Teaching computational methods for partial differential equations using the Web

Jaun, A.; Hedin, Johan; Johnson, Tracy

doi:10.1109/5992.919272

Cited by 8 publications

(5 citation statements)

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“…To solve this equation, the gradient on the faces of the cells is required. The non-orthogonal correction is adopted to obtain the gradient for unstructured meshes [64].…”

Section: Methodsmentioning

confidence: 99%

GiftBTE: an efficient deterministic solver for non-gray phonon Boltzmann transport equation

Hu,

Jia,

et al. 2023

J. Phys.: Condens. Matter

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Advances in nanotechnology have facilitated the exploration of submicron thermal transport. At this scale, Fourier's law is no longer applicable, and the governing equation for thermal transport is the phonon Boltzmann transport equation (BTE). However, the availability of open-source solvers for the phonon BTE is limited, impeding progress in this field. This study introduces an open-source package, GiftBTE, for numerically solving the non-gray phonon BTE. GiftBTE employs deterministic solutions and provides both steady-state and transient solvers. For the steady-state solver, GiftBTE employs the implicit discrete ordinates method (DOM) with second-order spatial accuracy and the synthetic iterative scheme. For the transient solver, GiftBTE employs the explicit DOM with second-order spatial accuracy. This package demonstrates excellent computational efficiency, enabling realistic three-dimensional simulations of devices and materials. By interfacing with first-principles calculations, this solver enables parameter-free computation of submicron thermal transport. The application of GiftBTE includes, but is not limited to, computing the thermal conductivity of nanostructures, predicting temperature rises in transistors, and simulating laser heating processes.

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“…To solve this equation, the gradient on the faces of the cells is required. The non-orthogonal correction is adopted to obtain the gradient for unstructured meshes [64].…”

Section: Methodsmentioning

confidence: 99%

GiftBTE: an efficient deterministic solver for non-gray phonon Boltzmann transport equation

Hu,

Jia,

et al. 2023

J. Phys.: Condens. Matter

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“…Como es posible que los puntos de la frontera no coincidan con los nodos de la discretización del dominio, se deberá realizar un manejo de error máximo permitido en la distancia de un nodo a la frontera, para que ese nodo pertenezca a la frontera. Con este método, se puede seguir utilizando el algoritmo de cálculo propuesto en la referencia [5,6] Utilizando el algoritmo derivado del modelo matemático explicado en la sección anterior y soportado en diferentes trabajos donde se puede consultar [1,2,3,4,5,6,7,8], se obtienen los siguientes resultados para geometrías que han sido presentadas en la literatura al respecto y en las referencias consultadas. En la figura 1 podemos observar los casos de aplicación de esta técnica sobre un dominio circular con condiciones de frontera del tipo Dirichlet por facilidad de trabajo y sobre un dominio en forma de L, con condiciones de frontera tipo Dirichlet y Neumann, para mostrar la versatilidad del algoritmo de trabajo fronterizo.…”

Section: Modelamiento Matemáticounclassified

Discretización en diferencias finitas de la ecuación de Laplace y Poisson. Aplicación a un anillo circular (dona)

Moreno

Hernández

Fonseca

2015

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Recepción: 05-ene-15 Aceptación: 24-jun-15 ResumenEntre los métodos de solución numérica más comunes para ecuaciones diferenciales parciales (EDP) se encuentran el método de las diferencias finitas y el método de los elementos finitos, que se acercan a la solución real a través de un algoritmo de convergencia de manera eficiente y acertada. Muchos de los fenómenos físicos que se pueden estudiar mediante estas técnicas obedecen su comportamiento a las EDP's de Laplace y de Poisson, sobre las que se pueden restringir diferentes condiciones iniciales y de frontera, para limitar las soluciones de la ecuación. Este trabajo muestra la aplicación del método de diferencias finitas con un manejo sencillo de la discretización del dominio y, así mismo, un manejo sencillo de las condiciones de frontera sobre varios dominios, principalmente sobre dominios del tipo anillo o "dona", mostrando resultados interesantes al seleccionar condiciones de frontera del tipo Dirichlet.Palabras clave: diferencias finitas, dona, Laplace, Poisson. AbstractAmong the more common numeric methods of solution for partial differential equations (PDE) we have the finite differences method and the finite elements method that approach the real solution through an efficient and accurate algorithm of convergence. Many of the physical phenomena that can be studied by means of these techniques obey their behavior to the EDP' s of Laplace and Poisson, on whom different initial and/or boundary conditions can be restricted, to limit the solutions of the equation. This work shows the application of the finite difference method with a simple handling of the domain discretization and a simple handling of the boundary conditions on several domains, mainly with the domain with shape of ring or "donut", showing interesting results when selecting border conditions of the Dirichlet kind.

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“…It fi ts well at the end of a sophomore-level numerical methods course and will help students be ready for more complicated models in heat transfer and other mechanical engineering courses in their third and fourth years. Also, 'computational methods [partial differential equations] are part of the problem solving skills that professionals working in quantitative fi elds need' [3].…”

Section: Introductionmentioning

confidence: 99%

Computer Experiments with Diffusion: Finite Differences, round-off Error and Animal Stripes?

Yerion

2013

International Journal of Mechanical Engineering Education

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This paper describes a module for the teaching of diffusion to mechanical engineering students that can be part of a course on numerical methods or advanced engineering mathematics and as a precursor to a heat transfer course. The purpose of the module is to pull together many of the mathematical concepts in the early education of mechanical engineering students while at the same time introducing them to other important topics -an application of fi nite differences to the numerical approximation of partial differential equations, the importance of controlling round-off error in computational algorithms, and the time complexity of some numerical models. Students learn these topics through the execution of computer experiments. The fi rst set of experiments cover the numerical approximation of heat conduction in a ring, and a second set cover the numerical approximation of the reaction and diffusion of chemicals that will produce the markings on an animal's tail. The learning outcomes of the module are evaluated with data from pre-and post-module surveys given to mechanical engineering students in several sections of a required course in numerical methods.

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Teaching computational methods for partial differential equations using the Web

Cited by 8 publications

References 0 publications

GiftBTE: an efficient deterministic solver for non-gray phonon Boltzmann transport equation

GiftBTE: an efficient deterministic solver for non-gray phonon Boltzmann transport equation

Discretización en diferencias finitas de la ecuación de Laplace y Poisson. Aplicación a un anillo circular (dona)

Computer Experiments with Diffusion: Finite Differences, round-off Error and Animal Stripes?

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