Bubble Functions Research Articles

A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials

This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials.

Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem

Abstract An optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in [14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

A P4 bubble enriched P3 divergence-free finite element on triangular grids

On triangular grids, the continuous Pk plus discontinuous Pk−1 mixed finite element is stable for polynomial degree k≥4. When k=3, the inf–sup condition fails and the mixed finite element converges at an order that is two orders lower than the optimal order. We enrich the continuous P3 by adding some P4 divergence-free bubble functions, to be exact, one P4 divergence-free bubble function each component each edge. We show that such an enriched P3–P2 mixed element is inf–sup stable, and converges at the optimal order. Numerical tests are presented, comparing the new element with the P4–P3 element and the unstable P3–P2 element.

A postprocessed flux conserving finite element solution

We propose a local postprocessing method to get a new finite element solution whose flux is conservative element‐wise. First, we use the so‐called polynomial preserving recovery (postprocessing) technique to obtain a higher order flux which is continuous across the element boundary. Then, we use special bubble functions, which have a nonzero flux only on one face‐edge or face‐triangle of each element, to correct the finite element solution element by element, guided by the above super‐convergent flux and the element mass. The new finite element solution preserves mass element‐wise and retains the quasioptimality in approximation. The method produces a conservative flux, of high‐order accuracy, satisfying the constitutive law. Numerical tests in 2D and 3D are presented.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1859–1883, 2017

A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model

In an earlier work of us, a new mixed finite element scheme was developed for the Boussinesq model describing natural convection. Our methodology consisted of a fixed-point strategy for the variational problem that resulted after introducing a modified pseudostress tensor and the normal component of the temperature gradient as auxiliary unknowns in the corresponding Navier-Stokes and advection-diffusion equations defining the model, respectively, along with the incorporation of parameterized redundant Galerkin terms. The well-posedness of both the continuous and discrete settings, the convergence of the associated Galerkin scheme, as well as a priori error estimates of optimal order were stated there. In this work we complement the numerical analysis of our aforementioned augmented mixed-primal method by carrying out a corresponding a posteriori error estimation in two and three dimensions. Standard arguments relying on duality techniques, and suitable Helmholtz decompositions are used to derive a global error indicator and to show its reliability. A globally efficiency property with respect to the natural norm is further proved via usual localization techniques of bubble functions. Finally, an adaptive algorithm based on a reliable, fully local and computable a posteriori error estimator induced by the aforementioned one is proposed, and its performance and effectiveness are illustrated through a few numerical examples in two dimensions.

Optimization of numerical orbitals using the Helmholtz kernel.

We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. The bubbles' part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H2, H2O, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10-4 to 10-7 Eh can be obtained with our approach.

Strain smoothing for compressible and nearly-incompressible finite elasticity

We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size.

Residual-baseda posteriorierror estimation for the Maxwell’s eigenvalue problem

We present an a posteriori estimator of the error in the L 2 -norm for the numerical approximation of the Maxwell's eigenvalue problem by means of N ed elec nite elements (see, for instance, (4)). As in (2), our analysis is based on a Helmholtz decomposition of the error, where, in particular, the L 2 -orthogonality property is used to derive a superconvergence result for the eigenfunction approximation. The analysis also makes use of a priori error estimates and the additional regularity of the eigenfunctions (see (3)). Inspired by (1), we prove a key result about superconvergence between the L 2 -orthogonal projection of the exact eigenfunction onto the curl of the N ed elec space and the eigenfunction approximation. Reliability of the a posteriori error estimator is proved up to higher order terms. Finally, the eciency of the error indicators is shown by using a standard bubble functions technique.

A posteriori error estimation for the Stokes–Darcy coupled problem on anisotropic discretization

This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in or 3. Korn's inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non‐conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non‐conforming case. With the obtained finite element solutions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. The lower error bound is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so‐called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media. Copyright © 2016 John Wiley & Sons, Ltd.

A posteriori error analysis of a fully-mixed formulation for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier–Stokes equations (with nonlinear viscosity) and the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite element subspaces employed are piecewise constants, Raviart–Thomas elements of lowest order, continuous piecewise linear elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the fluid, respectively, whereas Raviart–Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are considered in the porous medium. The proof of reliability of the estimator relies on a global inf–sup condition, suitable Helmholtz decompositions in the fluid and the porous medium, the local approximation properties of the Clément and Raviart–Thomas operators, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

Stabilized Mixed Finite Element Methods for Linear Elasticity on Simplicial Grids in ℝn

Abstract In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use 𝑯 ⁢ ( div , Ω ; 𝕊 ) ${\boldsymbol{H}(\operatorname{div},\Omega;\mathbb{S})}$ - P k ${P_{k}}$ and 𝑳 2 ⁢ ( Ω ; ℝ n ) ${\boldsymbol{L}^{2}(\Omega;\mathbb{R}^{n})}$ - P k - 1 ${P_{k-1}}$ to approximate the stress and displacement spaces, respectively, for 1 ≤ k ≤ n ${1\leq k\leq n}$ , and employ a stabilization technique in terms of the jump of the discrete displacement over the edges/faces of the triangulation under consideration; in the second class of elements, we use 𝑯 0 1 ⁢ ( Ω ; ℝ n ) ${\boldsymbol{H}_{0}^{1}(\Omega;\mathbb{R}^{n})}$ - P k ${P_{k}}$ to approximate the displacement space for 1 ≤ k ≤ n ${1\leq k\leq n}$ , and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini [19]. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis are two special interpolation operators, which can be constructed using a crucial 𝑯 ⁢ ( div ) ${\boldsymbol{H}(\operatorname{div})}$ bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.

A polytree-based adaptive approach to limit analysis of cracked structures

We in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures. The key idea is to develop a general refinement algorithm based on a so-called polytree mesh structure. The method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones, which are considered as special cases. Also, polytree meshes are conforming and can be regarded as a generalization of quadtree meshes. For the aim of this paper, we restrict our main interest in plane-strain limit analysis to von Mises-type materials, yet its extension to a wide class of other solid mechanics problems and materials is completely possible. To avoid volumetric locking, we propose an approximate velocity field enriched with bubble functions using Wachspress coordinates on a primal-mesh and design carefully strain rates on a dual-mesh level. An adaptive mesh refinement process is guided by an L2-norm-based indicator of strain rates. Through numerical validations, we show that the present method reaches high accuracy with low computational cost. This allows us to perform large-scale limit analysis problems favorably.

A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method

This article presents a general framework to estimate the pointwise error of linear partial differential equations. The error estimator is based on the variational multiscale theory, in which the error is decomposed in two components according to the nature of the residuals: element interior residuals and inter-element jumps. The relationship between the residuals (coarse scales) and the error components (fine scales) is established, yielding to a very simple model. In particular, the pointwise error is modeled as a linear combination of bubble functions and Green’s functions. If residual-free bubbles and the classical Green’s function are employed, the technology leads to an exact explicit method for the pointwise error. If bubble functions and free-space Green’s functions are employed, then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary. As a consequence, this gives a model for the so-called fine-scale Green’s functions. The numerical error is studied for the standard Galerkin and SUPG methods with application to the heat equation, the reaction–diffusion equation and the convection–diffusion equation. Numerical results show that stabilized methods minimize the propagation of pollution errors, which stay mostly locally.

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

Full-discrete adaptive FEM for quasi-parabolic integro-differential PDE-constrained optimal control problem

In this paper, we develop a full-discrete adaptive FEM for a quasi-parabolic integro-differential PDE-constrained optimal control problem. Firstly, we present weak formulations and optimal conditions and, by using the backward Euler scheme, we deduce equivalent a posteriori error estimators. Then we derive lower and upper bounds for the error estimates by bubble function techniques and use these indicators in adaptive finite element schemes. Furthermore, we carry out some numerical experiments to test our theoretical results.

Construction of high‐order complete scaled boundary shape functions over arbitrary polygons with bubble functions

SummaryThis manuscript presents the development of novel high‐order complete shape functions over star‐convex polygons based on the scaled boundary finite element method. The boundary of a polygon is discretised using one‐dimensional high order shape functions. Within the domain, the shape functions are analytically formulated from the equilibrium conditions of a polygon. These standard scaled boundary shape functions are augmented by introducing additional bubble functions, which renders them high‐order complete up to the order of the line elements on the polygon boundary. The bubble functions are also semi‐analytical and preserve the displacement compatibility between adjacent polygons. They are derived from the scaled boundary formulation by incorporating body force modes. Higher‐order interpolations can be conveniently formulated by simultaneously increasing the order of the shape functions on the polygon boundary and the order of the body force mode. The resulting stiffness‐matrices and mass‐matrices are integrated numerically along the boundary using standard integration rules and analytically along the radial coordinate within the domain. The bubble functions improve the convergence rate of the scaled boundary finite element method in modal analyses and for problems with non‐zero body forces. Numerical examples demonstrate the accuracy and convergence of the developed approach. Copyright © 2016 John Wiley & Sons, Ltd.

An explicit-extended penalty Galerkin method for solving an incompressible two-phase flow

In this paper, we present a stabilized explicit-extended penalty Galerkin method based on the implicit pressure and explicit saturation method to find the global solution for the two-phase flow in porous media at each time step. The bubble functions are employed as basis of the spatial dimensions for the extended penalty Galerkin method. The forward Euler method is applied to the temporal discretization. Since the accuracy of numerical simulations flow through porous media depends on the modeling of the injection and production well, we propose a new well model for the presented method. The details of the stability analysis for the proposed method are provided and suitable values of the penalty term and time steps are calculated. The efficiency of the method is illustrated by simulations of a waterflood in a heterogeneous oil reservoir. Comparisons are made with available literature which show the efficiency and accuracy of the proposed method.

Multiscale stabilization for convection-dominated diffusion in heterogeneous media

We develop a Petrov–Galerkin stabilization method for multiscale convection–diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not be sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems and thus, develop a systematic approach for finding reduced-order approximations of the solution. We start from a Petrov–Galerkin framework using optimal weighting functions. We introduce an auxiliary variable to a mixed formulation of the problem. The auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a dimensionally reduced space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution. To find the test space, we reformulate some recent mixed Generalized Multiscale Finite Element Methods. We introduce snapshots and local spectral problems that appropriately define local weight and trial spaces. In particular, we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable. The resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system. We discuss the stability and its relation to the approximation property of the test space. We design online basis functions, which accelerate convergence in the test space, and consequently, improve stability. We present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the Peclet number.

Two-level finite element variational multiscale method based on bubble functions for the steady incompressible MHD flow

ABSTRACTIn this paper, a finite element variational multiscale method (VMM) is developed for the stationary incompressible magnetohydrodynamics (MHD) problem and the corresponding stability and convergence are established. Our VMM is based on the polynomial bubble functions as a subgrid scale and the numerical implementation is based on the local Gauss integrations. Compared to the common VMM, our method does not introduce any extra variable. Furthermore, the two-level finite element VMM is also developed. Compared to the one-level VMM, the two-level VMM consists of solving a nonlinear MHD problem on a coarse mesh with mesh size H, and then a linearized magnetohydrodynamic problem based on the Oseen iteration on a fine mesh is solved with mesh size . Stability and convergence of numerical approximations in two-level VMM are presented. Finally, some numerical examples are provided to demonstrate the effectiveness of the developed numerical algorithms.

An adaptive-gridding lattice Boltzmann method with linked-list data structure for two-dimensional viscous flows

An adaptive mesh refinement technique for lattice Boltzmann method (LBM) is proposed in this paper. It combines hierarchical linked-list data structure and the LBM calculation. Based on uniform meshes, the adaptive algorithm refines the meshes by constructing the linked-lists of nodes, cells and levels for mesh levels refined. To guarantee the stability of numerical scheme, quadratic bubble function for the nodal momentum is used to interpolate in the LBM calculation of different mesh levels. For the flows of relatively higher Re, large Eddy simulation (LES) is adopted to solve turbulence problems. Because of the use of adaptive technique, the computational time can be cut and accurate flow field information can be captured. OpenMP parallel for linked-list data structure is used to improve computational efficiency. To verify the present method, flows over circular cylinder at Re = 40, 300, 500, 1,000 and 3,900 and NACA0012 airfoil at Re = 105 for AOA = 4° are simulated.