Numerical Models for the Sintering of Ceramics in a Multi-Mode Cavity

1994 ◽  
Vol 347 ◽  
Author(s):  
David C. Dibben ◽  
Wai B. Fu ◽  
Ricky A.C. Met Axas
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Numerical Models ◽  
Method Of Lines ◽  
Gradient Algorithm ◽  
Microwave Cavity ◽  
Algebraic Equations ◽  
Time Step ◽  
Linear Algebraic Equations ◽  
Multi Mode

ABSTRACTWe present two distinct approaches to the numerical determination of electromagnetic field intensities which must be known before the sintering of ceramics can be modelled in a multi-mode microwave cavity. In the first, a Finite Element Method, we employ edge elements to discretise Maxwell's Equations and apply the conjugate gradient algorithm to solve the resulting system of linear algebraic equations at each time step. The second, which is based on the Method of Lines, is a variant of the Finite Difference Time Domain technique and is used to transform Maxwell's Equations into a set of time-dependent ordinary differential equations. These methods are compared with the help of three examples: a small tray of mashed potatoes placed inside a cavity, a standard waveguide partially filled with a ceramic, and a cavity inhomogeneously loaded with the same material. The good agreements which we have found give us confidence in the soundness of either approach for use in numerical simulation work.

scholarly journals Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Algebraic Equations ◽  
Time Step ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Sinc Functions ◽  
Exponential Rate Of Convergence

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of orderα. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of3-αorder accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results.

scholarly journals Calculation of heat transfer in nanoscale heterostructures

2019 ◽  
Vol 21 (3) ◽  
pp. 175-181
Author(s):  
K. K. Abgarian ◽  
I. S. Kolbin
Keyword(s):  
Heat Transfer ◽  
Heat Dissipation ◽  
Parallel Implementation ◽  
Numerical Models ◽  
Hybrid Approach ◽  
Mean Free Path ◽  
Algebraic Equations ◽  
Mesh Free ◽  
Linear Algebraic Equations ◽  
Spatial Approximation

Abstract. The article discusses the calculation of the temperature regime in nanoscale AlAs/GaAs binary heterostructures. When modeling heat transfer in nanocomposites, it is important to take into account that heat dissipation in multilayer structures with layer sizes of the order of the mean free path of energy carriers (phonons and electrons) occurs not at the lattice, but at the layer boundaries (interfaces). In this regard, the use of classical numerical models based on the Fourier law is limited, because it gives significant errors. To obtain more accurate results, we used a model in which the heat distribution was assumed to be constant inside the layer, while the temperature was stepwise changed at the interfaces of the layers. A hybrid approach was used for the calculation: a finite−difference method with an implicit scheme for time approximation and a mesh−free model based on a set of radial basis functions for spatial approximation. The calculation of the parameters of the bases was carried out through the solution of the systems of linear algebraic equations. In this case, only weights of neuroelements were selected, and the centers and «widths» were fixed. As an approximator, a set of frequently used basic functions was considered. To increase the speed of calculations, the algorithm was parallelized. Calculation times were measured to estimate the performance gains using the parallel implementation of the method.

Space-Time Discontinuous Galerkin Method for Maxwell’s Equations

2013 ◽  
Vol 14 (4) ◽  
pp. 916-939 ◽  
Author(s):  
Ziqing Xie ◽  
Bo Wang ◽  
Zhimin Zhang
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Time Step ◽  
Fully Discrete ◽  
Temporal Domain ◽  
Grid Points ◽  
Spatial Approximation

AbstractA fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate is established under the L2-norm when polynomials of degree atmost r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

Tomographic inversion via the conjugate gradient method

Geophysics ◽  
1987 ◽  
Vol 52 (2) ◽  
pp. 179-185 ◽  
Author(s):  
John A. Scales
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Sparse Matrices ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Algebraic Equations ◽  
Tomographic Inversion ◽  
Linear Algebraic Equations ◽  
The Matrix

Tomographic inversion of seismic traveltime residuals is now an established and widely used technique for imaging the Earth’s interior. This inversion procedure results in large, but sparse, rectangular systems of linear algebraic equations; in practice there may be tens or even hundreds of thousands of simultaneous equations. This paper applies the classic conjugate gradient algorithm of Hestenes and Stiefel to the least‐squares solution of large, sparse systems of traveltime equations. The conjugate gradient method is fast, accurate, and easily adapted to take advantage of the sparsity of the matrix. The techniques necessary for manipulating sparse matrices are outlined in the Appendix. In addition, the results of the conjugate gradient algorithm are compared to results from two of the more widely used tomographic inversion algorithms.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

Sumudu Applications to Maxwell's Equations

PIERS Online ◽  
2009 ◽  
Vol 5 (4) ◽  
pp. 355-360 ◽  
Author(s):  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations

Maxwell’s Equations versus Newton’s Third Law

2018 ◽  
Author(s):  
Glyn Kennell ◽  
Richard Evitts
Keyword(s):  
Electric Fields ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Simulated Data ◽  
Numerical Data ◽  
Transport Equations ◽  
Concentration Gradients ◽  
Third Law

The presented simulated data compares concentration gradients and electric fields with experimental and numerical data of others. This data is simulated for cases involving liquid junctions and electrolytic transport. The objective of presenting this data is to support a model and theory. This theory demonstrates the incompatibility between conventional electrostatics inherent in Maxwell's equations with conventional transport equations. <br>

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