A finite element solution algorithm for the Navier-Stokes equations

A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame

Computational nonlinear vibration analysis for distributed geometrical nonlinearities in structural dynamics

The demand to reduce the impact of aviation on the environment is leading jet engine manu- facturers to increase the fuel and propulsion efficiency of the engines. This in turn is pushing materials to their physical limits by undergoing increasingly higher thermo-mechanical loads. In this regime, blades and other engine components are subjected to large deforma- tions generating nonlinearities that activate new failure mechanisms not dealt with before. Therefore, vibration analysis is essential to develop new methodologies for the accurate prediction of components’ behaviour. This research focuses on investigating the effect of the distributed geometric nonlinearities and rotational speed on the dynamic behaviour of three-dimensional structures. The Green-Lagrange strain measures are employed in this research to express the nonlinear relationship between the displacement and the strain. The nonlinear algorithms used for the numerical simulations are developed based on the Finite Element Method combined with the Harmonic Balance method. The complex geometries are discretised by using the geometric exact three-dimensional solid elements. The forced response functions and the backbone curves for the steady-state response of the nonlinear system can be computed. The research aims to develop and validate methodologies for the identification and control of undesired vibration modes which will inform new design choices. Finite element modelling of the blades generally involves an immense number of degree-of-freedoms, which could be infeasible to compute. The reduced order modelling (ROM) techniques are crucial for achieving an accurate prediction of the nonlinear behaviour in an efficient way. Detailed computation strategies for the intrusive ROM methods are delivered. ROM techniques based on the linear and nonlinear mapping between the full model and the reduced basis are presented. The capabilities and limitations of both methods are assessed. The projection method based on the linear eigenmodes only has a slow converge to the full system. On the other hand, the quadratic manifold method with the static modal derivatives involved in the reduced coordinates provides a fast convergence.Open Acces

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection-diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods

Eigenvalue problems for fully nonlinear elliptic partial differential equations with transport boundary conditions

Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations

Essays on Numerical Integration in Hamiltonian Monte Carlo

This thesis considers a variety of topics broadly unified under the theme of geometric integration for Riemannian manifold Hamiltonian Monte Carlo. In chapter 2, we review fundamental topics in numerical computing (section 2.1), classical mechanics (section 2.2), integration on manifolds (section 2.3), Riemannian geometry (section 2.5), stochastic differential equations (section 2.4), information geometry (section 2.6), and Markov chain Monte Carlo (section 2.7). The purpose of these sections is to present the topics discussed in the thesis within a broader context. The subsequent chapters are self-contained to an extent, but contain references back to this foundational material where appropriate. Chapter 3 gives a formal means of conceptualizing the Markov chains corresponding to Riemannian manifold Hamiltonian Monte Carlo and related methods; this formalism is useful for understanding the significance of reversibility and volume-preservation for maintaining detailed balance in Markov chain Monte Carlo. Throughout the remainder of the thesis, we investigate alternative methods of geometric numerical integration for use in Riemannian manifold Hamiltonian Monte Carlo, discuss numerical issues involving violations of reversibility and detailed balance, and propose new algorithms with superior theoretical foundations. In chapter 4, we evaluate the implicit midpoint integrator for Riemannian manifold Hamiltonian Monte Carlo, presenting the first time that this integrator has been deployed and assessed within this context. We discuss attributes of the implicit midpoint integrator that make it preferable, and inferior, to alternative methods of geometric integration such as the generalized leapfrog procedure. In chapter 5, we treat an empirical question as to what extent convergence thresholds play a role in geometric numerical integration in Riemannian manifold Hamiltonian Monte Carlo. If the convergence threshold is too large, then the Markov chain transition kernel will fail to maintain detailed balance, whereas a convergence threshold that is very small will incur computational penalties. We investigate these phenomena and suggest two mechanisms, based on stochastic approximation and higher-order solvers for non-linear equations, which can aid in identifying convergence thresholds or suppress its significance. In chapter 6, we consider a numerical integrator for Markov chain Monte Carlo based on the Lagrangian, rather than Hamiltonian, formalism in classical mechanics. Our contributions include clarifying the order of accuracy of this numerical integrator, which has been misunderstood in the literature, and evaluating a simple change that can accelerate the implementation of the method, but which comes at the cost of producing more serially auto-correlated samples. We also discuss robustness properties of the Lagrangian numerical method that do not materialize in the Hamiltonian setting. Chapter 7 examines theories of geometric ergodicity for Riemannian manifold Hamiltonian Monte Carlo and Lagrangian Monte Carlo, and proposes a simple modification to these Markov chain methods that enables geometric ergodicity to be inherited from the manifold Metropolis-adjusted Langevin algorithm. In chapter 8, we show how to revise an explicit integration using a theory of Lagrange multipliers so that the resulting numerical method satisfies the properties of reversibility and volume-preservation. Supplementary content in chapter E investigates topics in the theory of shadow Hamiltonians of the implicit midpoint method in the case of non-canonical Hamiltonian mechanics and chapter F, which treats the continual adaptation of a parameterized proposal distribution in the independent Metropolis-Hastings sampler

Solution of 3-dimensional time-dependent viscous flows. Part 3: Application to turbulent and unsteady flows

A numerical scheme is developed for solving the time dependent, three dimensional compressible viscous flow equations to be used as an aid in the design of helicopter rotors. In order to further investigate the numerical procedure, the computer code developed to solve an approximate form of the three dimensional unsteady Navier-Stokes equations employing a linearized block implicit technique in conjunction with a QR operator scheme is tested. Results of calculations are presented for several two dimensional boundary layer flows including steady turbulent and unsteady laminar cases. A comparison of fourth order and second order solutions indicate that increased accuracy can be obtained without any significant increases in cost (run time). The results of the computations also indicate that the computer code can be applied to more complex flows such as those encountered on rotating airfoils. The geometry of a symmetric NACA four digit airfoil is considered and the appropriate geometrical properties are computed

Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow

We present a computational framework for the simulation of blood flow with fully resolved red blood cells (RBCs) using a modular approach that consists of a lattice Boltzmann solver for the blood plasma, a novel finite element based solver for the deformable bodies and an immersed boundary method for the fluid-solid interaction. For the RBCs, we propose a nodal projective FEM (npFEM) solver which has theoretical advantages over the more commonly used mass-spring systems (mesoscopic modeling), such as an unconditional stability, versatile material expressivity, and one set of parameters to fully describe the behavior of the body at any mesh resolution. At the same time, the method is substantially faster than other FEM solvers proposed in this field, and has an efficiency that is comparable to the one of mesoscopic models. At its core, the solver uses specially defined potential energies, and builds upon them a fast iterative procedure based on quasi-Newton techniques. For a known material, our solver has only one free parameter that demands tuning, related to the body viscoelasticity. In contrast, state-of-the-art solvers for deformable bodies have more free parameters, and the calibration of the models demands special assumptions regarding the mesh topology, which restrict their generality and mesh independence. We propose as well a modification to the potential energy proposed by Skalak et al. 1973 for the red blood cell membrane, which enhances the strain hardening behavior at higher deformations. Our viscoelastic model for the red blood cell, while simple enough and applicable to any kind of solver as a post-convergence step, can capture accurately the characteristic recovery time and tank-treading frequencies. The framework is validated using experimental data, and it proves to be scalable for multiple deformable bodies

Time-dependent coupled-cluster for ultrafast spectroscopy

The ultimate reason for chemical reactivity is the electronic motion, occurring at an attosecond timescale. Until the last century, it was impossible to observe it directly, as the shortest available laser pulses had duration in the order of femtoseconds. Recent technological advances lead to sub-femtosecond laser pulses, making possible real-time observation and control of electron dynamics.My Ph.D. thesis aims to develop and implement a model for the interaction between ultrashort laser pulses and molecules. This is interesting as an extension of the theory and the computational tools available, to design experiments at laser facilities, and to predict and interpret their outcomes.The theoretical framework that we have chosen is the time-dependent coupled-cluster (TDCC) theory. We have implemented our code in the eT program, which represents the first released implementation of a TDCC method.After validating our procedures by comparison with the literature, we used our code to calculate the electronic response to a pump-probe sequence of laser pulses. We performed convergence tests of parameters on the LiH. Then, we observed and interpreted the effect of the delay between pump and probe pulses on the LiF transient absorption spectrum.We extended this implementation to a time-dependent equation-of-motion coupled-cluster (TD-EOM-CC) approach with the use of a reduced basis calculated with an asymmetric band Lanczos algorithm, and within the core-valence separation (CVS) approximation. This converged to the same spectral features as the TDCC but with much lower computational times, as we showed for LiF. We observed the limits of CVS approximation: for the LiH molecule, several peaks were not correctly retrieved. Finally, we modeled the transient absorption for the glycine molecule, which is a good candidate for experimental investigations.We also modeled the electronic impulsive stimulated Raman scattering (ISXRS) population transfer induced by an ultrashort laser pulse through the TD-EOM-CC model for Ne, CO, pyrrole, and p-aminophenol and visualized through a movie the real-time evolution of the electronic density of p-aminophenol.The significance of this work lies in the development of theoretical and computational tools to be used in attochemistry: one groundbreaking application can be the direct control of electrons, which would have a big impact on many research fields, like medicine, biology, and material science