Taylor Hood Research Articles

Augmenting the grad-div stabilization for Taylor–Hood finite elements with a vorticity stabilization

Abstract The least squares vorticity stabilization (LSVS), proposed in N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzmán, A. Linke, and C. Merdon (“A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation,” SIAM J. Numer. Anal., vol. 59, no. 5, pp. 2746–2774, 2021) for the Scott–Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor–Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott–Vogelius and Taylor–Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott–Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

A P 2 H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes

Abstract The P2 BDM H-div finite element is enriched by five P 5 P_{5} divergence-free bubbles on each triangle. The resulting finite element remains H-div and is also H-curl nonconforming. Thus the new finite element combined with the discontinuous P 1 P_{1} element for pressure, is stabilizer-free and still divergence-free in solving the Stokes equations on triangular meshes. The stabilization can also be done by three, four, or six P 5 P_{5} divergence-free bubble-enrichment each triangle, shown both in theory and in computation. Numerical tests show the advantage of the divergence-free element over the P 2 P_{2} Taylor–Hood finite element.

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.

Fast solution of incompressible flow problems with two-level pressure approximation

This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor–Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor–Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier–Stokes equations, by using a two-stage pressure convection–diffusion strategy. The codes used to generate the numerical results are available online.

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.

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.

Convergence of the Incremental Projection Method Using Conforming Approximations

Abstract We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.

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 priori error estimates of two monolithic schemes for Biot's consolidation model

AbstractThis paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate 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.

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

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.

Analysis of Drag Coefficients around Objects Created Using Log-Aesthetic Curves

A fair curve with exceptional properties, called the log-aesthetic curves (LAC) has been extensively studied for aesthetic design implementations. However, its implementation in terms of functional design, particularly hydrodynamic design, remains mostly unexplored. This study examines the effect of the shape parameter α of LAC on the drag generated in an incompressible fluid flow, simulated using a semi-implicit backward difference formula coupled with P2−P1 Taylor–Hood finite elements. An algorithm was developed to create LAC hydrofoils that were used in this study. We analyzed the drag coefficients of 47 LAC hydrofoils of three sizes with various shapes in fluid flows with Reynolds numbers of 30, 40, and 100, respectively. We found that streamlined LAC shapes with negative α values, of which curvature with respect to turning angle are almost linear, produce the lowest drag in the incompressible flow simulations. It also found that the thickness of LAC objects can be varied to obtain similar drag coefficients for different Reynolds numbers. Via cluster analysis, it is found that the distribution of drag coefficients does not rely solely on the Reynolds number, but also on the thickness of the hydrofoil.

Computational homogenization of unsteady flows with obstacles

AbstractModeling flows in domains containing obstacles becomes challenging when the obstacles are very small compared to the domain. This problem is often known as modeling flow through permeable or porous media. Darcy's law is well known to provide a robust solution to this problem for viscous flows, given that the permeability of the medium can be characterized. Solutions for inertial flows have not been proposed yet. In this article, a principle multiscale virtual power is formulated for the computational homogenization of unsteady incompressible flows in domains containing small obstacles. In this theory, the coarse scale of the domain is separated from the fine scale of the obstacles. The finite element (FE) implementation of this theory is developed in the form of a parallel (FE) algorithm with two‐way coupling between the two scales. The incompressibility constraint is handled with special care by introducing an independent pressure variable at both scales and relying on the Taylor–Hood P2/P1 pair. Simulations are conducted to analyze the accuracy and efficiency of the proposed multiscale approach. The results show that the method is robust and interesting from both the points of view of computational cost and accuracy when the ratio between the domain size and the obstacle size is larger than 100.

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.

Simplified finite element approximations for coupled natural convection and radiation heat transfer

This article focuses on the effect of radiative heat on natural convection heat transfer in a square domain inclined with an angle. The left vertical wall of the enclosure is heated while maintaining the vertical right wall at room temperature with both adiabatic upper and lower horizontal walls. The governing equations are Navier–Stokes equations subjected to Boussinesq approximation to account the change in density. The natural convection–radiation equations are solved continuously to obtain the temperature, velocity and pressure. Taylor–Hood finite element approach has been adopted to solve the equations using triangular mesh. Effects of Rayleigh number, Planck constant and optical depth on the results are considered, presented and analyzed. Results show that the adiabatic walls, Planck constant as well as the inclined angle play an important role in the distribution of heat transfer inside the cavity.

Characteristics/finite element analysis for two incompressible fluid flows with surface tension using level set method

In this paper, we present and analyze a finite element level set method based on the method of characteristics for two phase flow. Surface tension effects are taken into account by the CSF approach. We first write the variational formulation of the problem and investigate its well-posedness. Next, for the discretization, a first order method of characteristics approach for the evolution of the level set function and for the material derivative of the velocity is used. The velocity and pressure unknowns are discretized by P2−P1 Taylor–Hood finite elements. Then, in each time step, the interface transport is decoupled from the Navier–Stokes equations. Well–posedness results for subproblems in this decoupled discrete problem are derived. Furthermore, under high regularity assumptions, we state error estimates for our scheme. Ultimately, three computational examples illustrate the performance of the proposed method.

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.

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.