Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems

This paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems. In the monolithic heterogeneous multiscale method, the nonlinear macroscale problem is solved using the Newton--Raphson scheme. The resolution of many mesoscale problems per Gauss point allows to compute the homogenized constitutive law and its derivative by finite differences. In the proposed approach, the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations. The exchange of information between both problems is still carried out using the heterogeneous multiscale method. However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gauss point.Comment: submitted to JC

Adaptive multiresolution computations applied to detonations

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge--Kutta--Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allow to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte

High order resolution and parallel implementation on unstructured grids

The numerical solution of the two-dimensional inviscid Euler flow equations is given. The unstructured mesh is generated by the advancing front technique. A cell-centred upwind finite volume method has been adopted to discretize the Euler equations. Both explicit and point implicit time stepping algorithms are derived. The flux calculation using Roe's and Osher's approximate Riemann solvers are studied. It is shown that both the Roe and Osher's schemes produce an accurate representation of discontinuities (e.g. shock wave). It is also shown that better convergence performance has been achieved by the point implicit scheme than that by the explicit scheme. Validations have been done for subsonic and transonic flow over airfoils, supersonic flow past a compression corner and hypersonic flow past cylinder and blunt body geometries. An adaptive remeshing procedure is also applied to the numerical solution with the objective of getting improved results. The issue of high order reconstruction on unstructured grids has been discussed. The methodology of the Taylor series expansion is adopted. The calculation of the gradient at a reference point is carried out by the use of either the Green-Gauss integral formula or the least-square methods. Some recently developed limiter construction methods have been used and their performance has been demonstrated using the test example of the transonic flow over a RAE 2822 airfoil. It has been shown that similar pressure distributions are obtained by all limiters except for shock wave regions where the limiter is active. The convergence problem is illustrated by the mid-mod type limiter. It seems only the Venkatakrishnan limiter provides improved convergence. Other limiters do not appear to work as well as that shown in their original publications. Also the convergence history given by the least-square method appears better than that by the Green-Gauss method in the test

Coupling different discretizations for fluid structure interaction in a monolithic approach

In this thesis we present a monolithic coupling approach for the simulation of phenomena involving interacting fluid and structure using different discretizations for the subproblems. For many applications in fluid dynamics, the Finite Volume method is the first choice in simulation science. Likewise, for the simulation of structural mechanics the Finite Element method is one of the most, if not the most, popular discretization method. However, despite the advantages of these discretizations in their respective application domains, monolithic coupling schemes have so far been restricted to a single discretization for both subproblems. We present a fluid structure coupling scheme based on a mixed Finite Volume/Finite Element method that combines the benefits of these discretizations. An important challenge in coupling fluid and structure is the transfer of forces and velocities at the fluidstructure interface in a stable and efficient way. In our approach this is achieved by means of a fully implicit formulation, i.e., the transfer of forces and displacements is carried out in a common set of equations for fluid and structure. We assemble the two different discretizations for the fluid and structure subproblems as well as the coupling conditions for forces and displacements into a single large algebraic system. Since we simulate real world problems, as a consequence of the complexity of the considered geometries, we end up with algebraic systems with a large number of degrees of freedom. This necessitates the use of parallel solution techniques. Our work covers the design and implementation of the proposed heterogeneous monolithic coupling approach as well as the efficient solution of the arising large nonlinear systems on distributed memory supercomputers. We apply Newton’s method to linearize the fully implicit coupled nonlinear fluid structure interaction problem. The resulting linear system is solved with a Krylov subspace correction method. For the preconditioning of the iterative solver we propose the use of multilevel methods. Specifically, we study a multigrid as well as a two-level restricted additive Schwarz method. We illustrate the performance of our method on a benchmark example and compare the afore mentioned different preconditioning strategies for the parallel solution of the monolithic coupled system

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth