Solvers Available in Robot

Linear equation sets are solved with the following methods.

Frontal method

• Memory use: low
• Disk use: high
• Speed estimation: slow
• Application: up to 50000 equations; linear and non-linear statics, harmonic analysis
• Available analysis limitations: N/A
• Additional remarks: In many cases it enables obtaining numbers of nodes and degrees of freedom for equations leading to calculation problems such as incorrectly restricted structures.

A specific Gauss elimination procedure is applied to factorization of the matrix K = L * U when solving linear equation set Kx = b. This method is based on element-by-element aggregation of the matrix and the simultaneous elimination of fully assembled equations ^{1, 2} . Nele denotes a number of finite elements and Ke is an appropriate element matrix.

The frontal method does not implement a fully assembled matrix. A dense work array (front) consisting of fully assembled equations (top front) and partially assembled equations (bottom front) slides along the matrix. Decomposed equations present the upper triangle matrix U and are stored in the secondary storage immediately after completing the corresponding elimination.

The frontal method is available only if the load case list does not include an eigenvalue (modal and buckling) analysis.

Skyline method

• Memory use: low
• Disk use: high
• Speed estimation: slow
• Application: up to 50000 equations; all analysis types
• Available analysis limitations: N/A
• Additional remarks: In many cases it enables obtaining numbers of nodes and degrees of freedom for equations leading to calculation problems such as incorrectly restricted structures.

The skyline method is based on the Cuthill-McKee reordering method ^3,4 , a matrix profile scheme, and the Crout factorization technique [5]. It is applied when solving either a linearized equation set or an eigenvalue problem Kφ - λBφ = 0 (modal and buckling analysis). If the second matrix B is consistent (modal analysis with consistent mass matrix or buckling), then it is stored by means of the profile method (as is the K matrix). All required consistent matrices for different types of analysis are also stored by means of the profile method. For example, a stress-stiffened matrix for nonlinear and buckling analysis and a dynamic matrix K - λB for Sturm sequence check and harmonic analysis.

The skyline method is applied to all analysis types.

The following apply when the Auto option is enabled.

• The frontal method is applied if a load case list does not include an eigenvalue analysis. It is used for non-bar structures if a structure includes plane stress elements, shell elements or solid elements and does not include bars.
• Skyline method is applied if the eigenvalue (modal and buckling) analysis is performed, which differs from linear static analysis or if a structure includes bar elements.

Note: The sparse direct solver and iterative solver must be assigned explicitly in the Job Preferences dialog.

Sparse direct solvers

• Memory use: high
• Disk use: medium
• Speed estimation: medium/fast, depending on the reordering effectiveness
• Application: 10,000 - 200,000 equations; all analysis types except for modal analysis recognizing static forces
• Additional remarks: Sparse direct solvers (SPDS) are recommended for large 3D finite element models such as multi-story buildings, shell structures, and solid structures. It detects incorrectly conditioned structures, but it does not lead to obtaining the numbers of nodes and degrees of freedom for the equations where a calculation problem occurs. It is particularly recommended for incorrectly conditioned structures if there is no convergence of iterative methods.

SPDS are an effective calculation technique based on the factorization of matrix K = L * U with a significantly smaller number of non-zero matrix elements than frontal and skyline (Gauss method). SPDS are used for solving the eigenvalue problems Kφ - λBφ = 0 as well as a linear or linearized equation set.

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. Nested dissection method (NDM) [3] is used for reordering of equation numbers.

SparseM is a multifrontal solver which is applied with the nested dissection reordering method (NDM) and minimal degrees algorithm (MDA)³ (see also: Parameters of SparseM Iterative Solver ).

Normally, the size of the upper triangle matrix U (the number of non-zero matrix elements) obtained by means of SPDS method is 2-15 times smaller in comparison to the matrix size obtained by means of the frontal or skyline methods. Thus, the matrix factorization will be 2-15 times quicker. Effectiveness is still high in nonlinear problems requiring multiple applications of the factorization procedure.

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 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.

Sparse direct solvers are a good alternative for iterative methods if they are ill-conditioned.

Iterative solvers

See: Iterative Solver Parameters - General Information

Iterative ICCF

• Memory use: high
• Disk use: not used
• Speed estimation: quick (for well-conditioned problems)
• Application: 15 000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: N/A
• Additional remarks: ICCF is recommended for large-scale problems with a small number of right-hand sides. Correctness of structure restrictions is not checked. For incorrectly conditioned problems, the convergence is still slow.

Iterative diagonal

• Memory use: minimal
• Disk use: minimal
• Speed estimation: slow
• Application: 15 000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: N/A
• Additional remarks: Iterative diagonal is recommended for large-scale problems with a small number of right-hand sides. Correctness of structure restrictions is not checked. For incorrectly conditioned problems, the convergence is still slow.

Iterative Gauss-Cholesky

• Memory use: minimal
• Disk use: minimal
• Speed estimation: slow
• Application: 15,000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: N/A
• Additional remarks: Iterative Gauss-Cholesky is recommended for large-scale problems with a small number of right-hand sides and RAM is not sufficient for ICCF method. Correctness of structure restrictions is not checked. Verification of each element matrix is conducted by means of the Vinget regularization procedure ^6,7 . For incorrectly conditioned problems, the convergence is still slow.

Iterative multilevel ICCF

• Memory use: high
• Disk use: minimal
• Speed estimation: quick for well-conditioned problems
• Application: 15,000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: available structure types: 3D (bar, shell, solid and all special finite elements), 2D frame
• Additional remarks: Iterative multilevel ICCF is recommended for large-scale problems. Normally, it shows quicker convergence compared to the iterativie ICCF method. In some cases, it displays less stable convergence. Correctness of structure restrictions is not checked.

Iterative multilevel diagonal

• Memory use: minimal
• Disk use: minimal
• Speed estimation: slow
• Application: 15,000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: available structure types: 3D (bar, shell, solid and all special finite elements), 2D frame
• Additional remarks: this method is considerably slower than iterative multilevel ICCF. However, it requires much less RAM memory. Smoothing method 2 usually requires fewer iterations compared to the iterative multilevel ICCF. Correctness of structure restrictions is not checked. Verification of each element matrix is conducted by means of the Vinget regularization procedure ^6,7.

Iterative multilevel Gauss-Cholesky

• Memory use: minimal
• Disk use: minimal
• Speed estimation: medium (for well-conditioned problems)
• Application: 15,000 - 1,000,000 equations and more; linear statics, modal analysis, buckling
• Available analysis limitations: available structure types: 3D (bar, shell, solid and all special finite elements), 2D frame
• Additional remarks: this method is considerably slower than iterative multilevel ICCF. However, it requires much less RAM memory. Smoothing method 2 usually requires fewer iterations compared to the iterative multilevel ICCF. Correctness of structure restrictions is not checked. Verification of each element matrix is conducted by means of the Vinget regularization procedure ^6,7 .

References

1. Duff I.S., Reid J.K. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans. Math. Software, 1983, 9, N3, p.633-641.
2. B.Irons, A frontal solution of program for finite element analysis, Int. J. Numer. Methods Engrg. 2 (1970) 5-32.
3. George A., Liu J., Computer solution of large sparse positive definite systems, 1981.
4. Pissanetzky S. Sparse matrix technology, 1984.
5. Hughes T.R.J., Ferencz R.M., Raefsky A.M. The finite element method. DLEARN - A linear static and dynamic finite element analysis program.
6. Hughes T.J.R., Ferencz M. Implicit solution of large-scale contact and impact problems employing an EBE preconditioned iterative solver, IMPACT 87 Int. Conference on Effects of Fast Transient Loading in the Context of Structural Mechanics, Lausanne, Switzerland, August 26-27, 1987.
7. Hughes T.J.R., R.M.Ferencz, and j.O.Hallquist. Large-scale vectorized implicit calculations in solid mechanics on a CRAY X-MP/48 utilizing EBE preconditioned conjugate gradients, Comput. Meths. Appl. Mech. Engrg., 61