4.3 Refined Key Demands on the Assembly Module

After the decision to design and implement new modules has been made, the key demands on the assembly module can be refined. This also brings a row transformation as discussed in Section 2.2.1 into play. Due to its advantages, the assembly module shall provide the required capabilities. First of all, the following definitions are given in order to clarify some terms:

• Assemble: assemble means to bring together. Thus, assembling regards the adding of values to the linear equation system. Assembling is completed when all values have been added.
• Assembly: in this context, the assembly or the assembly module specifically provides all necessary features which allow a simulator to assemble those values. However, the assembly module is generally responsible for meeting all key demands given above.
• Compile: compile means to compose out of materials from other sources, to collect and edit into a volume, to run through a compiler, and to build up gradually. Since four matrices ( $\ensuremath{\mathbf{A}}_{\mathrm{b}}$, $\ensuremath{\mathbf{A}}_{\mathrm{s}}$, $\ensuremath{\mathbf{T}}_{\mathrm{b}}$, and $\ensuremath{\mathbf{T}}_{\mathrm{v}}$) and two sets of right-hand-side vectors ( $\ensuremath{\mathbf{b}}_{\mathrm{b}}$ and $\ensuremath{\mathbf{b}}_{\mathrm{s}}$) are assembled, the first task is to compile all of these parts to one linear equation system. Refer to equations (4.12) and (4.13) to see how the parts are compiled.
• Compilation result: the compilation result is one linear equation system $\ensuremath{\mathbf{A}} \ensuremath{\mathbf{x}} = \ensuremath{\mathbf{b}}$, the so-called complete linear equation system.
• Pre-elimination: after the complete linear equation system is compiled, all equations marked with an elimination flag are pre-eliminated. Hence, the assembly module also provides a solving capability based on Gaussian elimination (see Section 4.8).
• Inner linear equation system: one result of the pre-elimination is the so-called inner linear equation system. It consists of all equations which were not pre-eliminated.

The refined requirements can be summarized as follows:

1. Application Programming Interface (API) providing methods for
   • Adding values to the boundary system: $\ensuremath{\mathbf{A}}_{\mathrm{b}}$ and $\ensuremath{\mathbf{b}}_{\mathrm{b}}$.
   • Adding values to the segment system: $\ensuremath{\mathbf{A}}_{\mathrm{s}}$ and $\ensuremath{\mathbf{b}}_{\mathrm{s}}$.
   • Adding values to the transformation matrix $\ensuremath{\mathbf{T}}_{\mathrm{b}}$.
   • Deleting equations.
   • Administration of priority information required for the correct handling of grid-points which are part of several segments.
   • Setting elimination flags for pre-elimination: mark equations which are eliminated from the linear equation system before it is passed to the solver module.
   • Invoking the solving process.
   • Returning the solution: After reverting all transformations and back-substituting the pre-eliminated equations, the output of the assembly module is the complete solution vector (or vectors in the case of more than one right-hand-side vector). In addition, the right-hand-side vector(s) are returned which can be used for various norm calculations frequently required for the damping and update of the Newton method.
2. Usability
   • Random access during assembling: all parts of the linear equation system should be always accessible.
   • Centralized administration of control parameters.
3. Preparation of the linear equation system for solving:
   • Row transformation: linear combination of rows to extinguish large entries.
   • Variable transformation: reduce the coupling of the equations.
   • Pre-elimination: eliminate problematic equations by Gaussian elimination to improve the condition of the inner system matrix.
   • Scaling: Since some preconditioners (for example the Incomplete-LU factorization, see Section 5.2.5) use a threshold to decide whether to keep or skip an entry, the entries of the system matrix are normalized.
   • Sorting: Reduction of the bandwidth of a matrix in order to reduce the factorization fill-in and thus the memory consumption.
4. Handling of real-valued and complex-valued linear equation systems: In the context of an efficient small-signal mode, handling of complex-valued systems is particularly important.
5. Performance-related features:
   • Handling of multiple right-hand-side vectors: For some simulations, for example calculating the complex-valued admittance matrix (see Section 3.4), several linear equation systems differ only in the right-hand-side vector. Thus, the effort for assembling, compiling, pre-eliminating, sorting, scaling, and factorizing of the system matrix actually has to be done only once. This factored matrix can then be used for all right-hand-side vectors. For that reason the module is able to simultaneously assemble several right-hand-side vectors.
   • Reassembling of the imaginary part only: during a frequency stepping, only the imaginary part is changed. In order to speed up the simulation, the real-valued part should remain unmodified.
6. Efficient handling of the sparse linear equation systems: storing large linear equation systems in dense format requires a huge amount of memory. Since the system matrices contain relatively few non-zero entries (see Section 5.5.1), far less memory has to be allocated if sparse matrix formats, for example MCSR (see Section 4.6 and Appendix E), are used.

A plug-in concept has been implemented for scaling, sorting and solving the inner linear equation system, making it possible to adapt or replace these modules easily. The sorting and scaling modules obtain the system matrix on input and return the sorting and scaling (diagonal) matrices which are then applied by the assembly module. The solver module obtains the system matrix and all right-hand-side vectors on input and returns the solution vectors of all inner linear equation systems.