Fully-implicit log-conformation formulation of constitutive laws

Subject of this paper is the derivation of a new constitutive law in terms of the logarithm of the conformation tensor that can be used as a full substitute for the 2D governing equations of the Oldroyd-B, Giesekus and other models. One of the key features of these new equations is that - in contrast to the original log-conf equations given by Fattal and Kupferman (2004) - these constitutive equations combined with the Navier-Stokes equations constitute a self-contained, non-iterative system of partial differential equations. In addition to its potential as a fruitful source for understanding the mathematical subtleties of the models from a new perspective, this analytical description also allows us to fully utilize the Newton-Raphson algorithm in numerical simulations, which by design should lead to reduced computational effort. By means of the confined cylinder benchmark we will show that a finite element discretization of these new equations delivers results of comparable accuracy to known methods.Comment: 21 pages, 5 figure

On a reduced sparsity stabilization of grad-div type for incompressible flow problems

We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Preliminary finite element modeling of a piezoelectric actuated marine propulsion fin

New technologies surrounding composite materials and autonomous underwater vehicle (AUV) design have led to numerous studies involving the marine propulsion for these AUVs. AUVs traditionally are classified as highly efficient, payload capable, and can be utilized as reconnaissance or surveillance vehicles. Undullatory and oscillatory propulsion devices have been conceived to replace the present propulsion technologies, of propellers, with highly maneuverable, efficient, and quiet propulsion systems. Undullatory and oscillatory propulsion has been around for centuries employed by aquatic life, but only recently have the mini-technologies been available to present such propulsion devices economically and with enough materials research as to mimic biologic life on the same scale. Piezoelectric properties coupled with a thin plate allow for actuation properties, similar to bimetallic metals. Applying two piezoelectrics to the fixed end of a cantilevered beam or plate, on opposite sides, and actuating them with an opposite phase shift in electrical voltage potential results in transverse motion of the beam from the orthogonal plane to the vertical axis of the piezoelectric device. Coupling this property to a particular fiber orientation, composite thin plate, significantly increases the actuation properties. In addition, placing more than two piezoelectrics along the length of the thin composite plate gives the potential to increase actuation properties and change the motion from oscillatory to undullatory. These motions can again be increased by utilizing the natural vibration modes of the thin composite plate with piezoelectrics near resonance actuation. The current research is involved with modeling a piezoelectric actuated marine propulsion fin using the Galerkin finite element technique. An experimental proof of concept was developed to compare results. Using fluid-structure interaction (FSI) methods, it is proposed that the fluid and structure programs are resolved within one program. This is in contrast to traditional attempts at FSI problems that utilize a computational fluid dynamics (CFD) solver transferring load data between a structural dynamics/finite element (FE) program

The fully-implicit log-conformation formulation and its application to three-dimensional flows

The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation method as proposed by Fattal and Kupferman, the application of the efficient Newton-Raphson algorithm to the full monolithic system of governing equations, consisting of the log-conformation equations and the Navier-Stokes equations, has always posed a problem. In particular, it is the formulation of the constitutive equations by means of the spectral decomposition that hinders the application of further analytical tools. Therefore, up to now, a fully monolithic approach could only be achieved in two dimensions, as, e.g., recently shown in [P. Knechtges, M. Behr, S. Elgeti, Fully-implicit log-conformation formulation of constitutive laws, J. Non-Newtonian Fluid Mech. 214 (2014) 78-87]. The aim of this paper is to find a generalization of the previously made considerations to three dimensions, such that a monolithic Newton-Raphson solver based on the log-conformation formulation can be implemented also in this case. The underlying idea is analogous to the two-dimensional case, to replace the eigenvalue decomposition in the constitutive equation by an analytically more "well-behaved" term and to rely on the eigenvalue decomposition only for the actual computation. Furthermore, in order to demonstrate the practicality of the proposed method, numerical results of the newly derived formulation are presented in the case of the sedimenting sphere and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found that the expected quadratic convergence of Newton's method can be achieved.Comment: 21 pages, 9 figure