Jin Choi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Time Discretization ◽  
Second Order ◽  
Singular Kernel ◽  
Element Approximation ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Element Method

AbstractWe propose the second-order time discretization scheme with the finite-element approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the finite element method and the time discretization is based on the Crank-Nicolson scheme with a graded mesh. We show the stability of the scheme and obtain the second-order convergence result for the fully discretized scheme.

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