scholarly journals Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Gerhard Kirsten
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Interpolation Method ◽  
Three Dimensional ◽  
Order Reduction ◽  
Time Discretization ◽  
Three Dimensions ◽  
Implicit Time ◽  
Benchmark Problems ◽  
Tensor Form

<p style='text-indent:20px;'>We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.</p>

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

The Hopf bifurcation theorem in three dimensions

1977 ◽  
Vol 82 (3) ◽  
pp. 469-483 ◽  
Author(s):  
Peter Swinnerton-Dyer
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Differential Equations ◽  
Limit Cycle ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Bifurcation Theorem ◽  
First Order ◽  
Range Of Values ◽  
First Order Differential Equations

AbstractThe Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

A Three-Dimensional Transient Thermal Model for Machining

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Coskun Islam ◽  
Ismail Lazoglu ◽  
Yusuf Altintas
Keyword(s):  
Mathematical Model ◽  
Thermal Model ◽  
Three Dimensional ◽  
Measurement Data ◽  
Time Discretization ◽  
Temperature Fields ◽  
Implicit Time ◽  
Machining Processes ◽  
Transient Thermal ◽  
Interrupted Turning

This article presents an enhanced mathematical model for transient thermal analysis in machining processes. The proposed mathematical model is able to simulate transient tool, workpiece, and chip temperature fields as a function of time for interrupted processes with time varying chip loads such as milling and continuous machining processes such as turning and drilling. A finite difference technique with implicit time discretization is used for the solution of partial differential equations to simulate the temperature fields on the tool, workpiece, and chip. The model validations are performed with the experimental temperature measurement data available in the literature for the interrupted turning of Ti6Al6V–2Sn, Al2024, gray cast iron and for the milling of Ti6Al4V. The simulation results and experimental measurements agree well. With the newly introduced modeling approach, it is demonstrated that time-dependent dynamic variations of the temperature fields are predicted with maximum 12% difference in the validated cases by the proposed transient thermal model.

A path-independent integral for stress intensity factors in three dimensions

1997 ◽  
Vol 32 (6) ◽  
pp. 411-423 ◽  
Author(s):  
C M Bainbridge ◽  
M H Aliabadi ◽  
D P Rooke
Keyword(s):  
Stress Intensity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Function Technique ◽  
Reciprocal Theorem ◽  
Benchmark Problems ◽  
Surface Integral ◽  
Derivatives Of ◽  
Independent Integral ◽  
Good Agreement

A path-independent surface integral is derived for the calculation of the stress intensity factor in three dimensions. The integral is based upon a type of Green's function, which is used in conjunction with Betti's reciprocal theorem. Various benchmark problems have been solved using this technique. The results were found to be in good agreement with the values obtained in previous analyses, and the method appears to offer some efficiency advantages. For example, the method presented here does not require calculations of derivatives of the stress tensor, as does the J-integral for the three-dimensional case. Such calculations of derivatives of stress are inherently numerically difficult if good accuracy is required. It can also be used for mixed-mode problems without the need for decoupling as required by other techniques. This technique is complementary to the widely used weight function technique.

Three-Dimensional Elasticity Analysis of Thick Rectangular Laminated Composite Plates Using Meshless Local Petrov-Galerkin (MLPG) Method

2006 ◽  
Vol 5-6 ◽  
pp. 331-338 ◽  
Author(s):  
S.M.R. Alavi ◽  
Mohammad Mohammadi Aghdam ◽  
A. Eftekhari
Keyword(s):  
Interpolation Method ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Plates ◽  
Three Dimensions ◽  
Elasticity Analysis ◽  
Weak Formulation ◽  
Mlpg Method

This article presents apparently the first application of Meshless local Petrov-Galerkin (MLPG) method for 3-D elasticity analysis of moderately thick rectangular laminated plates. As with other Meshless methods, the problem domain is represented by a set of spread nodes in all three dimensions of the plate without configuration of elements. The Moving Least-Squares (MLS) method is applied to construct the required shape functions. A local asymmetric weak formulation of the problem is developed and MLPG is applied to solve the governing equations. Direct interpolation method is employed to enforce essential boundary conditions. Details of formulation, numerical procedure, convergence and accuracy characteristics of the method are investigated. Results are compared, where possible, with other analytical and numerical methods and show good agreement.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Turbulent three-dimensional dielectric electrohydrodynamic convection between two plates

2012 ◽  
Vol 696 ◽  
pp. 228-262 ◽  
Author(s):  
A. Kourmatzis ◽  
J. S. Shrimpton
Keyword(s):  
Spatial Data ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Budgets ◽  
Turbulent Regime ◽  
Scalar Flux ◽  
Wide Range ◽  
Scalar Variance

AbstractThe fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low (${\mathit{Re}}_{E} = 1$) and high electric Reynolds numbers (up to ${\mathit{Re}}_{E} = 120$). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field ${ E}_{i}^{\ensuremath{\prime} } $ in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that ${ E}_{i}^{\ensuremath{\prime} } $ is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

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