Finite element formulation for modelling nonlinear viscoelastic elastomers

Nonlinear viscoelastic response of reinforced elastomers is modeled using a three-dimensional mixed finite element method with a nonlocal pressure field. A general second-order unconditionally stable exponential integrator based on a diagonal Padé approximation is developed and the Bergström–Boyce nonlinear viscoelastic law is employed as a prototype model. An implicit finite element scheme with consistent linearization is used and the novel integrator is successfully implemented. Finally, several viscoelastic examples, including a study of the unit cell for a solid propellant, are solved to demonstrate the computational algorithm and relevant underlying physics

Condition number analysis and preconditioning of the finite cell method

The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method

High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality

In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context

Space-time hybridizable discontinuous Galerkin methods for free-surface wave problems

Free-surface problems arise in many real-world applications such as in the design of ships and offshore structures, modeling of tsunamis, and dam breaking. Mathematically, free-surface wave problems are described by a set of partial differential equations that govern the movement of the fluid together with certain boundary conditions that describe the free-surface. The numerical solution of such problems is challenging because the boundary of the computational domain depends on the solution of the problem. This implies that there is a strong coupling between the fluid and the free-surface, and the domain must be continuously updated to track the changes in the free-surface. In this thesis we explore and develop space-time hybridizable discontinuous Galerkin (HDG) methods for free-surface problems. First, we focus on a linear free-surface problem in which the amplitude of the waves is assumed to be small enough so that the domain can remain fixed. We initially consider a traditional approach for the numerical discretization of time-dependent partial differential equations: we discretize in space using, in this case, an HDG method to obtain an ordinary differential equation. Then, we use a second order backward differentiation formula to discretize in time. We see that in comparison to an interior penalty discontinuous Galerkin discretization, this HDG discretization results in smaller linear systems (in general), and produces better approximations to the velocity of the fluid. Next, we consider the solution of the same linear free-surface problem with a space-time hybridizable discontinuous Galerkin method. Unlike previous finite element discretizations of this problem, we consider a mixed formulation in which the velocity of the flow can be approximated with an optimal order of convergence. We develop a set of space-time analysis tools that allow us to obtain a priori error estimates in which the dependency on the spatial mesh size and the time step is explicit. This is in contrast to previous space-time error analyses in which the error bounds depend on the size of the space-time elements. Finally, we move on to incompressible nonlinear free-surface flow. We consider the two-fluid (gas and liquid) Navier-Stokes equations and use a level set method in which the flow and the level set equations are solved subsequently until a certain stopping criterion has been met. The flow equations are solved with a space-time HDG method which is exactly mass conserving. Furthermore, a space-time embedded discontinuous Galerkin method is employed for the solution of the level set equation. This discretization possesses the same conservation and stability properties as discontinuous Galerkin methods, but produces a continuous approximation to the free-surface elevation. When a discontinuous approximation to the free-surface elevation is obtained, smoothing techniques have to be applied in order to move the mesh and track the interface. It has been shown in the past that such techniques can lead to instabilities and stabilization terms have to be added to the discretization. Therefore, obtaining a continuous approximation to the free-surface elevation in our discretization is crucial: not only can the mesh be deformed in a straightforward manner, but it can also be done without introducing any potential sources of instabilities. We present two numerical results that demonstrate the capabilities of the method. In the first test case we compare against an analytical solution and we demonstrate how the mesh conforms to the interface between the two fluids. Finally, we present a simulation of waves generated by a submerged obstacle

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

In this work we exploit agglomeration based $h$-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature $h$-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element $L^2$ projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.Comment: 78 pages, 7 figure