Coupling time-stepping numerical methods and standard aerodynamics codes for instabilitiy analysis of flows in complex geometries

2011 ◽  
Author(s):  
Francisco Gomez Carrasco ◽  
Vassilios Theofilis ◽  
Raquel Gomez
Keyword(s):  
Numerical Methods ◽  
Complex Geometries ◽  
Time Stepping ◽  
Coupling Time

Rattlebacks for Undergraduate Engineers: Modelling Complex Behavior Using Introductory Dynamics and Numerical Methods

2018 ◽  
Author(s):  
Simon Jones ◽  
Kirby Kern
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Reference Frame ◽  
Engineering Students ◽  
Rotary Motion ◽  
Complex Dynamic ◽  
Time Stepping ◽  
Undergraduate Engineering ◽  
Clockwise Direction

Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin. If spun in, say, the clockwise direction, the rattleback will exhibit stable rotary motion. If spun in the counter-clockwise direction, the rattleback’s rotary motion will transition to a rattling motion, and then reverse its spin resulting in clockwise rotation. This counter-intuitive dynamic behavior has long been a favored subject of study in graduate-level dynamics classes. Previous literature on rattleback dynamics offer insight into a myriad of advanced topics, including three-dimensional motion, sliding and rolling friction models, stability regions, nondimensionalization, etc. However, it is the current authors’ view that focusing on these advanced topics clouds the students’ understanding of the fundamental kinetics of the body. The goal of this paper is to demonstrate that accurately simulating rattleback behavior need not be complicated; undergraduate engineering students can accurately model the behavior using concepts from introductory dynamics and numerical methods. The current paper develops an accurate dynamic model of a rattleback from first principles. All necessary steps are discussed in detail, including computing the mass moment of inertia, choice of reference frame, conservation of momenta equations, and application of kinematic constraints. Basic numerical techniques like Gaussian quadrature, Newton-Raphson root-finding, and Runge-Kutta time-stepping are employed to solve the necessary integrals, nonlinear algebraic equations, and ordinary differential equations. Since not all undergraduate engineering students are familiar with 3D dynamics, a simpler 2D rocking semi-ellipse example is first introduced to develop the transformation matrix between an inertial reference frame and a body-fixed reference frame. This provides the framework to transition seamlessly into 3D dynamics using roll, pitch, and yaw angles, concepts that are widely understood by engineering students. In fact, when written in vector notation, the governing equations for the rocking ellipse and the spin-biased rattleback are shown to be the same, enforcing the concept that 3D dynamics need not be intimidating. The purpose of this paper is to guide a typical undergraduate engineering student through a complex dynamic simulation, and to demonstrate that he or she already has the tools necessary to simulate complex dynamic behavior. Conservation of momenta will account for the dynamics, intimidating integrals and differentials can be tackled numerically, and classic time-stepping approaches make light work of nonlinear differential equations.

scholarly journals Preservation of Fine Structures in PDE-Based Image Denoising

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Hakran Kim ◽  
Velinda R. Calvert ◽  
Seongjai Kim
Keyword(s):  
Numerical Methods ◽  
Image Denoising ◽  
Numerical Technique ◽  
Significant Loss ◽  
Implicit Time ◽  
Alternating Direction Implicit ◽  
Time Stepping ◽  
Alternating Direction ◽  
Fine Structures ◽  
Efficiency And Reliability

Image denoising processes often lead to significant loss of fine structures such as edges and textures. This paper studies various innovative mathematical and numerical methods applicable for conventional PDE-based denoising models. The method of diffusion modulation is considered to effectively minimize regions of undesired excessive dissipation. Then we introduce a novel numerical technique for residual-driven constraint parameterization, in order for the resulting algorithm to produce clear images whose corresponding residual is as free of image textures as possible. A linearized Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to simulate the resulting model efficiently. Various examples are presented to show efficiency and reliability of the suggested methods in image denoising.

Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

2011 ◽  
Vol 66 (12) ◽  
pp. 735-744 ◽  
Author(s):  
Akbar Mohebbi ◽  
Zohreh Asgari ◽  
Alimardan Shahrezaee
Keyword(s):  
Numerical Methods ◽  
High Efficiency ◽  
High Accuracy ◽  
Processing Unit ◽  
Conservation Of Energy ◽  
One Dimensional ◽  
Central Processing ◽  
Time Stepping ◽  
Klein Gordon ◽  
The One

In this work we propose fast and high accuracy numerical methods for the solution of the one dimensional nonlinear Klein-Gordon (KG) equations. These methods are based on applying fourth order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. After transforming each equation to a system of ordinary differential equations, the linear operator is not diagonal, but we can implement the methods such as for the diagonal case which reduces the time in the central processing unit (CPU). In addition, the conservation of energy in KG equations is investigated. Numerical results obtained from solving several problems possessing periodic, single, and breather-soliton waves show the high efficiency and accuracy of the mentioned methods.

scholarly journals Comparison and Analysis of Neural Solver Methods for Differential Equations in Physical Systems

ELKHA ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 134
Author(s):  
Fabio M Sim ◽  
Eka Budiarto ◽  
Rusman Rusyadi
Keyword(s):  
Neural Networks ◽  
Numerical Methods ◽  
Differential Equations ◽  
Electric Fields ◽  
Test Cases ◽  
Mass System ◽  
Time Stepping ◽  
Computational Speed ◽  
Nonlinear Solutions ◽  
Near Future

Differential equations are ubiquitous in many fields of study, yet not all equations, whether ordinary or partial, can be solved analytically. Traditional numerical methods such as time-stepping schemes have been devised to approximate these solutions. With the advent of modern deep learning, neural networks have become a viable alternative to traditional numerical methods. By reformulating the problem as an optimisation task, neural networks can be trained in a semi-supervised learning fashion to approximate nonlinear solutions. In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation. Different methods, such as the naive and ansatz formulations, are contrasted, and their overall performance is analysed. Experimental data is also used to validate the neural solutions on test cases, specifically: the spring-mass system and Gauss’s law for electric fields. The errors of the neural solvers against exact solutions are investigated and found to surpass traditional schemes in certain cases. Although neural solvers will not replace the computational speed offered by traditional schemes in the near future, they remain a feasible, easy-to-implement substitute when all else fails.

A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows

2014 ◽  
Vol 24 (3) ◽  
pp. 735-759 ◽  
Author(s):  
Anis Younes ◽  
Ahmed Makradi ◽  
Ali Zidane ◽  
Qian Shao ◽  
Lyazid Bouhala
Keyword(s):  
Natural Convection ◽  
Numerical Methods ◽  
Analytical Solution ◽  
Discontinuous Galerkin ◽  
Truncation Error ◽  
Computational Time ◽  
Time Step ◽  
Content Type ◽  
Flow Problems ◽  
Time Stepping

Purpose – The purpose of this paper is to develop an efficient non-iterative model combining advanced numerical methods for solving buoyancy-driven flow problems. Design/methodology/approach – The solution strategy is based on two independent numerical procedures. The Navier-Stokes equation is solved using the non-conforming Crouzeix-Raviart (CR) finite element method with an upstream approach for the non-linear convective term. The advection-diffusion heat equation is solved using a combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods. To reduce the computational time due to the coupling, the authors use a non-iterative time stepping scheme where the time step length is controlled by the temporal truncation error. Findings – Advanced numerical methods have been successfully combined to solve buoyancy-driven flow problems on unstructured triangular meshes. The accuracy of the results has been verified using three test problems: first, a synthetic problem for which the authors developed a semi-analytical solution; second, natural convection of air in a square cavity with different Rayleigh numbers (103-108); and third, a transient natural convection problem of low Prandtl fluid with horizontal temperature gradient in a rectangular cavity. Originality/value – The proposed model is the first to combine advanced numerical methods (CR, DG, MPFA) for buoyancy-driven flow problems. It is also the first to use a non-iterative time stepping scheme based on local truncation error control for such coupled problems. The developed semi analytical solution based on Fourier series is also novel.

Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation

1998 ◽  
Vol 529 ◽  
Author(s):  
David J. Eyre
Keyword(s):  
Numerical Methods ◽  
Linear Equations ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Time Marching ◽  
Cahn Hilliard Equation

AbstractNumerical methods for time stepping the Cahn-Hilliard equation are given and discussed. The methods are unconditionally gradient stable, and are uniquely solvable for all time steps. The schemes require the solution of ill-conditioned linear equations, and numerical methods to accurately solve these equations are also discussed.

scholarly journals Metode Numerik Untuk Menentukan Harga Opsi Dengan Model Volatilitas Leland

2020 ◽  
Vol 2 (2) ◽  
pp. 41-51
Author(s):  
Arsyad L
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Transaction Cost ◽  
Option Price ◽  
Implicit Time ◽  
Implicit Methods ◽  
Time Stepping ◽  
Volatility Model ◽  
Upwind Finite Difference

Option price under transaction cost with leland volatility model is the solution of a non linear diferential equations. To solve this equation used numerical methods based on an upwind finite difference for spatial discretization as well as the use of explicit and implicit methods for discretizing time-stepping. upwind finite difference method with explicit time-stepping scheme proved to be unstable so as not konvegen. While the use of implicit time-stepping scheme is proved monotonous, consistent and stable so that converge to the viscosity solution.

scholarly journals Pattern recognition of the fluid flow in a 3D domain by combination of Lattice Boltzmann and ANFIS methods

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Meisam Babanezhad ◽  
Ali Taghvaie Nakhjiri ◽  
Azam Marjani ◽  
Saeed Shirazian
Keyword(s):  
Pattern Recognition ◽  
Fluid Flow ◽  
Numerical Methods ◽  
Flow Pattern ◽  
Fuzzy Inference System ◽  
Fuzzy Inference ◽  
Computational Time ◽  
Complex Geometries ◽  
Inference System ◽  
Anfis Method

Abstract Many numerical methods have been used to simulate the fluid flow pattern in different industrial devices. However, they are limited with modeling of complex geometries, numerical stability and expensive computational time for computing, and large hard drive. The evolution of artificial intelligence (AI) methods in learning large datasets with massive inputs and outputs of CFD results enables us to present completely artificial CFD results without existing numerical method problems. As AI methods can not feel barriers in numerical methods, they can be used as an assistance tool beside numerical methods to predict the process in complex geometries and unstable numerical regions within the short computational time. In this study, we use an adaptive neuro-fuzzy inference system (ANFIS) in the prediction of fluid flow pattern recognition in the 3D cavity. This prediction overview can reduce the computational time for visualization of fluid in the 3D domain. The method of ANFIS is used to predict the flow in the cavity and illustrates some artificial cavities for a different time. This method is also compared with the genetic algorithm fuzzy inference system (GAFIS) method for the assessment of numerical accuracy and prediction capability. The result shows that the ANFIS method is very successful in the estimation of flow compared with the GAFIS method. However, the GAFIS can provide faster training and prediction platform compared with the ANFIS method.

scholarly journals Microreactor Production by PolyJet Matrix 3D-Printing Technology

2019 ◽  
Vol 57 (2) ◽  
pp. 272-281 ◽  
Author(s):  
Mladen Šercer ◽  
Damir Godec ◽  
Božidar Šantek ◽  
Roland Ludwig ◽  
Martina Andlar ◽  
...  
Keyword(s):  
Numerical Methods ◽  
3D Printing ◽  
Material Properties ◽  
Single Phase ◽  
Acrylic Resin ◽  
Complex Geometries ◽  
Phase Flow ◽  
Single Phase Flow ◽  
3D Printing Technology ◽  
3D Printed

This work investigates the methodology of producing a 3D-printed microreactor from the acrylic resin by PolyJet Matrix process. The PolyJet Matrix technology employs different materials or their combinations to generate 3D-printed structures, from small ones to complex geometries, with different material properties. Experimental and numerical methods served for the evaluation of the geometry and production of the microreactor and its hydrodynamic characterization. The operational limits of the single-phase flow in the microchannels, further improvements and possible applications of the microreactor were assessed based on the hydrodynamic characterization.

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