An Arbitrary Order Variationally Consistent Integration Method for Galerkin Meshfree Methods
- Author(s): Hillman, Michael Charles
- Advisor(s): Chen, Jiun-Shyan
- et al.
The rate of convergence in Galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration. If insufficiently accurate domain integration is employed, the optimal convergence rate cannot be attained. For meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature, and the lack of accuracy in the domain integration may lead to sub-optimal convergence, or even solutions that diverge with refinement. The difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves. This dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature.
First, the conditions for achieving arbitrary order exactness in a boundary value problem using the Galerkin approximation with quadrature are derived. The conditions are derived in a general form and are applicable to all types of problems: the test function gradients in the Galerkin approximation should be consistent with the chosen numerical integration, and this is termed variational consistency. Specifically, integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature. Specific problems are then considered with the conditions explicitly stated, including elasticity, the Euler-Bernoulli beam, the Kirchhoff-Love plate, and the non-linear formulation of solid mechanics.
Treating the type of numerical integration as a given, test function gradients are then constructed to satisfy this condition. The resulting method is arbitrarily high order exact and applicable to all types of integration. The method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems. It is demonstrated that the error induced by numerical integration is greatly reduced, and optimal convergence is either partially or fully restored. Further, it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error.