23,686 research outputs found
Verification of Unstructured Grid Adaptation Components
Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. Laminar flow over a delta wing and turbulent flow over an ONERA M6 wing are verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and a moment. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. The wing adaptation cases in the current study document a clear improvement to existing grid adaptation procedures. The stage is set for the infusion of verified grid adaptation into production fluid flow simulations
Computational coarse graining of a randomly forced 1-D Burgers equation
We explore a computational approach to coarse graining the evolution of the
large-scale features of a randomly forced Burgers equation in one spatial
dimension. The long term evolution of the solution energy spectrum appears
self-similar in time. We demonstrate coarse projective integration and coarse
dynamic renormalization as tools that accelerate the extraction of macroscopic
information (integration in time, self-similar shapes, and nontrivial dynamic
exponents) from short bursts of appropriately initialized direct simulation.
These procedures solve numerically an effective evolution equation for the
energy spectrum without ever deriving this equation in closed form.Comment: 21 pages, 7 figure
Nonparametric likelihood based estimation of linear filters for point processes
We consider models for multivariate point processes where the intensity is
given nonparametrically in terms of functions in a reproducing kernel Hilbert
space. The likelihood function involves a time integral and is consequently not
given in terms of a finite number of kernel evaluations. The main result is a
representation of the gradient of the log-likelihood, which we use to derive
computable approximations of the log-likelihood and the gradient by time
discretization. These approximations are then used to minimize the approximate
penalized log-likelihood. For time and memory efficiency the implementation
relies crucially on the use of sparse matrices. As an illustration we consider
neuron network modeling, and we use this example to investigate how the
computational costs of the approximations depend on the resolution of the time
discretization. The implementation is available in the R package ppstat.Comment: 10 pages, 3 figure
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
We introduce an -version symmetric interior penalty discontinuous
Galerkin finite element method (DGFEM) for the numerical approximation of the
biharmonic equation on general computational meshes consisting of
polygonal/polyhedral (polytopic) elements. In particular, the stability and
-version a-priori error bound are derived based on the specific choice of
the interior penalty parameters which allows for edges/faces degeneration.
Furthermore, by deriving a new inverse inequality for a special class {of}
polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be
stable to incorporate very general polygonal/polyhedral elements with an
\emph{arbitrary} number of faces for polynomial basis with degree . The
key feature of the proposed method is that it employs elemental polynomial
bases of total degree , defined in the physical coordinate
system, without requiring the mapping from a given reference or canonical
frame. A series of numerical experiments are presented to demonstrate the
performance of the proposed DGFEM on general polygonal/polyhedral meshes
Preconditioned fully implicit PDE solvers for monument conservation
Mathematical models for the description, in a quantitative way, of the
damages induced on the monuments by the action of specific pollutants are often
systems of nonlinear, possibly degenerate, parabolic equations. Although some
the asymptotic properties of the solutions are known, for a short window of
time, one needs a numerical approximation scheme in order to have a
quantitative forecast at any time of interest. In this paper a fully implicit
numerical method is proposed, analyzed and numerically tested for parabolic
equations of porous media type and on a systems of two PDEs that models the
sulfation of marble in monuments. Due to the nonlinear nature of the underlying
mathematical model, the use of a fixed point scheme is required and every step
implies the solution of large, locally structured, linear systems. A special
effort is devoted to the spectral analysis of the relevant matrices and to the
design of appropriate iterative or multi-iterative solvers, with special
attention to preconditioned Krylov methods and to multigrid procedures.
Numerical experiments for the validation of the analysis complement this
contribution.Comment: 26 pages, 13 figure
- …