Bubble Functions Research Articles

Mixed finite element formulations with volume bubble functions for triangular elements

This paper deals with volume bubble functions for mixed finite triangular elements in geometrically linear elasticity. In two different versions these functions are used in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test are verified. In the numerical example, firstly the patch test is satisfied. Secondly, simulations of Cook’s membrane problem demonstrate that the proposed formulations avoid locking and reduce stress oscillations for incompressible materials.

A variational multiscale method for incompressible turbulent flows: Bubble functions and fine scale fields

This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re T = 395 and 590, and non-equilibrium turbulent flow around a cylinder at Re = 3,900.

A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements

AbstractWe construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

A stabilized mixed formulation for unsteady Brinkman equation based on the method of horizontal lines

AbstractIn this paper, we present a stabilized mixed formulation for unsteady Brinkman equation. The formulation is systematically derived based on the variational multiscale formalism and the method of horizontal lines. The derivation does not need the assumption that the fine‐scale variables do not depend on the time, which is the case with the conventional derivation of multiscale stabilized formulations for transient mixed problems. An expression for the stabilization parameter is obtained in terms of a bubble function, and appropriate bubble functions for various finite elements are also presented. Under the proposed formulation, equal‐order interpolation for the velocity and pressure (which is computationally the most convenient) is stable. Representative numerical results are presented to illustrate the performance of the proposed formulation. Spatial and temporal convergence studies are also performed, and the proposed formulation performed well. Copyright © 2011 John Wiley & Sons, Ltd.

A Robust Double Set Parameter Nonconforming Rectangular Element for Fourth Order Elliptic Singular Perturbation Problems

In this paper, with double set parameter method and bubble function technique, a robust C0 nonconforming rectangular element is proposed for solving fourth order singular perturbation problems. The properties of the new element is studied and the convergent result in the energy norm uniformly with respect to the perturbation parameter is presented. Numerical results are carried out to demonstrate the validity of the theoretical analysis.

A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.

Stabilized Mixed Tetrahedrals with Volume and Area Bubble Functions at Large Deformations

AbstractIn this contribution, stabilized mixed finite tetrahedral elements are presented in order to avoid volume locking and stress oscillations. Geometrically non‐linear elastic problems are addressed. The mixed method of incompatible modes is considered. As a key idea, volume and area bubble functions are used for the method of incompatible modes [1], thus giving it the interpretation of a mixed finite element method with stabilization terms. Concerning non‐linear problems these are non‐linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two‐field method, the method of incompatible modes in finite deformation problems is derived for a hyper elastic Neo‐Hookean material. In the representative examples Cook's membrane problem and a block under central pressure illustrate the good performance of the presented approaches compared to existing finite element formulations. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new robust C0-type nonconforming triangular element for singular perturbation problems

In this paper, a new robust C 0 triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique, and a general convergence theorem for C 0 nonconforming elements is presented. The convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter. Numerical experiments are also carried out to demonstrate the efficiency of the new element.

Stabilization procedures in coupled poromechanics problems: A critical assessment

AbstractNumerical solutions for problems in coupled poromechanics suffer from spurious pressure oscillations when small time increments are used. This has prompted many researchers to develop methods to overcome these oscillations. In this paper, we present an overview of the methods that in our view are most promising. In particular we investigate several stabilized procedures, namely the fluid pressure Laplacian stabilization (FPL), a stabilization that uses bubble functions to resolve the fine‐scale solution within elements, and a method derived by using finite increment calculus (FIC). On a simple one‐dimensional test problem, we investigate stability of the three methods and show that the approach using bubble functions does not remove oscillations for all time step sizes. On the other hand, the analysis reveals that FIC stabilizes the pressure for all time step sizes, and it leads to a definition of the stabilization parameter in the case of the FPL‐stabilization. Numerical tests in one and two dimensions on 4‐noded bilinear and linear triangular elements confirm the effectiveness of both the FPL‐ and the FIC‐stabilizations schemes for linear and nonlinear problems. Copyright © 2010 John Wiley & Sons, Ltd.

A variational multiscale method based on bubble functions for convection-dominated convection–diffusion equation

This work presents a variational multiscale method based on polynomial bubble functions as subgrid scale and a numerical implementation based on two local Gauss integrations. This method can be implemented easily and efficiently for the convection-dominated problem. Static condensation of the bubbles suggests the stability of the method and we establish its global convergence. Representative numerical tests are presented.

New finite element formulation based on bubble function interpolation for the transient compressible Euler equations

AbstractThe purpose of this paper is to present a stabilizing finite element method using a new bubble element for the unsteady fluid flow expressed by the compressible Euler equations to save efficient computational memory for computers. The formulation is based on the mixed interpolation for a transient term. The main idea of the present paper is to use the mixed interpolation for the transient term with bubble function as the weighting function and with linear function as the interpolation function. The nodal values at the bubble node can be eliminated even in the transient formulation. The stabilizing effect of the bubble function is effectively used. The formulation leads to the stabilizing effect, which is efficient to the unsteady fluid flow problems. The efficiency is verified based on the numerical studies, in which a rotating cone problem, a shock tube problem and a problem of flow around an airfoil are calculated. Copyright © 2008 John Wiley & Sons, Ltd.

A quadratic programming problem arising from the p-version for obstacle problems

Abstract We present a quadratic programming problem arising from the p -version for a finite element method with an obstacle condition prescribed in Gauss-Lobatto points. We show convergence of the approximate solution to the exact solution in the energy norm. We show an a-priori error estimate and derive an a-posteriori error estimate based on bubble functions which is used in an adaptive p -version. Numerical examples show the superiority of the p -version compared with the h -version.

Examination for adjoint boundary conditions in initial water elevation estimation problems

AbstractI present here a method of generating a distribution of initial water elevation by employing the adjoint equation and finite element methods. A shallow‐water equation is employed to simulate flow behavior. The adjoint equation method is utilized to obtain a distribution of initial water elevation for the observed water elevation. The finite element method, using the stabilized bubble function element, is used for spatial discretization, and the Crank–Nicolson method is used for temporal discretizations. In addition to a method for optimally assimilating water elevation, a method is presented for determining adjoint boundary conditions. An examination using the observation data including noise data is also carried out. Copyright © 2009 John Wiley & Sons, Ltd.

Two dimensional shape optimization using partial control and finite element method for compressible flows

The objective of this study is to determine the two dimensional shape of a body located in a compressible viscous flow, where the applied fluid force is minimized. The formulation to obtain the optimal shape is based on an optimal control theory. An optimal state is defined as a state, in which the performance function defined as the integration of the square sum of the applied fluid forces is minimized due to a reduction in the applied fluid forces. Compressible Navier–Stokes equations are treated as constraint equations. In other words, the body is considered to have a shape that minimizes the fluid forces under the constraint of the Navier–Stokes equations. The gradient of the performance function is computed using the adjoint variables. A weighted gradient method is used as the minimization algorithm. The volume of the body is assumed to be the same as that of the initial body. In the case of the algorithm used in this study, both the creation of a structured mesh around the surface of the body and the smoothing procedure are employed for the computation of gradient. In this study, a remeshing technique based on the structured mesh around the body changing its configuration in the iteration cycle is employed. For the correction to keep the volume constant, the surface coordinates are moved along the radial direction. For the discretization of both the state and adjoint equations, the efficient bubble function interpolation presented previously by the authors [18] is employed. The algorithm, which is known as the partial control algorithm, is applied to the numerical procedure to determine the movement of the coordinates. In the case of the gradient method, in order to avoid the convergence of the final shape to the local minimum shape, the new algorithm, which is called the partial control algorithm, is presented in this study. In numerical studies, the shape determination of a body in a uniform flow field is carried out in 2D domains. The initial shape of the body is assumed to be an elliptical cylinder. The shape is modified by minimizing the applied fluid forces. Finally, the desired shape of a body, whose performance function is reduced and converged to a constant value, is obtained. By carrying out a procedure that involves the use of the partial control algorithm, the desired shape of a body, whose performance function is reduced further, is obtained. Stable shape determination of a body in a compressible viscous flow is carried out by using the presented method. It is indicated that the optimal shape can be obtained by using the partial control algorithm.

A Unified Analysis of Several Mixed Methods for Elasticity with Weak Stress Symmetry

We give a unified error analysis of several mixed methods for linear elasticity which impose stress symmetry weakly. We consider methods where the rotations are approximated by discontinuous polynomials. The methods we consider are such that the approximate stress spaces contain standard mixed finite element spaces for the Laplace equation and also contain divergence free spaces that use bubble functions.

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier-Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.

Analysis of the cell boundary element methods for convection dominated convection–diffusion equations

We introduce two kinds of the cell boundary element (CBE) methods for convection dominated convection–diffusion equations: one is the CBE method with the exact bubble function and the other with inexact bubble functions. The main focus of this paper is on inexact bubble CBE methods. For inexact bubble CBE methods we introduce a family of numerical methods depending on two parameters, one for control of interior layers and the other for outflow boundary layers. Stability and convergence analysis are provided and numerical tests for inexact bubble CBEs with various choices of parameters are presented.

A Novel Least-Squares Finite Element Method Enriched with Residual-Free Bubbles for Solving Convection-Dominated Problems

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient $\mathbf{p}:=-\kappa\nabla u$ as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables $u$ and $\mathbf{p}$ is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This enriched LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving stability parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed enriched LSFEM. The accuracy and computational cost of this enriched LSFEM are also compared with those of the primitive LSFEM. We find that for a small diffusivity $\kappa$, the enriched LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the enriched LSFEM provides a significant improvement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems.

Using bubble functions in the multi-scale finite element modeling of the convection–diffusion–reaction equation

The Convection–diffusion–reaction (CDR) equation shows multi-scale behaviour in cases where it represents convection or reaction dominated transport processes. Bubble function enriched finite elements are used to generate stable and accurate solutions for this equation. To validate the approach, the numerical results obtained for a benchmark problem are compared with their corresponding analytical solution for both exponential and propagation regimes.

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C 0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.