Local Projection Stabilization Research Articles

Simulation of viscoelastic two-phase flows with insoluble surfactants

A stabilized finite element scheme is developed for computations of buoyancy driven 3D-axisymmetric viscoelastic two-phase flows with insoluble surfactants. The numerical scheme solves the Navier–Stokes equations for the fluid flow, Giesekus constitutive equation for the effects of viscoelasticity and simultaneously an evolution equation for the surfactant concentration on the interface. The interface is tracked by the coupled arbitrary Lagrangian–Eulerian (ALE) and Lagrangian approach. The interface-resolved moving meshes allow accurate incorporation of the interfacial tension force, Marangoni forces and the jumps in the material properties. Further, the tangential gradient operator technique is used to handle the curvature approximation in a semi-implicit manner. An one-level Local Projection Stabilization (LPS), which is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh is used to stabilize the model equations. The stabilized numerical scheme allows us to use isoparametric second order conforming finite elements enriched with cubic bubble functions for velocity and viscoelastic stress, second order finite elements for surfactant concentration and discontinuous first order finite element for pressure. A number of computations are performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column with insoluble surfactants on the interface. The influence of the Marangoni number, initial surfactant concentration and Peclet number on the dynamics of the rising drop are analyzed. The numerical study shows that a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape without insoluble surfactants. The presence of insoluble surfactants forces the drop to rise slowly but the drop at the tail end is pulled up more. However, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape without insoluble surfactants. The presence of surfactants pulls the tail end of the drop up slightly and makes the tail flatter with/without small undulations depending on the magnitude of the surfactant concentrations.

Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number

In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared using a high Reynolds number vortex dynamics simulation. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov–Galerkin (SUPG) method together with grad–div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulation of a high Reynolds number flow with vortical structures on relatively coarse grids are showcased, by focusing on a two-dimensional plane mixing-layer flow. Both Inf–Sup Stable (ISS) and Equal Order (EO) H1-conforming FE pairs are explored using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies conducted, it is concluded that the SUPG method using EO–FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend the SUPG method by the higher order terms of the RB-VMS method.

Equal-Order Stabilized Finite Element Approximation of the p-Stokes Equations on Anisotropic Cartesian Meshes

Abstract The p-Stokes equations with power-law exponent p ∈ ( 1 , 2 ) {p\in(1,2)} describes non-Newtonian, shear-thinning, incompressible flow. In many industrial applications and natural settings, shear-thinning flow takes place in very thin domains. To account for such anisotropic domains in simulations, we here study an equal-order bi-linear anisotropic finite element discretization of the p-Stokes equations, and extend a non-linear Local Projection Stabilization to anisotropic meshes. We prove an a priori estimate and illustrate the results with two numerical examples, one confirming the rate of convergence predicted by the a-priori analysis, and one showing the advantages of an anisotropic stabilization compared to an isotropic one.

An efficient time-splitting approximation of the Navier–Stokes equations with LPS modeling

In this work, the solution of the Navier–Stokes equations (NSE) is addressed by a Finite Element (FE) Local Projection Stabilization (LPS) method combined with an efficient time-splitting approximation strategy. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an alternative interpolation-stabilized structure. The main contribution is on numerically analyzing for the cited LPS method the proposed time approximation via stable velocity–pressure segregation, using semi-implicit Backward Differentiation Formulas (BDF). An overview about theoretical results from the numerical analysis of the proposed method is given, by highlighting the used mathematical tools. Numerical studies support the analytical results and illustrate the potential of the method for the efficient and accurate simulation of turbulent flows on relatively coarse grids. Smooth unsteady flows are simulated with optimal order of accuracy.

Computational modeling of impinging viscoelastic droplets

A numerical study on the impingement and spreading of an isothermal viscoelastic droplet on a solid surface is presented in this work. The time-dependent incompressible Navier–Stokes equations are used to describe the fluid flow in the liquid droplet, whereas the viscoelasticity in the moving droplet is described by the Giesekus constitutive equation. A finite element scheme with the arbitrary Lagrangian–Eulerian (ALE) approach is proposed to solve the coupled time-dependent incompressible Navier–Stokes equation and the Giesekus constitutive equation in a time-dependent domain. In addition, a three-field formulation based on the Local Projection Stabilization (LPS) is used in the numerical scheme. The stabilized scheme allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. The coupled system is solved by a monolithic approach in a 3D-axisymmetric configuration. In addition to the mesh convergence study, parametric studies of the Weissenberg number, Newtonian solvent ratio, polymeric viscosity, Reynolds number and the equilibrium contact angle are performed to demonstrate the effects of viscoelasticity on the flow dynamics of the droplet on wetting surfaces. The numerical study shows that the wetting diameter of the droplet increases with an increase in the viscoelasticity of the fluid. Further, the viscoelastic effects during the spreading process increases with an increase in the Reynolds number. Moreover, the viscoelastic effects on the flow dynamics are not influenced by the equilibrium contact angle.

Local projection stabilization for convection–diffusion–reaction equations on surfaces

The numerical solution of convection–diffusion–reaction equations in two and three dimensional domains Ω is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known on convection–diffusion–reaction equations on surfaces Γ=∂Ω. We propose a fitted finite element local projection stabilization (LPS) for convection–diffusion–reaction equations on surfaces based on a linear surface approximation and continuous, piecewise linear finite elements. Unique solvability of the continuous and discrete problem is established. A-priori error estimates show the O(h3∕2) convergence in mesh-dependent norms for smooth solutions in the convection-dominated case. The associated bilinear form satisfies an inf–sup condition in a norm stronger than the standard LPS-norm leading to improved stability properties. Numerical test examples demonstrate the properties and the potential of the proposed method.

Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization

Abstract This paper studies non inf-sup stable finite element approximations to the evolutionary Navier–Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty (0,T;L^2(\varOmega ))$ decays with rate $l+1/2$ in the case that $\nu \le h$, with $\nu$ being the dimensionless viscosity and $h$ being the mesh width. In the analysis of another method it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.

Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem

Abstract In finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix–Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche’s technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and 3 2 \frac{3}{2} in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings.

Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows

In this work, we study the performance of some local projection-based solvers in the Large Eddy Simulation (LES) of laminar and turbulent flows governed by the incompressible Navier–Stokes Equations (NSE). On one side, we focus on a high-order term-by-term stabilization Finite Element (FE) method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard Local Projection Stabilization (LPS) methods is replaced by an interpolation-stabilized structure. The interest of LPS methods is that they ensure a self-adapting high accuracy in laminar regions of turbulent flows, which turns to be of overall optimal high accuracy if the flow is fully laminar. On the other side, we propose a new Reduced Basis (RB) Variational Multi-Scale (VMS)-Smargorinsky turbulence model, based upon an empirical interpolation of the sub-grid eddy viscosity term. This method yields dramatical improvements of the computing time for benchmark flows. An overview about known results from the numerical analysis of the proposed methods is given, by highlighting the used mathematical tools. In the numerical study, we have considered two well known problems with applications in industry: the (3D) turbulent flow in a channel and the (2D/3D) recirculating flow in a lid-driven cavity.

Construction of L2-orthogonal elements of arbitrary order for Local Projection Stabilization

We construct L2-orthogonal conforming elements of arbitrary order for the Local Projection Stabilization (LPS). L2-orthogonal basis functions lead to a diagonal mass matrix which can be advantageous for time discretizations. We prove that the constructed family of finite elements satisfies a local inf-sup condition. Additionally, we investigate the size of the local inf-sup constant with respect to the polynomial degree. Our numerical tests show that the discrete solution is oscillation-free and of optimal accuracy in the regions away from the boundary or interior layers.

A local projection stabilization/continuous Galerkin–Petrov method for incompressible flow problems

A local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous-in-time case for both the velocity and pressure approximation. In addition, the fully discrete case in combination with higher order continuous Galerkin–Petrov (cGP) methods is studied. Error estimates of order k+1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.

Local projection stabilization for the Stokes equation with Neumann condition

The local projection stabilization (LPS) method is already an established method for stabilizing saddle-point problems and convection–diffusion problems. The a priori error analysis is usually done for homogeneous Dirichlet data. It turns out that without Dirichlet conditions the situation is more involved, because additional boundary terms appear in the analysis. The standard approach can be modified by using additional stabilization terms on the non-Dirichlet boundary parts. We show that such terms lead to a similar a priori estimate as the classical LPS method but in a stronger norm.

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.

Efficient and scalable discretization of the Navier–Stokes equations with LPS modeling

In this work, we address the solution of the Navier–Stokes equations (NSE) by a Finite Element (FE) Local Projection Stabilization (LPS) method. The focus is on a LPS method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure, which only acts on the high frequency components of the flow. As a main contribution, we propose and test an efficient discretization of the model via a stable velocity–pressure segregation, using semi-implicit Backward Differentiation Formulas (BDF) in time. On the one hand, numerical studies illustrate that the solver accurately reproduces first and second-order statistics of benchmark turbulent flows for relatively coarse meshes. On the other hand, they show that the solver works in an efficient (i.e., robust and fast) way, especially when interfaced with scalable domain decomposition methods. Such scalability results are obtained on up to 16,384 cores with a near-ideal speedup.

A three-field local projection stabilized formulation for computations of Oldroyd-B viscoelastic fluid flows

A new three-field formulation based on the Local Projection Stabilization (LPS) is developed for computations of the coupled Navier–Stokes and Oldroyd-B viscoelastic constitutive equations at high Weissenberg numbers. One-level LPS is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh. It allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. Further, the coupled system of equations are solved in a monolithic approach. Since the stabilization terms in LPS are assembled only once, the proposed scheme is computationally efficient in comparison with residual based stabilized numerical schemes. Numerical studies using method of manufactured solutions show an optimal order of convergence in the respective norms. Further, the proposed scheme is validated using two benchmark problems: flow past a cylinder in a rectangular channel and lid-driven cavity flow. Moreover, the numerical results are compared with the results in the literature and the effects of elasticity and inertia are studied.

A High-Order Local Projection Stabilization Method for Natural Convection Problems

In this paper, we propose a local projection stabilization (LPS) finite element method applied to numerically solve natural convection problems. This method replaces the projection-stabilized structure of standard LPS methods by an interpolation-stabilized structure, which only acts on the high frequencies components of the flow. This approach gives rise to a method which may be cast in the variational multi-scale framework, and constitutes a low-cost, accurate solver (of optimal error order) for incompressible flows, despite being only weakly consistent. Numerical simulations and results for the buoyancy-driven airflow in a square cavity with differentially heated side walls at high Rayleigh numbers (up to $$Ra=10^7$$ ) are given and compared with benchmark solutions. Good accuracy is obtained with relatively coarse grids.

Linearity-preserving monotone local projection stabilization schemes for continuous finite elements

This paper presents some new tools for enforcing discrete maximum principles and/or positivity preservation in continuous piecewise-linear finite element approximations to convection-dominated transport problems. Using a linear first-order advection equation as a model problem, we construct element-level bilinear forms associated with first-order artificial diffusion operators and their two-scale counterparts. The underlying design philosophy is similar to that behind local projection stabilization (LPS) techniques and variational multiscale (VMS) methods. The difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected. The proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation. The value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots. The result is a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The use of variational gradient recovery makes it possible to add high-order background dissipation leading to improved approximation properties in smooth regions. The numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time-dependent test problems in two space dimensions.

Convergence of local projection stabilisation finite element methods for convection–diffusion problems on layer-adapted meshes

We present a minor modification of of the stabilisation term in the local projection stabilisation (LPS) method for stabilising singularly perturbed problems on layer-adapted meshes, such that the error can be estimated uniformly and in optimal order in the associated LPS-norm, which is stronger than the standard energy norm. Numerical results confirm the theoretical findings.

Blending low-order stabilised finite element methods: A positivity-preserving local projection method for the convection–diffusion equation

In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.

Stabilization of low-order mixed finite elements for the plane elasticity equations

In this paper, we consider the mixed finite element method of the plane elasticity equations based on the Hellinger–Reissner variational principle. Low order mixed finite element spaces are used to approximate the stress and the displacement. Based on the local polynomial projection stabilization method, we present a stabilization scheme for the pairs to overcome the lack of the inf–supcondition. The stability is proved and the error estimate is derived. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method. These pairs are very convenient in the numerical implementation for the mixed form of the plane elasticity equations.