Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and Fluid-Structure Interaction Problems

Efficient methods for the approximation of solutions to incompressible fluid flow and fluid-structure interaction problems are presented. In particular, partial differential equations (PDEs) are derived from basic conservation principles. First, the incompressible Navier-Stokes equations for Newtonian fluids are introduced. This is followed by a consideration of solid mechanical problems. Both, the fluid equations and the equation for solid problems are then coupled and a fluid-structure interaction problem is constructed. Furthermore, a discretization by the finite element method for weak formulations of these problems is described. This spatial discretization of variables is followed by a discretization of the remaining time-dependent parts. An implementation of the discretizations and problems in a parallel C++ software environment is described. This implementation is based on the software package Trilinos. The parallel execution of a program is the essence of High Performance Computing (HPC). HPC clusters are, in general, machines with several tens of thousands of cores. The fastest current machine, as of the TOP500 list from November 2019, has over 2.4 million cores, while the largest machine possesses over 10 million cores. To achieve sufficient accuracy of the approximate solutions, a fine spatial discretization must be used. In particular, fine spatial discretizations lead to systems with large sparse matrices that have to be solved. Iterative preconditioned Krylov methods are among the most widely used and efficient solution strategies for these systems. Robust and efficient preconditioners which possess good scaling behavior for a parallel execution on several thousand cores are the main component. In this thesis, the focus is on parallel algebraic preconditioners for fluid and fluid-structure interaction problems. Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes problems are presented. Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on block factorizations. In order to obtain numerically scalable algorithms, coarse spaces obtained from the Generalized Dryja-Smith-Widlund (GDSW) and the Reduced dimension GDSW (RGDSW) approach are used. These coarse spaces can be constructed in an essentially algebraic way. Numerical results of the parallel implementation are presented for various incompressible fluid flow problems. Good scalability for up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, were achieved. A comparison of these monolithic approaches and commonly used block preconditioners with respect to time-to-solution is made. Similarly, the most efficient construction of two-level overlapping Schwarz preconditioners with GDSW and RGDSW coarse spaces for solid problems is reported. These techniques are then combined to efficiently solve fully coupled monolithic fluid-strucuture interaction problems

Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Saddle Point Problems

Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes, Navier-Stokes, and mixed linear elasticity ty e are presented. For the first time, coarse spaces obtained from the GDSW (Generalized Dryja-Smith-Widlund) approach are used in such a setting. Numerical results of our parallel implementation are presented for several model problems. In particular, cases are considered where the problem cannot or should not b e reduced using local static condensation, e.g., Stokes, Navier-Stokes or mixed elasticity problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the FROSch (Fast and Robust Overlapping Schwarz) library, which is part of the Trilinos package ShyLU. The implementation is algebraic in the sense that the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Parallel scalability results for several thousand cores for Stokes, Navier-Stokes, and mixed linear elasticity model problems are reported. Each of the local problems is solved using a direct solver in serial mo de, whereas the coarse problem is solved using a direct solver in serial or MPI-parallel mode or using an MPI-parallel iterative Krylov solve

Mo-9Si-8B alloys with additons of Zr – microstructure and creep properties

Three phase Mo-9Si-8B (at.%) alloys are a prominent example for a potential new high temperature structural material. Due to their high melting point and excellent creep resistance. In this study the effect of Zr addition (0…4 at.%) on the microstructure and creep properties of Mo-9Si-8B (at.%) alloys is investigated. Two powder metallurgical processes, hot isostatic pressing (HIP) and spark plasma sintering (SPS), are used to prepare specimens. The resulting microstructures are examined using SEM and TEM analysis. SPS alloys exhibit smaller grain sizes and fewer oxides compared to the HIP alloys, because of the oxygen availability during HIP. The more Zr is present in the alloys, the more and finer the observed particles are. With addition of Zr the formation of SiO2 on the grain boundaries can be prevented completely, due to the formation of ZrO2. High temperature tensile creep tests are carried out under vacuum to determine the influence of the microstructure on creep properties. The creep rates are one order of magnitude lower for the Zr containing alloys. However with a level of 4 at.% Zr the minimum creep rates increase again. Please click Additional Files below to see the full abstract

Labor selection over the business cycle: An empirical assessment

This paper is the first to analyze how much the probability of selecting a worker from a pool of applicants fluctuates over the business cycle. We use the German Job Vacancy Survey to construct the selection rate on the regional, industry, and national level and show that it is negatively correlated with unemployment. In addition, panel estmations reveal a positive comovement between the selection rate and market tightness, which is in line with the theoretical prediction from labor selection models

Fully algebraic two-level overlapping Schwarz preconditioners for elasticity problems

Different parallel two-level overlapping Schwarz preconditioners with Generalized Dryja-Smith-Widlund (GDSW) and Reduced dimension GDSW (RGDSW) coarse spaces for elasticity problems are considered. GDSW type coarse spaces can be constructed from the fully assembled system matrix, but they additionally need the index set of the interface of the corresponding nonoverlapping domain decomposition and the null space of the elasticity operator, i.e., the rigid body motions. In this paper, fully algebraic variants, which are constructed solely from the uniquely distributed system matrix, are compared to the classical variants which make use of this additional information; the fully algebraic variants use an approximation of the interface and an incomplete algebraic null space. Nevertheless, the parallel performance of the fully algebraic variants is competitive compared to the classical variants for a stationary homogeneous model problem and a dynamic heterogenous model problem with coefficient jumps in the shear modulus; the largest parallel computations were performed on 4,096 MPI (Message Passing Interface) ranks. The parallel implementations are based on the Trilinos package FROSch

Reduced Dimension GDSW Coarse Spaces for Monolithic Schwarz Domain Decomposition Methods for Incompressible Fluid Flow Problems

Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this paper, several techniques to further improve the convergence speed as well as the computing time are applied to monolithic two-level Generalized Dryja–Smith–Widlund (GDSW) preconditioners. In particular, reduced dimension GDSW (RGDSW) coarse spaces, restricted and scaled versions of the first level, hybrid and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time-dependent Navier-Stokes problem on 240 MPI ranks, a reduction of 86 % of the time-to-solution can be obtained. Even without applying recycling strategies, the time-to-solution can be reduced by more than 50% for a larger steady Stokes problem on 4 608 MPI ranks. For the largest problems with 11979 MPI ranks the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved

Revisiting the matching function

There is strong empirical evidence for Cobb-Douglas matching functions. We show in this paper that this widely found relation between matches on the one hand and unemployment and vacancies on the other hand can be the result of different underlying mechanisms. Obviously, it can be generated by assuming a Cobb-Douglas matching function. Less obvious, the same relationship results from a vacancy free entry condition and idiosyncratic productivity shocks. A positive aggregate productivity shock leads to more vacancy posting, a shift of the idiosyncratic selection cutoff and thereby more hiring. We calibrate a model with both mechanisms to administrative German labor market data and show that idiosyncratic productivity for new contacts is an important driver of the elasticity of the job-finding rate with respect to market tightness. Accounting for idiosyncratic productivity can explain the observed negative time trend in estimated matching efficiency and asymmetric business cycle responses to large aggregate shocks