GaspiLS – Software Solution for CFD and FEM Simulations

GaspiLS is our scalable linear solver library, which has already proven itself in industrial companies in times of exascale computing. Many simulations in engineering are based on Computational Fluid Dynamics and Finite Element Methods (CFD and FEM methods), for example the determination of aerodynamic properties of aircraft or the analysis of building statics. A large part of the computational time is required for solving the underlying equations by iterative methods such as the Krylov Subspace. The performance of the iterative solvers used thus has a large impact on the overall runtime of such simulations. To gain faster insights from these simulations, we have developed the linear solver library GaspiLS.
 

GaspiLS Accelerates Transient Simulations

In time-dependent simulations, similar systems of equations must be solved repeatedly over many time steps. GaspiLS specifically exploits this similarity and can thus reuse previously computed structures for subsequent calculation steps. This reduces computation time, particularly in complex transient simulations. If the structure of the problem to be solved remains the same between time steps and only individual values change, the AMG preconditioner in GaspiLS can reuse parts of its previously generated operator hierarchy. This eliminates the need to rebuild the entire structure from scratch at every time step. The extent to which structures are reused can be controlled adaptively or statically, depending on the specific application.

A reservoir simulation illustrates how this affects computational performance: As an AMG-based CPRW preconditioner integrated into OPM, GaspiLS achieves a speed advantage of up to a factor of two in the SPE11a benchmark – which involves approximately one million cells – across the tested process numbers compared to the default ISTL and Hypre-BoomerAMG implementations.

This makes GaspiLS particularly well-suited for computationally intensive applications in which similar systems of equations must be solved over numerous time steps, such as in flow and reservoir simulations.

We do this without global synchronization points and, in combination with the large amount of generated subproblems, compensate for inequalities in computation time and latency caused by the exchange of data. Each individual core is maximally utilized at all times.
 

Optimal Convergence for a Wide Range of Problems

Preconditioners are designed to improve the convergence of iterative Krylov subspace methods. They reduce the number of required iterations or even enable the solution of a given problem in the first place. Preconditioners are therefore an equally central component for GaspiLS, along with the underlying high-performance matrix vector multiplication, to efficiently solve sparse linear systems of equations.

GaspiLS now provides two different classes of black-box preconditioners:

1. An Algebraic Multigrid Method (AMG) for symmetric positive definite problems, such as those encountered in CFD applications.
2. A multi-level incomplete LU decomposition (Multi-Level ILU) in combination with different domain decomposition methods for general problems.

Both implementations use a scalable, efficient, and hybrid parallel Gaspi-based programming model.

AMG, as a multigrid method, has linear numerical complexity, i.e., the effort to solve increases only proportionally to the number of unknowns to be solved. This is optimal and is used for symmetric positive definite problems. A hierarchy of operators is constructed directly from the matrix without requiring explicit knowledge of the underlying geometry or the partial differential equation.

Multilevel ILU are robust, accurate and, as black-box preconditioners, optimally suited for general sparse systems of equations. ILU essentially consists of a Gaussian elimination procedure in which less important contributions are discarded. Depending on the parameters used and the resources available, our multi-level ILU implementation can interpolate between a full Gaussian elimination (exact inverse, high resource requirement) and a simple Jacobi preconditioner (low resource requirement, low approximation accuracy). In combination with different overlapping or non-overlapping domain decomposition methods available for parallelization, our implementation provides a comprehensive arsenal of methods to implement different preconditioners optimized for the problem at hand. Thus, even poorly conditioned problems can be solved in an optimal way.