1,307 research outputs found
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
WAVEx: Stabilized Finite Elements for Spectral Wind Wave Models Using FEniCSx
Several potential FEM discretizations of the Wave Action Balance Equation are
discussed. The methods, which include streamline upwind Petrov-Galerkin (SUPG),
least squares, and discontinuous Galerkin, are implemented using the open
source finite element library FEniCSx for simplified 2-D cases. Open source
finite element libraries, such as FEniCSx, typically only support geometries up
to dimension of 3. The Wave Action Balance Equation is 4 dimensions in space so
this presents difficulties. A method to use a FEM library, such as FEniCSx, to
solve problems in domains with dimension larger than 4 using the product basis
is discussed. A new spectral wind wave model, WAVEx, is formulated and
implemented using the new finite element library FEniCSx. WAVEx is designed to
allow for construction of multiple FEM discretizations with relatively small
modifications in the Python code base. An example implementation is then
demonstrated with WAVEx using continuous finite elements and SUPG stabilization
in geographic/spectral space. For propagation in time, a generalized one step
implicit finite difference method is used. When source terms are active, the
second order operator splitting scheme known as Strang splitting is used. In
the splitting scheme, propagation is solved using the aforementioned implicit
method and the nonlinear source terms are treated explicitly using second order
Runge-Kutta. Several test cases which are part of the Office for Naval Research
Test Bed (ONR Test Bed) are demonstrated both with and without 3rd generation
source terms and results are compared to analytic solutions, observations, and
SWAN output
A controllability method for Maxwell's equations
We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached
Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems
In this work, we consider space-time goal-oriented a posteriori error
estimation for parabolic problems. Temporal and spatial discretizations are
based on Galerkin finite elements of continuous and discontinuous type. The
main objectives are the development and analysis of space-time estimators, in
which the localization is based on a weak form employing a partition-of-unity.
The resulting error indicators are used for temporal and spatial adaptivity.
Our developments are substantiated with several numerical examples.Comment: Changes in v2: - Updated the title - Reworked space-time function
spaces - Added cG(1) in time partition-of-unity - Added links to the now
published codes used for this work - Added further reference
A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
We present a comparison between hybridized and non-hybridized discontinuous
Galerkin methods in the context of target-based hp-adaptation for compressible
flow problems. The aim is to provide a critical assessment of the computational
efficiency of hybridized DG methods. Hybridization of finite element
discretizations has the main advantage, that the resulting set of algebraic
equations has globally coupled degrees of freedom only on the skeleton of the
computational mesh. Consequently, solving for these degrees of freedom involves
the solution of a potentially much smaller system. This not only reduces
storage requirements, but also allows for a faster solution with iterative
solvers. Using a discrete-adjoint approach, sensitivities with respect to
output functionals are computed to drive the adaptation. From the error
distribution given by the adjoint-based error estimator, h- or p-refinement is
chosen based on the smoothness of the solution which can be quantified by
properly-chosen smoothness indicators. Numerical results are shown for
subsonic, transonic, and supersonic flow around the NACA0012 airfoil.
hp-adaptation proves to be superior to pure h-adaptation if discontinuous or
singular flow features are involved. In all cases, a higher polynomial degree
turns out to be beneficial. We show that for polynomial degree of approximation
p=2 and higher, and for a broad range of test cases, HDG performs better than
DG in terms of runtime and memory requirements
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