Browsing by Author "Scovazzi, Guglielmo"

In this work, a stabilized finite element framework is developed to simulate small and large deformation solid mechanics problems involving complex geometries and complicated constitutive models. In particular, the focus is on solid dynamics problems involving nearly and fully incompressible materials. The work is divided into three main themes, the first is concerned with the development of stabilized finite element algorithms for hyperelastic materials, the second handles the case of viscoelastic materials, and the third focuses on algorithms for J2-plastic materials. For all three cases, problems in the small and large deformation regime are considered, and for the J2-plasticity case, both quasi-static and dynamic problems are examined.

Some of the key features of the algorithms developed in this work is the simplicity of their implementation into an existing finite element code, and their applicability to problems involving complicated geometries. The former is achieved by using a mixed formulation of the solid mechanics equations where the velocity and pressure unknowns are represented by linear shape functions, whereas the latter is realized by using triangular elements which offer numerous advantages compared to quadrilaterals, when meshing complicated geometries. To achieve the stability of the algorithm, a new approach is proposed in which the variational multiscale approach is applied to the mixed form of the solid mechanics equations written down as a first order system, whereby the pressure equation is cast in rate form.

Through a series of numerical simulations, it is shown that the stability properties of the proposed algorithm is invariant to the constitutive model and the time integrator used. By running convergence tests, the algorithm is shown to be second order accurate, in the $L^2$-nrom, for the displacements, velocities, and pressure. Finally, the robustness of the algorithm is showcased by considering realistic test cases involving complicated geometries and very large deformation.

Numerical simulation of hyperbolic systems remains a challenge, particularly in the case of complex geometries. In particular, the need to construct meshes for complicated geometries is a bottleneck in many cases. This is especially evident when doing rapid prototyping and design optimization, where generating a new mesh for every trial geometry is prohibitive. These difficulties can be obviated by employing an embedded/immersed boundary method, in which boundary conditions are enforced weakly.

In my Ph.D. work, a new Nitsche-type approach is proposed for the weak enforcement of Dirichlet and Neumann boundary conditions in the context of time-domain wave propagation problems in mixed form. A peculiar feature of the proposed method is that, due to the hyperbolic structure of the problem considered, two penalty parameters are introduced, corresponding to Dirichlet and Neumann conditions, respectively. A stability and convergence estimate is also provided, in the case of a discontinuous-in-time Galerkin space–time integrator. The spatial discretization used is based on a stabilized method with equal order interpolation for all solution components. In principle, however, the proposed methodology is not confined to stabilized methods. An extensive set of tests are provided to validate the robustness and accuracy of the proposed approach.

The proposed Nitsche method is then extended to embedded domain computations of hyperbolic systems, using as models the equations of acoustic wave propagation and shallow water flows, through Shifted Boundary Method (SBM). The SBM belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on grids that conform to the geometry of the computational domain boundaries are also presented.

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.

Numerical computations involving complex geometries have a wide variety of applications in both science and engineering, including the simulation of fractures, melting and solidification, multiphase flows, biofilm growth, etc. Classical finite element methods rely on computational grids that are adapted (fitted) to the geometry, but this approach creates fundamental computational challenges, especially when considering evolving interfaces/boundaries. Embedded methods facilitate the treatment of complex geometries by avoiding fitted grids in favor of immersing the geometry on pre-existing grids.

The first part of this thesis work introduces a new embedded finite element interface method, the shifted interface method (SIM), to simulate partial differential equations over domains with internal interfaces. Our approach belongs to the family of surrogate/approximate interface methods and relies on the idea of shifting the location and value of interface jump conditions, by way of Taylor expansions. This choice has the goal of preserving optimal convergence rates while avoiding small cut cells and related problematic issues, typical of traditional embedded methods. In this part, SIM is applied to internal interface computations in the context of the mixed Poisson problem, also known as the Darcy flow problem and is extended to linear isotropic elasticity interface problems. An extensive set of numerical tests is provided to demonstrate the accuracy, robustness and efficiency of the proposed approach.

In the second part of the thesis, we propose a new framework for linear fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. The approach evolves from SIM, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry. Similar to SIM, the interface conditions on the surrogate fracture are modified with Taylor expansions to achieve a prescribed level of accuracy. This shifted fracture method (SFM) does not require cut cell computations or complex data structures, since the behavior of the true fracture is mimicked with standard integrals on the approximate fracture. Furthermore, the energetics of the true fracture are represented within the prescribed level of accuracy and independently of the grid topology. We present a general computational framework and then apply it in the specific context of cohesive zone models, with an extensive set of numerical experiments in two and three dimensions.

In the third and final part, we develop a shifted boundary method (SBM), originated from Main and Scovazzi (2018), for the thermoelasticity equations. SBM requires to shift the location and value of both Dirichlet and Neumann boundary conditions to surrogate boundaries with Taylor expansions. In such a way, an opti- mal convergence rate can be preserved for both temperature and displacement. An extensive of numerical examples in both two and three dimensions are presented in this part to demonstrate the performance of SBM.