3,444 research outputs found
Spectral/hp element methods for plane Newtonian extrudate swell
Spectral/hp element methods and an arbitrary Lagrangian-Eulerian (ALE)
moving-boundary technique are used to investigate planar Newtonian extrudate
swell. Newtonian extrudate swell arises when viscous liquids exit long die
slits. The problem is characterised by a stress singularity at the end of the
slit which is inherently difficult to capture and strongly influences the
predicted swelling of the fluid. The impact of inertia (0 <Re < 100) and slip
along the die wall on the free surface profile and the velocity and pressure
values in the domain and around the singularity are investigated. The high
order method is shown to provide high resolution of the steep pressure profile
at the singularity. The swelling ratio and exit pressure loss are compared with
existing results in the literature and the ability of high-order methods to
capture these values using significantly fewer degrees of freedom is
demonstrated
A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains
We propose and analyze a new discretization technique for a linear-quadratic
optimal control problem involving the fractional powers of a symmetric and
uniformly elliptic second oder operator; control constraints are considered.
Since these fractional operators can be realized as the Dirichlet-to-Neumann
map for a nonuniformly elliptic equation, we recast our problem as a
nonuniformly elliptic optimal control problem. The rapid decay of the solution
to this problem suggests a truncation that is suitable for numerical
approximation. We propose a fully discrete scheme that is based on piecewise
linear functions on quasi-uniform meshes to approximate the optimal control and
first-degree tensor product functions on anisotropic meshes for the optimal
state variable. We provide an a priori error analysis that relies on derived
Holder and Sobolev regularity estimates for the optimal variables and error
estimates for an scheme that approximates fractional diffusion on curved
domains; the latter being an extension of previous available results. The
analysis is valid in any dimension. We conclude by presenting some numerical
experiments that validate the derived error estimates
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
We present a class of spline finite element methods for time-domain wave
propagation which are particularly amenable to explicit time-stepping. The
proposed methods utilize a discontinuous Galerkin discretization to enforce
continuity of the solution field across geometric patches in a multi-patch
setting, which yields a mass matrix with convenient block diagonal structure.
Over each patch, we show how to accurately and efficiently invert mass matrices
in the presence of curved geometries by using a weight-adjusted approximation
of the mass matrix inverse. This approximation restores a tensor product
structure while retaining provable high order accuracy and semi-discrete energy
stability. We also estimate the maximum stable timestep for spline-based finite
elements and show that the use of spline spaces result in less stringent CFL
restrictions than equivalent piecewise continuous or discontinuous finite
element spaces. Finally, we explore the use of optimal knot vectors based on L2
n-widths. We show how the use of optimal knot vectors can improve both
approximation properties and the maximum stable timestep, and present a simple
heuristic method for approximating optimal knot positions. Numerical
experiments confirm the accuracy and stability of the proposed methods
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
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