scholarly journals Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

2021 ◽  
Author(s):  
Dinshaw S. Balsara ◽  
Roger Käppeli ◽  
Walter Boscheri ◽  
Michael Dumbser
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Higher Order ◽  
Two Dimensions ◽  
Field Equations ◽  
Von Neumann ◽  
Riemann Solvers ◽  
Scheme Design ◽  
First Order ◽  
Pde Systems

AbstractSeveral important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Vibroacoustic Comparisons and Acoustic Power Insulation Mechanisms of Multidirectional Stiffened Laminated Plates

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Xiongtao Cao ◽  
Hongxing Hua
Keyword(s):  
Fourier Transform ◽  
High Efficiency ◽  
Three Dimensional ◽  
Acoustic Power ◽  
Higher Order ◽  
Laminated Plates ◽  
Transverse Displacement ◽  
Third Order ◽  
First Order ◽  
Implicit Equations

Vibroacoustic characteristics of multidirectional stiffened laminated plates with or without compliant layers are explored in the wavenumber and spatial domains with the help of the two-dimensional continuous Fourier transform and discrete inverse fast Fourier transform. Implicit equations of motion for the arbitrary angle ply laminated plates are derived from the three-dimensional higher order and Reddy third order shear deformation plate theories. The expressions of acoustic power of the stiffened laminated plates with or without complaint layers are formulated in the wavenumber domain, which is a significant method to calculate acoustic power of the stiffened plates with multiple sets of cross stiffeners. Vibroacoustic comparisons of the stiffened laminated plates are made in terms of the transverse displacement spectra, forced responses, acoustic power, and input power according to the first order, Reddy third order, and three-dimensional higher order plate theories. Sound reduction profiles of compliant layers are further examined by the theoretical deductions. This study shows the feasibility and high efficiency of the first order and Reddy third order plate theories in the broad frequency range and allows a better understanding the principal mechanisms of acoustic power radiated from multidirectional stiffened laminated composite plates with compliant layers, which has not been adequately addressed in its companion paper. (Cao and Hua, 2012, “Sound Radiation From Shear Deformable Stiffened Laminated Plates With Multiple Compliant Layers,” ASME J. Vib. Acoust., 134(5), p. 051001.)

Enhanced First-Order Shear Deformation Theory for Laminated and Sandwich Plates

2005 ◽  
Vol 72 (6) ◽  
pp. 809-817 ◽  
Author(s):  
Jun-Sik Kim ◽  
Maenghyo Cho
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Plate Theory ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Laminated Plates ◽  
Sandwich Plates ◽  
First Order ◽  
Transverse Shear Strains

A new first-order shear deformation theory (FSDT) has been developed and verified for laminated plates and sandwich plates. Based on the definition of Reissener–Mindlin’s plate theory, the average transverse shear strains, which are constant through the thickness, are improved to vary through the thickness. It is assumed that the displacement and in-plane strain fields of FSDT can approximate, in an average sense, those of three-dimensional theory. Relationship between FSDT and three-dimensional theory has been systematically established in the averaged least-square sense. This relationship provides the closed-form recovering relations for three-dimensional variables expressed in terms of FSDT variables as well as the improved transverse shear strains. This paper makes two main contributions. First an enhanced first-order shear deformation theory (EFSDT) has been developed using an available higher-order plate theory. Second, it is shown that the displacement fields of any higher-order plate theories can be recovered by EFSDT variables. The present approach is applied to an efficient higher-order plate theory. Comparisons of deflection and stresses of the laminated plates and sandwich plates using present theory are made with the original FSDT and three-dimensional exact solutions.

Simulation of Ideal Grain Growth Using the Multi-Phase-Field Model

2007 ◽  
Vol 558-559 ◽  
pp. 1177-1181 ◽  
Author(s):  
Philippe Schaffnit ◽  
Markus Apel ◽  
Ingo Steinbach
Keyword(s):  
Grain Size ◽  
Grain Growth ◽  
Phase Field ◽  
Large Scale ◽  
Field Model ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Two Dimensions ◽  
Von Neumann ◽  
Average Grain Size

The kinetics and topology of ideal grain growth were simulated using the phase-field model. Large scale phase-field simulations were carried out where ten thousands grains evolved into a few hundreds without allowing coalescence of grains. The implementation was first validated in two-dimensions by checking the conformance with square-root evolution of the average grain size and the von Neumann-Mullins law. Afterwards three-dimensional simulations were performed which also showed fair agreement with the law describing the evolution of the mean grain size against time and with the results of S. Hilgenfeld et al. in 'An Accurate von Neumann's Law for Three-Dimensional Foams', Phys. Rev. Letters, 86(12)/2685, March 2001. Finally the steady state grain size distribution was investigated and compared to the Hillert theory.

scholarly journals On the relativistic invariance of quantized field theories

1948 ◽  
Vol 195 (1042) ◽  
pp. 365-376 ◽  
Keyword(s):  
Present Author ◽  
Higher Order ◽  
Field Theories ◽  
Relativistic Invariance ◽  
Field Equations ◽  
First Order ◽  
Generalized Poisson ◽  
General Relativistic ◽  
Quantized Field ◽  
Derivatives Of

Heisenberg & Pauli (1929) have shown how to quantize field theories derived from Lagrangians containing first-order derivatives of the field quantities. They showed their quantization to be Lorentz invariant. Fuchs (1939) subsequently showed that the quantized theory was in fact invariant under general transformations of co-ordinates. The present author in another paper has shown how the theory of Heisenberg & Pauli can be extended to field equations derived from higher order Lagrangians, i. e. Lagrangians containing higher deri­vatives than the first of the field quantities. In the present paper the general relativistic invariance of the higher order quantized theories is established, making use of the generalized Poisson brackets introduced by Weiss.

The vacuum polarization in two-dimensional noncommutative QED

2003 ◽  
Vol 81 (8) ◽  
pp. 997-1003 ◽  
Author(s):  
F T Brandt ◽  
D.G.C. McKeon
Keyword(s):  
Vector Field ◽  
Gauge Field ◽  
Vacuum Polarization ◽  
Two Dimensions ◽  
Tree Level ◽  
Polarization Tensor ◽  
Two Dimensional ◽  
First Order ◽  
The One ◽  
Vacuum Polarization Tensor

The one-loop vacuum polarization tensor in noncommutative spinor QED in two dimensions is computed. A first-order formalism is used to simplify the interaction vertices for the vector field. Although the form of the gauge field-spinor interaction is chosen so as to vanish at tree level, as the noncommuting parameter goes to zero, the vacuum polarization is nontrivial in this limit. PACS No.: 12.20.–m

scholarly journals T-dualization of Gödel string cosmologies via Poisson–Lie T-duality approach

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Ali Eghbali ◽  
Reza Naderi ◽  
Adel Rezaei-Aghdam
Keyword(s):  
Lie Groups ◽  
Effective Action ◽  
Three Dimensional ◽  
Target Space ◽  
Loop Order ◽  
Field Equations ◽  
First Order ◽  
Self Duality ◽  
Duality Approach ◽  
Dual Models

AbstractUsing the homogeneous Gödel spacetimes we find some new solutions for the field equations of bosonic string effective action up to first order in $$\alpha '$$ α ′ including both dilaton and axion fields. We then discuss in detail the (non-)Abelian T-dualization of Gödel string cosmologies via the Poisson–Lie (PL) T-duality approach. In studying Abelian T-duality of the models we get seven dual models in such a way that they are constructed by one-, two- and three-dimensional Abelian Lie groups acting freely on the target space manifold. The results of our study show that the Abelian T-dual models are, under some of the special conditions, self-dual; moreover, by applying the usual rules of Abelian T-duality without further corrections, we are still able to obtain two-loop solutions. We also study the Abelian T-duality of Gödel string cosmologies up to $$\alpha '$$ α ′ -corrections by using the T-duality rules at two-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian duals of the Gödel spacetimes are constructed by two- and three-dimensional non-Abelian Lie groups such as $$A_2$$ A 2 , $$A_2 \oplus A_1$$ A 2 ⊕ A 1 and $$SL(2, \mathbb {R})$$ S L ( 2 , R ) . In this way, the PL self-duality of $$AdS_3 \times \mathbb {R}$$ A d S 3 × R space is discussed.

SUCCA3D—An Alternative Scheme to Reduce False Diffusion in Three-Dimensional Flows

1993 ◽  
Vol 207 (5) ◽  
pp. 307-313 ◽  
Author(s):  
T J Scanlon ◽  
C Carey ◽  
S M Fraser
Keyword(s):  
Three Dimensional ◽  
Higher Order ◽  
Upwind Scheme ◽  
Scalar Transport ◽  
Benchmark Test ◽  
Alternative Scheme ◽  
First Order ◽  
Quick Scheme ◽  
Scalar Variable ◽  
Convection Dominated

An alternative flow-oriented convection algorithm is presented which acts as a replacement for the first-order accurate UPWIND scheme in three-dimensional scalar transport. The scheme, formally titled SUCCA3D (skew upwind corner convection algorithm 3D), attempts to follow local streamlines, thus directly reducing the multi-dimensional false diffusion of the conserved scalar. In a standard benchmark test of pure convection across a three-dimensional cavity the SUCCA3D scheme was found to compare favourably with alternative schemes such as UPWIND and the higher-order QUICK scheme. The results highlight the potential of the SUCCA3D code for the reduction of three-dimensional false diffusion of a scalar variable in convection-dominated flows.

On the effectiveness of functional language features: NAS benchmark FT

1997 ◽  
Vol 7 (1) ◽  
pp. 103-123 ◽  
Author(s):  
J. HAMMES ◽  
S. SUR ◽  
W. BÖHM
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Functional Language ◽  
One Dimensional ◽  
First Order ◽  
Space And Time ◽  
The One ◽  
Dimensional Object

In this paper we investigate the effectiveness of functional language features when writing scientific codes. Our programs are written in the purely functional subset of Id and executed on a one node Motorola Monsoon machine, and in Haskell and executed on a Sparc 2. In the application we study – the NAS FT benchmark, a three-dimensional heat equation solver – it is necessary to target and select one-dimensional sub-arrays in three-dimensional arrays. Furthermore, it is important to be able to share computation in array definitions. We compare first order and higher order implementations of this benchmark. The higher order version uses functions to select one-dimensional sub-arrays, or slices, from a three-dimensional object, whereas the first order version creates copies to achieve the same result. We compare various representations of a three-dimensional object, and study the effect of strictness in Haskell. We also study the performance of our codes when employing recursive and iterative implementations of the one-dimensional FFT, which forms the kernel of this benchmark. It turns out that these languages still have quite inefficient implementations, with respect to both space and time. For the largest problem we could run (323), Haskell is 15 times slower than Fortran and uses three times more space than is absolutely necessary, whereas Id on Monsoon uses nine times more cycles than Fortran on the MIPS R3000, and uses five times more space than is absolutely necessary. This code, and others like it, should inspire compiler writers to improve the performance of functional language implementations.

scholarly journals Parallel finite-element implementation for higher-order ice-sheet models

2012 ◽  
Vol 58 (207) ◽  
pp. 76-88 ◽  
Author(s):  
Mauro Perego ◽  
Max Gunzburger ◽  
John Burkardt
Keyword(s):  
Finite Element ◽  
Computational Models ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Integrated Model ◽  
Higher Order ◽  
Order Model ◽  
Length Scales ◽  
First Order ◽  
Finite Element Approximations

AbstractHigher-order models represent a computationally less expensive alternative to the Stokes model for ice-sheet modeling. In this work, we develop linear and quadratic finite-element methods, implemented on parallel architectures, for the three-dimensional first-order model of Dukowicz and others (2010) that is based on the Blatter-Pattyn model, and for the depth-integrated model of Schoof and Hindmarsh (2010). We then apply our computational models to three of the ISMIP-HOM benchmark test cases (Pattyn and others, 2008). We compare results obtained from our models with those obtained using a reliable Stokes computational model, showing that our first-order model implementation produces reliable and accurate solutions for almost all characteristic length scales of the test geometries considered. Good agreement with the reference Stokes solution is also obtained by our depth-integrated model implementation in fast-sliding regimes and for medium to large length scales. We also provide a comprehensive comparison between results obtained from our first-order model implementation and implementations developed by ISMIP-HOM participants; this study shows that our implementation is at least as good as the previous ones. Finally, a comparison between linear and quadratic finite- element approximations is carried out, showing, as expected, the better accuracy of the quadratic finite-element method.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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