Mathematics Colloquia and Seminars

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Over the past decade, meshfree Galerkin methods have shown promise over traditional finite elements for large deformation simulations in solid mechanics. As opposed to finite elements, the basis functions in meshfree methods are constructed using just the nodal coordinates, without the need for any notion of element structure and nodal connectivity. Recently, maximum-entropy (max-ent) approximants have come to the forefront---these basis functions are obtained via a constrained optimization problem in which the Shannon-Jaynes entropy functional is maximized subject to the linear reproducing conditions as the constraints. In the numerical implementation, the Lagrangian dual problem (unconstrained convex minimization) is solved using a Newton method to compute the basis functions. In this talk, I will present some of our recent work on max-ent meshfree method, with focus on applications in solid mechanics and computer graphics. We devise a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction. Secondly, a locking-free small-strain elasticity formulation for max-ent meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. Various benchmark problems in two-dimensional near-incompressible small strain elasticity and Stokes flow will be presented to demonstrate the accuracy and optimal convergence in the energy norm of the meshfree formulation.