Semiannual final report, 1 October 1991 - 31 March 1992

A summary of research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period 1 Oct. 1991 through 31 Mar. 1992 is presented

Simulations of Unsteady Shocks via a Finite-Element Solver with High-Order Spatial and Temporal Accuracy

This research aims to improve the modeling of stationary and moving shock waves by adding an unsteady capability to an existing high-spatial-order, finite-element, streamline upwind/Petrov-Galerkin (SU/PG), steady-state solver and using it to examine a novel shock capturing technique. Six L-stable, first- through fourth-order time-integration methods were introduced into the solver, and the resulting unsteady code was employed on three canonical test cases for verification and validation purposes: the two-dimensional convecting inviscid isentropic vortex, the two-dimensional circular cylinder in cross ow, and the Taylor-Green vortex. Shock capturing is accomplished in the baseline solver through the application of artificial diffusion in supersonic cases. When applied to inviscid problems, especially those with blunt bodies, numerical errors from the baseline shock sensor accumulated in stagnation regions, resulting in non-physical wall heating. Modifications were made to the solver\u27s shock capturing approach that changed the calculation of the artificial diffusion flux term (Fad) and the shock sensor. The changes to Fadwere designed to vary the application of artificial diffusion directionally within the momentum equations. A novel discontinuity sensor, derived from the entropy gradient, was developed for use on inviscid cases. The new sensor activates for shocks, rapid expansions, and other ow features where the grid is insufficient to resolve the high-gradient phenomena. This modified shock capturing technique was applied to three inviscid test cases: the blunt-body bow shock of Murman, the planar Noh problem, and the Mach 3 forward-facing step of Colella and Woodward

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988

Méthodes numériques adaptées à la résolution des équations de Navier-Stokes

The research group "Instabilités et Méthodes Numériques Spéci-fiques" operates in the development of numerical tools for solving nonlinear problems by using, in particluar, the Asymptotic Numer- ical Method (ANM). Based on coupling a perturbation method and a spatial discretization, the ANM is effective and makes it possible to precisely determine the transitions such as, for example, loss of uniqueness of the solution. The objective of this thesis is to offer al- ternative numerical methods both robust and effective, for solving the Navier-Stokes equations. We are interested in steady bifurcation analysis, and in time dependent flow simulation .Initially, numerical bifurcation analysis techniques for steady flow problems in very large number of degrees of freedom are de- scribed. These techniques, based on the ANM, are implemented in the multiphysics ELMER open-source software. We detail the im- plementation of the steady bifurcation analysis methods such as continuation of solution branches, detection of load parameter critical values and branch switching at steady bifurcation point. The emer- gence of a geometric progression in ANM series terms in the vicinity of a singularity is described. Discussions are proposed for the case of symmetry breaking bifurcations. The methods described in this the- sis are validated using reference cases of the literature, such as flow in pipe with sudden expansion/contraction. New results for three- dimensional flow in a sudden expansion, are obtained according to a parametric study. The use of high performance computing libraries makes possible the bifurcation analysis for models with high number of degrees of freedom, in affordable computing times. Secondly, high-order solvers are proposed for the simulation of un- steady flows. Homotopy with convex combination and a perturba- tion technique, are coupled to a time integration scheme in order to solve the unsteady Navier-Stokes equations. The case of two- dimensional flow around a fixed cylinder is studied. This reference problem allows us to validate and discuss proposed improvements. In this way, we confirm, in the numerical tests, that it is possible to reduce the computation time by avoiding operators assembly and resolution of unuseful linear systems in respect to the solution quality. In addition, new lighting is provided on the use of Padé approximants over previous work. The use of these nonlinear solvers allows us to significantly reduce the number of matrix factorization retaining them valid for many time steps, and sometimes on the complete time do- main. Many opportunities are envisaged, in particular the analysis of ANM series for the case of limit point, the Hopf bifurcation, the study of other cases of three-dimensional flow, the fluid-structure interaction. Similarly, the combination of ANM models with reduction techniques f stable periodic orbits are possible.Le groupe de recherche Instabilités et Méthodes Numériques Spécifiques mène ses activités dans le développement d’outils numé- riques pour la résolution de problèmes non linéaires en utilisant, no- tamment, la Méthode Asymptotique Numérique (MAN). Basée sur le couplage d’une méthode de perturbation et de discrétisation spa- tiale, la MAN est efficace et permet de déterminer précisément les transitions telles que, par exemple, la perte d’unicité de la solution. L’objectif de ce travail de thèse est de proposer des méthodes numé- riques alternatives à la fois robustes, performantes pour la résolution des équations de Navier-Stokes. Nous nous intéressons à l’analyse de bifurcation stationnaire, mais aussi à la simulation d’écoulement dépendant du temps. Dans un premier temps, des techniques d’analyse de bifurcation nu- mérique pour des problèmes stationnaires à très grand nombre de degrés de liberté sont décrites. Nous implémentons ces techniques, basées sur la MAN, dans le logiciel open-source multi-physique ELMER . Nous détaillons l’implémentation des méthodes d’analyse de bifurcation stationnaire telles que la continuation de branches solutions, les techniques de détection des valeurs critiques du pa- ramètre de charge et les changements de branche en un point de bifurcation stationnaire. L’émergence d’une progression géométrique dans les termes de séries MAN à l’approche d’une singularité est dé- crite. Des discussions sont proposées pour le cas de bifurcations par brisure de symétrie. Les méthodes proposées dans ce travail sont validées en utilisant des cas référencés dans la littérature, tels que des écoulements dans des conduites à expansion/contraction sou- daine. Une étude paramétrique permet de présenter de nouveaux ré- sultats pour les écoulements tridimensionnels dans une expansion brusque. L’utilisation de librairies de calculs intensifs rend possible la réalisation d’analyse de bifurcation pour des modèles à très grand nombre de degrés de liberté, en des temps de calcul abordables. Dans un deuxième temps, des solveurs d’ordre élevé sont proposés pour la simulation d’écoulements instationnaires. Une technique d’homotopie à combinaison convexe et une technique de pertur- bation, sont couplées à un schéma d’intégration temporelle pour résoudre les équations instationnaires de Navier-Stokes. Le cas d’un écoulement bidimensionnel autour d’un cylindre fixe est étudié. Ce problème de référence nous permet de valider et discuter des amélio- rations proposées. De cette manière, nous confirmons, au cours des essais numériques, qu’il est possible de réduire les temps de cal- cul en évitant des assemblages d’opérateurs et des résolutions de systèmes linéaires qui n’apportent aucune information supplémen- taire pour la qualité des solutions. De plus, un nouvel éclairage est apporté sur l’utilisation des approximants de Padé par rapport aux travaux antérieurs. L’utilisation de ces solveurs non linéaires nous permet de réduire significativement le nombre de factorisations de matrice en les conservant valides pour un grand nombre de pas de temps, et parfois sur le domaine temporel complet. De nombreuses perspectives sont envisagées, notamment pour l’analyse des séries pour le cas d’un point limite, la bifurcation de Hopf, l’étude d’autre cas d’écoulements tridimensionnels, le couplage fluide-structure. De même, l’association des techniques MAN aux techniques de réductions de modèles et l’analyse de stabilité des orbites périodiques sont envisageables

Numerical Advances for Fluid-Structure Interaction in Entangled Polymer Solutions with Applications to Active Microbead Rheology

Active microbead rheology is an important counterpart to passive microbead rheology. Both techniques have proven essential for exploring the viscoelastic properties of soft materials that yield at extremely low stress and strain thresholds. Many soft materials, especially arising in biology, are furthermore only available in small volumes that are not amenable to classical rheometers. In passive microbead rheology, thermal fluctuations of microbeads reveal the linear, equilibrium viscous and elastic moduli of the material across the frequency range that can be resolved by the microscope. In active microbead rheology, viscoelastic fluids can be driven out of equilibrium by controlled forces applied to magnetic microbeads, and the materials then exhibit a range of responses: the so-called linear response regime, where responses are proportional to the magnetic force at sufficiently low levels, and then a transition to a variety of nonlinear responses that are unique to different types of viscoelastic fluids. Understanding this rheological phenomenon is important in the study of dynamics of many biological systems involving flexible structures, such as ciliary transport of mucus in the human lung. Despite ongoing developments in modeling such systems, there is still a lack of accurate and efficient numerical methods and software packages that can describe such nonlinear phenomena quantitatively. Modeling viscoelastic fluids usually requires high spatial resolution and time-consuming simulations, and the interactions between fluids and flexible structures introduce additional numerical and computational challenges. The main goal of this dissertation is to develop and analyze robust numerical methods for viscoelastic fluid-structure interaction (FSI) with applications to active microbead rheology, and in particular, the transition of a linear to nonlinear response exhibited by a specific class of viscoelastic fluids, entangled polymer solutions. We employ the immersed boundary (IB) method to model fluid-structure interaction and use the open-source software IBAMR to implement the simulations. The simulations are guided and validated by experimental data. Motivated by numerical issues we encountered in the microbead simulations, we propose and implement a novel implicit solver for the constrained IB formulation provided by IBAMR, and we investigate its accuracy and efficiency with extensive numerical tests. Lastly, we develop adaptive mesh refinement (AMR) capabilities for solving the Stokes problem to enable more efficient simulations of high-resolution FSI problems at low Reynolds numbers.Doctor of Philosoph

Finite Element Modeling and Optimization of High-Speed Aerothermoelastic Systems

The design of supersonic and hypersonic aerospace vehicles is by nature a multi-disciplinary problem requiring the close integration of compressible fluid dynamics, heat transfer, and structural dynamics. The transient flow around the body must be accurately characterized in order to assess its effect on the thermal and structural responses; conversely, the thermal and structural behavior may significantly alter the aerodynamic performance. The core of this dissertation effort is concerned with the development and demonstration of an analysis and design capability for the aerothermoelastic behavior of high-speed aerospace vehicles. This nominally involves coupling of the compressible Navier-Stokes equations for the fluid dynamics, the transient heat equation for the thermal response, and the elastodynamic equations for the structural dynamics. The streamline upwind Petrov-Galerkin (SUPG) stabilized finite element method is used for solving the compressible flow problem. Both a standard Galerkin and stabilized Galerkin gradient least squares (GGLS) finite element method are utilized for solving the heat equation, and a standard Galerkin method is used for solving the elastodynamic equations. The transient and steady-state responses of a problem are determined via a single, simultaneously coupled nonlinear system, thus bypassing accuracy and stability issues of classical staggered multi-physics coupling strategies. A gradient-based optimization framework is developed for designing transient coupled aerothermoelastic systems via adjoint-based sensitivity analysis. This framework is used to optimize the design of a structure in regard to thermal and structural performance. The efforts of this thesis have yielded a state-of-the-art approach for coupled aerothermoelastic analysis and design optimization