Least-squares Finite Element Method Research Articles

On a least‐squares finite element formulation for sea ice dynamics

AbstractIn this contribution a mixed least‐squares finite element method (LSFEM) for the modeling of sea ice motion including a viscous‐plastic (VP) sea ice rheology is investigated. The simulation of sea ice motion goes back to the findings in Hibler III [4], where a numerical model for the simulation of sea ice circulation and thickness evolution over a seasonal cycle is introduced. Both the ice thickness and ice concentration distribution are explicitly described on the basis of evolution equations. Recent research in this field is devoted to finite element formulations based on the Galerkin variational approach. Here, special focus lies on the stabilization of the numerically complex scheme. It is therefore desirable to establish a least‐squares formulation to overcome possible numerical drawbacks. The least‐squares variational approach is well established in finite element formulations in the branch of fluid mechanics, see [2], [3] and [6], for instance. A great advantage of the method is its applicability to first‐order systems, such that it results in stable and robust formulations also for not self‐adjoint operators like in the Navier‐Stokes equations, for instance. A box test case, see [7], is investigated for the least‐squares formulation.

Numerical Simulations of Viscoelastic Fluid Flows Past a Transverse Slot Using Least-Squares Finite Element Methods

This paper presents a least-squares (LS) finite element method for linear Phan-Thien–Tanner (PTT) viscoelastic fluid flows. We consider stabilized weights in the LS method for the viscoelastic model and prove that the LS approximation converges to the linearized solutions of the linear PTT model; the convergence is at the optimal rate for the velocity in the $$H^{1}$$ -norm and at suboptimal rates for the stress and pressure in the $$L^{2}$$ -norm, respectively. For numerical experiments we first consider the flow through a planar channel to illustrate our theoretical results. The LS method is then applied to a flow through the slot channel with two depth ratios and the effects of physical parameters are discussed. Numerical solutions of the channel problem indicate that flow characteristics of the viscoelastic polymer solution are described by the results obtained using the method. Furthermore, we present the hole pressure for various Weissenberg numbers, and compare with that derived from the Higashitani–Pritchard (HP) theory.

Simulation of Fluid-Thermal Field in Oil-Immersed Transformer Winding Based on Dimensionless Least-Squares and Upwind Finite Element Method

In order to study the coupling fluid and thermal problems of the local winding in oil-immersed power transformers, the least-squares finite element method (LSFEM) and upwind finite element method (UFEM) are adopted, respectively, to calculate the fluid and thermal field in the oil duct. When solving the coupling problem by sequential iterations, the effect of temperature on the material property and the loss density of the windings should be taken into account. In order to improve the computation efficiency for the coupling fields, an algorithm, which adopts two techniques, the dimensionless LSFEM and the combination of Jacobi preconditioned conjugate gradient method (JPCGM) and the two-side equilibration method (TSEM), is proposed in this paper. To validate the efficiency of the proposed algorithm, a local winding model of a transformer is built and the fluid field is computed by the conventional LSFEM, dimensionless LSFEM, and the Fluent software. While the fluid and thermal computation results of the local winding model of a transformer obtained by the two LSFEMs are basically consistent with those of the Fluent software, the stiffness matrix, which is formed by the dimensionless scheme of LSFEM and preconditioned by the JPCGM and TSEM, has a smaller condition number and a faster convergence rate of the equations. Thus, it demonstrates a broader applicability.

Application of local least squares finite element method (LLSFEM) in the interface capturing of two-phase flow systems

A least squares finite element method (LSFEM) is applied locally for the prediction of the interface advection in a two-phase flow system. Moreover, a phantom element is introduced, which determines the nodal values of the volume fractions at the onset of the flow, and provides a highly satisfactory reconstructed interface at the same time. The quality of the interface reconstruction is also examined after advection and the interface tangent, as an important parameter in the surface tension calculations, is estimated by the derivatives of the volume fraction on the Gaussian points in the vicinity of the interface intersections. The validity of the method is examined by comparing the results of the drop deformation for various Reynolds numbers with available data in the literature. Finally, some explanatory examples are provided to evaluate our suggested procedure.

A discontinuous least-squares finite-element method for second-order elliptic equations

ABSTRACTIn this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes.

A weak Galerkin least-squares finite element method for div–curl systems

In this paper, we introduce a weak Galerkin least-squares method for solving div–curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method.

Comparison between results of solution of Burgers\u2019 equation and Laplace\u2019s equation by Galerkin and least-square finite element methods

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers’ equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace’s equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace’s equation in an earth dam is considered. By solution of Laplace’s equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers’ equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers’ equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace’s equation, Galerkin and least square methods can show water table correctly in earth dam.

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.

DATA ASSIMILATION FOR NAVIER-STOKES USING THE LEAST-SQUARES FINITE-ELEMENT METHOD

We investigate theoretically and numerically the use of the Least-Squares Finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stress-velocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces $H({\rm div}) \times H^1 \times L^2$ respectively. Resolution of the system is via minimization of a least-squares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data-assimilation is to add a data-discrepancy term to the functional. Whereas most data-assimilation techniques require a large number of evaluations of the forward-simulations and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that - in the linear case - the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes - in particular with respect to mass-conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary-conditions.

Asymptotic Exactness of the Least-Squares Finite Element Residual

The discrete minimal least-squares functional $LS(f;U)$ is equivalent to the squared error $|| u-U ||^2$ in least-squares finite element methods and so leads to an embedded reliable and efficient a...

Adaptive weights for mass conservation in a least-squares finite element method

ABSTRACTIn this paper, we develop a least-squares (LS) finite element method for the Stokes equations. The LS method uses the -norm of the residuals of the continuity equation multiplied by appropriately adjusted weights. To adjust the weight, we employ an adaptive weight iteration approach based on a residual type a posteriori error estimator for the LS functional. This LS method is applied to flows through a planar channel, lid-driven cavity flows, and flows past a transverse slot. The results demonstrate that mass conservation of the LS method in fluid simulations can be significantly improved by appropriately adjusting weights, and that this can be accomplished using low-order basis functions, without substantial complications in coding. We provide a posteriori error estimates for the linearized velocity–vorticity–pressure first-order system and show numerical results supporting the estimate. Numerical results reveal that the mass conservation constant is problem dependent.

Discrete least-squares finite element methods

A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h−1) for a variety of formulations of Poisson’s equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson’s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson’s equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here is an outgrowth of the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan (2015). The strength of DPG is highlighted throughout, however, the majority of theory presented is more general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

Measuring thermal conductivity of soils based on least squares finite element method

Soil thermal conductivity is an important parameter in the thermal simulations of geo-environmental structures. It is, however, difficult to measure the thermal conductivity as it is affected by several parameters, especially the soil water content. The objective of this paper was to present a measurement methodology based on least squares finite element method (LSFEM). The LSFEM algorithm was developed to directly solve the heat conduction equation, in which the measured temperatures served as input data and the thermal parameters were the unknown variables. The appeal of the LSFEM algorithm lied in its high efficiency and accuracy because no iterations were needed. To obtain the temperature data used for LSFEM solution, we designed a laboratory experiment to produce measurable temperature gradients. The spatial and temporal differences in the temperature measurements were identified to ensure the uniqueness and nonsingularity of the LSFEM solution. The corresponding water content can be simultaneously obtained from the methodology. The measured results suggest that the proposed methodology is promising for measuring the soil thermal conductivity.

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

Abstract We consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

Modeling of high pressure arc-discharge with a fully-implicit Navier–Stokes stabilized finite element flow solver

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier–Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman–Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart–Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov–Galerkin stabilized finite element formulation is employed to solve the Navier–Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

Formula omitted]-adaptive least-squares finite element methods for the 2D Stokes equations of any order with optimal convergence rates

The adaptive least-squares finite element method (LS-FEM) for the Stokes equations has recently been based on alternative error estimators in Bringmann and Carstensen (2016) for the lowest-order case. Since the first-order div LS-FEM measures the flux errors in H(div), the data resolution error measures the L2 norm of the right-hand side f minus the piecewise polynomial approximation Πkf without a mesh-size factor. This enforces separate marking with an overall abstract theory (Carstensen and Rabus, 2016). This paper contributes (a) a discussion of the scaling of the LS-FEM, (b) optimal rates of the adaptive h-version of any order k, and (c) comparing numerical results in 2D with all details on inhomogeneous Dirichlet data covered by the analysis.

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

In this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. The numerical examples cover a wide range of applied problems, including singularly...

Comparison of different least‐squares mixed finite element formulations for hyperelasticity

AbstractThe main goal of this contribution is the solution of geometrically nonlinear problems using the mixed least‐squares finite element method (LSFEM). An investigation of a hyperelastic material law based on logarithmic deformation measures is performed. The basis for the proposed LSFEM is a div‐grad first‐order system consisting of the equilibrium condition and the constitutive equation, see e.g. Cai and Starke [1]. For the interpolation of the solution variables vector‐valued Raviart‐Thomas functions for the approximation of the stresses and standard Lagrange polynomials for the displacements are used. In order to show the performance of the presented formulations a numerical example is investigated, where we compare the different interpolation combinations used. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

SU-G-TeP1-15: Toward a Novel GPU Accelerated Deterministic Solution to the Linear Boltzmann Transport Equation

Purpose: To develop a Graphic Processor Unit (GPU) accelerated deterministic solution to the Linear Boltzmann Transport Equation (LBTE) for accurate dose calculations in radiotherapy (RT). A deterministic solution yields the potential for major speed improvements due to the sparse matrix-vector and vector-vector multiplications and would thus be of benefit to RT. Methods: In order to leverage the massively parallel architecture of GPUs, the first order LBTE was reformulated as a second order self-adjoint equation using the Least Squares Finite Element Method (LSFEM). This produces a symmetric positive-definite matrix which is efficiently solved using a parallelized conjugate gradient (CG) solver. The LSFEM formalism is applied in space, discrete ordinates is applied in angle, and the Multigroup method is applied in energy. The final linear system of equations produced is tightly coupled in space and angle. Our code written in CUDA-C was benchmarked on an Nvidia GeForce TITAN-X GPU against an Intel i7-6700K CPU. A spatial mesh of 30,950 tetrahedral elements was used with an S4 angular approximation. Results: To avoid repeating a full computationally intensive finite element matrix assembly at each Multigroup energy, a novel mapping algorithm was developed which minimized the operations required at each energy. Additionally, a parallelized memory mapping for the kronecker product between the sparse spatial and angular matrices, including Dirichlet boundary conditions, was created. Atomicity is preserved by graph-coloring overlapping nodes into separate kernel launches. The one-time mapping calculations for matrix assembly, kronecker product, and boundary condition application took 452±1ms on GPU. Matrix assembly for 16 energy groups took 556±3s on CPU, and 358±2ms on GPU using the mappings developed. The CG solver took 93±1s on CPU, and 468±2ms on GPU. Conclusion: Three computationally intensive subroutines in deterministically solving the LBTE have been formulated on GPU, resulting in two orders of magnitude speedup. Funding support from Natural Sciences and Engineering Research Council and Alberta Innovates Health Solutions. Dr. Fallone is a co-founder and CEO of MagnetTx Oncology Solutions (under discussions to license Alberta bi-planar linac MR for commercialization).

An adaptive least-squares FEM for the Stokes equations with optimal convergence rates

This paper introduces the first adaptive least-squares finite element method (LS-FEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowest-order Raviart-Thomas and conforming $$P_1$$P1 discrete spaces for the divergence least-squares formulation in 2D. Although the least-squares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residual-based a posteriori error control with exact solve. Particular interest is on the treatment of the data approximation error which requires a separate marking strategy. The paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non-linear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:43---62 2015) from the Poisson model problem to the Stokes equations.