Classical Galerkin Finite Element Method Research Articles

A SUPS formulation for simulating unsteady natural/mixed heat convection phenomena in square cavities under intense magnetic forces

This paper presents stabilized finite element simulations of unsteady natural and mixed heat convection phenomena occurring in square enclosures. It is assumed that the enclosures are subject to intense uniform magnetic fields applied perpendicular to the vertical walls. It is well known that the classical Galerkin finite element method (GFEM) is prone to yielding nonphysical oscillations when simulating convection-dominated flows. Therefore, in order to handle such flows occurring at high Hartmann (0≤Ha≤100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le {\\rm Ha} \\le 100$$\\end{document}) and Rayleigh (103≤Ra≤108\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{3} \\le {\\rm Ra} \\le 10^{8}$$\\end{document}) numbers, the stabilization of the GFEM is accomplished with the streamline-upwind/Petrov–Galerkin and pressure-stabilizing/Petrov–Galerkin formulations. Temporal discretizations are performed with the backward Euler formula. At each time step, the nonlinear systems are linearized with the Newton–Raphson (N–R) method, and the linearized systems are solved directly using the sparse lower–upper decomposition. The incompressible flow solvers are developed in-house, run in parallel within the FEniCS ecosystem, and tested on a comprehensive set of numerical experiments with various thermal boundary conditions. Numerical simulations and comparisons reveal that the proposed formulation performs quite well at high Hartmann and Rayleigh numbers without exhibiting any significant localized or globally spread numerical instabilities. Moreover, it achieves this by using uniform meshes and only linear elements, making the formulation much more computationally efficient. Mesh convergence analyses are also performed to demonstrate the reliability of the numerical results obtained.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

Computational analysis of acoustic metamaterials: An outsider’s perspective

Numerical simulation has become an increasingly important tool in the study and design of acoustic metamaterials. However, most of the work to date seems limited to the classical Galerkin finite element method as implemented in commercial off-the-shelf software. In this talk, the limitations of this conventional approach are outlined and alternative computational techniques that can and should be brought to bear for the advanced analysis of acoustic metamaterials are discussed.

A high order conservative flux optimization finite element method for steady convection-diffusion equations

This article presents a high order conservative flux optimization (CFO) finite element method that preserve the important mass conservation property locally for the convection-diffusion equations. The numerical scheme is an extension of the first order CFO scheme proposed in [25], it is based on the classical Galerkin finite element method enhanced by a flux approximation on the boundary of a prescribed set of arbitrary control volumes (either the finite element partition itself or dual voronoi mesh, etc). The numerical approximations can be characterized as the solution of a constrained-minimization problem with constraints given by the flux conservation equations on each control volume. The discrete linear system is a typical saddle-point problem, but with less number of degrees of freedom than the Raviart-Thomas (for k>1) and the Brezzi-Douglas-Marini (for k>3) mixed finite element methods, where k denotes the degree of polynomial approximation spaces. Moreover, the numerical solution of the proposed scheme is of super-closeness with the finite element solution. Error estimates of optimal order are established for the numerical flux as well as the primary variable approximations. We present several numerical studies in order to verify convergence of the CFO schemes. Test cases under the presence of boundary layers, internal discontinuities, and a simplified two-phase flow in highly heterogeneous porous media model problem will also be presented. The numerical results show obvious advantages of applying high order CFO schemes.

A parallel monolithic approach for fluid-structure interaction in a cerebral aneurysm

Abstract A parallel fully-coupled numerical algorithm has been developed for the fluid-structure interaction problem in a cerebral artery with aneurysm. For the fluid part of the problem, an Arbitrary Lagrangian-Eulerian formulation based on the side-centered unstructured finite volume method is employed for the governing incompressible Navier–Stokes equations. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant-Kirchhoff material and the classical Galerkin finite element method is used to discretise the governing equations in a Lagrangian frame. The time integration method for the structure domain is based on the Newmark type generalized − α method while the second-order backward difference (BDF2) is used in the fluid domain. A special attention is given to construct an algorithm obeying the local/global discrete geometric conservation laws (DGCL) in order to conserve fluid volume at machine precision when the fluid domain is entirely enclosed by solid domain boundary. Therefore, a compatible kinematic boundary condition is applied at the interface between the solid and fluid domains. The parallel implementation of the present fully coupled unstructured fluid-structure solver is based on the PETSc library and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned sub-domains is utilized for the resulting fully coupled system. The proposed algorithm is initially validated for a pressure pulse propagating in a flexible tube and the mass conservation accuracy is tested for a thin elastic sphere filled with an incompressible fluid in a circular tube. Then the numerical method is applied to a complicated problem involving unsteady pulsatile blood flow in a cerebral artery with aneurysm as a realistic fluid-structure interaction problem encountered in biomechanics. Various hemodynamic quantities of interest like fluid velocities, blood pressure and wall shear stresses (WSS) are computed as well as the time dependent artery wall deformations.

Augmenting numerical stability of the Galerkin finite element formulation for electromagnetic flowmeter analysis

Due to the complexities in handling liquid metals, theoretical evaluation of the sensitivity of magnetic flowmeters forms an attractive and preferred choice. The classical Galerkin finite element formulation is generally opted for the required evaluation. However, it is known to lead to numerical oscillations at higher flow rates. To overcome this, modified methods like upwind/Petrov–Galerkin schemes are generally suggested in allied areas like fluid dynamics. However, it requires the evaluation of stabilisation parameter and this parameter is not readily available for elements of order beyond quadratic. After a careful analysis of the numerical instability through a reduced one-dimensional problem, an elegant and stable approach is devised. In this scheme, the input magnetic field is restated in terms of the associated vector potential and the classical Galerkin finite element method is employed without any modification. The analytical solution of the associated difference equation is employed to show: (i) the stability of the proposed approach at higher flow rates and (ii) quantification of the small oscillations remnant at intermediate flow rates. It is then applied to the original flowmeter problem and the stability of the numerical solution is clearly demonstrated.

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

SummaryA novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.

The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber

Numerical dispersion, or what is often referred to as the pollution effect, presents a challenge to an efficient finite element discretization of the Helmholtz equation in the medium frequency regime. To alleviate this effect and improve the unsatisfactory pre-asymptotic convergence of the classical Galerkin finite element method based on piecewise polynomial basis functions, several discretization methods based on plane wave bases have been proposed. Among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical Galerkin finite element method for a number of constant wavenumber Helmholtz problems and has also outperformed two representative methods that use plane waves – the partition of unity and the ultra-weak variation formulation methods. In this paper, the discontinuous enrichment method is extended to the variable wavenumber Helmholtz equation. To this effect, the concept of enrichment functions based on free-space solutions of the homogeneous form of the governing differential equation is enlarged to include free-space solutions of approximations of this equation obtained in this case by successive Taylor series expansions of the wavenumber around a reference point. This leads to plane wave enrichment functions based on the piece-wise constant approximation of the wavenumber, and to Airy wave enrichment functions. Several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound-hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space. All these elements are shown to outperform by a substantial margin their continuous polynomial counterparts.

GFEM AND LSFEM IN THE SOLUTION OF THE TRANSIENT BIDIMENSIONAL CONVECTION-DIFFUSION EQUATION

Convection dominated flows result in a hyperbolic system of equations which leads to ill-conditioned matrices and oscillatory approximations when using the classical Galerkin Finite Element Method (GFEM). In this paper, the Least Square Finite Method (LSFEM) is introduced in the study of transient bidimensional convection-diffusion problems. The differentiated equation of second order which describes the convective-diffusive phenomenon is transformed into an equivalent system of partial differentiated equations of first order which is discretized by the formulation of the LSFEM resulting in a defined algebraic, symmetrical and positive system. The performance of the method is verified by the solution of a test- problem.

Reduced basis method for linear elasticity problems with many parameters

The reduced basis (RB) methods are proposed here for the solution of parametrized equations in linear elasticity problems. The fundamental idea underlying RB methods is to decouple the generation and projection stages (offline/online computational procedures) of the approximation process in order to solve parametrized equations in a rapid, inexpensive and reliable way. The method allows important computational savings with respect to the classical Galerkin-finite element method, ill suited to a repetitive environment like the parametrized contexts of optimization, many queries and sensitivity analysis. We consider different parametrization for the systems: either physical quantities – to model the materials and loads – and geometrical parameters – to model different geometrical configurations. Then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads.

Modelling of diffuse failure mechanisms of catastrophic landslides

This paper presents a theoretical and numerical framework to model the initiation mechanisms of catastrophic landslides. Attention is paid to the mechanism of failure referred to as diffuse , characteristic of soils presenting very loose or metastable structures, as in the case of the liquefaction of a sand layer induced by an earthquake, where the effective stress approaches zero as the pore pressure increases. The equations describing the coupling between the solid skeleton and the pore fluids are presented following an Eulerian approach based on mixture theory that can provide a unified formulation for both initiation and propagation phases. The system of partial differential equations is then discretized using the classical Galerkin finite element method and neglecting the convective terms. A simple improvement of the Generalized Plasticity model is introduced to deal with bonded geomaterials and to represent the mechanical collapse of such materials. The use of an implicit integration scheme guarantees the efficiency of the numerical convergence.

Modelling of diffuse failure mechanisms of catastrophic landslides

This paper presents a theoretical and numerical framework to model the initiation mechanisms of catastrophic landslides. Attention is paid to the mechanism of failure referred to as diffuse, characteristic of soils presenting very loose or metastable structures, as in the case of the liquefaction of a sand layer induced by an earthquake, where the effective stress approaches zero as the pore pressure increases. The equations describing the coupling between the solid skeleton and the pore fluids are presented following an Eulerian approach based on mixture theory that can provide a unified formulation for both initiation and propagation phases. The system of partial differential equations is then discretized using the classical Galerkin finite element method and neglecting the convective terms. A simple improvement of the Generalized Plasticity model is introduced to deal with bonded geomaterials and to represent the mechanical collapse of such materials. The use of an implicit integration scheme guarantees the efficiency of the numerical convergence.

Two-dimensional viscous flow computations on the Connecti on Machine: Unstructured meshes, upwind schemes and massively parallel computations

Here we report on our effort in simulating two-dimensional viscous flows on the Connection Machine, using a second-order accurate monotomic upwind scheme for conservation laws (MUSCL) on fully unstructured grids. The spatial approximation combines an upwind finite volume method for the discretization of the convective fluxes with a classical Galerkin finite element method for the discretization of the diffusive fluxes. The resulting semi-discrete equations are time integrated with a second-order low-storage explicit Runge-Kutta method. A communication efficient strategy for mapping thousands of processors onto an arbitrary mesh is presented and proposed as an alternative to the fast north-east-west-south (NEWS) communication mechanism, which is restricted to structured grids. Measured performance results for the simulation of low Reynolds number chaotic flows indicate that an 8K CM-2 (8192 processors) with single precision floating point arithmetic is at least as fast as one CRAY-2 processor.