Sparse Direct Solvers

Sparse direct solvers are recommended for large 3D finite element models such as multi-story buildings, shell structures, and solid structures.

The sparse method involves solving the linear equation system K * x = b or the equation system of eigenvalue problems Kφ - λBφ = 0.

The direct sparse solvers (SPDS) are efficient computational techniques based on decomposition of the matrix K = L * U with a considerably less number of matrix elements different from zero.

Non-zero matrix elements

The size of the upper matrix U, i.e. the number of non-zero matrix elements, is 2-15 times smaller when using the Sparse Direct Solvers, than it is with the frontal or skyline methods. The matrix factorization is also 2 to 15 times faster.

Sparse direct solvers have a significantly smaller number of non-zero matrix elements than the frontal and skyline methods (Gauss method) because:

• The frontal and skyline methods store all matrix elements from the diagonal to the last non-zero element, including all zero elements.
• The sparse method stores only non-zero elements of the stiffness matrix during the matrix decomposition process.

Effectiveness is still high in nonlinear problems requiring multiple applications of the factorization procedure.

Efficiency of the sparse direct solvers method for solving eigenvalue problems

The efficiency of the SPDS technique for solving eigenvalue problems results not only from quick matrix factorization, but also from the quick resolutions of L * U * x = b and the matrix-vector product calculation B * x. These procedures repeat many times while solving eigenvalue problems. The first requirement is satisfied due to the reduction of the non-zero element number in the factorized matrix compared with the skyline method. Implementation of the quick matrix-vector product procedure is still important for the consistent B matrix (modal analysis with a consistent mass matrix or buckling). The compact format technique is used for fast computation of B*x. Matrix B is stored in RAM with only non-zero entries.

Appropriate data structures are developed to allow fast computations of the matrix-vector product. If the size of the problem does not allow allocation of compact format data structures, then the element-by-element procedure is switched on automatically. It enables you to avoid storing the large-scale matrix B on the disk and disregards I/O operations during calculations B * x. This ensures significant calculation acceleration compared to the skyline technique for large-scale problems.

Reordering Methods

Two reordering methods are available:

• Nested dissection method (NDM)
• Minimal degrees algorithm (MDA).

Both methods can be used to reduce the non-zero fill-ins in the equation matrix during factorization. These reordering methods produce sparse matrices which do not have band structure. The factorization methods for such matrices are called sparse methods.

Sparse approach

The Sparse approach is the substructure-by-substructure (super-element) approach with deep step-by-step multi-level nesting. Substructure-by-substructure subdivision of a source structure is performed automatically.

Sparse solvers always use the Nested dissection method (NDM) for the reordering of equation numbers.

SparseM approach

SparseM is a multifrontal solver which is applied with the nested dissection reordering method (NDM) and minimal degrees algorithm (MDA)¹ (see also: Parameters of SparseM Iterative Solver ).

SparseM solvers can use either the NDM or the MDA method.

Note: It is impossible to predict a-priori which reordering method is better. At present, the application of SparseM solver with MDA is recommended for the majority of building structures. For solids, the NDM method is usually more appropriate.

Specifications

• Memory use: high
• Disk use: medium
• Speed estimation: medium/fast, depending on the reordering effectiveness
• Quantity of equations: 10,000 - 200,000 equations.
• Supported analyses: All except Modal Analysis Recognizing Static Forces.
• Additional remarks:

  Sparse Direct Solvers

  • Solve the eigenvalue problems Kφ - λBφ = 0 as well as a linear or linearized equation set.
  • Are particularly recommended for incorrectly conditioned structures if there is no convergence of iterative methods.
  • Detect incorrectly conditioned structures, but they do not lead to obtaining the numbers of nodes and degrees of freedom for the equations where a calculation problem occurs.

Sparse Method calculations

The calculations dialog for the Sparse method shows the following phases:

• • Model reduction (renumbering of nodes and elements)
  • Definition of the stiffness matrix for individual structure elements
  • Matrix decomposition K = L * U
  • Problem solving for successive load cases.