Computing the null space of finite element problems
We present a method for computing the null space of finite element models, including models with equality constraints. The method is purely algebraic; it requires access to the element matrices, but not to the geometry or material properties of the model. Theoretical considerations show that under certain conditions, both the amount of computation and the amount of memory required by our method scale linearly with model size; memory scales linearly but computation scales quadratically with the dimension of the null space. Our experiments confirm this: the method scales extremely well on 3-dimensional model problems. In general, large industrial models do not satisfy all the conditions that the theoretical results assume; however, experimentally the method performs well and outperforms an established method on industrial models, including models with many equality constraints. The accuracy of the computed null vectors is acceptable, but the method is usually less accurate than a more naive (and computationally much more expensive) method.
null space finite element models rigid-body motions linear constraints singular linear systems fretsaw preconditioner