Least-squares Finite Element Formulation Research Articles

3D modeling of generalized Newtonian fluid flow with data assimilation using the least-squares finite element method

In this contribution we present a least-squares finite element formulation to model steady-state flow of incompressible, non-Newtonian fluids in three dimensions including data assimilation. The approach is based on the incompressible Navier–Stokes equations and the nonlinear viscosity is considered by means of the Carreau–Yasuda viscosity model, which can account for shear-thickening and shear-thinning behavior of generalized Newtonian fluids. We outline the procedure how to integrate given data into the numerical solution of flow problems without additional computational cost using the least-squares FEM. Assimilation of experimental data provides the opportunity to reduce model errors resulting in a solution which more closely approximates reality. Furthermore, the preprocessing of the available data using Kriging interpolation is also described briefly.The presented formulation is first validated by investigating the flow in a cube with an exact solution without data assimilation. Convergence is evaluated based on the error in velocities and pressure compared to the exact solution. Then the effect of data assimilation is shown by modeling blood flow through a carotid bifurcation model and integrating data either along lines or over entire cross-sectional areas. The improvement of the numerical solution by means of data assimilation is revealed by comparing the calculated velocity profiles with experimental and numerical reference values.

Implicit time discretization schemes for mixed least-squares finite element formulations

This work is an extension of the ideas in Averweg et al. (2019) with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow, based on the incompressible Navier–Stokes equations, and linear elastodynamic structural behavior. The variational approaches for fluid and solid mechanics are based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the constructed first-order systems of the governing differential equations is based on two-field stress–velocity (SV) functionals. For the time discretization of the SV-fluid formulation, four different types of implicit integration schemes are investigated, namely the Houbolt method, the Crank–Nicolson method and two explicit, singly diagonally implicit Runge–Kutta methods (ESDIRK). The SV-formulation for the solid is discretized applying the Houbolt method. The presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark. Since both (fluid and solid) SV formulations are discretized using conforming finite element spaces in H(div) and H1, respectively, the inherent fulfillment of coupling conditions, when modeling fluid–structure interaction problems, is given a priori. Therefore, the applicability is also examined by two simplified FSI problems for small deformations, in order to represent the main characteristics of the presented approach.

Efficient stress–velocity least-squares finite element formulations for the incompressible Navier–Stokes equations

In this contribution three different mixed least-squares finite element methods (LSFEMs) are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies. The deliberations are made for the incompressible Navier–Stokes equations. The generally known first-order div–grad system in terms of the (total) stress, velocity and pressure (SVP) formulation is the basis for two further div–grad least-squares formulations in terms of (total) stress and velocity (SV), whereby both formulations are derived from different fountainheads. The extended SV formulation is an enrichment of the known stress–velocity formulation proposed by Cai et al. (2004). The second SV formulation is based on the substitution of the pressure in the SVP formulation, thus it is based on a discontinuous pressure interpolation. Advantage of the SV formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom. Nevertheless, the approximation quality has to last the demand of the established least-squares formulations. The drawback of a poor mass conservation is well-investigated for the LSFEM and known to overcome by using high-order interpolations for all solution variables. Therefore, the main attention of this contribution is focused on accuracy, while aiming for high efficiency, which is intrinsic for the fundamental idea in here. In the light of efficient LSFEM solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h-type adaptive mesh refinement.

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.

Enhancing Least-Squares Finite Element Methods Through a Quantity-of-Interest

In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-of-interest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest. We establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions. Central to the approach is that it offers an element-level estimate of the error in the quantity-of-interest. As a result, we introduce an adaptive approach that yields efficient, adaptively refined approximations. Several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology. Notably, the results show that the new approach is effective at improving the accuracy per total computational cost.

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L 2 norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.

A first‐order system least‐squares finite element method for the Poisson‐Boltzmann equation

The Poisson-Boltzmann equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann equation. We expose the flux directly through a first-order system form of the equation. Using this formulation, we propose a system that yields a tractable least-squares finite element formulation and establish theory to support this approach. The least-squares finite element approximation naturally provides an a posteriori error estimator and we present numerical evidence in support of the method. The computational results highlight optimality in the case of adaptive mesh refinement for a variety of molecular configurations. In particular, we show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole, which highlights robustness of our approach.

A collocation penalty least-squares finite element formulation for incompressible flows

We present a penalty least-squares finite element formulation for incompressible flows. The pressure degree of freedom is eliminated using the classical penalty approach and the least-squares model is formed in terms of velocity, vorticity, and dilatation. An iterative penalization is implemented, which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix. The h-convergence is verified using the exact solution of Kovasznay’s flow. Numerical results are presented for a number of benchmark problems, e.g. steady flow past a large circular cylinder in a channel, transient flow over a backward facing step, unsteady flow past a circular cylinder, and buoyancy driven natural convection in a square cavity. For all numerical examples, the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters (10–100). These problems are also solved on coarse meshes to show that good mass conservation is achieved.

A stress‐based least‐squares finite‐element model for incompressible Navier–Stokes equations

AbstractIn this paper we present a stress‐based least‐squares finite‐element formulation for the solution of the Navier–Stokes equations governing flows of viscous incompressible fluids. Stress components are introduced as independent variables to make the system first order. Continuity equation becomes an algebraic equation and is eliminated from the system with suitable modifications. The h and p convergence are verified using the exact solution of Kovasznay flow. Steady flow past a large circular cylinder in a channel is solved to test mass conservation. Transient flow over a backward‐facing step problem is solved on several meshes. Results are compared with that obtained using vorticity‐based first‐order formulation for both benchmark problems. Copyright © 2007 John Wiley & Sons, Ltd.

Stabilized 3D finite elements for the numerical solution of the Navier–Stokes equations in semiconductors

In this article, we deal with the three-dimensional numerical simulation of semiconductor devices using the Viscous-Hydrodynamic (VHD) transport model. A reformulation of the VHD system using entropy variables allows to end up with a quasi-linear form that is symmetric and for which a stability result (in form of Clausius–Duhem inequality) is proved to hold. The numerical approximation of the VHD model is then performed using a Time-Discontinuous Galerkin Least-Squares finite element formulation including a discontinuity shock-capturing operator and based on a fully unstructured tetrahedral decomposition of the device domain. The approach combines in a unified framework the stability and optimality features of the standard Galerkin method with the ability of the scheme in effectively coping with the strong variations attained by the solution throughout the semiconductor device, as is demonstrated by numerical results in the simulation of several benchmark problems subject to quite different boundary conditions.

Elastic Wave Propagation in Laminated Composites Using the Space-Time Least-Squares Formulation in h,p,k Framework

In this paper, numerical simulation of one-dimensional elastic wave propagation in periodically laminated composites using space-time least-squares finite element processes in h, p, k mathematical framework in [1, 2, 3, 4] is presented. It is demonstrated that in space-time coupled processes, if one uses time marching procedure to simulate the evolution in which the solution is computed for a space-time strip for one increment of time, then time marched, the space-time local approximations of class C0 and degree p in time yield global differentiability of order p in time for the space-time strip. However, the inter-strip global differentiability in time remains limited by the global differentiability of the local approximations in time. Four different space-time least-squares finite element formulations are presented here and their merits and drawbacks, convergence rates, and their ability to incorporate single material physics as well as bi-material interface macro physics of wave propagation, transmission, reflection and interaction in laminated composites are discussed. Numerical results are presented to illustrate mathematical features of each formulation for wave propagation in a single medium as well as in domains containing bi-material interfaces. Numerical results are also computed for an impact experiment and compared with experimental data.

A least-squares finite element formulation for unsteady incompressible flows with improved velocity–pressure coupling

In the weak form Galerkin formulation for incompressible flows, the pressure has a well-understood role. At all times, it may be interpreted as a Lagrange multiplier that enforces the divergence-free constraint on the velocity field. This is not the case in least-squares formulations for incompressible flows, where the divergence-free constraint is enforced in a least-squares sense in a variational setting of residual minimization. Thus, the role of the pressure in a least-squares formulation is rather vague. We find that this lack of velocity–pressure coupling in least-squares formulations may induce spurious temporal pressure oscillations when using the non-stationary form of the equations. We present a least-squares formulation with improved velocity–pressure coupling, based on the use of a regularized divergence-free constraint. A first-order system least-squares (FOSLS) approach based on velocity, pressure and vorticity is used to allow the use of practical C 0 element expansions in the finite element model. We use high-order spectral element expansions in space and second- and third-order time stepping schemes. Excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results.

Spectral/ hp penalty least-squares finite element formulation for the steady incompressible Navier–Stokes equations

In this paper, we present spectral/ hp penalty least-squares finite element formulation for the numerical solution of steady incompressible Navier–Stokes equations. The continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method, which is equivalent to replacing the pressure variable p with p n = p n − 1 − γ(∇ · u), where u is the velocity field, γ is the penalty parameter and superscript ‘ n’ is non-linear iteration number. The modified Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity–dilatation or stresses as additional independent variables and the least-squares method is used to develop the finite element model. High-order basis functions are used to construct the discrete model. Unlike other penalty finite element formulations, equal order integration is used for all terms of the coefficient matrix. Schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver (banded Cholesky factorization) or iterative solver (preconditioned conjugate gradient). Present formulation produces very accurate results for even very low penalty parameters (10–40). In this way present formulation (and implementation) fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations. Proposed formulation is verified using the exact solution of Kovasznay flow. Numerical results are also presented for a number of two-dimensional benchmark problems, e.g., 2D lid-driven cavity flow, flow over a backward-facing step, steady flow past a circular cylinder, and 2D flow past a large circular cylinder in a channel. Finally, the proposed formulation is extended to coupled velocity–temperature problems, and buoyant flow inside a square enclosure is solved to test the penalty least-squares finite element model developed herein.

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.

Space–time coupled spectral/ hp least-squares finite element formulation for the incompressible Navier–Stokes equations

We consider least-squares finite element models for the numerical solution of the non-stationary Navier–Stokes equations governing viscous incompressible fluid flows. The paper presents a formulation where the effects of space and time are coupled, resulting in a true space–time least-squares minimization procedure, as opposed to a space–time decoupled formulation where a least-squares minimization procedure is performed in space at each time step. The formulation is first presented for the linear advection-diffusion equation and then extended to the Navier–Stokes equations. The formulation has no time step stability restrictions and is spectrally accurate in both space and time. To allow the use of practical C 0 element expansions in the resulting finite element model, the Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity as an additional independent variable and the least-squares method is used to develop the finite element model of the governing equations. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton's method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix-free form. Spectral convergence of the L 2 least-squares functional and L 2 error norms in space–time is verified using a smooth solution to the two-dimensional non-stationary incompressible Navier–Stokes equations. Numerical results are presented for impulsively started lid-driven cavity flow, oscillatory lid-driven cavity flow, transient flow over a backward-facing step, and flow around a circular cylinder; the results demonstrate the predictive capability and robustness of the proposed formulation. Even though the space–time coupled formulation is emphasized, we also present the formulation and numerical results for least-squares space–time decoupled finite element models. The numerical results show that the space–time coupled formulation has superior predictive capabilities for flows demanding high space–time resolution, exemplified here by the transient flow over a backward-facing step.

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

Spectral/ hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/ hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/ hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L 2 least-squares functional and L 2 error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.

THE k-VERSION OF FINITE ELEMENT METHOD FOR SELF-ADJOINT OPERATORS IN BVP

In this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems.

Least-squares finite element formulation for hydrodynamic modeling of semiconductor devices

A least-squares finite element formulation is presented for the hydrodynamic modeling of semiconductor devices. The formulation results in a symmetric and positive definite algebraic system. Its capability of adopting equal-order interpolations for all unknown variables makes it simple to formulate and easy to program. The semiconductor hydrodynamic equations coupled with the Poisson equation are formulated as one unified equation system in this least-squares finite element scheme. The developed method is examined on 1-D and 2-D deep-submicron semiconductor device structures to demonstrate its capability of handling the large gradients of variables and highly nonlinear source terms in the semiconductor hydrodynamic equations.