Stable Embedded Grid Techniques in Computational Mechanics
Engineering mechanics problems involving interfaces, whether physical or introduced by numerical methodologies, are commonplace. Just a few examples include fracture and fault mechanics, classical contact-impact, phase boundary propagation, and fluid-structure interaction. This dissertation focuses on issues of numerical stability and accuracy that must be addressed when such interfaces are included in a realistic simulation of a physical system.
We begin by presenting a novel numerical method of fluid/structure interaction that may be applied to the problem of movable devices and ocean waves. The work is done with finite differences, large motion Lagrangian mechanics, and an eye towards creating a model in which complex rigid body dynamics can be incorporated.
We then review the many advantages of embedded mesh techniques for interface representation, and investigate a completely finite element based strategy for embedding domains. The work is seen as a precursor to robust multi-physics simulation capabilities. Special attention must be given to these techniques in terms of stable and convergent representation of surface fluxes. Mesh locking and over-constraint are particularly addressed. We show that traditional methods for enforcing continuity at embedded interfaces cannot adequately guarantee flux stability, and show a less traditional method, known as Nitsche's method, to be a stable alternative. We address the open problem of applying Nitsche's method to non-linear analysis by drawing parallels between the embedded mesh and discontinuous Galerkin (DG) methods, and propose a DG style approach to Nitsche's method. We conclude with stable interfacial fluxes for a continuity constraint for a case of embedded finite element meshes in large deformation elasticity. The general conclusion is drawn that stability must be addressed in the choice of interface treatment in computational mechanics.
finite element method
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