An ultraweak DPG method for viscoelastic fluids

Keith, Brendan; Knechtges, Philipp; Roberts, Nathan V.; Elgeti, Stefanie; Behr, Marek; Demkowicz, Leszek

doi:10.1016/j.jnnfm.2017.06.006

Cited by 21 publications

(23 citation statements)

References 48 publications

(79 reference statements)

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“…The resulting drag forces for different Weissenberg numbers are plotted in Figure 5, listed in Table 4, and compared to selected literature 14,20,21 . For smaller Weissenberg numbers (Wi ≤ 0.6), we are in good agreement with the values obtained in other studies.…”

Section: Numerical Resultssupporting

confidence: 86%

“…One of these is the confined cylinder problem in which the flow around a cylinder immersed in a narrow channel with a blocking ratio of 2:1 is investigated numerically. This benchmark has been analysed in numerous works, for example, References 7,16‐21, but still, the numerical treatment of this test case is challenging because there are problems arising in resolving steep gradients in velocity and stress and very thin boundary layers as well as a very fine nearly singular beam in the wake of the cylinder 20 . Without stabilizing techniques, most numerical approaches diverge for Weissenberg numbers (Wi) greater than 0.7, for example, Reference 14.…”

Section: Introductionmentioning

confidence: 99%

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A fully coupled high‐order discontinuous Galerkin solver for viscoelastic fluid flow

Kikker

Kummer

Oberlack

2021

Numerical Methods in Fluids

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A fully coupled high order discontinuous Galerkin (DG) solver for viscoelastic Oldroyd B fluid flow problems is presented. Contrary to known methods combining DG for the discretization of the convective terms of the material model with standard finite element methods (FEM) and using elastic viscous stress splitting (EVSS) and its derivatives, a local discontinuous Galerkin (LDG) formulation first described for hyperbolic convection‐diffusion problems is used. The overall scheme is described, including temporal and spatial discretization as well as solution strategies for the nonlinear system, based on incremental increase of the Weissenberg number. The solvers suitability is demonstrated for the two‐dimensional confined cylinder benchmark problem. The cylinder is immersed in a narrow channel with a blocking ratio of 1:2 and the drag force of is compared to results from the literature. Furthermore, steady and unsteady calculations give a brief insight into the characteristics of instabilities due to boundary layer phenomena caused by viscoelasticity arising in the narrowing between channel and cylinder.

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Section: Numerical Resultssupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A fully coupled high‐order discontinuous Galerkin solver for viscoelastic fluid flow

Kikker

Kummer

Oberlack

2021

Numerical Methods in Fluids

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“…In choosing the best algorithm to solve the least-squares problem coming from a DLS method, many factors are important to consider. For instance, the normal equation have been demonstrated to be adequate when the methodology has been applied to many DPG problems [64,65,21,67,33,37,49,62,30,40,38,35]. Indeed, in many reasonable circumstances, the round-off error in the solution from the associated linear solve cannot be expected to be nearly as large as the truncation error due to the finite element discretization.…”

Section: Solution Algorithmsmentioning

confidence: 99%

“…Because G can be efficiently inverted with DPG, in that setting, much smaller problems, posed solely in the primal variable u, can be solved directly. This has been performed for many problems of engineering interest [65,21,67,33,37,49,62,30,40,38,35].…”

Section: Introductionmentioning

confidence: 99%

Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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ABSTRACT. 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 was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

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“…For example, the equations of static linear elasticity have been flexibly solved with the methodology using several different variational formulations [43], which were then coupled to capitalize on the fact that some formulations were robust in the incompressible limit while others were computationally more efficient [27]. Similarly, different features of the methodology have been taken advantage of while solving problems related to fluid flow [59,8,26,44], wave propagation [69,31,18], electromagnetism [6], elasticity [43,7,5], transmission problems in unbounded domains [38,29], and even optical fibers [19], among others.…”

Section: Introductionmentioning

confidence: 99%

Using a DPG method to validate DMA experimental calibration of viscoelastic materials

Fuentes

Demkowicz

Wilder

2017

Computer Methods in Applied Mechanics and Engineering

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A discontinuous Petrov-Galerkin (DPG) method is used to solve the time-harmonic equations of linear viscoelasticity. It is based on a "broken" primal variational formulation, which is very similar to the classical primal variational formulation used in Galerkin methods, but has additional "interface" variables at the boundaries of the mesh elements. Both the classical and broken formulations are proved to be well-posed in the infinite-dimensional setting, and the resulting discretization is proved to be stable. A full hp-convergence analysis is also included, and the analysis is verified using computational simulations. The method is particularly useful as it carries its own natural arbitrary-p a posteriori error estimator, which is fundamental for solving problems with localized solution features. This proves to be useful when validating calibration models of dynamic mechanical analysis (DMA) experiments. Indeed, different DMA experiments of epoxy and silicone resins were successfully validated to within 5% of the quantity of interest using the numerical method.

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An ultraweak DPG method for viscoelastic fluids

Cited by 21 publications

References 48 publications

A fully coupled high‐order discontinuous Galerkin solver for viscoelastic fluid flow

A fully coupled high‐order discontinuous Galerkin solver for viscoelastic fluid flow

Discrete least-squares finite element methods

Using a DPG method to validate DMA experimental calibration of viscoelastic materials

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