Numerical differential equations
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, the method is generally distinguished by its inclusion of rotational degrees-of-freedom as well as stateful contact and often complicated geometries (including polyhedra). With advances in computing power and numerical algorithms for nearest neighbor sorting, it has become possible to numerically simulate millions of particles on a single processor. Today DEM is becoming widely accepted as an effective method of addressing engineering problems in granular and discontinuous materials, especially in granular flows, powder mechanics, and rock mechanics. DEM has been extended into the Extended Discrete Element Method taking heat transfer, chemical reaction and coupling to CFD and FEM into account. Discrete element methods are relatively computationally intensive, which limits either the length of a simulation or the number of particles. Several DEM codes, as do molecular dynamics codes, take advantage of parallel processing capabilities (shared or distributed systems) to scale up the number of particles or length of the simulation. An alternative to treating all particles separately is to average the physics across many particles and thereby treat the material as a continuum. In the case of solid-like granular behavior as in soil mechanics, the continuum approach usually treats the material as elastic or elasto-plastic and models it with the finite element method or a mesh free method. In the case of liquid-like or gas-like granular flow, the continuum approach may treat the material as a fluid and use computational fluid dynamics. Drawbacks to homogenization of the granular scale physics, however, are well-documented and should be considered carefully before attempting to use a continuum approach. (Wikipedia).
Jacobi, Gauss-Seidel and SOR Methods | Lecture 66 | Numerical Methods for Engineers
Iterative methods for solving the discrete Laplace equation. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?
From playlist Numerical Methods for Engineers
Discrete Math - 2.6.1 Matrices and Matrix Operations
Characteristics of a matrix, finding the sum, product and transpose of a matrix. Identity matrix is also introduced. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz
From playlist Discrete Math I (Entire Course)
[Discrete Mathematics] Set Operations Examples #2
In this video we do some examples with set operations. For instance, given some operators, can we find the original sets? We also do a proof with subsets. LIKE AND SHARE THE VIDEO IF IT HELPED! Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playl
From playlist Discrete Math 1
Set Definitions and Operations: Sample Problems
This video contains solutions to some of the sample problems from our section on set definitions and operations.
From playlist Discrete Mathematics
Discrete Laplace Equation | Lecture 62 | Numerical Methods for Engineers
Derivation of the discrete Laplace equation using the central difference approximations for the partial derivatives. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscr
From playlist Numerical Methods for Engineers
[Discrete Mathematics] Mathematical Induction with Derivatives and Matrices
We do two examples of mathematical induction. One with derivatives, and one with matrix multiplication. LIKE AND SHARE THE VIDEO IF IT HELPED! Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playlists--* Discrete Mathematics 1: https://www.youtube
From playlist Discrete Math 1
Introduction to Discrete and Continuous Functions
This video defines and provides examples of discrete and continuous functions.
From playlist Introduction to Functions: Function Basics
Discrete Fourier Transform - Example
We do a very simple example of a Discrete Fourier Transform by hand, just to get a feel for it. We quickly realize that using a computer for this is a good idea...
From playlist Mathematical Physics II Uploads
Sets might contain an element that can be identified as an identity element under some binary operation. Performing the operation between the identity element and any arbitrary element in the set must result in the arbitrary element. An example is the identity element for the binary opera
From playlist Abstract algebra
Convergent Evolving Surface Finite Element Algorithms for Geometric Evolution Equations
Professor Christian Lubich University of Tübingen, Germany
From playlist Distinguished Visitors Lecture Series
Lec 1 | MIT Finite Element Procedures for Solids and Structures, Linear Analysis
Lecture 1: Some basic concepts of engineering analysis Instructor: Klaus-Jürgen Bathe View the complete course: http://ocw.mit.edu/RES2-002S10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT Linear Finite Element Analysis
Erik Burman: Combining cut element methods and hybridization
Recently there has been a surge in interest in cut, or unfitted, finite element methods. In this class of methods typically the computational mesh is independent of the geometry. Interfaces and boundaries are allowed to cut through the mesh in a very general fashion. Constraints on the bou
From playlist Numerical Analysis and Scientific Computing
Marta D'Elia: A coupling strategy for nonlocal and local models with applications ...
The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local Partial Differential Equation (PDE) models motivates a
From playlist HIM Lectures: Trimester Program "Multiscale Problems"
Keynote: Solving Engineering Problems with Mathematica's PDE Tools
Oliver Rübenkönig To learn more about the Wolfram Technologies, visit http://www.wolfram.com The European Wolfram Technology Conference featured both introductory and expert sessions on all major technologies and many applications made possible with Wolfram technology. Learn to achieve
From playlist European Wolfram Technology Conference 2015
8ECM Invited Lecture: Ilaria Perugia
From playlist 8ECM Invited Lectures
Numerical Homogenization by Localized Orthogonal Decomposition (Lecture 3) by Daniel Peterseim
DISCUSSION MEETING Multi-Scale Analysis: Thematic Lectures and Meeting (MATHLEC-2021, ONLINE) ORGANIZERS: Patrizia Donato (University of Rouen Normandie, France), Antonio Gaudiello (Università degli Studi di Napoli Federico II, Italy), Editha Jose (University of the Philippines Los Baño
From playlist Multi-scale Analysis: Thematic Lectures And Meeting (MATHLEC-2021) (ONLINE)
Martin Vohralik: A posteriori error estimates and solver adaptivity in numerical simulations
Abstract: We review how to bound the error between the unknown weak solution of a PDE and its numerical approximation via a fully computable a posteriori estimate. We focus on approximations obtained at an arbitrary step of a linearization (Newton-Raphson, fixed point, ...) and algebraic s
From playlist Numerical Analysis and Scientific Computing
Ari Stern: Hybrid finite element methods preserving local symmetries and conservation laws
Abstract: Many PDEs arising in physical systems have symmetries and conservation laws that are local in space. However, classical finite element methods are described in terms of spaces of global functions, so it is difficult even to make sense of such local properties. In this talk, I wil
From playlist Numerical Analysis and Scientific Computing
Discrete Structures, Oct 20: Counting
Combinations, Permutations, Pigeonhole Principle
From playlist Discrete Structures
MFEM Workshop 2021 | Unstructured Finite Element Neutron Transport using MFEM
The LLNL-led MFEM (Modular Finite Element Methods) project provides high-order mathematical calculations for large-scale scientific simulations. The project’s first community workshop was held virtually on October 20, 2021, with participants around the world. Learn more about MFEM at https
From playlist MFEM Community Workshop 2021