The Second Generation Shifted Boundary Method with Applications to Porous Media Flow and Solid Mechanics


Scovazzi, Guglielmo

Atallah, Nabil





Civil and Environmental Engineering


Complex geometries has been a challenge to numerical algorithms. For classical body-fitted computational techniques, the challenge manifests itself in the time consuming and labor intensive grid generation phase. While for standard embedded/immersed methods, the challenge is mainly in the complicated and computationally intensive geometric construction of the partial elements cut by the embedded boundary.

Recently, the shifted boundary method (SBM) was proposed by Main and Scovazzi within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids the geometric construction of the cut elements and maintains accuracy by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions.

The first part of this thesis is devoted to the development and numerical analysis of the enhanced variational SBM formulations for the Poisson and Stokes problems over their original counterparts. First, we show that these second-generation SBM implementations can be proved asymptotically stable and convergent without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L2-estimates without the cumbersome assumption - of earlier proofs - that the surrogate domain is convex.

As for the second part of the thesis, we adopt the second generation formulations as a reference point and propose a new SBM framework for the flow in porous media (Darcy flow) equations. In particular, we develop equal-order discontinuous Galerkin (DG) in addition to continuous Galerkin (CG) discretizations to accurately capture the velocity and pressure fields under highly anisotropic and/or heterogeneous porous media. We corroborate our CG and DG schemes with a full analysis of stability and convergence in addition to extensive tests in two and three dimensions. The value of this approach is clearly visible from the 3D simulation of water flow around tree roots which was otherwise not possible with a body-fitted approach.

In third and final part of this thesis, we develop a SBM framework for the solid mechanics equations; in particular, the equations of linear isotropic elastostatics. The main challenge was with handling traction boundary conditions for a displacement-based, irreducible formulation of the SBM in combination with piecewise linear finite element spaces. We circumvented this problem by transforming the displacement-based equation into a mixed strain-displacement one; resulting in a system of equations akin to the Darcy flow one. The net result is a more accurate approximation of stresses and strains in exchange for an increase in the computational and storage costs. If this tradeoff is deemed unacceptable, the mixed formulation is restricted to a layer of elements (of unit depth) in proximity of the surrogate (approximate) boundary, while applying a standard primal formulation everywhere else. The net result is an enhanced formulation that maintains the bulk cost of the base primal formulation, but allows for an accurate imposition of boundary conditions. A full analysis of stability and convergence of the method is presented and complemented with an extensive set of computational experiments in two and three dimensions, for progressively more complex geometries.



Applied mathematics


Complex geometries


Darcy flow


Embedded/Immersed method


Finite Element Method


Numerical Analysis


Solid mechanics


The Second Generation Shifted Boundary Method with Applications to Porous Media Flow and Solid Mechanics




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