Taylor-Hood Elements Research Articles

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

An efficient and easy-to-implement recovery-based a posteriori error estimator for isogeometric analysis of the Stokes equation

In this article, we present an efficient recovery-based a posteriori error estimator for adaptive isogeometric analysis (IGA) of the Stokes equation, which is straightforward to implement. The increased regularity of splines in IGA versus Lagrange polynomials used in the classical finite element method plays a significant role for the success of the presented recovery technique. We consider LR B-splines (Johannessen et al., 2014) for the adaptive mesh refinement, and the isogeometric Taylor–Hood element, Sub-Grid element (Buffa et al., 2011), and Raviart–Thomas element (divergence-conforming) (Buffa et al., 2010; Evans and Hughes, 2013; Johannessen et al., 2015) for the mixed discretization of the Stokes equation. Three benchmark problems with analytical solutions are tested, and they demonstrate clearly the great performance of the recovery-based error estimator compared to classical residual-based error estimator.

New error analysis of charge-conservative finite element methods for stationary inductionless MHD equations

In this paper, we present a new error analysis of a class of charge-conservative finite element methods for stationary inductionless magnetohydrodynamics (MHD) equations. The methods use the standard inf-sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas face element for solving the current density. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimate in negative norms, we establish new and optimal error estimates. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L2-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor-Hood element supplies the optimal accuracy for the velocity in H1-norm and the pressure in L2-norm. Furthermore, we propose a simple recovery technique to obtain a new numerical current density of one order higher accuracy by re-solving a mixed Poisson equation. Numerical results are provided to verify the theoretical analysis.

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

<abstract><p>Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.</p></abstract>

Derivation and simulation of a two-phase fluid deformable surface model

To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

Mean zero artificial diffusion for stable finite element approximation of convection in cellular aggregate formation

We develop and implement finite element approximations for the coupled problem of cellular aggregate formation. The equation governing evolution of cell density is convective in nature, necessitating a modification of standard approaches to circumvent the instabilities associated with standard finite element approximations. To this end, a novel mean zero artificial diffusion approach is proposed, in which the classical artificial diffusion term is replaced by one that is orthogonal to its projection on to continuous functions. The resulting approach for the convective equation is shown to be well-posed. A range of numerical results illustrate the stability and accuracy of the new approach, and its behaviour in comparison with an alternative approach using Taylor–Hood elements. The results also provide insights into the behaviour of cellular aggregates in the context of the model studied here.

New error analysis and recovery technique of a class of fully discrete finite element methods for the dynamical inductionless MHD equations

In this paper, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic (MHD) equations. The methods use the semi-implicit backward Euler scheme in time and use the standard inf–sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L∞(0,T;L2(Ω))-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor–Hood element supplies the optimal accuracy for the velocity in L2(0,T;H1(Ω))-norm and the pressure in L1(0,T;L2(Ω))-norm. Moreover, we propose a simple recovery technique to obtain a new numerical current density of first-order higher accuracy in the spatial direction by re-solving a mixed Poisson equation. Numerical experiments are performed to verify the theoretical analysis.

A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist several different Hamiltonians introduced for quasi-incompressible elastodynamics based on different multifield variational formulations, there is not much study on the fully incompressible materials in the literature. The adopted mixed formulation can be viewed as a finite-strain generalization of Herrmann variational formulation, and it naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient and scaled mid-point formulas, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. Our analysis and numerical evidence also reveal that the scaled mid-point formula is non-robust numerically. The generalized Taylor–Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano’s formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.

A numerical approach for fluid deformable surfaces with conserved enclosed volume

We consider surface finite elements and a semi-implicit time stepping scheme to simulate fluid deformable surfaces. Such surfaces are modeled by incompressible surface Navier-Stokes equations with bending forces. Here, we consider closed surfaces and enforce conservation of the enclosed volume. The numerical approach builds on higher order surface parameterizations, a Taylor-Hood element for the surface Navier-Stokes part, appropriate approximations of the geometric quantities of the surface, mesh redistribution and a Lagrange multiplier for the constraint. The considered computational examples highlight the solid-fluid duality of fluid deformable surfaces and demonstrate convergence properties that are known to be optimal for different sub-problems.

Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch IgA

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor–Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.

Influence of alternating magnetic field on non-newtonian blood perfusion and transport of nanoparticles in tissues with embedded blood vessel during hyperthermia

In this paper, we consider the impact of an alternating magnetic field on axisymmetric non-Newtonian nanofluid flow and nanoparticle (magnetite or iron-platinum) transport through a blood vessel and surrounding tissue during hyperthermia therapy. The geometrical domain of the problem is divided into four regions; the blood vessel region, cancerous and non-cancerous muscle or prostate regions, and a fat region. Each of the tissue regions is considered to be a saturated porous material that admits fluid flow, which is given by the Darcy-Brinkman-Forchheimer model. The Quemada model is used to describe the non-Newtonian nature of blood in each region. The resulting unsteady coupled governing equations and corresponding boundary and initial conditions are numerically solved by using the finite element method with Taylor-Hood elements. A physically realistic range of parameter values are used when numerically solving the governing equations. Using the obtained numerical solution, the effects of magnetic field amplitude and oscillation frequency, inlet nanoparticle solid volume fraction and nanoparticle diameter on nanofluid velocity, temperature, pressure and nanoparticle distribution are examined. The results determined that more heat generation is achieved with a larger volume fraction of injected nanoparticles and higher magnetic field amplitude. Also, heat generation is maximized by using nanoparticles with a diameter near 15nm and a magnetic field oscillation frequency near 4×104s-1. Thus, the present study provides an improved understanding of the factors influencing the effectiveness of intravenous magnetic hyperthermia cancer therapy in muscle and prostate tissues. The heat generated by the considered nanoparticles under an alternating magnetic field can be more accurately predicted and controlled using the two-phase non-Newtonian model of fluid flow and convective heat and mass transfer that is investigated herein.

A simple nonconforming tetrahedral element for the Stokes equations

In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous P2–P1 Taylor–Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn’s inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.

Non-isothermal two-phase flow in deformable porous media: systematic open-source implementation and verification procedure

We propose a formulation for non-isothermal two-component two-phase flow through deformable porous media. The approach covers phase transitions among both phases, i.e. liquid phase components evaporate into the gas phase while gas phase components dissolve or condensate into the liquid phase. These phase transitions always take place in thermodynamic equilibrium. The set of model equations is thereby largely independent of the specific constitutive relations. Starting from general equilibrium equations, we show the evolution of the system of weak formulations of all governing equations, which are then discretised with Taylor-Hood elements in a standard finite element approach. The model equations and the construction of the constitutive equilibria are implemented in the open-source simulator OpenGeoSys, which can be freely used and modified. To verify the implementation, we have selected a number of complementary test cases covering a wide range of process couplings. The numerical model is compared with analytical and semi-analytical solutions of these problems as well as with experimental results. It is shown in the paper that by including thermodynamic effects, solid mechanics, and phase transition processes, the proposed numerical model covers many characteristic features of unsaturated geomaterials and can be employed for the description of a broad range of problems encountered in geotechnical engineering.Article highlightsAn open-source FEM tool for non-isothermal two-phase flow in deformable porous or fractured media is presented in detail.The model features phase transitions across both fluid phases based on simple equilibrium conditions.A variety of benchmark tests is presented and compared to other software results and to exact solutions.

On the choice of finite element for applications in geodynamics

Abstract. Geodynamical simulations over the past decades have widely been built on quadrilateral and hexahedral finite elements. For the discretization of the key Stokes equation describing slow, viscous flow, most codes use either the unstable Q1×P0 element, a stabilized version of the equal-order Q1×Q1 element, or more recently the stable Taylor–Hood element with continuous (Q2×Q1) or discontinuous (Q2×P-1) pressure. However, it is not clear which of these choices is actually the best at accurately simulating “typical” geodynamic situations. Herein, we provide a systematic comparison of all of these elements for the first time. We use a series of benchmarks that illuminate different aspects of the features we consider typical of mantle convection and geodynamical simulations. We will show in particular that the stabilized Q1×Q1 element has great difficulty producing accurate solutions for buoyancy-driven flows – the dominant forcing for mantle convection flow – and that the Q1×P0 element is too unstable and inaccurate in practice. As a consequence, we believe that the Q2×Q1 and Q2×P-1 elements provide the most robust and reliable choice for geodynamical simulations, despite the greater complexity in their implementation and the substantially higher computational cost when solving linear systems.

Solutions of Poisson-Nernst Planck Equations with Ion Interaction

This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.

Fortin operator for the Taylor\u2013Hood element

We design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the P_2–P_0 and the augmented Taylor–Hood element in 3D.

Topology optimization applied to the design of Tesla-type turbine devices

Tesla-type turbine devices are turbine devices which convert fluid motion into the rotating motion of a rotor, but without the need of any blades. This is possible due to the interaction of the boundary layer with the rotating fluid motion. In order to successfully optimize this type of fluid flow device through the topology optimization method, some relevant aspects need to be considered. The first one is due to its unique configuration, meaning that the 2D swirl flow model may be considered, which is much less computationally expensive than considering a “full” 3D model. Moreover, since higher mesh resolutions may be necessary in the design of Tesla-type turbines from to the possible appearence of smaller disk-like structures, this may increase the overall optimization cost if the traditional Taylor-Hood elements (quadratic finite elements for the velocity) are considered. This additional computational cost may be reduced by considering MINI elements (linear finite elements with bubble enrichment for the velocity) instead. Another modification that may be useful in its design is augmenting the traditional Brinkman model used in topology optimization with an additional inertial term (Brinkman-Forchheimer model), which may lead to better optimized Tesla turbine designs. Another factor to be considered is the multi-objective function, which may be defined to minimize the relative energy dissipation and maximize the power transferred from the fluid. In such case, the power objective function may be augmented with an additional porosity (material model) term. Numerical examples are presented, taking into account some aspects of the design of Tesla-type turbine devices.

A continuum and computational framework for viscoelastodynamics: I. Finite deformation linear models

This work concerns the continuum basis and numerical formulation for deformable materials with viscous dissipative mechanisms. We derive a viscohyperelastic modeling framework based on fundamental thermomechanical principles. Since most large deformation problems exhibit isochoric properties, our modeling work is constructed based on the Gibbs free energy in order to develop a continuum theory using pressure-primitive variables, which is known to be well-behaved in the incompressible limit. A set of general evolution equations for the internal state variables is derived. With that, we focus on a family of free energies that leads to the so-called finite deformation linear model. Our derivation elucidates the origin of the evolution equations of that model, which was originally proposed heuristically and thus lacked formal compatibility with the underlying thermodynamics. In our derivation, the thermodynamic inconsistency is clarified and rectified. A classical model based on the identical polymer chain assumption is revisited and is found to have non-vanishing viscous stresses in the equilibrium limit, which is counter-intuitive in the physical sense. Because of that, we then discuss the relaxation property of the non-equilibrium stress in the thermodynamic equilibrium limit and its implication on the form of free energy. A modified version of the identical polymer chain model is then proposed, with a special case being the model proposed by G. Holzapfel and J. Simo. Based on the consistent modeling framework, a provably energy stable numerical scheme is constructed for incompressible viscohyperelasticity using inf–sup stable elements. In particular, we adopt a suite of smooth generalization of the Taylor–Hood element based on Non-Uniform Rational B-Splines (NURBS) for spatial discretization. The temporal discretization is performed via the generalized-α scheme. We present a suite of numerical results to corroborate the proposed numerical properties, including the nonlinear stability, robustness under large deformation, and the stress accuracy resolved by the higher-order elements. Additionally, the pathological behavior of the original identical polymer chain model is numerically identified with an unbounded energy decaying. This again underlines the importance of demanding vanishing non-equilibrium stress in the equilibrium limit.