Iterative Solver for Linear Equation Systems
Direct factorization techniques have been the norm in standard commercial
finite element software. However, over the last several years,
iterative solution schemes for linear systems of equations
have made their way into commercial finite element
programs due to the two benefits
they offer: less CPU time and less memory.
The iterative solver from Visual Kinematics
is a standard feature in the VfsTools component software library.
It is based on a Preconditioned Conjugate Gradient method, and
is suitable for all symmetric, positive definite matrices. Our many years
of experience in the finite element arena has resulted in an extremely fast
iterative solver especially designed for finite element problems. The
performance is very competitive
for both continuum elements (e.g. 10-node tetrahedra),
and shell elements (e.g. 8-node quadrilaterals).
The performance characteristics are retained even in the thin shell limit.
Two examples below illustrate the iterative solver performance.
Each simulation was run on a 175MHz MIPS R10000
Silicon Graphics Workstation. The quoted times for the iterative solver
include only the CPU time spent in the analysis performing the solution of
the linear system of equations for a single load vector.
This includes all time required to form the preconditioner and perform
the iterative solution.
It specifically does not include the time for element stiffness
formulation and
stress computation.
Both problems used less than 512 Mbytes of memory and no disk space.
10-Node Tetrahedron Example
The pressure loaded model is shown below. It contains 74,478 nodes,
and 39,630 elements.
After supressing 4,374 zero boundary conditions, the remaining linear
system consists of 219,057 equations.
The linear equation solution time
was 181 seconds,
Displacement magnitude contours are shown in the figure below.
8-Node Quadrilateral Shell Example
Unlike continuum elements, shell-based problems are known for leading to
ill-conditioned systems, as membrane, bending, and transverse shear effects
are orders of magnitude apart.
The problem analyzed is shown below. Its planar dimensions are 1,319mm by
1,575mm, with a thickness of 3mm. The model consists of 37,302 nodes and
12,275 elements, and is loaded via a uniform pressure. After suppressing
2,124 zero boundary conditions, the resulting linear system consists of
221,688 equations.
The linear equation solution time was 315 seconds.
Displacement magnitude contours are shown below.
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