scholarly journals Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 293
Author(s):  
Nurlybek Kasimov ◽  
Eric Dymkoski ◽  
Giuliano De Stefano ◽  
Oleg V. Vasilyev
Keyword(s):  
Boundary Conditions ◽  
Adaptive Mesh Refinement ◽  
A Priori ◽  
Approximation Error ◽  
Adaptive Mesh ◽  
Moving Boundaries ◽  
Penalization Method ◽  
Invariant Characteristic ◽  
Volume Penalization Method ◽  
Galilean Invariant

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

FE Meshing Tool for the Dynamic Analysis of Beam-Like Structures

1991 ◽  
Author(s):  
M. De Smet ◽  
H. Van Brussel ◽  
P. Sas
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Adaptive Mesh Refinement ◽  
Dynamic Behaviour ◽  
A Priori ◽  
Adaptive Mesh ◽  
Structural Dynamic ◽  
Resonant Frequencies ◽  
Tapered Beam ◽  
Mesh Density

Abstract This paper outlines a methodology for the automatic creation of finite element meshes used for calculating structural dynamic behaviour. Starting from the idealized geometry of the structure, the proposed Intelligent Meshing Tool determines a mesh density that a priori guarantees the resonant frequencies to be calculated within predetermined accuracy limits. Unlike most existing techniques for automatic meshing, this technique is not inherently adaptive, but does not exclude an extra adaptive mesh refinement step, either. Implementation of this methodology is discussed for structures composed of straight uniform, curved and tapered beam-like components. The feasibility of the meshing strategy is illustrated by means of an example.

scholarly journals Anisotropic polygonal and polyhedral discretizations in finite element analysis

2019 ◽  
Vol 53 (2) ◽  
pp. 475-501 ◽  
Author(s):  
Steffen Weißer
Keyword(s):  
Finite Element Analysis ◽  
Adaptive Mesh Refinement ◽  
A Priori ◽  
Adaptive Mesh ◽  
Linear Mapping ◽  
Three Dimensions ◽  
Element Analysis ◽  
Reference Element ◽  
Polyhedral Meshes ◽  
Interpolation Operators

Interpolation and quasi-interpolation operators of Clément- and Scott-Zhang-type are analyzed on anisotropic polygonal and polyhedral meshes. Since no reference element is available, an appropriate linear mapping to a reference configuration plays a crucial role. A priori error estimates are derived respecting the anisotropy of the discretization. Finally, the found estimates are employed to propose an adaptive mesh refinement based on bisection which leads to highly anisotropic and adapted discretizations with general element shapes in two- and three-dimensions.

scholarly journals PDE eigenvalue iterations with applications in two-dimensional photonic crystals

2020 ◽  
Vol 54 (5) ◽  
pp. 1751-1776
Author(s):  
Robert Altmann ◽  
Marine Froidevaux
Keyword(s):  
Photonic Crystals ◽  
Eigenvalue Problem ◽  
Adaptive Mesh Refinement ◽  
Eigenvalue Problems ◽  
A Priori ◽  
Adaptive Mesh ◽  
Inverse Power ◽  
Two Dimensional ◽  
Arnoldi Method ◽  
The Matrix

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude–Lorentz model, we propose the direct application of a Newton-type iteration. Assuming some a priori knowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

scholarly journals Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions

2020 ◽  
Author(s):  
Max Jensen ◽  
Axel Målqvist ◽  
Anna Persson
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Joule Heating ◽  
Adaptive Mesh Refinement ◽  
Mixed Boundary ◽  
Adaptive Mesh ◽  
Mixed Boundary Conditions ◽  
Time Dependent ◽  
Discrete Maximum Principle ◽  
Backward Euler

Abstract We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.

scholarly journals New boundary conditions for simulating the filling stage of the injection molding process

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Wagner de Campos Galuppo ◽  
Ana Magalhães ◽  
Luís Lima Ferrás ◽  
João Miguel Nóbrega ◽  
Célio Fernandes
Keyword(s):  
Injection Molding ◽  
Boundary Conditions ◽  
Adaptive Mesh Refinement ◽  
Complex Geometry ◽  
Real Life ◽  
Adaptive Mesh ◽  
Injection Molding Process ◽  
Molding Process ◽  
Content Type ◽  
Filling Stage

Purpose The purpose of this paper is to develop new boundary conditions for simulating the injection molding process of polymer melts. Design/methodology/approach The boundary conditions are derived and implemented to simulate real-life air vents (used to allow the air escape from the mold). The simulations are performed in the computational library OpenFOAM® by considering two different fluid models, namely, Newtonian and generalized Newtonian (Bird–Carreau model). Findings A detailed study on the accuracy of the solver interFoam for simulating the filling stage is presented, by considering simple geometries and adaptive mesh refinement. The verified code is then used to study the three-dimensional filling of a more complex geometry. Originality/value The results obtained showed that the numerical method is stable and allows one to model the filling process, simulating the real injection molding process.

Adaptive mesh refinement for two-phase slug flows with an a priori indicator

2013 ◽  
Vol 49 ◽  
pp. 83-98 ◽  
Author(s):  
M. Gourma ◽  
N. Jia ◽  
C. Thompson
Keyword(s):  
Adaptive Mesh Refinement ◽  
A Priori ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Two Phase ◽  
Slug Flows

scholarly journals Error analysis for a vorticity/Bernoulli pressure formulation for the Oseen equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Verónica Anaya ◽  
David Mora ◽  
Amiya K. Pani ◽  
Ricardo Ruiz-Baier
Keyword(s):  
Error Analysis ◽  
Adaptive Mesh Refinement ◽  
A Priori ◽  
Adaptive Mesh ◽  
Momentum Balance ◽  
Error Estimator ◽  
Oseen Equations ◽  
Post Process ◽  
Pressure Formulation ◽  
New Formulation

Abstract A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L2-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an a posteriori error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and also confirm the theoretical findings.

scholarly journals A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations

2021 ◽  
Author(s):  
Omar Lakkis ◽  
Amireh Mousavi
Keyword(s):  
Least Squares ◽  
Elliptic Equations ◽  
Adaptive Mesh Refinement ◽  
Least Squares Method ◽  
A Priori ◽  
Adaptive Mesh ◽  
A Posteriori ◽  
Nondivergence Form ◽  
Convergence Results ◽  
Discrete Spaces

Abstract We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding a priori and a posteriori convergence results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.

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