Discover what is the Cell Method and how to apply it to physical problems. This presentation will guide you through some interesting computational implementations.
Youtube video: https://youtu.be/ZyyRovtibjo
Time Independent Perturbation Theory, 1st order correction, 2nd order correctionJames Salveo Olarve
The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
Perturbation theory allows physicists to approximate how small changes to a quantum system's potential will affect it. It involves treating the changed part of the Hamiltonian as a perturbation and solving the perturbed eigenvalue problem order-by-order. The first order energy correction is the expectation value of the perturbing potential in the unperturbed eigenstate. The first order eigenstate correction is a superposition of unperturbed eigenstates weighted by the perturbing potential's matrix elements.
application of differential equation and multiple integraldivya gupta
This document discusses differential equations and their applications. It begins by defining differential equations as mathematical equations that relate an unknown function to its derivatives. There are two types: ordinary differential equations involving one variable, and partial differential equations involving two or more variables. Applications are given for modeling physical systems involving mass, springs, dampers, fluid dynamics, heat transfer, and rigid body dynamics. The document also discusses surface and volume integrals involving vectors, with examples of calculating fluid flow rates and mass of water in a reservoir. Differential equations and multiple integrals find diverse applications in engineering fields.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
This presentation discusses perturbation theory in physical chemistry. Perturbation theory allows solving problems where the potential energy function is slightly different than a problem that can be solved exactly. It provides approximations for how small changes or perturbations to the potential energy function affect the system's energy levels and eigenfunctions. The presentation provides examples of using regular and singular perturbation theory to solve differential equations that model perturbed systems. It also outlines the general process of deriving the first-order energy correction term using perturbation theory.
Applications of differential equation in Physics and BiologyAhamed Yoonus S
This document discusses several applications of differential equations in physics. It provides examples of how differential equations are used to model radioactive decay, linear and projectile motion, harmonic oscillations, and more. Solving these differential equations provides insights into the physical processes being modeled and has allowed technological progress across many scientific disciplines. Differential equations are necessary to describe most physical phenomena accurately because real-world relationships are typically non-linear rather than linear.
The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.
Time Independent Perturbation Theory, 1st order correction, 2nd order correctionJames Salveo Olarve
The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
Perturbation theory allows physicists to approximate how small changes to a quantum system's potential will affect it. It involves treating the changed part of the Hamiltonian as a perturbation and solving the perturbed eigenvalue problem order-by-order. The first order energy correction is the expectation value of the perturbing potential in the unperturbed eigenstate. The first order eigenstate correction is a superposition of unperturbed eigenstates weighted by the perturbing potential's matrix elements.
application of differential equation and multiple integraldivya gupta
This document discusses differential equations and their applications. It begins by defining differential equations as mathematical equations that relate an unknown function to its derivatives. There are two types: ordinary differential equations involving one variable, and partial differential equations involving two or more variables. Applications are given for modeling physical systems involving mass, springs, dampers, fluid dynamics, heat transfer, and rigid body dynamics. The document also discusses surface and volume integrals involving vectors, with examples of calculating fluid flow rates and mass of water in a reservoir. Differential equations and multiple integrals find diverse applications in engineering fields.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
This presentation discusses perturbation theory in physical chemistry. Perturbation theory allows solving problems where the potential energy function is slightly different than a problem that can be solved exactly. It provides approximations for how small changes or perturbations to the potential energy function affect the system's energy levels and eigenfunctions. The presentation provides examples of using regular and singular perturbation theory to solve differential equations that model perturbed systems. It also outlines the general process of deriving the first-order energy correction term using perturbation theory.
Applications of differential equation in Physics and BiologyAhamed Yoonus S
This document discusses several applications of differential equations in physics. It provides examples of how differential equations are used to model radioactive decay, linear and projectile motion, harmonic oscillations, and more. Solving these differential equations provides insights into the physical processes being modeled and has allowed technological progress across many scientific disciplines. Differential equations are necessary to describe most physical phenomena accurately because real-world relationships are typically non-linear rather than linear.
The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
Introduction to perturbation theory, part-1Kiran Padhy
Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations that contain derivatives of unknown functions of several variables and one or more partial derivatives. The solutions to PDEs are differentiable functions that satisfy boundary or initial conditions. PDEs are often used to express laws of physics. Examples of common PDEs discussed include the Laplace equation, Poisson equation, wave equation, heat equation, and diffusion equation.
This document provides information about a Statics course including the course goals, objectives, content, assessment, teaching strategies, textbook, and lecture times. The course aims to introduce concepts of forces, couples and moments in two and three dimensions and develop relevant analytical skills. Upon completion, students should be able to determine force resultants, centroids, draw free body diagrams, and apply equilibrium principles and analytical techniques to engineering problems. The course will be taught via lectures and tutorials using a specified textbook and will include assignments, tests, and an exam for assessment.
The document discusses applications of first order differential equations. It provides examples in several domains:
1) Cooling/warming laws can be modeled using differential equations, like the temperature of cooling coffee over time.
2) Population growth and decay models use differential equations, like calculating time for a population to double at a growth rate.
3) Determining geometric properties of curves can involve solving differential equations, like finding the equation of a curve based on its tangent slope.
4) Applications also include radioactive decay, electrical circuits, mixture of solutions, and modeling motion. First order differential equations have widespread uses in physics, statistics, chemistry, engineering, and other fields.
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an overview of engineering mechanics statics. It covers topics including:
- Defining mechanics as the science dealing with bodies at rest or in motion under forces.
- Dividing mechanics into statics, dynamics, and other subfields. Statics deals with bodies at rest.
- Introducing fundamental concepts of forces, units of measurement, and representing forces as vectors that add according to the parallelogram law.
- Providing examples of adding forces graphically using the parallelogram law and triangle rule to determine the resultant force.
- Discussing problems involving determining the magnitude and direction of resultant forces from multiple forces acting on structures, stakes, and brackets
This document provides an overview of the topics that will be covered in the TSPGECET-2018 exam. It discusses the following engineering subjects: engineering mechanics, strength of materials, fluid mechanics, heat transfer, materials science, metallurgical thermodynamics, advanced materials science, and nanotechnology. Specific concepts covered include stresses and strains, fluid dynamics, heat transfer principles, crystal structures, phase equilibria, electrical and optical properties of materials, polymers, ceramics, and composite materials. The document also briefly outlines the challenges in nanotechnology and its applications.
This document discusses time-dependent perturbation theory. It begins by introducing the concept of applying a time-dependent perturbation to a quantum system to induce transitions between its energy eigenstates. It then describes how the interaction picture can be used to focus on the slow evolution induced by the perturbation, without considering the rapid oscillation from the unperturbed Hamiltonian. The interaction picture defines a transformed state vector and operators such that the perturbation Hamiltonian governs the evolution operator in a Schrodinger equation.
A talk presented at the University of New South Wales on the occasion of Ian Sloan's 80th birthday, remembering our work together and thinking about how math is used in science.
This chapter focuses on objects in static equilibrium, where the net force and net torque on the object are both zero. Solving static equilibrium problems involves drawing free body diagrams showing all external forces acting on the object, then resolving forces into components and setting the sums of forces in each direction equal to zero. Three examples are given of solving static equilibrium problems involving particles under the influence of multiple forces. The problems are solved by resolving forces into horizontal and vertical or parallel and perpendicular components, setting the component force equations equal to zero, and solving the equations to determine the magnitudes of unknown forces. Key steps include drawing diagrams, resolving forces, setting force sums to zero, and solving the resulting equations.
This document discusses key concepts relating to work, energy, and power in mechanics. It defines work as the force multiplied by the distance moved, and explains that work is equal to the magnitude of the force component in the direction of motion. It also discusses different types of energy (kinetic, potential, elastic, etc.), the principle of conservation of energy, and defines power as the rate of doing work.
Oscillation of Solutions to Neutral Delay and Advanced Difference Equations w...inventionjournals
In this article we give infinite-sum conditions for the oscillation of all solutions of the following first order neutral delay and advanced difference equations with positive and negative coefficientsof the forms and where is a sequence of nonnegative real numbers, and are sequences of positive real numbers, and are positive integers. We derived sufficient conditions for oscillation of all solutions of and . AMS Subject Classification 2010: 39A10, 39A12
hermodynamics is the branch of physics that has to do with heat and temperature and their relation to energy and work. The behavior of these quantities is governed by the four laws of thermodynamics, irrespective of the composition or specific properties of the material or system in question. The laws of thermodynamics are explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, chemical engineering and mechanical engineering.
Oscillation of Solutions to Neutral Delay and Advanced Difference Equations w...inventionjournals
In this article we give infinite-sum conditions for the oscillation of all solutions of the following first order neutral delay and advanced difference equations with positive and negative coefficientsof the forms and where is a sequence of nonnegative real numbers, and are sequences of positive real numbers, and are positive integers. We derived sufficient conditions for oscillation of all solutions of and . AMS Subject Classification 2010: 39A10, 39A12
THE FEYNMAN-STUCKELBREG INTERPRETATION OF E<0 SOLUTIONSHusnain Ali
The document discusses relativistic quantum mechanics and the Klein-Gordon equation. It describes the Feynman-Stuckelberg interpretation of negative energy solutions to the Klein-Gordon equation, which interprets them as representing positive energy anti-particles moving forward in time. This resolves issues with negative probabilities and allows a consistent treatment of antiparticles. The interpretation is illustrated using a diagram of double electron scattering, showing it accounts for both time orderings through particle-antiparticle pair production and annihilation.
This document discusses the macroscopic and microscopic approaches in thermodynamics. The macroscopic approach examines the total energy a system possesses with respect to external frames of reference, such as kinetic and potential energies. The microscopic approach considers the molecular structure and activity within a system independent of outside influences. Energy can exist in many forms, both within a system and between a system and its environment. Temperature is defined as the average kinetic energy of molecules in a system and is a measure of heat. Pressure is defined as the external force per unit area on the boundaries of a system, which results from the transfer of molecular momentum at the boundaries.
1. The Ising model is a statistical mechanics model of ferromagnetism. It represents magnetic dipole moments as "spins" on a lattice that can point in one of two directions and interact with neighboring spins.
2. The Ising model can explain phase transitions like ferromagnetism, anti-ferromagnetism, gas-liquid transitions, and liquid-solid transitions.
3. The statistical mechanics of the Ising model are studied using the Hamiltonian, which includes terms for spin-spin interaction energy and the energy of an external magnetic field interacting with the magnetic moments. Partition functions are then used to calculate thermodynamic properties.
The document provides instructions for a lab on Newton's Second Law of Motion. It instructs students to predict what will happen to the acceleration of a cart under different forces and masses. It lists the materials needed and measurements that will be taken, which include qualitative and quantitative observations. It also reviews key concepts like the relationships between force, mass and acceleration defined by F=ma, and how to account for significant figures in calculations and measurements.
This document discusses thermodynamic principles and concepts. It defines key thermodynamic terms like system, environment, isolated system, open system, closed system, state parameters, and equations of state. The first law of thermodynamics states that the change in internal energy of a system equals the heat transferred plus work done on the system. The second law states that the entropy of any isolated system always increases and approaches a maximum value. Entropy is a measure of disorder in a system and is related to the number of microscopic arrangements that can produce a given macrostate.
This document provides an overview of the course MCT-114: Fundamentals of Thermal Sciences. The objectives of the course are to provide a solid grounding in engineering thermodynamics and its fundamental concepts. Topics covered include the basic concepts, laws of energy, ideal gas model, entropy, and power/refrigeration cycles. The course also introduces heat transfer concepts. The document outlines the suggested textbooks, course learning objectives, and provides an introduction to thermal-fluid sciences, thermodynamics, heat transfer, and fluid mechanics.
The document discusses first order perturbation theory. It begins by introducing perturbation theory as an approximate method to solve the Schrodinger equation for complex quantum systems where the Hamiltonian cannot be solved exactly. It then presents the key equations of first order perturbation theory. The first order correction to the energy is given by the expectation value of the perturbation operator over the unperturbed ground state wavefunction. The first order correction to the wavefunction is expressed as a linear combination of unperturbed eigenstates.
Introduction to perturbation theory, part-1Kiran Padhy
Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations that contain derivatives of unknown functions of several variables and one or more partial derivatives. The solutions to PDEs are differentiable functions that satisfy boundary or initial conditions. PDEs are often used to express laws of physics. Examples of common PDEs discussed include the Laplace equation, Poisson equation, wave equation, heat equation, and diffusion equation.
This document provides information about a Statics course including the course goals, objectives, content, assessment, teaching strategies, textbook, and lecture times. The course aims to introduce concepts of forces, couples and moments in two and three dimensions and develop relevant analytical skills. Upon completion, students should be able to determine force resultants, centroids, draw free body diagrams, and apply equilibrium principles and analytical techniques to engineering problems. The course will be taught via lectures and tutorials using a specified textbook and will include assignments, tests, and an exam for assessment.
The document discusses applications of first order differential equations. It provides examples in several domains:
1) Cooling/warming laws can be modeled using differential equations, like the temperature of cooling coffee over time.
2) Population growth and decay models use differential equations, like calculating time for a population to double at a growth rate.
3) Determining geometric properties of curves can involve solving differential equations, like finding the equation of a curve based on its tangent slope.
4) Applications also include radioactive decay, electrical circuits, mixture of solutions, and modeling motion. First order differential equations have widespread uses in physics, statistics, chemistry, engineering, and other fields.
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an overview of engineering mechanics statics. It covers topics including:
- Defining mechanics as the science dealing with bodies at rest or in motion under forces.
- Dividing mechanics into statics, dynamics, and other subfields. Statics deals with bodies at rest.
- Introducing fundamental concepts of forces, units of measurement, and representing forces as vectors that add according to the parallelogram law.
- Providing examples of adding forces graphically using the parallelogram law and triangle rule to determine the resultant force.
- Discussing problems involving determining the magnitude and direction of resultant forces from multiple forces acting on structures, stakes, and brackets
This document provides an overview of the topics that will be covered in the TSPGECET-2018 exam. It discusses the following engineering subjects: engineering mechanics, strength of materials, fluid mechanics, heat transfer, materials science, metallurgical thermodynamics, advanced materials science, and nanotechnology. Specific concepts covered include stresses and strains, fluid dynamics, heat transfer principles, crystal structures, phase equilibria, electrical and optical properties of materials, polymers, ceramics, and composite materials. The document also briefly outlines the challenges in nanotechnology and its applications.
This document discusses time-dependent perturbation theory. It begins by introducing the concept of applying a time-dependent perturbation to a quantum system to induce transitions between its energy eigenstates. It then describes how the interaction picture can be used to focus on the slow evolution induced by the perturbation, without considering the rapid oscillation from the unperturbed Hamiltonian. The interaction picture defines a transformed state vector and operators such that the perturbation Hamiltonian governs the evolution operator in a Schrodinger equation.
A talk presented at the University of New South Wales on the occasion of Ian Sloan's 80th birthday, remembering our work together and thinking about how math is used in science.
This chapter focuses on objects in static equilibrium, where the net force and net torque on the object are both zero. Solving static equilibrium problems involves drawing free body diagrams showing all external forces acting on the object, then resolving forces into components and setting the sums of forces in each direction equal to zero. Three examples are given of solving static equilibrium problems involving particles under the influence of multiple forces. The problems are solved by resolving forces into horizontal and vertical or parallel and perpendicular components, setting the component force equations equal to zero, and solving the equations to determine the magnitudes of unknown forces. Key steps include drawing diagrams, resolving forces, setting force sums to zero, and solving the resulting equations.
This document discusses key concepts relating to work, energy, and power in mechanics. It defines work as the force multiplied by the distance moved, and explains that work is equal to the magnitude of the force component in the direction of motion. It also discusses different types of energy (kinetic, potential, elastic, etc.), the principle of conservation of energy, and defines power as the rate of doing work.
Oscillation of Solutions to Neutral Delay and Advanced Difference Equations w...inventionjournals
In this article we give infinite-sum conditions for the oscillation of all solutions of the following first order neutral delay and advanced difference equations with positive and negative coefficientsof the forms and where is a sequence of nonnegative real numbers, and are sequences of positive real numbers, and are positive integers. We derived sufficient conditions for oscillation of all solutions of and . AMS Subject Classification 2010: 39A10, 39A12
hermodynamics is the branch of physics that has to do with heat and temperature and their relation to energy and work. The behavior of these quantities is governed by the four laws of thermodynamics, irrespective of the composition or specific properties of the material or system in question. The laws of thermodynamics are explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, chemical engineering and mechanical engineering.
Oscillation of Solutions to Neutral Delay and Advanced Difference Equations w...inventionjournals
In this article we give infinite-sum conditions for the oscillation of all solutions of the following first order neutral delay and advanced difference equations with positive and negative coefficientsof the forms and where is a sequence of nonnegative real numbers, and are sequences of positive real numbers, and are positive integers. We derived sufficient conditions for oscillation of all solutions of and . AMS Subject Classification 2010: 39A10, 39A12
THE FEYNMAN-STUCKELBREG INTERPRETATION OF E<0 SOLUTIONSHusnain Ali
The document discusses relativistic quantum mechanics and the Klein-Gordon equation. It describes the Feynman-Stuckelberg interpretation of negative energy solutions to the Klein-Gordon equation, which interprets them as representing positive energy anti-particles moving forward in time. This resolves issues with negative probabilities and allows a consistent treatment of antiparticles. The interpretation is illustrated using a diagram of double electron scattering, showing it accounts for both time orderings through particle-antiparticle pair production and annihilation.
This document discusses the macroscopic and microscopic approaches in thermodynamics. The macroscopic approach examines the total energy a system possesses with respect to external frames of reference, such as kinetic and potential energies. The microscopic approach considers the molecular structure and activity within a system independent of outside influences. Energy can exist in many forms, both within a system and between a system and its environment. Temperature is defined as the average kinetic energy of molecules in a system and is a measure of heat. Pressure is defined as the external force per unit area on the boundaries of a system, which results from the transfer of molecular momentum at the boundaries.
1. The Ising model is a statistical mechanics model of ferromagnetism. It represents magnetic dipole moments as "spins" on a lattice that can point in one of two directions and interact with neighboring spins.
2. The Ising model can explain phase transitions like ferromagnetism, anti-ferromagnetism, gas-liquid transitions, and liquid-solid transitions.
3. The statistical mechanics of the Ising model are studied using the Hamiltonian, which includes terms for spin-spin interaction energy and the energy of an external magnetic field interacting with the magnetic moments. Partition functions are then used to calculate thermodynamic properties.
The document provides instructions for a lab on Newton's Second Law of Motion. It instructs students to predict what will happen to the acceleration of a cart under different forces and masses. It lists the materials needed and measurements that will be taken, which include qualitative and quantitative observations. It also reviews key concepts like the relationships between force, mass and acceleration defined by F=ma, and how to account for significant figures in calculations and measurements.
This document discusses thermodynamic principles and concepts. It defines key thermodynamic terms like system, environment, isolated system, open system, closed system, state parameters, and equations of state. The first law of thermodynamics states that the change in internal energy of a system equals the heat transferred plus work done on the system. The second law states that the entropy of any isolated system always increases and approaches a maximum value. Entropy is a measure of disorder in a system and is related to the number of microscopic arrangements that can produce a given macrostate.
This document provides an overview of the course MCT-114: Fundamentals of Thermal Sciences. The objectives of the course are to provide a solid grounding in engineering thermodynamics and its fundamental concepts. Topics covered include the basic concepts, laws of energy, ideal gas model, entropy, and power/refrigeration cycles. The course also introduces heat transfer concepts. The document outlines the suggested textbooks, course learning objectives, and provides an introduction to thermal-fluid sciences, thermodynamics, heat transfer, and fluid mechanics.
The document provides an overview of statistical thermodynamics including:
- Its historic background beginning with Bernoulli's work in the 18th century and contributions from Maxwell, Boltzmann, Gibbs, and others.
- The key difference between classical thermodynamics and statistical thermodynamics is that the latter links microscopic properties to macroscopic behaviors.
- Statistical thermodynamics is needed to explain thermodynamic parameters in terms of molecular properties and interactions since classical thermodynamics does not address this microscopic level.
- Central topics in statistical thermodynamics include the partition function, degrees of freedom, heat capacity, and Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics.
The document discusses key concepts from kinetic theory of gases and thermodynamics. It defines kinetic theory of gases as describing gas as particles in random motion that collide with each other and container walls. This explains macroscopic gas properties like pressure. It then outlines Maxwell-Boltzmann distribution and related equations that describe the distribution of molecular speeds at a given temperature. The document also summarizes the four laws of thermodynamics, including definitions of entropy, Carnot cycle efficiency, and applications of thermodynamic concepts.
The document discusses ab initio molecular dynamics simulation methods. It begins by introducing molecular dynamics and Monte Carlo simulations using empirical potentials. It then describes limitations of empirical potentials and the need for ab initio molecular dynamics which calculates the potential from quantum mechanics. The document outlines several ab initio molecular dynamics methods including Ehrenfest molecular dynamics, Born-Oppenheimer molecular dynamics, and Car-Parrinello molecular dynamics. It provides details on how these methods treat the quantum mechanical potential and classical nuclear motion.
1. The second law of thermodynamics states that entropy, a measure of disorder, always increases over time for an isolated system.
2. Entropy has two components - internal entropy generated within a system and external entropy generated by interactions with the outside environment.
3. The second law can be restated as: entropy can never decrease for a closed system, and the external entropy change is associated with the change in heat per temperature for the system. Increases in entropy correspond to irreversible processes.
The document discusses several real-world applications of differential and integral calculus. It provides examples of first-order differential equations being used to model jumping motions in video games and the cooling of objects. Surface and volume integrals are applied in fields like electrostatics, fluid dynamics, and continuity equations. Matrix determinants can estimate areas like that of the Bermuda Triangle. Overall, calculus has wide applications in science, engineering, economics and other domains.
The document provides an overview of key concepts in heat transfer, including:
1) It defines heat transfer and the three main modes of heat transfer: conduction, convection, and radiation.
2) It explains the relationship between heat transfer and thermodynamics, noting that heat transfer studies the rate and distribution of temperature over time.
3) It provides definitions and examples of key terms used in heat transfer problems, such as steady state, control mass/volume, and uncertainty.
Vasil Penchev. Cyclic mechanics. The principle of cyclicityVasil Penchev
1) The document discusses a theory of cyclic mechanics intended to unify quantum mechanics and general relativity.
2) It proposes several foundational principles, including that the universe can return to any point, time is not uniformly flowing, and all laws must be invariant to discrete and continuous transformations.
3) A key concept is introducing the notion of "quantum measure" and "quantum information" to provide a common measure to equate equations from different theories, with the goal of unifying them.
This document discusses the development of thermodynamics and its application to small systems and molecular machines. It summarizes that thermodynamics laws have been generalized to account for non-equilibrium states through developments like the fluctuation theorem. The fluctuation theorem allows for negative entropy production along single stochastic trajectories, helping to explain why molecular machines can have higher efficiencies than macroscale machines despite operating in a high noise environment. Further developments are still needed to understand non-equilibrium thermodynamics for biological systems where the Jarzynski equality may not apply.
The new emerging technology which is under research but when will come into practice, it will change the era of computing.
Its is based on changing the concept of inputs received by the machine.
till now the machine works with 0 and 1,however it will implement an input b/w 0 and 1 i.e 1/2.
The speed of processing will raise up-to 8 times and things will be beyond our expectations.
Linear elasticity theory assumes small deformations and the existence of an unstressed reference state. It relates stress and strain through generalized Hooke's law, with the elastic constants tensor. The stress power is the time derivative of the internal energy density, which defines an elastic potential. The stress is the derivative of this potential with respect to strain, and the elastic constants tensor is the second derivative of the potential.
Rise and fall of the clockwork universe - matter in extremes r2 OCR Physics BGab D
- Kinetic theory explains gas behavior using a probabilistic and mechanical approach, considering gases as large numbers of randomly moving particles. These ideas are extended to high and low temperatures, introducing the Boltzmann factor as the beginnings of thermodynamic thinking. Understanding matter in these terms is fundamental.
Unit 1 thermodynamics by varun pratap singh (2020-21 Session)Varun Pratap Singh
Free Download Link (Copy URL):
https://sites.google.com/view/varunpratapsingh/teaching-engagements
Dear Students,
Please find the Basic Mechanical Engineering (TME-101, 2020-21 Session) Unit One notes in this section.
Topic cover in this section are:
UNIT-1: Fundamental Concepts and Definitions
Definition of thermodynamics, System, Surrounding and universe, Phase, Concept of continuum, Macroscopic & microscopic point of view. Density, Specific volume, Pressure, temperature. Thermodynamic equilibrium, Property, State, Path, Process, Cyclic and non-cyclic processes, Reversible and irreversible processes, Quasi-static process, Energy and its forms, Enthalpy.
The document discusses heat and mass transfer. It outlines objectives related to understanding thermodynamics and heat transfer mechanisms. The key mechanisms of heat transfer are conduction, convection and radiation. Heat transfer occurs through these three modes simultaneously in many practical systems. Fourier's law, Newton's law of cooling, and the Stefan-Boltzmann law govern conductive, convective and radiative heat transfer respectively.
This document discusses differential equations and their application. It begins by defining what a differential equation is and provides examples of first order differential equations. It then discusses Newton's Law of Cooling, providing the derivation and formulation of the law. Several applications of Newton's Law of Cooling are presented, including using it to estimate time of death from temperature readings and determining cooling system specifications for computer processors. Other topics covered include the Mean Value Theorem, precalculus concepts, and examples of how calculus is applied in various fields such as credit cards, biology, engineering, architecture, and more.
This is a lecture is a series on combustion chemical kinetics for engineers. The course topics are selections from thermodynamics and kinetics especially geared to the interests of engineers involved in combusition
This document provides an introduction to thermodynamics. It defines thermodynamics as the science dealing with heat, work, and their relation to properties of matter and energy change. The document outlines the four laws of thermodynamics and describes the zeroth law regarding thermal equilibrium, the first law regarding conservation of energy and internal energy, the second law regarding limits on heat conversion and direction of processes, and the third law defining absolute zero entropy. Examples of engineering applications are given in areas like heat engines, refrigeration, and air conditioning. Key concepts discussed include system, surroundings, state, path, process, equilibrium, intensive/extensive properties, and reversible/irreversible processes.
This document discusses statistical thermodynamics and key concepts such as:
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The Cell Method
1. The Cell Method
for Computational
Physics
Prof. Enzo Tonti Dr. Giovanni Rinaldin
1
.
2. The quantitative description of physical properties, typical of physical
theories, leads to the introduction of physical quantities.
The measurement of a physical quantity implies always a tolerance:
the term ”exact value” has no place in applied physics. Also the
result of a calculation on physical variables is requested within a
prescribed tolerance.
These statements sound provocative to hears of a mathematician for
which the term “exact” is essential, but we are speaking of “applied”
mathematics not of “pure” mathematics.
The mathematical formulation of physics is made, usually, in terms of
differential calculus, hence the fundamental problem is expressed by a
differential equation.
Since the computer cannot deal with the notion of limit, exclusive feature
of the human mind, in order to solve the fundamental problem of a
physical theory we are obliged to discretize the differential equations.
Premise
2
3. When we read the indication of an amperometer or of a caliber we read
the first two or three significant values. This means that the value of the
physical quantity is approximate.
For this reason also the solution of a physical problem is approximate.
The important fact is that the result satisfies a prescribed tolerance.
Hence the passage from a differential formulation of the fundamental
equation to a discrete formulation in order to solve physical problem, by
computers become a tortuous process that can be avoided starting
directly with a discrete (=algebraic) formulation of the fundamental
equation.
3
Premise
Premise
4. Whatdowewanttodo?
This is the Cell Method. To describe it, let's look at some types of
problems that we may encounter in physics and engineering.
We want to present a method that allows to give an approximate
algebraic formulation to problems which are usually described by exact
differential equations.
In fact, the differential equations, in order to be treated with the computer,
they must be discretized (i.e. transformed into algebraic equations).
This procedure is typical of the traditional methods, like FEM, FVM, BEM,
FDM, LSQ, etc.
The method we present provides a formulation directly discrete
(or algebraic, or finite) without having to pass through the differential
formulation.
4
5. Thefundamentalproblemofafield
The fundamental problem of a physical field, such as:
the thermal field,
the elastic field,
the electromagnetic field
is to determine the “configuration” of the field
once its “sources” are assigned.
Let us look at three physical theories: 5
6. 1)Thermalconduction
The heat Q emitted from heat generators propagates from regions of higer
temperature to those of lower temperature.
The fundamental problem of the thermal conduction is to find
the temperature (=potential) at each point of the region we are studying
once the heat generators, their position and their intensity are assigned.
( , , , ) ?T t x y z temperature
The main physical variable of a thermal field is the temperature: this physical
variable is introduced in order to describe how cold and how worm is any
part of the region or of the body.
The temperature can be considered as the “potential” of the thermal field.
Let us consider a space region or a body: we want to consider its thermal
status.
6
7. 1)Thermalconduction
A law states that heat goes from regions to higher temperatures to those of
lower temperature.
Another law states that the amount of heat that passes through one surface
element depends on the difference in temperature between the two points
who are riding the surface.
This law is espressed by a constitutive equation, and is known as the
Fourier Elementary Law.
What are the laws of thermal conduction?
Another law is the energy balance. This law states that the increase of the
internal energy of a system is the sum of the heat and the work entering
the system.
Increase of internal energy = work entering + heat entering
Combining these laws on obtains the fundamental equation of the
thermal conduction, the Fourier law.
7
8. Consider the deformation of a solid body.
The cause of deformation are the forces acting on the solid.
Since deformation is transmitted from point to point, there must some laws
that govern the way in which it is propagated: this is expressed by a
constitutive equation. If the material is elastic it is Hooke's law.
2) Deformation of a solid body
Once a force is applied, the deformation is transmitted along the body
determining the displacement (effect) of the body's points.
The fundamental problem of deformation theory is to find
the displacement that undergoes every point of the body, starting from an
initial configuration, when are assigned the forces that generate it.
One of these laws imposes the equilibrium, of every piece of body if the
deformation is static; or the balance of momentum if the deformation is
dynamic.
8
9. Consider the electromagnetic field that is generated by electric charges
at rest and in motion. The description of the field is made by the scalar
electric potential and the vector magnetic potential.
3)Electromagnetism
( , , , ) !
( , , , ) !
t x y z
t x y z
J
electric charge density
electric current density
The fundamental problem of electromagnetism is the following:
assigned the spatial and temporal distribution of charges and currents,
determine the electric an the magnetic potential at every point of the
field at every moment:
( , , , ) ?
( , , , ) ?
t x y z
t x y z
A
scalar potential
vector potential
9
10. Fieldfunctions
We said that in each field the goal is to determine the configuration of the field.
The configuration is described by place and time functions of scalar
or vector type.
( , , , ) ?t x y z v
( , , , ) ?t x y z u
( , , , ) ?T t x y z
A(t,x, y,z)=?
( , , , ) ?t x y z
temperature
displacement
velocity
electric scalar-potential
magnetic vector potential
For historical reasons,
these laws are expressed by
partial differential equations.
We raise the question:
is it easy to solve
the partial differential equations
that we encounter
in Physics and Engineering?
10
11. ...solvethefundamentalequations
it is not easy at all !
Approximations need to be made
and to use numerical techniques
which force the use of the computer.
This requires transforming the differential equations
into approximated algebraic equations
through one of the many methods of discretization.
The answer is:
11
13. use the four
fundamental
operations:
sums
differences
products
divisions
From that moment they
introduce:
the derivative (limit of a ratio)
and the integral (limit of a sum)
and therefore the whole
differential calculus;
the ordinary differential equations
and those with partial derivatives
algebra infinitesimal analysis
introduces the notion of
limit
mathematics
13
The two great brances of mathematics
14. Why can not a computer treat infinity?
1/2
1/4
1/8
1/16
1/32
1 1 1 1 1
1 ...
2 4 8 16 32
the notion of limit is an ideal notion, that belongs to men,
not to the computers.
the computer would provide: 1.99999999999909
1
Let's propose to evaluate the sum
2
14
15. Thedifferentialformulationofphysicallaws
It is thought to discretize the differential equations to obtain
an appproximate algebraci formulation.
At the beginning using the finite differences, later other using
other discretization methods.
What happened then?
-occurred about three centuries ago (1687)
-physical laws have been mathematically formulated
in terms of differential equations.
The advent of computers required an algebraic
description of physical laws because the computer
ignores the notion of limit
It was not thought to go back to experimental facts
to obtain a direct algebraic formulation of the physical laws.
Philosophiae
Naturalis
Principia
Mathematica
I.Newton
5+7=12
15
16. from discrete to the differential ...
and then to go back to the discrete!
...
This lecture
BEM
FDM
physical
field
approximated
solution
algebraic
equations
FVM
FEM
differential
equations
16
17. It's possible
a DIRECT algebraic formulation
of physical fields?
The variety of discretization methods of differential equations
it raises the following question:
Finiteformulation(=discrete=algebraicformulation)
17
is easy !
is possible !
is intuitive !
is ready for numerical solution !
We want to prove that a DIRECTALGEBRAIC formulation of field laws ...
18. Apologyofthe”balance"
The fundamental equations of physical phenomena comes from a balance:
• in the static of deformable solids, the balance of forces is essential;
• in the thermal conduction the energy balance is essential;
• in the fluid dynamics, the balance of momentum is essential;
as well as the balance of mass;
• chemical reactions expresses the mass balance;
• in the theory of electric circuits the balance of the currents at the nodes
is essential;
• in electromagnetism the balance of charge is essential;
• etc.
18
19. Think for example to
a bottle factory.
The freshly formed bottles
come out of the furnace:
some of these are stored
on the shelves of the plant
and a part is sent out.
Whatisabalance?
A balance is a relatiom about some of extensive physical variables for
which we can talk about storage, production and outgoing.
Bproduced
= DtBstored
+ Bsent out
Bproduced
19
20. For the equilibrium of a body, the sum of the of volume and surface forces
must vanish.
A balance is valid for any size any shape of the body portion to which
it is applied, there is no need to shrink to an infinitesimal volume.
Hence, why we apply it to an infinitesimal region
to get a differential equation?
A balance is valid in the finite setting
as well as in the infinitesimal
20
surface forces
volume forces
22. Stationary thermal conduction 1/7
22
Let us consider three rooms in each the temperature is uniform.
They are separated by a wall of thickness d and area A. Each room contain
an heater and let us denote with Pk the heat emitted for unit time.
Suppose that the side walls, the ceiling and the floor are insulating.
The heat for unit time passing throug each wall is given by the Fourier
first law, the constitutive equation
T
1P 2P 3P
20° 16T2
dT1
T1 T2
T3 T3
G1 = T1 - 20o
G2 = T2 -T1 G3 = T3 -T2 G4 =16o
-T3
d d d d
23. fundamental equation
23
1 2 3, ,P P P
3 heat generation rates
assigned
1 2 3, ,T T T
3 temperatures
unknowns
Stationary thermal conduction 2/7
We propose to find the temperature Tk in each room.
F
T = temperature;
G = temperature difference
P = heat generation rate
= heat current
T
1P 2P 3P
20° 16T2
dT1
T1 T2
T3 T3
G1 = T1 - 20o
G2 = T2 -T1 G3 = T3 -T2 G4 =16o
-T3
d d d d
As is common in thermodynamics, we consider positive the heat entering
into the system (here each room).
24. 1 2 3T ,T ,T
1 2 3 4G ,G ,G ,G
G1
:=T1
- T0
G2
:=T2
-T1
G3
:=T3
-T2
G4
:=T4
-T3
ì
í
ï
ï
î
ï
ï
1 2 3P ,P ,P
1 2 3 4, , ,
2 1 1
3 2 2
4 3 3
P
P
P
F1
= -l
A
d
G1
F2
= -l
A
d
G2
F3
= -l
A
d
G3
F4
= -l
A
d
G4
ì
í
ïï
î
ï
ï
24
Stationary thermal conduction 3/7
The fundamental equation is obtained
combining three kinds of equations:
the equations that define the temperature
differences; the constitutive equations and
the equations of balance.
constitutive equations
equationsofDefining
equationsofbalance
T
1P 2P 3P
20° 16T2
dT1
T1 T2
T3 T3
G1 := T1 - 20o
G2 := T2 -T1 G3 := T3 -T2 G4 :=16o
-T3
d d d d
25. 1
1
2
2
3
3
4
1 1 0 0
0 1 1 0
0 0 1 1
P
P
P
1 1 0
2 2 1
3 3 2
4 4 3
=T - T
G =T -T
G =T -T
G =T -T
G
2 1 1
3 2 2
4 3 3
P
P
P
1 1 2 2
3 3 4 4
A A
G G
d d
A A
G G
d d
G1
G2
G3
G4
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
1 0 0
-1 1 0
0 -1 1
0 0 -1
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
T1
T2
T3
ì
í
ï
î
ï
ü
ý
ï
þ
ï
+
-20
0
0
16
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
1 1
2 2
3 3
4 4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
G
GA
Gd
G
defining matrix
balance matrix
constitutive
matrix
25
Stationary thermal conduction 4/7T
1P 2P 3P
20° 16T2
dT1
T1 T2
T3 T3
G1 := T1 - 20o
G2 := T2 -T1 G3 := T3 -T2 G4 :=16o
-T3
d d d d
26. F1
F2
F3
F4
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
= -l
A
d
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
G1
G2
G3
G4
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
1
1
2
2
3
3
4
1 1 0 0
0 1 1 0
0 0 1 1
P
P
P
defining matrix balance matrix
Here some general properties are revealed:
constitutive matrix
minus signG1
G2
G3
G4
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
:=
1 0 0
-1 1 0
0 -1 1
0 0 -1
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
T1
T2
T3
ì
í
ï
î
ï
ü
ý
ï
þ
ï
+
-20
0
0
16
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
2) the constitutive matrix is symmetric.
26
transpose with the minus sign
symmetric
1) the balance matrix is the transpose of the defining matrix with the minus sign.
Stationary thermal conduction 5/7
27. defining matrix
minus sign
0G = T +GD
G C
T
= P D
constitutive matrix
balance matrix
The fact that the matrix on the right side is the transpose of the matrix on the
left side, with the minus sign, is a property common to all physical theories.
The symmetry of the constitutive matrix is another property common to many
physical theories: C = CT.
From these two properties it follows that the matrix of the fundamental
equation is symmetric. This symmetry is the root of the variational principles
of physics.
27
source
“vector”
temperature
“vector”
T
0( T +G ) =P D C D
fundamental equation
Stationary thermal conduction 6/7
algebraic formulation (useful for computational purposes)
28. By connecting the source with the configuration variable by means of the
balance-constitutive-defining equations, we obtain the fundamental equation.
constitutive equation
(Fourier)
balance
equation
defining
equation
temperature
thermal
gradient
heat
source
heat current
density
P
L
T
g:= ÑT
-q g
Ñi q=s
qg
div qdiv -lg( )=sdiv -lgrad T( )=s-ldiv grad T( )=s
28
Stationary thermal conduction 7/7
differential formulation
2
- T
Fundamental equation
Fourier
30. Energy balance (first principle of thermodynamics):
the increase in internal energy over a period of time
it is equal to the sum of the heat and of the work supplied to the system
in the same time interval.
F = - k A
T2
-T1
d
The two laws of thermal conduction
given to the system produced
ingoing
Q Q Q
30
2) Constitutive equation: (Fourier elementary law)
the heat that crosses a “wall” for unit time
(heat current) is proportional to the area of the
“wall”, to temperature jump across the “wall” and
is inversely proportional to the thikness of the
“wall” and proceeds from heat to cold:
F
DtU= Qgiven to the system
+ Wentering
F
A
d
31. Thefundamentalproblemofstationarythermalconduction
thermal
source
assigned a body;
specified the physical nature of the materials that make up;
specify the thermal sources in extension and intensity;
specify the conditions at the boundary of the region;
determine the temperature at each point of the region.
temperature
assigned
temperature
assigned
unknown
temperature
temperature
unlnown
unknown heat
unknown heat
heat assigned
heat
assigned
heath
assigned
31
32. We build in the region a complex of triangular-shaped cells.
Let's triangulate, so that the triangles lean against the surface of
separation between the two materials. Let us assign a number to each
vertex and a number to each cell. This is automatically done by the mesh
generator used.
1
2
3 5 7
4
6
8
9
10
11
12
13
14
15
16
17
Complex of triangular cells
32
temperature
assigned
temperature
assigned
heat assigned
heat
assigned
1
2
7
11
3
9
4
5
6
8
1012
13
14
15
16
17
18
19
20
33. A cell complex in two dimensions imply to consider it as a transversal
section of a cilindrical solid. Along each straigth fiber the temperature
is uniform so that heat propagates only in the transversal direction with
respect to the fibers.
What really means a two dimensional problem?
33
Hence every triangle is really a triangular prism, as shown below.
h
i
j
s
We will denote with s the thickness of the layer.
34. A first type of nodal influence areas
is made up of polygons whose sides are the axes of the sides of the triangles.
The resulting polygons (in red) are called polygons of Voronoi.
To write the energy balance, for each node we choose a polygon that
contains the node. These polygons give rise to a second subdivision of the
region that is called dual of the first one.
Node influence areas: Voronoi polygons
34
1
2
3 5 7
4 6
8
9
10
11
12
13
14
15
16
17
1 2
7
11
3
9
4
5
6
81213
14
15
16
17
18
19
20
10
36. 36
Physical variables are associated
with “space elements”
All geometrical objects are composed of four “space elements”:
points, lines, surfaces, and volumes.
Physical quantities can be divided into two classes (with few ambiguities):
physical constants and physical variables. To the first class belong the
universal constants, such as the Plank constant, the speed of light in vacuum,
the Avogadro number, etc. To this class belong also all material constants
such as the elastic modulus of a material, the thermal conductivity of a material,
the permittivity of a material, etc.
Physical variables are those quantities that may assume whatever value inside
a prescribed interval. Such is the temperature, the speed, the displacement, the
mass, the momentum, the force, etc. Of course the physical variables may be
scalars, vectors, tensors, etc.
The index of refraction of a transparent material is usually a constant, but for a gas
depends on its pressure, hence in this case it behaves as a physical variable.
37. 37
Mass, enegy, entropy are associated with volumes; fluxes of energy, of electric
charge, heat are associated with surfaces; elongation is associated with a line;
temperature, electric potential, gravitational potential are associated with points
etc.
Every space element can be endowed with orientation. There are two kinds of
orientation, the inner orientation and the outer orientation. So a line segment
connecting two points, A and B may be oriented from A to B or from B to A. These
are the two opposite inner orientations. Line segments that belong to a same
plane may also endowed with a transversal (or outer) orientation: this is the
inner orientations of a line segment that crosses the original line segment.
A A
B B
inner orientations
A A
B B
transversal (=outer) orientation
Inner and outer orientation
38. 38
Inner and outer orientation in space 1/2
The notions of inner and outer orientation of space elements in the
threedimensional space are illustrated in the next two slides
It is important to remark that a physical variable associated with
an oriented space element changes its sign when we reverse the orientation
of the space element.
Then in fluid dynamics the vorticity is linked to a small surface element and
is associated with the inner orientation of the surface element.
The same happens for magnetic flux. On the contrary, a flow of matter and
an energy flow are associated with a small surface element with an external
orientation and change sign when we invert the external orientation of the
surface element.
The voltage along a line from A to B has the opposite sign of the
voltage from B to A. In a fluid flow the velocity circulation changes sign when
let's go from B to A.
39. 39
Inner and outer orientation in space 2/2
If we consider a point P and a point Q, in this order, the temperature
difference is T(Q)-T(P). If we consider as first the point Q and as second
the point P, the temperature difference is T(P)-T(Q). The same happens
with the potential difference between two points.
The voltage along a line from A to B has the opposite sign of the
voltage from B to A. In a fluid flow the velocity circulation changes sign
when we go from B to A instead that from A to B.
The notion of inner and outer orientation is valid also for points.
This means that a line connecting P with Q the inner orientation is
from P to Q then the arrow that denotes the orientation direction is
”outgoing” from P and ”ingoing" in point Q, then Q is considered as
a well and P as a source.
40. Inner orientation
inner orientation of a line
is an order of its bounding points.
A
B
A
B
inner orientation of a volume = a compatible (*)
inner orientation of its bounding faces.
inner orientation of a surface = a compatible (*)
inner orientation of its bounding edges;
or a direction of rotation.
40
P
L
S
V
(*) – “compatible” means that two adjacent elements induce opposite
orientation on the common boundary.
inner orientation of a point; the
lines pointing inwards being positive, or viceversa
(the points are “sinks” or “sources”).
“sinks” “sources”
The space elements endowed with inner orientation will be denoted with
a bar over the letter.
41. Outer orientation
outer orientation of a surface:
an order of its faces (rear / front)
outer orientation of a volume:
from the inside to the outside or viceversa
outer orientation of a line:
a direction of rotation around the line
41
outer orientation of a point:
a consistent orientation of the lines
with one extreme to the point;
i.e. lines endowed with outer orientation.
part 3/8
The space elements endowed with outer orientation will be denoted with
a tilde over the letter.
42. 42
Triangulation = “simplicial complex”
We oganize our presentation of the Cell Method by a division of the plane region
considered into triangles: the complex arising is called “simplicial complex”.
Since we wont to consider the intersection of the axes of the sides of each
triangle, i.e. its circumcenter, we require that the circumcircles of every triangle
do not contain other vertices of the mesh. Such a mesh is called Delaunay mesh.
There are mesh generators that authomatically generate a Delaunay complex for
a given two-dimensional region and also for a three dimensional region.
43. 43
Numeration of the nodes
The mesh generetor assigned a number to the vertices of a cell complex (= nodes)
and a number to every triangle.
The side of a (two-dimensional) mesh will be endowed with inner orientation:
the orientation of each side goes from the node of lower number to the one
of greater number.
1
2 3
5
6
7 8
9
10
42
1 3
6
4
8
5
10
9
7
11
44. 44
Now, let us consider a triangle in a cartesian system. Let us denote with
h,i,j its vertices, with Lh, Li, Lj its sides, with Ah, Ai, Aj three vectors normal
to the sides and of equal length of the corresponding side.
h
i
j
y
Lh
Li
Lj
Ah
Aj
Ai
We approximate the temperature inside each triangle by a function with
linear behaviour, i.e. with an affine function:
a + gx xh + gy yh = Th
a + gx xi + gy yi = Ti
a + gx xj + gy yj = Tj
ì
í
ï
î
ï
Imposing that in the three nodes the
funtion T(x,y) assumes the nodal
temperatures, we can find the coefficients
a, gx, gy:
T(x,y) = a+ gx x + gy y
We have used the letters gx an gy because
they are the cartesian components of the
gradient of the function: gx =
¶T
¶x
gy =
¶T
¶y
x
Defining equation 1/4
45. ...
45
a + gx xh + gy yh = Th
a + gx xi + gy yi = Ti
a + gx xj + gy yj = Tj
ì
í
ï
î
ï
subtracting the second equation from the third
one; the first equation from the second one and
putting
Lhx = xj - xi Lhy = yj - yi
Ljx = xi - xh Ljy = yi - yh
we obtain
Ljx Ljy
Lhx Lhy
é
ë
ê
ù
û
ú
gx
gy
é
ë
ê
ù
û
ú
Ti -Th
Tj -Ti
é
ë
ê
ù
û
ú=
At this point we must invert the matrix
gx
gy
é
ë
ê
ù
û
ú
Ljx Ljy
Lhx Lhy
é
ë
ê
ù
û
ú=
Ti -Th
Tj -Ti
é
ë
ê
ù
û
ú
-1
Defining equation 2/4
46. 46
Let us remember how to invert a 2 x 2 matrix.
a b
c d
é
ë
ê
ù
û
úM = the transpose is
a c
b d
é
ë
ê
ù
û
ú the minor are
d - b
-c a
é
ë
ê
ù
û
ú
putting D = det(M) = ad -bc the inverse matrix is M-1
=
1
D
d - b
-c a
é
ë
ê
ù
û
ú
We can test the result:
M M-1
=
1
D
a b
c d
é
ë
ê
ù
û
ú
d - b
-c a
é
ë
ê
ù
û
ú =
1
ad - bc
ad - bc 0
0 ad - bc
é
ë
ê
ù
û
ú =
1 0
0 1
é
ë
ê
ù
û
ú
that is the unit matrix.
Defining equation 3/4
47. 47
gx
gy
é
ë
ê
ù
û
ú
1
D
Lhy - Ljy
-Lhx Ljx
é
ë
ê
ù
û
ú=
Ti - Th
Tj -Ti
é
ë
ê
ù
û
úif we apply the previous formula we get
gx
gy
é
ë
ê
ù
û
ú
1
D
+Lhy Ti -Th( )- Ljy Tj -Ti( )
-Lhx Ti -Th( )+ Ljx Tj -Ti( )
é
ë
ê
ê
ù
û
ú
ú
=
1
D
Lhy + Ljy( )Ti - LhyTh - LjyTj
Lhx + Ljx( )Ti + LhxTh + LjxTj
é
ë
ê
ê
ù
û
ú
ú
=
gx
gy
é
ë
ê
ù
û
ú =
1
D
-LhyTh - LiyTi - LjyTj
+LhxTh + Lix Ti +LjxTj
é
ë
ê
ù
û
ú =
1
D
-Lhyh - Liy - Ljy
+Lhx + Lix +Ljxj
é
ë
ê
ù
û
ú
Th
Ti
Tj
é
ë
ê
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú
since Lh +Li +LJ = 0 it follows that
We remark that the determinant is double of the area A of the triangle,
hence we can write
D
D = 2A.
Defining equation 4/4
48. 48
The heat current
Let us start with an analogy: let us consider a river and a region in which
the water flow is uniform. If we introduce in the region a plate that is free
to rotatate around a vertical axis, a force is exerted by the water on it.
Of course the force depends on the inclination of the plate and si maximum
when the plate is orthogonal to the lines of heat current.
lines of current
In analogous way, if we consider a uniform heat flow and a plane surface
(not a plate) the amount of heat that crosses the plate depends on the
inclination of the plate.
Let us consider the direction in which the heat crossing the surface is
maximum. Let us consider the heat current (in Watt) crossing the surface
and divide it by the area of the surface. Let us denote by q such a ratio.
There will be a direction for which q is maximum. We can consider a vector
q whose direction is orthogonal to the plane surface for which q is
maximum:
q = qmaxn
49. A first type of nodal influence areas
is made up of polygons whose sides are the axes of the sides of the triangles.
The resulting polygons (in red) are called polygons of Voronoi.
To write the energy balance, for each node we choose a polygon that
contains the node. These polygons give rise to a second subdivision of the
region that is called dual of the first one.
Node influence areas: Voronoi polygons
49
1
2
3 5 7
4 6
8
9
10
11
12
13
14
15
16
17
1 2
7
11
3
9
4
5
6
81213
14
15
16
17
18
19
20
10
50. We will write the heat balance for each dual polygon.
Voronoi polygons have the advantage of having the sides of the polygons
orthogonal to the sides of the triangles: this makes it very simple
the writing of constitutive equations.
Let us first consider internal polygons (whole)
Whole interior polygons
50
1
2
3 5 7
4 6
8
9
10
11
12
13
14
15
16
17
1 2
7
11
3
9
4
5
81213
14
15
16
17
18
19
20
10
51. We will limit ourselves to the stationary case: the energy accumulation is
null and therefore the outgoing heat is equal to that produced.
Qoutgoing
=Qproduced
Energy balance on dual polygons
51
We will denote with P the heat generated for unit time.
1
2
3 5 7
4 6
8
9
10
11
12
13
14
15
16
17
1 2
7
1
1
3
9
4
5
81213
14
15
16
17
18
19
20
1
02
Q
2,1
0
Q2,11
Q2,3
Q2,4
Q1.2
Q2,15
Q2,14
+ Q1.2 - Q2,3 - Q2,4- Q2,15 - Q2,14- Q2,10 - Q2,11 = P2poligono 2:
52. Estimatetheheatcomingout
The heat crossing one side of the Voronoi polygon, considered
positive when it enters, depends on the temperature difference
measured on the two sides of the faces.
L2,1
A2,1
The Fourier Heat Elementary Law states that in a uniform flow region, the
heat passing through a surface in the unit of time is proportional to the
temperature difference per unit of length according to the formula
We will make the approximation to consider
the surrounding region on each side of
the polygon as a region of uniformity.
This will be the only approximation we will make!
1
2
4
10
11
14
15
3
52
54. We will now write the equation of balance for each dual edge polygon
(broken).
1
...
2
4
10
11
14
15
3
Basicequationfortheonboardpolygons
The term B indicates the heat coming from the edge of the broken polygon
for unit time (B stands for boundary)
54
55. For each edge polygon we write that the
sum of the output streams is equal to the
incoming flow rate B from the edge.
Also using comstitequations here is the approximate
algebraic equation
Balance on polygon 1
1B 1
2
4
10
11
15
3
55
Energy balance on split dual polygons
56. Thefundamentalalgebraicsystem
In this way you get to an algebraic system
of n equations in n unknowns containing
the nodal temperatures:
Since we have a dual polygon
for each node, writing a balance equation
for each dual polygon, we will have so
many equations how many knots are and
how many are the unknowns.
56
1
2
4
10
11
15
3
58. Insteadthedifferentialformulation...
But then there is a need to make the limit, consider infinitesimal volumes
and so come to a differential equation that we can not solve?
gives us two beautiful differential equations, one for each material
(assumed homogeneous), forces us to write the equations of junction on
separation surfaces between two different materials.
C1
C1
C2
C2
A
B
58
59. From where is the approximation born?
It is evident that the smaller the size of the cells,
the more the field can be considered uniform in each of them.
All it was worth to avoid passing to the limit and keeping us
in algebraic formulation from the beginning!
At this point comes the temptation to make the limit, or to reduce volumes
so that the law of constitution becomes ”exact".
Doing so we fall back into the differential formulation!
And since the calculator needs an algebraic formulation,
we are forced to discretize the differential equation...
Here it is the only approximation of numeric solution
59
Because a constitutive equation is valid in regions of uniform field,
when applied in non-uniform field regions, becomes approximate.
60. Simplicialcomplex
20020 40 60 80 100 120 140 160 180 200
concentrated source
distribuite source
Region with material 2
Region with Material 1
60
Cell method can work with:
Different materials
Different kind of sources
Complex meshes
61. ItsVoronoidual
20 40 60 80 100 120 140 160 180 20061
It’s always possible to
find the Voronoi dual of
a primal mesh.
Dual mesh always allows
equibrium for any kind
of sources.
62. 62
A detailed presentation of the problem of finding the temperature in
a stationary thermal conduction can be found in the author paper
http://www.discretephysics.org/papers/TONTI/The%20Cell%20Method.pdf
Tonti E., A Direct Discrete Formulation of Field Laws: The Cell Method
CMES, Vol. I, no .1, p.11-26
The paper can be freely downloadesd from the web site
64. 64
Using Cell Method for elasto-static problems
The formulations with CM (Cell Method) of a regular plane-stress
quadrilateral element is shown in the following.
From an overall point of view, the entire procedure follows the
diagram depicted below:
For the stress matrix, a Voigt’s notation is used, such to have a 3 components
vector.
65. 65
Using Cell Method for elasto-static problems
Formulation of a regular quadrilateral cell for
mechanical plane stress analysis (i.e.
membrane behaviour):
1. Choose a displacement interpolation
function (bilinear in this case) as to describe all
the possible deformation into the element:
Deformation matrix
( , )
( , )
u x y a bx cy dxy
v x y a bx cy dxy
The internal stresses
are evaluated
in points A,B,C,D
Such matrix connects
deformations and
displacements:
u
U
v
BU
66. 66
Sample problem: plane stress analysis of a wall
2. Use constitutive equations
With:
E as elastic (Young’s) modulus
G as shear modulus
n as Poisson’s coefficient
as deformation component
Deformation matrix is the
equivalent, in FEM, of the matrix
collecting the derivatives of
shape functions
67. 67
Sample problem: plane stress analysis of a wall
3. Write equilibrium in internal points (A,B,C,D) as to obtain
stiffness matrix. Let be:
• t the thickness of the membrane
• p the length of half the edge
We can write the equilibrium on the dual cells (red edges) as
follows:
In which:
N is the number of dual cells
T is the resultant of surface forces acting from the outside
F is the resultant of volume forces (e.g. self-weight)
B is the extenal surface force eventually present if the node in
which we write the equilibrium lies on the boundary.
68. 68
Sample problem: plane stress analysis of a wall
We can write the equilibrium as follows: Ac: area of
the primal cell
h,i,j,k are the
vertices
x,y reference
planar axis
Tx,Ty are the
resulting
forces
69. 69
Sample problem: plane stress analysis of a wall
Stiffness matrix
We wrote the equilibrium as:
Hence the local stiffness matrix is:
We must turn to the global stiffness matrix to assemble:
where Lc is a location matrix, which allows to transform a
local stiffness matrix kc in global coordinates.
Hence, we get the algebraic equivalent of Navier’s differential
equation of elasto-static.
70. 70
Sample problem: plane stress analysis of a wall
Wall mesh into
quadrilateral cells
Choosing a certain mesh
will influence the accuracy
of results
=>
thicker is the mesh, more
accurate will be the result
The results obtained will be compared with ones from Finite Element
Method (FEM).
Restrained nodes
Applied nodal loadsLet’s consider a practical case:
71. 71
Sample problem: plane stress analysis of a wall
displacement contour
and deformed shape
The whole procedure has been
implemented in a Matlab code,
leading the depicted result.
Boundary conditions were applied
by reducing the global (assembled)
stiffness matrix. Bounded vertices
are obtained by removing the
associated rows and columns.
Result are compared with FEM
FEM (Abaqus): stress Von Mises stresses and deformed shape
73. 73
Sample problem: plane stress analysis of a wall
Results completely agree with FEM approaches with linear shape functions.
For triangular mesh, the result completely
agrees with FEM:
Method
X displacement Y displacement Diff. in X Diff. in Y
[mm] [mm] % %
CEM tria 0.032202701 -0.015655325 0.000% 0.000%
FEM CPS3 0.0322027 -0.0156553 0.000% 0.000%
Method
X displacement Y displacement Diff. in X Diff. in Y
[mm] [mm] % %
CEM quad 0.025277997 -0.012230291 0.000% 0.000%
FEM CPS4R 0.0270812 -0.0134509 7.133% 9.980%
FEM CPS4 0.0255565 -0.0124236 1.102% 1.581%
FEM CPS8 0.026532 -0.0130891 4.961% 7.022%
For quadrilateral meshes, the result presents a
good agreement with FEM, even if the
presented formulation was made for regualar
quadrilateral cells.
CPS4R are reduced integration 4 nodes plane stress
elements,
CPS4 are 4 nodes plane stress elements,
CPS8 are 8 nodes plane stress elements.
74. 74
Sample problem: 3D analysis of a wall
With the same procedure, a regular hexahedral cell has been implemented.
Displacement interpolation functions:
The same load and
boundary conditions as
before has been
applied.
The Matlab code has
been adjusted to
manage exahedral
elements with the
shape functions above.
75. 75
Sample problem: 3D analysis of a wall
Method
X displacement Y displacement Z displacement Diff. in X Diff. in Y
Diff. in
Z
[mm] [mm] [mm] % % %
FEM C3D8 0.002571950 0.000001296 -0.001254150
CEM esa 0.002525352 0.000001258 -0.001221776 1.85% 2.99% 2.65%
A regular hexahedral cell has been implemented.
Results and comparisons with FEM are reported below.
77. 77
Planar and solid mesh can be obtained using free tools like:
NETGEN: https://ngsolve.org/showcases/netgen
TetGen: http://wias-berlin.de/software/index.jsp?id=TetGen&lang=1
In solid mechanics, using Cell Method you obtain the same stiffness of Finite
Elements approach. Assembly of matrices and solving can be conducted in the
same way as FEM, using optimized algorithm.
The presented approach is able to reproduce the behaviour of linear elements in
FEM. Cell Method can be enriched by adding Gauss points to improve accuracy
Cell Method is particularly suitable for teaching and for numerical
implementations, see for instance:
Usefulhints
https://www.researchgate.net/publication/224157318_Multiphysics_Problems_via_the_Cell_Method_
The_Role_of_Tonti_Diagrams
79. the algebraic formulation applies to:
regions of any shape, with holes, tips, cracks, grooves, etc.
regions containing different materials
anisotropic materials
nonlinear materials
sintered materials
it treats naturally concentrated sources
does not present infinite
Convergence orders of order higher than the second
it is applied with simplicity.
...
Characteristicsofalgebraicformulation
79