## CYLSHELL: Finite element analysis of cylindrical shells

### from the Independent Sets and Generators

Set CYLSHELL
Source: Reijo Kouhia, Helsinki University of Technology, reijo.kouhia@hut.fi
Discipline: Structural mechanics
Accession: Summer 1997

These matrices result from finite element discretization of an octant of a cylindrical shell. The ends of the cylinder are free. A summary of the problem is given below; further details, including plots of the finite element meshes can be found in a separate description.

Notation: T indicates the Radius to thickness ratio, R/t = 10T; MO indicates the model type: DK = Discrete-Kirchhoff, RM = Reissner-Mindlin; EL indicates element type: T3 = 3-node triangular, Q4 = 4-node quadrilateral, etc.; MX indicates the mesh type: M1 = 30x30 uniform, M2 = 100x150 uniform, M3 = graded mesh of 1666 triangles (grading near one boundary and one vertex). The naming convention, cTMOELMX, is based on these.

The finite element is a facet type shell element (3/4 nodes) with drilling rotation incorporated by the Hughes-Brezzi technique and using the penalty parameter value of G/1000 (regularization parameter) , where G is the shear modulus. In order to improve the coarse mesh accuracy the membrane interpolation is amended by the Allman type quadratic modes linked to the drilling rotation. The bending formulation utilizes the stabilized MITC technique with the stabilization parameter equal to 0.4.

For iterative conjugate gradient type solvers, the problem gets harder when the radius to thickness ratio R/t increases. For quadrilateral meshes using the IC(0) preconditioner, iteration counts are about 100 (for R/t=10) and 180 (for R/t=1000) in reaching the relative residual norm of 10-9 depending slightly on the right-hand side vector. For matrices corresponding to triangular meshes the number of iterations doubles in comparison to quadrilateral ones.

The matrices are all symmetric positive definite.

The following estimates of largest and smallest eigenvalues and spectral condition numbers were provided by the author. These were computed using the Lanczos algorithm from QMRPACK (subroutine DSLAL) with 100 steps.

MatrixMax eigenvalueMin eigenvalueCondition No.
S1RMQ4M16.8743215039276076E+053.7969627252353000E-011.810479E+06
S2RMQ4M16.8743398166879822E+043.8746327712024738E-041.774191E+08
S3RMQ4M16.8745248885112433E+033.8936812501755643E-071.765559E+10
S1RMT3M19.6684220885668334E+053.7976650205204426E-012.545886E+06
S2RMT3M19.6681138121311058E+043.8747124280659988E-042.495182E+08
S3RMT3M19.6688590706791765E+033.8956266582265278E-072.481978E+10
S3DKQ4M24.6016534362482462E+032.4268618877322254E-081.896133E+11
S3DKT3M28.7984363691286471E+032.4269393992599454E-083.625322E+11
S3RMT3M39.5986080894852857E+033.9983547883056739E-072.400639E+10

The following matrices have been withdrawn from this set: C1RMQ4M1, C2RMT3M1, C3RMQ4M1, C1RMT3M1, C3DKQ4M2, C3RMT3M1, C2RMQ4M1, C3DKT3M2, C3RMT3M3. These matrices are identical to the current matrices with names starting with S, except that the current matrices are represented using 17 digits of precision. Some of the previous matrices were indefinite in the 6 digits in which they were represented.

## Matrices in this set:

• S1RMQ4M1 (real symmetric positive definite, 5489 by 5489, 143300 entries), Cylindrical shell, uniform 30x30 quadrilateral mesh, stabilized MITC4 elements, R/t=10
• S2RMQ4M1 (real symmetric positive definite, 5489 by 5489, 143300 entries), Cylindrical shell, uniform 30x30 quadrilateral mesh, stabilized MITC4 elements, R/t=100
• S3RMQ4M1 (real symmetric positive definite, 5489 by 5489, 143300 entries), Cylindrical shell, uniform 30x30 quadrilateral mesh, stabilized MITC4 elements, R/t=1000
• S1RMT3M1 (real symmetric positive definite, 5489 by 5489, 112505 entries), Cylindrical shell, uniform 30x30 triangular mesh, stabilized MITC3 elements, R/t=10
• S2RMT3M1 (real symmetric positive definite, 5489 by 5489, 112505 entries), Cylindrical shell, uniform 30x30 triangular mesh, stabilized MITC3 elements, R/t=100
• S3RMT3M1 (real symmetric positive definite, 5489 by 5489, 112505 entries), Cylindrical shell, uniform 30x30 triangular mesh, stabilized MITC3 elements, R/t=1000
• S3DKQ4M2 (real symmetric positive definite, 90449 by 90449, 2455670 entries), Cylindrical shell, uniform 150x100 quadrilateral mesh, R/t=1000
• S3DKT3M2 (real symmetric positive definite, 90449 by 90449, 1921955 entries), Cylindrical shell, uniform 150x100 triangular mesh, R/t=1000
• S3RMT3M3 (real symmetric positive definite, 5357 by 5357, 106526 entries), Cylindrical shell, graded mesh with 1666 triangles, R/t=1000

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