Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 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 vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques

Parallel Overlapping Schwarz Preconditioners and Multiscale Discretizations with Applications to Fluid-Structure Interaction and Highly Heterogeneous Problems

Accurate simulations of transmural wall stresses in artherosclerotic coronary arteries may help to predict plaque rupture. Therefore, a robust and efficient numerical framework for Fluid-Structure Interaction (FSI) of the blood flow and the arterial wall has to be set up, and suitable material laws for the modeling of the fluid and the structural response have to be incorporated. In this thesis, monolithic coupling algorithms and corresponding monolithic preconditioners are used to simulate FSI using highly nonlinear anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models for the arterial wall. An MPI-parallel FSI software from the LifeV library is coupled to the software FEAP in order to enable access to the structural material models implemented in FEAP. To define a benchmark test for highly nonlinear material models in FSI, a simple geometry corresponding to a section of an idealized coronary artery, suitable boundary conditions, and material parameters adapted to experimental data are used. In particular, the geometry is chosen to be nonsymmetric to make effects due to the anisotropy of the structure visible. An initialization phase and several heartbeats are simulated, and systematical studies with meshes of increasing refinement and different space discretizations are carried out. The results indicate that, for the highly nonlinear material models, piecewise quadratic or F-bar element discretizations lead to significantly better results than piecewise linear shape functions. The results using piecewise linear shape functions are less accurate with respect to the displacements and, in particular, to the approximation of the stresses. To improve the performance of the FSI simulations, a more robust preconditioner for the highly nonlinear structural material models has to be used. Therefore, a parallel implementation of the GDSW (Generalized Dryja-Smith-Widlund) preconditioner, which is a geometric two-level overlapping Schwarz preconditioner with energy-minimizing coarse space, is presented. The implementation, which is based on the software library Trilinos, is held flexible to make further extensions of the preconditioner possible. Even though the dimension of its coarse space is comparably large, parallel scalability for two and three dimensional scalar elliptic and linear elastic problems for thousands of cores is demonstrated. Also for unstructured domain decompositions and for a hybrid version of the preconditioner, convincing scalability is presented. When used as a preconditioner for the structure block in FSI simulations, the GDSW preconditioner shows excellent performance as well: scalability for up to 512 cores and a significant reduction of the simulation time and of the number of iterations with respect to the previously used preconditioner, IFPACK, are observed. IFPACK is an algebraic one-level overlapping Schwarz preconditioner. Finally, highly heterogeneous (multiscale) problems are investigated. Since the GDSW coarse space is not robust for general problems of this type, spaces based on Approximate Component Mode Synthesis (ACMS) are considered. On the basis of the ACMS space, coarse spaces for overlapping Schwarz methods are constructed, and a parallel implementation of a special finite element method is presented. For the coarse spaces, preliminary results indicating numerical scalability and robustness are discussed. For the parallel implementation of the special finite element method, very good parallel weak scalability is observed with respect to the construction of the basis functions and to the solution of the resulting linear system using the FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) method

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC 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 clearly shows its rapid trend to further diversity and depth

Finite element schemes for elliptic boundary value problems with rough coefficients

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the task of computing reliable numerical approximations of the solutions of elliptic equations and systems where the coefficients vary discontinuously, rapidly, and by large orders of magnitude. Such problems, which occur in diffusion and in linear elastic deformation of composite materials, have solutions with low regularity with the result that reliable numerical approximations can be found only in approximating spaces, invariably with high dimension, that can accurately represent the large and rapid changes occurring in the solution. The use of the Galerkin approach with such high dimensional approximating spaces often leads to very large scale discrete problems which at best can only be solved using efficient solvers. However, even then, their scale is sometimes so large that the Galerkin approach becomes impractical and alternative methods of approximation must be sought. In this thesis we adopt two approaches. We propose a new asymptotic method of approximation for problems of diffusion in materials with periodic structure. This approach uses Fourier series expansions and enables one to perform all computations on a periodic cell; this overcomes the difficulty caused by the rapid variation of the coefficients. In the one dimensional case we have constructed problems with discontinuous coefficients and computed the analytical expressions for their solutions and the proposed asymptotic approximations. The rates at which the given asymptotic approximations converge, as the period of the material decreases, are obtained through extensive computational tests which show that these rates are fundamentally dependent on the level of regularity of the right hand sides of the equations. In the two dimensional case we show how one can use the Galerkin method to approximate the solutions of the problems associated with the periodic cell. We construct problems with discontinuous coefficients and perform extensive computational tests which show that the asymptotic properties of the approximations are identical to those observed in the one dimensional case. However, the computational results show that the application of the Galerkin method of approximation introduces a discretization error which can obscure the precise asymptotic rate of convergence for low regularity right hand sides. For problems of two dimensional linear elasticity we are forced to consider an alternative approach. We use domain decomposition techniques that interface the subdomains with conjugate gradient methods and obtain algorithms which can be efficiently implemented on computers with parallel architectures. We construct the balancing preconditioner, M,, and show that it has the optimal conditioning property k(Mh(^-1)Sh) = 0 is a constant which is independent of the magnitude of the material discontinuities, H is the maximum subdomain diameter, and h is the maximum finite element diameter. These properties of the preconditioning operator Mh allow one to use the computational power of a parallel computer to overcome the difficulties caused by the changing form of the solution of the problem. We have implemented this approach for a variety of problems of planar linear elasticity and, using different domain decompositions, approximating spaces, and materials, find that the algorithm is robust and scales with the dimension of the approximating space and the number of subdomains according to the condition number bound above and is unaffected by material discontinuities. In this we have proposed and implemented new inner product expressions which we use to modify the bilinear forms associated with problems over subdomains that have pure traction boundary conditions.This work is funded by the Engineering and Physical Sciences Research Council

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, 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 clearly shows its rapid trend to further diversity and depth