**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Periodic boundary conditions

Summary

Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell. PBCs are often used in computer simulations and mathematical models. The topology of two-dimensional PBC is equal to that of a world map of some video games; the geometry of the unit cell satisfies perfect two-dimensional tiling, and when an object passes through one side of the unit cell, it re-appears on the opposite side with the same velocity. In topological terms, the space made by two-dimensional PBCs can be thought of as being mapped onto a torus (compactification). The large systems approximated by PBCs consist of an infinite number of unit cells. In computer simulations, one of these is the original simulation box, and others are copies called images. During the simulation, only the properties of the original simulation box need to be recorded and propagated. The minimum-image convention is a common

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (31)

Related concepts (4)

Molecular dynamics

Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giv

Water model

In computational chemistry, a water model is used to simulate and thermodynamically calculate water clusters, liquid water, and aqueous solutions with explicit solvent. The models are determined from

Comparison of software for molecular mechanics modeling

This is a list of computer programs that are predominantly used for molecular mechanics calculations.
See also
*Car–Parrinello molecular dynamics
*Comparison of force-field implement

Related courses (71)

ME-373: Finite element modelling and simulation

L'objectif de ce cours est d'apprendre à réaliser de manière rigoureuse et critique des analyses par éléments finis de problèmes concrets en mécanique des solides à l'aide d'un logiciel CAE moderne.

CH-351: Molecular dynamics and Monte-Carlo simulations

Introduction to molecular dynamics and Monte-Carlo simulation methods.

PHYS-309: Solid state physics I

This lecture gives an introduction to Solid State Physics, namely to their crystal and electronic structure, their magnetic properties, as well as to their thermal and electric conductance. The level is that of the book by Ashcroft & Mermin. The lecture is for Physics Students in their 3rd year

Related publications (100)

Related units (17)

Related lectures (173)

Loading

Loading

Loading

In this work, two problems linked to glacier modeling are investigated. We propose an optimisation method for studying the flow of the ice and we present a numerical study about glacier thermal phenomena. In the first chapter of this thesis, we expose the models of these two problems. On one hand, we note that the boundary conditions on the bedrock are misunderstood, which explains it is difficult to obtain an accurate simulation of the motion of the ice. Also we establish a mathematical model where the bedrock boundary conditions depend on a control parameter. The aim of this study is to minimize a cost functional describing the difference between the computed velocity at the surface and the measure done. We study the cost functional with respect to the control parameter and we detail an optimisation method to solve the optimal control problem. On the other hand, we introduce two thermodynamical model governing the temperature and the water content field. The models correspond to a Stefan problem for the temperature and a convection-diffusion equation for the water content. The second chapter deals with the numerical resolution of the optimisation problem. First, a Finite Element Method (FEM) is described to solve the partial differential equations. Then, the algorithms used for the optimal control problem are detailed. Finally, this techniques are applied on two glaciers : Griesgletcher for 2D and Storglaciaren for 3D. The third chapter deals with the numerical resolution of the temperature and the water content models. A FEM is used for each problem. Concerning the temperature problem, the Stefan problem is numerically solved and the results allow to detect a free surface between the temperated ice and the cold ice. The water content field is also simulated. Numerical results are discussed on the Storglaciaren.

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020The miniaturization trend in industry requires micro devices for handling and sensing in a micro environment. As the size of the handled objects decreases and their fragility increases, the demand for sensors with higher force sensitivity and resolution grows. Force measurement as a technical process is always the conversion of the quantity force into another measurable quantity. The most commonly used "transducing"-principle for force measurement is the conversion of force to elastic deformation i.e. mechanical strain. To yield a high sensitivity, the force has to induce a large strain, which makes the sensor very compliant. The biggest problem of the high compliance force sensors is that the concept works only for forces which are not position dependent (i.e. gravitational forces). To provide force measurement for highly position dependent forces, a system is needed that is very stiff but also very sensitive. The concept of choice is a force sensor system where the force induced deflection is actively compensated, e.g. by a piezo-electric actuator. This work describes the development of the analytical models for two kinds of piezo-electric sensors and actuators. In the first part, the direct-drive-actuators (e.g. stacked actuators) are discussed, and in the second part the models for beam shaped actuators are subject of investigation. The developed models represent a new modeling approach which is open to any geometrical variations and different boundary conditions. The suitability of the presented models for optimization is shown later in this work and a procedure for optimization is developed. In the last, part the function of the two considered systems is demonstrated experimentally. The concept of the direct-drive-system is demonstrated with a charge-sensing stacked actuator-sensor system. For the beam-shaped-actuator system, a 1-degree of freedom was set up. In these experiments strain gauges measured the force and the position.