Preprint: Consistent Weak Forms for Meshfree Methods: Full Realization of h-refinement, p-refinement, and a-refinement in Strong-type Essential Boundary Condition Enforcement
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Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the approximation such that this is possible. However, with these methods, test and trial functions do not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency results in the loss of Galerkin orthogonality and best approximation property, and correspondingly, failure to pass the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order boundary value problems that is independent of the order of completeness in the approximation. As a result, this places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement with p th order rates with strong enforcement of boundary conditions beyond linear (p > 1) for the first time. In addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing the kernel measure in meshfree approximations, previously unavailable.