scholarly journals Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods

2021 ◽  
Vol 4 (3) ◽  
pp. 1-24
Author(s):  
Ben J. Ransom ◽  
◽  
Dean R. Wheeler
Keyword(s):  
Spectral Method ◽  
Analytic Solution ◽  
Effective Properties ◽  
Effective Conductivity ◽  
Analytic Solutions ◽  
Equivalent Circuits ◽  
Phase 2 ◽  
Primitive Cell ◽  
Effective Transport ◽  
Spectral Coefficients

<abstract><p>This work presents models for homogenizing or finding the effective transport or mechanical properties of microscale composites formed from highly contrasting phases described on a grid. The methods developed here are intended for engineering applications where speed and geometrical flexibility are a premium. A canonical case that is mathematically challenging and yet can be applied to many realistic materials is a 4-phase 2-dimensional periodic checkerboard or tiling. While analytic solutions for calculating effective properties exist for some cases, versatile methods are needed to handle anisotropic and non-square grids. A reinterpretation and extension of an existing analytic solution that utilizes equivalent circuits is developed. The resulting closed-form expressions for effective conductivity are shown to be accurate within a few percent or better for multiple cases of interest. Secondly a versatile and efficient spectral method is presented as a solution to the 4-phase primitive cell with a variety of external boundaries. The spectral method expresses the solution to effective conductivity in terms of analytically derived eigenfunctions and numerically determined spectral coefficients. The method is validated by comparing to known solutions and can allow extensions to cases with no current analytic solution.</p></abstract>

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

scholarly journals Dynamic Characteristics of Deeply Buried Spherical Biogas Digesters in Viscoelastic Soils

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Hongwei Hou ◽  
Shihu Gao ◽  
Qianqian Guo ◽  
Long Chen ◽  
Bing Wu ◽  
...  
Keyword(s):  
Cyclic Loading ◽  
Fractional Derivative ◽  
Elastic Medium ◽  
Shell Structure ◽  
Fractional Derivatives ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Dynamic Responses ◽  
Significant Difference ◽  
The Continuum

The harmonic vibration characteristics of a deeply buried spherical methane tank in viscoelastic soil subjected to cyclic loading in the frequency domain are investigated. The dynamic behavior of the soil is described based on the theory of fractional derivatives. By introducing potential functions, the closed-form expressions for the displacement and the stress of the viscoelastic soil surrounding the deeply buried spherical methane tank are obtained. Two die structures are considered: a homogeneous elastic medium and a shell structure. Based on the theory of elastic motion and the Flügge theory, analytic solutions for the dynamic responses of the spherical methane tank in a fractional-derivative viscoelastic soil are derived explicitly. Analytic solution expressions of the undetermined coefficients are determined by using the continuum boundary conditions. The system dynamic responses to the homogeneous elastic medium and the shell structure and the influences of the parameters of the fractional derivative, soil, and die on the dynamic characteristic of the system are compared and analyzed. The results indicate a significant difference between the dynamic responses of the die structures for the two models.

Construction of a Code Verification Matrix for Heat Conduction With Finite Element Code Applications

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
Aysenur Toptan ◽  
Nathan W. Porter ◽  
Jason D. Hales ◽  
Benjamin W. Spencer ◽  
Martin Pilch ◽  
...  
Keyword(s):  
Finite Element ◽  
Heat Conduction ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Simulation Tool ◽  
Code Verification ◽  
Simulation Tools ◽  
Steady State Heat ◽  
The One ◽  
Selection Of

Abstract When establishing the pedigree of a simulation tool, code verification is used to ensure that the implemented numerical algorithm is a faithful representation of its underlying mathematical model. During this process, numerical results on various meshes are systematically compared to a reference analytic solution. The selection of analytic solutions can be a laborious process, as it is difficult to establish adequate code confidence without performing redundant work. Here, we address this issue by applying a physics-based process that establishes a set of reference problems. In this process, code simulation options are categorized and systematically tested, which ensures that gaps in testing are easily identified and addressed. The resulting problems are primarily intended for code verification analysis but may also be useful for comparison to other simulation codes, troubleshooting activities, or training exercises. The process is used to select fifteen code verification problems relevant for the one-dimensional steady-state heat conduction equation. These problems are applicable to a wide variety of simulation tools, but, in this work, a demonstration is performed using the finite element-based nuclear fuel performance code BISON. Convergence to the analytic solution at the theoretical rate is quantified for a selection of the problems, which establishes a baseline pedigree for the code. Not only can this standard set of conduction solutions be used for verification of other codes, but also the physics-based process for selecting problems can be utilized to quantify and expand testing for any simulation tool.

Nonlinear behavior for periodically excited brushless motor

2019 ◽  
Vol 38 (2) ◽  
pp. 522-535
Author(s):  
Jianzhe Huang ◽  
Xingzhong Xiong
Keyword(s):  
Working Condition ◽  
Analytic Solution ◽  
Nonlinear Behavior ◽  
Analytic Solutions ◽  
Strong Nonlinearity ◽  
Periodic Motions ◽  
Content Type ◽  
Strongly Nonlinear ◽  
Brushless Motor ◽  
Analytic Expressions

Purpose Due to the coupling between the direct-axis current, quadrature-axis current and rotor speed, the dynamic response could be strongly nonlinear. Besides, if the working condition is severe, the loading is no longer constant and multiple harmonics could be introduced. In this paper, the periodic motions for brushless motor will be solved, and accurate analytic solution will be obtained through the proposed method. The purpose of this study is to provide accurate analytic solution of periodic motions for brushless motor with fluctuated loading, which is a dynamic system with strong nonlinearity. Design/methodology/approach A newly developed semi-analytic algorithm called discrete implicit maps is used to give analytic solutions for both stable and unstable motions for such a motor. Findings The accurate analytic expressions for stable and unstable periodic motions have been obtained. For unstable motion, it can stay on the unstable orbit for many periods without any controller. Through bifurcation analysis, the parameter sensitivity has been obtained which can be a suggestion for design and operation. Originality/value This paper provides all possible analytical solutions for period-1 motion as well as the unstable motions in a range of system parameters. It offers a chance to control the unstable motion for such a motor.

Analytic solutions for calcium ion fertilisation waves on the surface of eggs

2019 ◽  
Vol 36 (4) ◽  
pp. 549-562 ◽  
Author(s):  
Bronwyn H Bradshaw-Hajek ◽  
Philip Broadbridge
Keyword(s):  
Calcium Ion ◽  
Calcium Concentration ◽  
Analytic Solution ◽  
Reaction Diffusion ◽  
Approximate Equation ◽  
Analytic Solutions ◽  
Nonlinear Reaction ◽  
Nonclassical Symmetry ◽  
Sperm Entry ◽  
Insight Into

Abstract The evolution of calcium fertilisation waves on the cortex of amphibian eggs can be described by a nonlinear reaction-diffusion process on the surface of a sphere. Here, we use the nonclassical symmetry technique to find an exact analytic solution that describes the evolution of the calcium concentration. The solutions presented compare well with published experimental results. The analytic solution can be used to give insight into the processes governing the fertilisation wave, such as the flow of calcium ions from the sperm entry point. By finding a spiral solution to an approximate equation linearised near saturation, we also demonstrate how solutions with other properties may be constructed using this technique.

scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

Computational Evaluation of Effective Conductivity of Particulate Composites with Imperfect Interfaces

2009 ◽  
Vol 631-632 ◽  
pp. 35-40
Author(s):  
M. Zhang ◽  
Peng Cheng Zhai ◽  
Qing Jie Zhang
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Effective Properties ◽  
Effective Conductivity ◽  
Three Dimensional ◽  
Particulate Composite ◽  
Imperfect Interface ◽  
Particulate Composites ◽  
Imperfect Interfaces ◽  
Volume Fractions

This paper is aimed to numerically evaluate the effective thermal conductivity of randomly distributed spherical particle composite with imperfect interface between the constituents. A numerical homogenization technique based on the finite element method (FEM) with representative volume element (RVE) was used to evaluate the effective properties with periodic boundary conditions. Modified random sequential adsorption algorithm (RSA) is applied to generate the three dimensional RVE models of randomly distributed spheres of identical size with the volume fractions up to 50%. Several investigations have been conducted to estimate the influence of the imperfect interfaces on the effective conductivity of particulate composite. Numerical results reveal that for the given composite, due to the existence of an interfacial thermal barrier resistance, the effective thermal conductivity depends not only on the volume fractions of the particle but on the particle size.

scholarly journals A composite material with inextensible-membrane-type interface

2018 ◽  
Vol 24 (2) ◽  
pp. 499-510
Author(s):  
LP Castro ◽  
E Pesetskaya
Keyword(s):  
Composite Material ◽  
Analytic Solution ◽  
Effective Conductivity ◽  
Heat Conduction Problem ◽  
Linear Boundary ◽  
Temperature Dependent ◽  
The Matrix ◽  
Membrane Type ◽  
Conductive Properties ◽  
Linear Boundary Value Problem

We consider a model of a composite material with “inextensible membrane type” interface conditions. An analytic solution of a stationary heat conduction problem in an unbounded doubly periodic two-dimensional composite whose matrix and inclusions consist of isotropic temperature-dependent materials is given. In the case where the conductive properties of the inclusions are proportional to those of the matrix, the problem is transformed into a fully linear boundary value problem for doubly periodic analytic functions. The solution makes it possible to calculate the average properties over the unit cell and discuss the effective conductivity of the composite. We present numerical examples to indicate some peculiarities of the solution.

scholarly journals A new analytic solution of complex Langevin differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rabha Ibrahim
Keyword(s):  
Differential Equations ◽  
Analytic Solution ◽  
Function Theory ◽  
Special Functions ◽  
Analytic Solutions ◽  
Geometric Function Theory ◽  
Content Type ◽  
Complex Differential Equation ◽  
Geometric Function ◽  
Carathéodory Functions

PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

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