Cédric Chauvière

Computational Modeling of Uncertainty in Time-Domain Electromagnetics

We discuss computationally efficient ways of accounting for the impact of uncertainty, e.g., lack of detailed knowledge about sources, materials, shapes, etc., in computational time-domain electromagnetics. In contrast to classic statistical Monte Carlo--based methods, we explore a probabilistic approach based on high-order accurate expansions of general stochastic processes. We show this to be highly efficient and accurate on both one- and two-dimensional examples, enabling the computation of global sensitivities of measures of interest, e.g., radar-cross-sections (RCS) in scattering applications, for a variety of types of uncertainties.

A fast solver for Fokker--Planck equation applied to viscoelastic flows calculations: 2D FENE model

A residual current circuit breaker 1 has a partition wall 3 which separates an over-current protection device 4 from a residual current detection circuit 5. A plunger rod 24 extends through a bore 23 in the armature 21 of a coil 20 and is moved independently through the coil 20 to trip the breaker if a residual current is detected by the circuit 5. The plunger rod 24 is moved by a drive rod 31, the operation of which is controlled by a permanent magnet which retains the drive rod 31 retracted and an electromagnet which allows the drive rod 31 to drive forwardly under the action of a spring 32 in the event of a residual current being detected. The plunger rod 30 is reset by a reset lever 40 which is moved when an operating handle 18 of the breaker moves from a non-tripped to a tripped position upon tripping of the breaker.

Efficient Computation of RCS From Scatterers of Uncertain Shapes

We propose a way of accounting for the lack of detailed knowledge about material shapes in computational time-domain electromagnetics. We use Legendre-Gauss-Lobatto, Stroud-2 and Stroud-3 quadrature formulas to solve the resulting stochastic equation and we show the efficiency of the proposed method over statistical Monte Carlo simulations. We also show how the radar cross section (RCS) in scattering is affected by the uncertainty in shape of the objects and by the direction of the incident field

A locally-upwinded spectral technique (LUST) for viscoelastic flows

Abstract In this paper, we develop an SUPG spectral element scheme suitable for computations of viscoelastic flows at high Deborah numbers. The novelty of the scheme lies in the derivation of the upwinding factors used in the perturbed test tensors of the weak form of the equations. These factors are related to the Deborah number and to the local mesh spacing and have been derived using the superconsistency ideas of Funaro [SIAM J. Numer. Anal. 30 (1993) 1664–1676; J. Sci. Comp. 12 (1997) 385–394; Comput. Math. Appl. 33 (1997) 95–103]. Results of applying the method to the benchmark problem of flow past a single confined cylinder demonstrate the superior stability and accuracy properties of the new scheme when compared with the SUPG spectral element method used previously by the authors [J. Non-Newtonian Fluid Mech. 95 (2000) 1–33; Comput. Meth. Appl. Mech. Engrg. 190 (2001) 3999–4018]. The new method gives excellent agreement with reference results in the literature.

Fokker–Planck simulations of fast flows of melts and concentrated polymer solutions in complex geometries

In 1999, Ottinger introduced a thermodynamically admissible reptation model incorporating chain stretching, anisotropic tube cross sections, double reptation, and the convective constraint release mechanism. In this paper, we describe and use a new high-order Fokker–Planck-based numerical method for the simulation of the Ottinger model in complex geometries. Evidence, in the case of startup homogeneous flows, of the significant CPU time advantage (for comparable levels of accuracy) of our method over a stochastic simulation [Fang et al. (2000)], is presented. For the confined cylinder benchmark problem, differences in the drag behavior observed between the Ottinger model and those of Doi and Edwards (1978a, 1978b, 1978c) and Mead et al. (1998) are explained in terms of double reptation and the differing relaxation spectra.

A new spectral element method for the reliable computation of viscoelastic flow

Abstract A new stabilised spectral element method has been developed by the authors for the accurate integration of the mixed elliptic–hyperbolic system of partial differential equations governing certain viscoelastic flows. The method is illustrated by solving the benchmark problem of the flow of an Oldroyd-B and a PTT fluid past a cylinder in a channel. Results are presented to demonstrate the advantages of the proposed method over traditional Galerkin-type procedures in terms of accuracy and stability.

How accurate is your solution?: Error indicators for viscoelastic flow calculations

Abstract The present work is an extension of earlier results of Owens [R.G. Owens, Comput. Methods Appl. Mech. Eng. 164 (1998) 375–395] and is an attempt to provide some theoretical undergirding to the question of appropriate error indicators for numerical solutions of flows of viscoelastic fluids having a differential constitutive equation. In particular, it is shown that the local elemental residual for the elastic stresses only accounts for the so-called stress cell error and as a consequence is, in general, an inadequate measure of the local error. An improved error indicator is then proposed which takes account of the transmitted error. Acknowledging that the stress errors form the major contribution to our error indicator for sufficiently large Deborah numbers we have devised a method of computing the error indicator on an element-by-element basis. Numerical results are presented to show how the approximate error and the ‘exact’ error obtained by calculating the difference between the numerical solution and a reference calculation decrease with mesh refinement. As an illustration of the use of our error indicator we proceed to describe an adaptive spectral element method for flow of an Oldroyd-B fluid past a sphere in a tube. The use of consistent upwinding is shown to result in greater accuracy and stability than is possible with a Galerkin approach. The results are compared with those in the literature.

An Efficient Technique for Simulations of Viscoelastic Flows, Derived from the Brownian Configuration Field Method

In this paper we present a new approach for calculating viscoelastic flows. This method is derived from the Brownian configuration field method [M. A. Hulsen, A. P. G. Van Heel, and B. H. A. A. Van Den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79--101] for the simulation of flows of dilute polymeric solutions. The method presented here has the same attractive features as those in a previous work [C. Chauviere, SIAM J. Sci. Comput., 23 (2002), pp. 2123--2140], i.e., improved stability in comparison with traditional methods starting from closed-form constitutive equations. Moreover, the method developed here has the additional advantage of being significantly more CPU efficient. Results are presented by solving the benchmark problem of the flow of an Oldroyd-B fluid past a cylinder in a channel using a spectral element method to demonstrate the advantages of the proposed method.

A New Method for Micro-Macro Simulations of Viscoelastic Flows

In this paper we develop a robust numerical method derived from the Brownian configuration field method [M. A. Hulsen, A.P. G. Van Heel, and B.H.A. A. Van Den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79--101] for the simulation of flows of dilute polymeric solutions. The statistical properties of the Wiener stochastic process in the stochastic differential equation describing the evolution of the configuration vector are exploited in order to derive a simple expression for the polymeric contribution to the stress. The method is tested numerically by solving the benchmark problem of the flow of an Oldroyd B fluid past a cylinder in a channel using a spectral element method. Results are presented to demonstrate the advantages of the proposed method.

A Robust Spectral Element Method for Simulations of Time-Dependent Viscoelastic Flows, Derived from the Brownian Configuration Field Method

This paper is an extension of previous work [4], where a robust numerical method derived from the Brownian configuration field method [8] was introduced in order to simulate the flows of dilute polymeric solutions. In [4], we limited our study to solutions of dumbbells having infinite extensibility (Oldroyd-B model), whereas in this paper, we tackle the more difficult problem of dumbbells having finite extensibility (FENE-P model).