Getting a stiffness matrix out of OpenSees is straightforward using printA(). But what about getting a stiffness matrix into OpenSees? This is the situation I faced recently testing BennySparse with linear systems from the SuiteSparse Matrix Collection. I had no way of building an OpenSees model that could recreate those matrices. After some trial and error, I found that … Continue reading Bring Your Own Matrix
Tag: Equation solver
Choose Your Own Topology
I've been working on a sparse linear equation solver. Not anything new, just implementing the methods presented by Timothy Davis in Direct Methods for Sparse Linear Systems. Why? Because I want to learn how sparse matrix solvers work. So for the sake of discussion, let's call my solver BennySparse. Aside from implementing a standalone sparse … Continue reading Choose Your Own Topology
The Good, the Not So Bad, and the Full General
Just like shopping for a new refrigerator, picking a linear equation solver in OpenSees (via the system command) can lead to paralysis of choice. And while you can consult Consumer Reports for the pros and cons of refrigerators A, B, and C, the only way to figure out the pros and cons of OpenSees solvers … Continue reading The Good, the Not So Bad, and the Full General
Murum, cura te ipsum
OpenSees has its fair share of element implementations that are computationally inefficient. Fortunately, most of those elements are never used. But among elements that are used, SFI-MVLEM is the undisputed champion. Whereas the standard MVLEM element uses a uniaxial material in each fiber, the SFI-MVLEM element accounts for the interaction of axial and shear stress ($latex \sigma_{11}$ … Continue reading Murum, cura te ipsum
Like Spinning Nodes
After posting on reasons that the solution to Ax=b fails, I realized I omitted an important case: truss nodes in a frame model. Although this post might be a stretch for an LBU (least bloggable unit), the blogging equivalent of an LPU, there are important factors to consider for structural models comprised of truss and … Continue reading Like Spinning Nodes
Failure to Solve
Solving a system of simultaneous linear equations, canonically referred to as solving Ax=b in math speak, is at the heart of every equilibrium solution algorithm for nonlinear analysis. In the context of OpenSees, A is the effective tangent stiffness matrix, x is the vector of displacement increments, and b is the residual force vector. However, … Continue reading Failure to Solve
Don’t Try This at Home
The central difference method is an explicit integrator that forms a linear combination of the mass and damping matrices to advance the solution to the next time step. So, if the mass matrix is lumped and there's no damping, or only mass-proportional damping, the left-hand side matrix is diagonal. Then, you can solve the system … Continue reading Don’t Try This at Home
Closing the Loop on Direct Assembly
All structural engineering students learn the direct assembly method, where you fix all degrees of freedom (DOFs) in a structural model, then impose a unit value of displacement at and in the direction of the $latex j^{th}$ DOF in order to get the $latex j^{th}$ column of the stiffness matrix from the fixed-end forces of … Continue reading Closing the Loop on Direct Assembly
Most Solvers Can Be Marplots
Have you ever tried to replicate the familiar beam stiffness coefficients $latex 12EI/L^3$, $latex 6EI/L^2$, $latex 4EI/L$, and $latex 2EI/L$ (there's a poem about them here) by imposing unit displacements and rotations at fixed supports? It should be one of the first sanity checks you make when using or developing new structural analysis software. You … Continue reading Most Solvers Can Be Marplots
Handle Your Constraints with Care
Manipulating the nodal equilibrium equations is necessary to enforce constraints between degrees of freedom (DOFs) at two or more nodes in a structural model. These multi-point constraints arise from assumptions of axial and flexural rigidity of frame elements, e.g., rigid diaphragms, and also between two nodes at the same location where some of the DOFs … Continue reading Handle Your Constraints with Care
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