scholarly journals A carbuncle cure for the Harten-Lax-van Leer contact (HLLC) scheme using a novel velocity-based sensor

2021 ◽  
Author(s):  
U. S. Vevek ◽  
B. Zang ◽  
T. H. New
Keyword(s):  
Data Structures ◽  
Numerical Experiments ◽  
Additional Data ◽  
Shear Layers ◽  
Hybrid Scheme ◽  
Local Pressure ◽  
Numerical Flux ◽  
Shear Sensor ◽  
Non Local

AbstractA hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.

A Two-Level Method for Pressure Projection Stabilized P1 Nonconforming Approximation of the Semi-Linear Elliptic Equations

2016 ◽  
Vol 8 (3) ◽  
pp. 386-398
Author(s):  
Sufang Zhang ◽  
Hongxia Yan ◽  
Hongen Jia
Keyword(s):  
Data Structures ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Linear Elliptic Equation ◽  
Local Pressure ◽  
Level Method ◽  
Support Sets ◽  
Pressure Projection ◽  
Edge Based ◽  
Computational Properties

Abstract.In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP1–P1 triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.

A New Approach for Simultaneous Shape and Topology Optimization Based on Implicit Topology Description Functions

2004 ◽  
Author(s):  
Xu Guo ◽  
Kang Zhao ◽  
Michael Yu Wang
Keyword(s):  
Topology Optimization ◽  
Numerical Experiments ◽  
Shape Derivative ◽  
Structural Topology ◽  
Numerical Instability ◽  
New Approach ◽  
Shape And Topology Optimization ◽  
Topology Description Function ◽  
And Topology ◽  
Non Local

In the present paper, a new approach for structural topology optimization based on implicit topology description function (TDF) is proposed. TDF is used to describe the shape/topology of a structure, which is approximated in terms of the nodal values. Then a relationship is established between the element stiffness and the values of the topology description function on its four nodes. In this way and with some non-local treatments of the design sensitivities, not only the shape derivative but also the topological derivative of the optimal design can be incorporated in the numerical algorithm in a unified way. Numerical experiments demonstrate that by employing this approach, the computational efforts associated with TDF (and level set) based algorithms can be saved. Clear optimal topologies and smooth structural boundaries free from any sign of numerical instability can be obtained simultaneously and efficiently.

scholarly journals On the numerical approximation of vectorial absolute minimisers in $$L^\infty $$

2020 ◽  
Vol 27 (6) ◽  
Author(s):  
Nikos Katzourakis ◽  
Tristan Pryer
Keyword(s):  
Calculus Of Variations ◽  
Boundary Data ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Lipschitz Mappings ◽  
Open Set ◽  
Locally Lipschitz ◽  
Non Local ◽  
A New Technique ◽  
Local Nature

AbstractLet $$\Omega $$ Ω be an open set. We consider the supremal functional $$\begin{aligned} \text {E}_\infty (u,{\mathcal {O}})\, {:}{=}\, \Vert \text {D}u \Vert _{L^\infty ( {\mathcal {O}} )}, \ \ \ {\mathcal {O}} \subseteq \Omega \text { open}, \end{aligned}$$ E ∞ ( u , O ) : = ‖ D u ‖ L ∞ ( O ) , O ⊆ Ω open , applied to locally Lipschitz mappings $$u : \mathbb {R}^n \supseteq \Omega \longrightarrow \mathbb {R}^N$$ u : R n ⊇ Ω ⟶ R N , where $$n,N\in \mathbb {N}$$ n , N ∈ N . This is the model functional of Calculus of Variations in $$L^\infty $$ L ∞ . The area is developing rapidly, but the vectorial case of $$N\ge 2$$ N ≥ 2 is still poorly understood. Due to the non-local nature of (1), usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $$n,N\ge 2$$ n , N ≥ 2 . We present numerical experiments aimed at understanding the behaviour of minimisers through a new technique involving p-concentration measures.

scholarly journals On the Oval Shapes of Beach Stones

AppliedMath ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 16-38
Author(s):  
Theodore P. Hill
Keyword(s):  
Continuous Time ◽  
Basic Equation ◽  
Numerical Experiments ◽  
Physical Mechanism ◽  
Wave Process ◽  
Random Wave ◽  
Local Shape ◽  
Standard Curve ◽  
Non Local ◽  
Curve Shortening

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

Turbulent shear flow over active and passive porous surfaces

2001 ◽  
Vol 442 ◽  
pp. 89-117 ◽  
Author(s):  
JAVIER JIMÉNEZ ◽  
MARKUS UHLMANN ◽  
ALFREDO PINELLI ◽  
GENTA KAWAHARA
Keyword(s):  
Shear Flow ◽  
Pressure Fluctuations ◽  
Shear Layers ◽  
Linear Instability ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Flow Mechanism ◽  
Helmholtz Instability ◽  
Local Pressure ◽  
The Mean

The behaviour of turbulent shear flow over a mass-neutral permeable wall is studied numerically. The transpiration is assumed to be proportional to the local pressure fluctuations. It is first shown that the friction coefficient increases by up to 40% over passively porous walls, even for relatively small porosities. This is associated with the presence of large spanwise rollers, originating from a linear instability which is related both to the Kelvin–Helmholtz instability of shear layers, and to the neutral inviscid shear waves of the mean turbulent profile. It is shown that the rollers can be forced by patterned active transpiration through the wall, also leading to a large increase in friction when the phase velocity of the forcing resonates with the linear eigenfunctions mentioned above. Phase-lock averaging of the forced solutions is used to further clarify the flow mechanism. This study is motivated by the control of separation in boundary layers.

On the Reynolds number dependence of velocity-gradient structure and dynamics

2018 ◽  
Vol 861 ◽  
pp. 163-179 ◽  
Author(s):  
Rishita Das ◽  
Sharath S. Girimaji
Keyword(s):  
Reynolds Number ◽  
Velocity Gradient ◽  
Probability Distributions ◽  
Small Scale ◽  
Gradient Structure ◽  
Local Pressure ◽  
Non Local ◽  
Normalized Gradient ◽  
Scalar Invariants ◽  
Reynolds Number Dependence

We seek to examine the changes in velocity-gradient structure (local streamline topology) and related dynamics as a function of Reynolds number ($Re_{\unicode[STIX]{x1D706}}$). The analysis factorizes the velocity gradient ($\unicode[STIX]{x1D608}_{ij}$) into the magnitude ($A^{2}$) and normalized-gradient tensor ($\unicode[STIX]{x1D623}_{ij}\equiv \unicode[STIX]{x1D608}_{ij}/\sqrt{A^{2}}$). The focus is on bounded $\unicode[STIX]{x1D623}_{ij}$ as (i) it describes small-scale structure and local streamline topology, and (ii) its dynamics is shown to determine magnitude evolution. Using direct numerical simulation (DNS) data, the moments and probability distributions of $\unicode[STIX]{x1D623}_{ij}$ and its scalar invariants are shown to attain $Re_{\unicode[STIX]{x1D706}}$ independence. The critical values beyond which each feature attains $Re_{\unicode[STIX]{x1D706}}$ independence are established. We proceed to characterize the $Re_{\unicode[STIX]{x1D706}}$ dependence of $\unicode[STIX]{x1D623}_{ij}$-conditioned statistics of key non-local pressure and viscous processes. Overall, the analysis provides further insight into velocity-gradient dynamics and offers an alternative framework for investigating intermittency, multifractal behaviour and for developing closure models.

scholarly journals On the small-scale structure of turbulence and its impact on the pressure field

2018 ◽  
Vol 861 ◽  
pp. 422-446 ◽  
Author(s):  
Dimitar G. Vlaykov ◽  
Michael Wilczek
Keyword(s):  
Velocity Gradient ◽  
Pressure Field ◽  
Reynolds Numbers ◽  
Scale Structure ◽  
Small Scale ◽  
Local Pressure ◽  
Small Scale Structure ◽  
Kolmogorov Scale ◽  
Non Local ◽  
The Impact

Understanding the small-scale structure of incompressible turbulence and its implications for the non-local pressure field is one of the fundamental challenges in fluid mechanics. Intense velocity gradient structures tend to cluster on a range of scales which affects the pressure through a Poisson equation. Here we present a quantitative investigation of the spatial distribution of these structures conditional on their intensity for Taylor-based Reynolds numbers in the range [160, 380]. We find that the correlation length of the second invariant of the velocity gradient is proportional to the Kolmogorov scale. It is also a good indicator for the spatial localization of intense enstrophy and strain-dominated regions, as well as the separation between them. We describe and quantify the differences in the two-point statistics of these regions and the impact they have on the non-locality of the pressure field as a function of the intensity of the regions. Specifically, across the examined range of Reynolds numbers, the pressure in strong rotation-dominated regions is governed by a dissipation-scale neighbourhood. In strong strain-dominated regions, on the other hand, it is determined primarily by a larger neighbourhood reaching inertial scales.

scholarly journals An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

2021 ◽  
Vol 89 (1) ◽  
Author(s):  
R. A. M. van Gestel ◽  
M. J. H. Anthonissen ◽  
J. H. M. ten Thije Boonkkamp ◽  
W. L. IJzerman
Keyword(s):  
Phase Space ◽  
Boundary Conditions ◽  
Ray Tracing ◽  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Local Boundary ◽  
Non Local ◽  
Liouville’S Equation

AbstractLiouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of $$10^{-6}$$ 10 - 6 .

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