Numerical investigation and application of immersogeometric and direct discontinuous Galerkin methods
Date
2021-08
Authors
Muchowski, Heather Marie
Major Professor
Advisor
Hsu, Ming-Chen
Yan, Jue
Passalacqua, Alberto
Rossmanith, James
Krishnamurthy, Adarsh
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Abstract
Discontinuous Galerkin (DG) methods were introduced to numerically solve partial differential equations. Since then, they have been prominent in many applications. Nitsche's method is a commonly studied finite element method that was later proposed as a DG method specifically for diffusion problems. It has become widely used to weakly enforce Dirichlet boundary conditions in various applications. The weak imposition of no-slip conditions has been beneficial particularly in immersed methods. In this work, we utilize a specific immersed method called immersogeometric analysis that makes use of Nitsche's method to weakly impose the no-slip condition.
Immersogeometric analysis has been an attractive technology as it allows one to discretize the fluid and structure subproblems separately. When using this method, the fluid mesh is arbitrarily cut by the structural boundary, thus producing a boundary layer discretization that is much too coarse for smooth solutions near the immersed surface. Hence, weak imposition of Dirichlet boundary conditions is crucial to obtaining accurate flow solutions. However, when using this Nitsche-type formulation for thin shell problems, the method degenerates to pure penalty method and the disadvantages for penalty methods become apparent. Penalty methods require sufficiently large penalty parameter but not too large to cause ill-conditioning of the matrices. This disadvantage of penalty methods has motivated the development of new techniques that require much lower penalty.
In this work, we discuss a particular subclass of DG methods called the direct discontinuous Galerkin (DDG) method, which has been shown to need much lower penalty values for convergence. The DDG method introduces a new definition for the numerical flux term that penalizes the solution jump across discontinuous element boundaries. The formulation also utilizes the average of the spatial gradient and the second order jump term. The DDG method has been shown to require much lower penalty to obtain high order accurate solutions and has a unique super convergence feature when studying the convergence under a weak norm. We numerically investigate the super convergence of the DDG methods and discuss how the penalty parameters may be chosen to achieve optimal convergence.
We study the DDG method in hopes to incorporate this method into our immersogeometric framework to study a new application of endovascular aneurysm repair (EVAR). EVAR is a minimally invasive repair technique for aortic aneurysms.
The safe anchoring of the endograft in the patient's anatomy is crucial for the success of the procedure. Due to the long-term complications of this procedure, we develop and apply an immersogeometric fluid--structure interaction framework for the modeling and simulation of the EVAR procedure to determine the likelihood of migration. We propose a geometrically flexible geometry modeling procedure in which we utilize periodic curves and surfaces to impose higher continuity on the baseline geometry of the endograft. The endograft is then crimped and deployed into an idealized aortic aneurysm. We use physiologically realistic boundary conditions to fully simulate the endograft through the cardiac cycle. Finally, we determine the likelihood of migration given the current configuration and aorta geometry.
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dissertation