Toward a Simple, Accurate Lagrangian Hydrocode.

Lagrangian hydrocodes play an important role in the computation of transient, compressible, multi-material flows. This research was aimed at developing a simply constructed cell-centered Lagrangian method for the Euler equations that respects multidimensional physics while achieving second-order accuracy. Algorithms that can account for the multidimensional physics associated with acoustic wave propagation and vorticity transport are needed in order to increase accuracy and prevent mesh imprinting. Many of the building blocks of traditional finite volume schemes, such as Riemann solvers and spatial gradient limiters, have their foundations in one-dimensional ideas and so were not used here. Instead, multidimensional point estimates of the fluxes were computed with a Lax-Wendroff type procedure and then nonlinearly modified using a temporal flux limiting mechanism. The linear acoustic equations were used as a simplified test environment for the Lagrangian Euler system. Here Lax-Wendroff methods that exactly preserve vorticity were investigated and found to resist mesh imprinting. However, the dispersion properties of the schemes were poor and so third-order accurate vorticity preserving methods were developed to remedy the problem. The third-order methods guided the construction of a temporal limiting mechanism, which was then used in a vorticity preserving flux-corrected transport scheme. While the acoustic work was interesting in its own right, it also proved to be a useful stepping stone to Lagrangian hydrodynamics. The acoustics algorithms were extended to produce the Simple Lagrangian Method (SLaM). Standard test problems have shown that a first-order accurate version of the method is able to resist mesh imprinting and spurious vorticity despite its minimalistic structure. SLaM is capable of second-order accuracy with a simple parameter change and some preliminary work was done to extend the temporal flux limiting ideas from acoustics to the Lagrangian case. The limited SLaM method converges at second-order for smooth data and is able to capture shocks without producing large unphysical oscillations.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113577/1/tblung_1.pd

Two-dimensional evaluation of atham-fluidity, a nonhydrostatic atmospheric model using mixed continuous/discontinuous finite elements and anisotropic grid optimization

AbstractThis paper presents the first attempt to apply the compressible nonhydrostatic Active Tracer High-Resolution Atmospheric Model–Fluidity (ATHAM-Fluidity) solver to a series of idealized atmospheric test cases. ATHAM-Fluidity uses a hybrid finite-element discretization where pressure is solved on a continuous second-order grid while momentum and scalars are computed on a first-order discontinuous grid (also known as ). ATHAM-Fluidity operates on two- and three-dimensional unstructured meshes, using triangular or tetrahedral elements, respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run time. The solver is evaluated using two-dimensional-only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large-scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite–Weighted Essentially Nonoscillatory (Hermite-WENO) slope limiter are discussed. These cases are also used to test the grid optimization algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a sixfold decrease in computational time and a fivefold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step toward the development of a new comprehensive limited area atmospheric model.This research has received funding from the European Union Seventh Framework Program (FP7/2007-2013) under Grant agreement 603663 for the research project PEARL (Preparing for Extreme And Rare events in coastaL regions). The EPSRC multiphase program grant MEMPHIS is also acknowledged.This is the author accepted manuscript. The final version is available from the American Meteorological Society via http://dx.doi.org/10.1175/MWR-D-15-0398.

Adaptive mesh simulations of compressible flows using stabilized formulations

This thesis investigates numerical methods that approximate the solution of compressible flow equations. The first part of the thesis is committed to studying the Variational Multi-Scale (VMS) finite element approximation of several compressible flow equations. In particular, the one-dimensional Burgers equation in the Fourier space, and the compressible Navier-Stokes equations written in both conservative and primitive variables are considered. The approximations made for the VMS formulation are extensively researched; the design of the matrix of stabilization parameters, the definition of the space where the subscales live, the inclusion of the temporal derivatives of the subscales, and the non-linear tracking of the subscales are formulated. Also, the addition of local artificial diffusion in the form of shock capturing techniques is included. The accuracy of the formulations is studied for several regimes of the compressible flow, from aeroacoustic flows at low Mach numbers to supersonic shocks. The second part of the thesis is devoted to make the solution of the smallest fluctuating scales of the compressible flow affordable. To this end, a novel algorithm for $h-$refinement of computational physics meshes in a distributed parallel setting, together with the solution of some refinement test cases in supercomputers are presented. The definition of an explicit a-posteriori error estimator that can be used in the adaptive mesh refinement simulations of compressible flows is also developed; the proposed methodology employs the variational subscales as a local error estimate that drives the mesh refinement. The numerical methods proposed in this thesis are capable to describe the high-frequency fluctuations of compressible flows, especially, the ones corresponding to complex aeroacoustic applications. Precisely, the direct simulation of the fricative [s] sound inside a realistic geometry of the human vocal tract is achieved at the end of the thesis.Esta tesis investiga métodos numéricos que aproximan la solución de las ecuaciones de flujo compresible. La primera parte de la tesis está dedicada al estudio de la aproximación numérica del flujo compresible por medio del método multiescala variacional (VMS) en elementos finitos. En particular, se consideran la ecuación de Burgers unidimensional descrita en el espacio de Fourier y las ecuaciones de Navier-Stokes de flujo compresible escritas en variables conservativas y primitivas. Las aproximaciones hechas para plantear la formulación VMS son ampliamente investigadas; el diseño de la matriz de parámetros de estabilización, la definición del espacio donde viven las subescalas, la inclusión de las derivadas temporales de las subescalas y el seguimiento no lineal de las subescalas son particularidades de la formulación que se analizan para cada una de las ecuaciones consideradas. Además, se incluye la adición de difusión artificial local en forma de técnicas de captura de choque. La precisión de las formulaciones se estudia para varios regímenes del flujo compresible, desde flujos aeroacústicos a bajos números de Mach hasta choques supersónicos. La segunda parte de la tesis está dedicada a hacer asequible la solución de las escalas fluctuantes más pequeñas del flujo compresible. Con este fin, se presenta un algoritmo novedoso para el refinamiento $h$ de las mallas de física computacional usadas en computación distribuida en paralelo. Además, se demuestra la solución en superordenadores de algunos casos de prueba del refinamiento de mallas. También se desarrolla la definición de un estimador de error explícito a posteriori que se puede usar en las simulaciones adaptativas de refinamiento de malla de flujos compresibles; la metodología propuesta emplea las subescalas variacionales como una estimación de error local que induce el refinamiento de la malla. Los métodos numéricos propuestos en esta tesis son capaces de describir las fluctuaciones de alta frecuencia de los flujos compresibles, especialmente los correspondientes a aplicaciones aeroacústicas complejas. Precisamente, la simulación directa del sonido consonántico fricativo [s] dentro de una geometría realista del tracto vocal humano se demuestra al final de la tesis

High-Order Schemes for Compressible Viscous Flows

Ph.DDOCTOR OF PHILOSOPH

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy-Based Artificial Viscosity Stabilization

This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization

Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method

The accurate simulation of supersonic and hypersonic flows is well suited to higher-order (p > 1), adaptive computational fluid dynamics (CFD). Since these cases involve flow velocities greater than the speed of sound, an appropriate shock capturing for higher-order, adaptive methods is necessary. Artificial viscosity can be combined with a higher-order discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a non-smooth artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a new, higher-order, state-based artificial viscosity with an associated governing partial differential equation (PDE).In the governing PDE, the shock sensor acts as a forcing term, driving the artificial viscosity to a non-zero value where it is necessary. The decay rate of the higher-order solution modes and edge-based jumps are both shown to be reliable shock indicators. This new approach leads to a smooth, higher-order representation of the artificial viscosity that evolves in time with the solution. For applications involving the Navier-Stokes equations, an artificial dissipation operator that preserves total enthalpy is introduced. The combination of higher-order, PDE-based artificial viscosity and enthalpy-preserving dissipation operator is shown to overcome the disadvantages of the non-smooth artificial viscosity. The PDE-based artificial viscosity can be used in conjunction with an automated grid adaptation framework that minimizes the error of an output functional. Higher-order solutions are shown to reach strict engineering tolerances with fewer degrees of freedom. The benefit in computational efficiency for higher-order solutions is less dramatic in the vicinity of the shock where errors scale with O(h/p). This includes the near-field pressure signals necessary for sonic boom prediction. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDE-based artificial viscosity is less susceptible to errors introduced by poor shock-grid alignment. Surface heating can also drive the output-based grid adaptation framework to arrive at the same heat transfer distribution as a well-designed structured mesh.This work was made possible by the following sources, • National Defense Science and Engineering Graduate Fellowship • U. S. Air Force Research Laboratory (USAF-3306-03-SC-0001) • The Boeing Company • National Aeronautics and Space Administration (Co-op Agreement NNX07AC70A