Extensions of the software package SAMG for the solution of large sparse systems of equations

SAMG for systems of equations with purely algebraic constraints

The object of this project is to extend the application areas of the SAMG solver package. SAMG already offers Uzawa variants as smoothers in multigrid cycling for saddle point problems in the Navier-Stokes area. For many important problems in the areas of mixed finite element methods, contact and obstacle problems in mechanics, geomechanics or in the area of continuum scale material design (microstructure optimization), the matrices to be solved have a similar structure, but this Uzawa smoothier cannot be used successfully because the physical background of the constraints or saddle point structure is completely different. Here SAMG has to treat certain equations as real algebraic constraints.

It is well known that many simulation codes set up their equations to be solved with such Lagrange multipliers as additional algebraic equations. Typically, however, all iterative methods have greatest difficulties to solve such equations at all - let alone efficiently. So far, direct solvers in large simulation codes have been considered as indispensable standard solvers. We want to change this by our new developments for the mentioned application classes. Therefore, new strategies and algorithms have to be developed that integrate these equations into the multigrid hierarchy in a suitable way. Also new transfer operators for AMG cycling, suitable approximations for a Schur complement as well as suitable methods must be developed, which have good smoothing properties even in the presence of algebraic equations, which do not stem directly from the discretization of partial differential equations.

Project duration: 12/2018 - 11/2021

Extension of simulation packages for the efficient handling of industrial applications in mechanical process engineering and electrical engineering

The aim of this project is the development of special problem-adapted simulation tools, which are suitable as plugin extensions for standard simulation packages.

Today's simulation applications are characterized by ever increasing problem sizes and the use of increasingly complex models.  As a result, the demands on the efficiency and scalability of the simulation environment are becoming more and more important.  In particular, many applications require the use of domain-specific or problem-adapted simulation tools and correspondingly adapted mathematical methods to solve the resulting large systems of equations.  An example for this is the oil interface of SAMG developed by SCAI, which takes care of out the highly specialized pre-processing of the linear equation systems, that result from the discretization of the complex models of the oil reservoir simulation, in order to successfully apply SAMG to the entire coupled system.

In practice, open source simulation packages are increasingly being used.  The aim of the project is therefore to develop specialized extensions of such open source simulation packages in combination with a corresponding solver interface for SAMG.  For this purpose, a series of OpenFOAM extensions will be developed, which are only possible in special application areas by combining solver and discretizer expertise.  Another open source package that we want to extend in a similar way is CSC/Elmer.  

Project duration: 12/2018 - 11/2021