1,557 research outputs found
Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods
Computational science, including computational fluid dynamics (CFD), has become an indispensible tool for scientific discovery and engineering design, yet a key remaining challenge is to simultaneously ensure accuracy, efficiency, and robustness of the calculations. This research focuses on advancing a class of high-order finite element methods and develops a set of algorithms to increase the accuracy, efficiency, and robustness of calculations involving convection and diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses high-order discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for simultaneous mesh and order adaptation to reduce the error in a scalar functional of the approximate solution to the discretized equations. Contributions are made in key aspects of these methods applied to general systems of equations, addressing the scalability and memory requirements, accuracy of HDG methods, and efficiency and robustness with new adaptation methods.
First, this work generalizes existing HDG methods to systems of equations, and in so doing creates a new primal formulation by applying DG stabilization methods as the viscous stabilization for HDG. The primal formulation is shown to be even more computationally efficient than the existing methods. Second, by instead keeping existing viscous stabilization methods and developing a new convection stabilization, this work shows that additional accuracy can be obtained, even in the case of purely convective systems. Both HDG methods are compared to DG in the same computational framework and are shown to be more efficient.
Finally, the set of adaptation frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options. The second framework intended for simplex meshes extends the mesh optimization via error sampling and synthesis (MOESS) method to incorporate order adaptation.
Collectively, the results from this research address a number of key issues that currently are at the forefront of high-order CFD methods, and particularly to output-based hp-adaptation for DG and HDG methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/137150/1/jdahm_1.pd
The LifeV library: engineering mathematics beyond the proof of concept
LifeV is a library for the finite element (FE) solution of partial
differential equations in one, two, and three dimensions. It is written in C++
and designed to run on diverse parallel architectures, including cloud and high
performance computing facilities. In spite of its academic research nature,
meaning a library for the development and testing of new methods, one
distinguishing feature of LifeV is its use on real world problems and it is
intended to provide a tool for many engineering applications. It has been
actually used in computational hemodynamics, including cardiac mechanics and
fluid-structure interaction problems, in porous media, ice sheets dynamics for
both forward and inverse problems. In this paper we give a short overview of
the features of LifeV and its coding paradigms on simple problems. The main
focus is on the parallel environment which is mainly driven by domain
decomposition methods and based on external libraries such as MPI, the Trilinos
project, HDF5 and ParMetis.
Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar
Optimum experimental design for parameter estimation with 2D partial differential equation models
In this thesis, we investigate optimization problems with partial differential equation (PDE) constraints. In particular we are concerned with the efficient numerical solution of optimum experimental design (OED) problems for parameter estimation (PE) with PDE models, among them sampling design problems. We consider two dimensional (2D) stationary diffusion advection reaction PDE models, including the challenging case of an advection dominated PDE.
For the simulation of the PDE boundary value problem, we utilize discontinuous Galerkin finite element methods and adaptive spatial grid refinement. We solve the optimization problems with derivative-based algorithms. For the optimization algorithms to converge fast and to converge to the âtrueâ optimum, we need to provide accurate sensitivities. It is a challenge to evaluate the sensitivities, which correspond to the approximate solution of the primal PDE model and are in this sense consistent. In this thesis we develop efficient and accurate methods for sensitivity generation. We transfer the principle of internal numerical differentiation (IND) from ordinary differential equations (ODE)s to PDEs. That means, we incorporate the sensitivity generation in the solution process. The standard upwind discontinuous Galerkin method is not differentiable. Therefore, we propose a differentiable discontinuous Galerkin method and give a rigorous convergence analysis of it. We develop methods for structure exploitation of the primal and tangential discretization schemes to efficiently generate the sensitivities with automatic differentiation (AD). Furthermore, we establish methods for frozen adaptivity to generate consistent sensitivities. We are especially concerned with frozen spatial grid refinement and the adaptive step number of the linear solver.
We implement the developed methods in the software SeafaND-Optimizer, short for structure exploiting and frozen adaptivity numerical differentiation optimizer. It is a software for efficient and accurate simulation, PE and OED with PDE models. We perform numerical case studies for PE and OED problems with advection dominated 2D diffusion advection PDE models. With the structure exploiting techniques developed in this thesis, the example problems are solved with efficient memory usage. Due to the frozen adaptivity methods, we computed efficiently the consistent sensitivities. We test the PE algorithm with different noise levels. We perform a case study with different diffusion coefficients for sequential OED. Finally, we investigate, whether the developed methods are stable under mesh refinements
A numerical approach to the optimal control of thermally convective flows
The optimal control of thermally convective flows is usually modeled by an
optimization problem with constraints of Boussinesq equations that consist of
the Navier-Stokes equation and an advection-diffusion equation. This optimal
control problem is challenging from both theoretical analysis and algorithmic
design perspectives. For example, the nonlinearity and coupling of fluid flows
and energy transports prevent direct applications of gradient type algorithms
in practice. In this paper, we propose an efficient numerical method to solve
this problem based on the operator splitting and optimization techniques. In
particular, we employ the Marchuk-Yanenko method leveraged by the
projection for the time discretization of the Boussinesq equations so
that the Boussinesq equations are decomposed into some easier linear equations
without any difficulty in deriving the corresponding adjoint system.
Consequently, at each iteration, four easy linear advection-diffusion equations
and two degenerated Stokes equations at each time step are needed to be solved
for computing a gradient. Then, we apply the Bercovier-Pironneau finite element
method for space discretization, and design a BFGS type algorithm for solving
the fully discretized optimal control problem. We look into the structure of
the problem, and design a meticulous strategy to seek step sizes for the BFGS
efficiently. Efficiency of the numerical approach is promisingly validated by
the results of some preliminary numerical experiments
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
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