This document discusses two methods for developing the underlying equations of finite element analysis (FEA): the method of weighted residuals and calculus of variations. It focuses on describing the method of weighted residuals, which takes the governing equations in their strong form and transforms them into weaker statements. This is done by approximating the solution over the problem domain using shape functions with unknown constants, then minimizing the residuals through an integral approach to determine the constants. Two specific weighted residual methods are outlined: collocation, which sets residuals to zero at discrete points, and subdomain, which integrates the residual over subdivided regions.
In this paper, we give several new fixed point theorems to extend results [3]-[4] ,and we apply
the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are
proceed to examine some a class of integral-differential equations and some partial differential equation, to
illustrate the effectiveness and convenience of this method(see[7]). Finally we have also discussed Berge type
equation with exact solution
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
1. The document discusses tensor analysis and its use in studying the Einstein field equations. It defines key tensors such as the Riemann-Christoffel curvature tensor and its properties including the antisymmetric and cyclic properties.
2. Bianchi identities are derived using a geodesic coordinate system. Taking the covariant derivative of the curvature tensor leads to the Bianchi identities.
3. Other concepts discussed include the Ricci tensor, gradient and divergence of tensors, and the Einstein tensor obtained by contracting the Bianchi identities. The Einstein tensor is related to the Ricci tensor and metric tensor.
The document discusses the Gauss-Seidel iterative method for solving systems of linear equations. It begins by describing how Gauss-Seidel improves upon the Jacobi method by using the most recently calculated values. An example applying Gauss-Seidel to a system of 4 equations is shown. The solution converges rapidly, requiring only 5 iterations versus 10 for Jacobi. Finally, the Gauss-Seidel method is expressed in matrix form.
This document outlines the course contents for PMATH 351: Real Analysis. It covers topics including the axiom of choice and cardinality, metric spaces, and the space (C(X), · ∞). The course will examine notions such as the topology of metric spaces, Cauchy sequences, completeness, compactness, and the Stone-Weierstrass theorem. Students will complete 5 homework assignments and a midterm exam. The final exam will constitute 55% of the final grade.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This document discusses the application of partial differential equations. It begins by classifying partial differential equations according to their mathematical form as either boundary value problems or steady-state equations. Some common partial differential equations are then presented, including the wave equation, heat equations, and Laplace's equation. Solution methods like separation of variables are introduced. Specific examples of the 1D wave equation and 1D heat equation are then covered. Finally, the document discusses the Laplace equation in 2D and 3D.
In this paper, we give several new fixed point theorems to extend results [3]-[4] ,and we apply
the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are
proceed to examine some a class of integral-differential equations and some partial differential equation, to
illustrate the effectiveness and convenience of this method(see[7]). Finally we have also discussed Berge type
equation with exact solution
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
1. The document discusses tensor analysis and its use in studying the Einstein field equations. It defines key tensors such as the Riemann-Christoffel curvature tensor and its properties including the antisymmetric and cyclic properties.
2. Bianchi identities are derived using a geodesic coordinate system. Taking the covariant derivative of the curvature tensor leads to the Bianchi identities.
3. Other concepts discussed include the Ricci tensor, gradient and divergence of tensors, and the Einstein tensor obtained by contracting the Bianchi identities. The Einstein tensor is related to the Ricci tensor and metric tensor.
The document discusses the Gauss-Seidel iterative method for solving systems of linear equations. It begins by describing how Gauss-Seidel improves upon the Jacobi method by using the most recently calculated values. An example applying Gauss-Seidel to a system of 4 equations is shown. The solution converges rapidly, requiring only 5 iterations versus 10 for Jacobi. Finally, the Gauss-Seidel method is expressed in matrix form.
This document outlines the course contents for PMATH 351: Real Analysis. It covers topics including the axiom of choice and cardinality, metric spaces, and the space (C(X), · ∞). The course will examine notions such as the topology of metric spaces, Cauchy sequences, completeness, compactness, and the Stone-Weierstrass theorem. Students will complete 5 homework assignments and a midterm exam. The final exam will constitute 55% of the final grade.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This document discusses the application of partial differential equations. It begins by classifying partial differential equations according to their mathematical form as either boundary value problems or steady-state equations. Some common partial differential equations are then presented, including the wave equation, heat equations, and Laplace's equation. Solution methods like separation of variables are introduced. Specific examples of the 1D wave equation and 1D heat equation are then covered. Finally, the document discusses the Laplace equation in 2D and 3D.
Tensor algebra and tensor analysis for engineersSpringer
This document discusses vector and tensor analysis in Euclidean space. It defines vector- and tensor-valued functions and their derivatives. It also discusses coordinate systems, tangent vectors, and coordinate transformations. The key points are:
1. Vector- and tensor-valued functions can be differentiated using limits, with the derivatives being the vector or tensor equivalent of the rate of change.
2. Coordinate systems map vectors to real numbers and define tangent vectors along coordinate lines.
3. Under a change of coordinates, components of vectors and tensors transform according to the Jacobian of the coordinate transformation to maintain geometric meaning.
I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
This document discusses power series solutions and the Frobenius method for solving ordinary differential equations with variable coefficients. It explains that power series can be used to find solutions around ordinary points, while the Frobenius method extends this approach to regular singular points through generalized power series involving an index term. The Frobenius method involves making an ansatz for the solution as a power series with an unknown index, then determining the index and coefficients by substituting into the differential equation and setting terms of different powers of x equal to zero.
The document provides an overview of power series methods for solving differential equations. It defines key concepts like radius of convergence, region of convergence, and ordinary points. It also outlines the basic steps of the power series method: (1) assume a power series solution; (2) derive the recurrence relation; (3) determine the coefficients to obtain particular solutions. Examples are provided to illustrate these steps in finding series solutions to ordinary differential equations about ordinary points.
This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, LU decomposition, and Cholesky decomposition. It provides the algorithms and formulas for each method. As an example, it applies the Jacobi and Gauss-Seidel methods to solve a system of 3 equations with 3 unknowns, showing how Gauss-Seidel converges faster by immediately using updated values at each step.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
This document provides an introduction to tensor calculus. It begins with definitions of tensors and coordinate systems. Section 1 defines tensors and contravariant and covariant indices. Section 2 focuses on Cartesian tensors and introduces tensor notation rules. It defines tensor terms, expressions, and operations. Section 3 will cover general curvilinear coordinates and covariant differentiation. The document establishes the foundation for working with tensors and their transformation properties.
Cap 2_Nouredine Zettili - Quantum Mechanics.pdfOñatnom Adnara
This document discusses mathematical tools used in quantum mechanics. It introduces Hilbert spaces and their properties including linear vector spaces, scalar products, completeness, and bases. Vectors in a Hilbert space belong to an abstract vector space and satisfy properties like orthonormality. Operators acting on these vectors are linear. Examples of finite and infinite dimensional Hilbert spaces are given for the 3D Euclidean space and space of complex functions. Linear independence of sets of vectors and functions is also discussed.
The document discusses Fourier series and two of its applications. It provides an overview of Fourier series, including its definition as an infinite series representation of periodic functions in terms of sine and cosine terms. It also discusses two key applications of Fourier series: (1) modeling forced oscillations, where a Fourier series is used to represent periodic forcing functions; and (2) solving the heat equation, where Fourier series are used to represent temperature distributions over time.
This document discusses differential equations and their origins and applications. It begins by defining differential equations as equations containing derivatives of dependent variables with respect to independent variables. It notes that differential equations involving ordinary derivatives are called ordinary differential equations. Examples are provided of first order, second order, linear and non-linear ordinary differential equations. The document also discusses the physical applications of differential equations, such as modeling simple harmonic motion, oscillations of springs, and rotational dynamics of shafts.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. The time and radial solutions are then expressed in terms of outgoing and ingoing coordinates, which leads to outgoing and ingoing waves with different analytic properties on either side of the event horizon.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. These solutions are then expressed in terms of outgoing and ingoing coordinates, which have distinct analytic properties inside and outside the event horizon.
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
The document presents information about differential equations including:
- A definition of a differential equation as an equation containing the derivative of one or more variables.
- Classification of differential equations by type (ordinary vs. partial), order, and linearity.
- Methods for solving different types of differential equations such as variable separable form, homogeneous equations, exact equations, and linear equations.
- An example problem demonstrating how to use the cooling rate formula to calculate the time of death based on measured body temperatures.
This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
This document presents a framework for analyzing the convergence of Galerkin approximations for a class of noncoercive operators. It begins by introducing assumptions on the operators and establishing well-posedness of the continuous problem. It then analyzes a "GAP" condition on the finite element discretization that is sufficient for stability and quasi-optimal convergence. Finally, it discusses two applications of the theory: Maxwell's equations with variable coefficients, and a boundary integral formulation for electromagnetic wave propagation.
The document discusses ordinary differential equations, including exponential growth/decay models, separation of variables, numerical and hybrid numerical-symbolic solving techniques, orthogonal curves, Newton's law of heating and cooling, and medical modeling examples. Specific examples are provided to illustrate concepts like families of solutions, implicit solutions, direction fields, and determining parameter values from initial conditions.
This document discusses various methods for modeling shallow water flows and waves using numerical techniques. It covers topics like wave theories, wave modeling approaches, meshfree Lagrangian methods, smoothed particle hydrodynamics (SPH), and the use of graphics processing units (GPUs) for real-time simulations. SPH is presented as a meshfree Lagrangian technique for modeling wave breaking processes. The document outlines the governing SPH equations, kernel approximations, time stepping approaches, and submodels for viscosity and turbulence. Validation examples are shown comparing SPH simulations to experimental data.
Tensor algebra and tensor analysis for engineersSpringer
This document discusses vector and tensor analysis in Euclidean space. It defines vector- and tensor-valued functions and their derivatives. It also discusses coordinate systems, tangent vectors, and coordinate transformations. The key points are:
1. Vector- and tensor-valued functions can be differentiated using limits, with the derivatives being the vector or tensor equivalent of the rate of change.
2. Coordinate systems map vectors to real numbers and define tangent vectors along coordinate lines.
3. Under a change of coordinates, components of vectors and tensors transform according to the Jacobian of the coordinate transformation to maintain geometric meaning.
I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
This document discusses power series solutions and the Frobenius method for solving ordinary differential equations with variable coefficients. It explains that power series can be used to find solutions around ordinary points, while the Frobenius method extends this approach to regular singular points through generalized power series involving an index term. The Frobenius method involves making an ansatz for the solution as a power series with an unknown index, then determining the index and coefficients by substituting into the differential equation and setting terms of different powers of x equal to zero.
The document provides an overview of power series methods for solving differential equations. It defines key concepts like radius of convergence, region of convergence, and ordinary points. It also outlines the basic steps of the power series method: (1) assume a power series solution; (2) derive the recurrence relation; (3) determine the coefficients to obtain particular solutions. Examples are provided to illustrate these steps in finding series solutions to ordinary differential equations about ordinary points.
This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, LU decomposition, and Cholesky decomposition. It provides the algorithms and formulas for each method. As an example, it applies the Jacobi and Gauss-Seidel methods to solve a system of 3 equations with 3 unknowns, showing how Gauss-Seidel converges faster by immediately using updated values at each step.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
This document provides an introduction to tensor calculus. It begins with definitions of tensors and coordinate systems. Section 1 defines tensors and contravariant and covariant indices. Section 2 focuses on Cartesian tensors and introduces tensor notation rules. It defines tensor terms, expressions, and operations. Section 3 will cover general curvilinear coordinates and covariant differentiation. The document establishes the foundation for working with tensors and their transformation properties.
Cap 2_Nouredine Zettili - Quantum Mechanics.pdfOñatnom Adnara
This document discusses mathematical tools used in quantum mechanics. It introduces Hilbert spaces and their properties including linear vector spaces, scalar products, completeness, and bases. Vectors in a Hilbert space belong to an abstract vector space and satisfy properties like orthonormality. Operators acting on these vectors are linear. Examples of finite and infinite dimensional Hilbert spaces are given for the 3D Euclidean space and space of complex functions. Linear independence of sets of vectors and functions is also discussed.
The document discusses Fourier series and two of its applications. It provides an overview of Fourier series, including its definition as an infinite series representation of periodic functions in terms of sine and cosine terms. It also discusses two key applications of Fourier series: (1) modeling forced oscillations, where a Fourier series is used to represent periodic forcing functions; and (2) solving the heat equation, where Fourier series are used to represent temperature distributions over time.
This document discusses differential equations and their origins and applications. It begins by defining differential equations as equations containing derivatives of dependent variables with respect to independent variables. It notes that differential equations involving ordinary derivatives are called ordinary differential equations. Examples are provided of first order, second order, linear and non-linear ordinary differential equations. The document also discusses the physical applications of differential equations, such as modeling simple harmonic motion, oscillations of springs, and rotational dynamics of shafts.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. The time and radial solutions are then expressed in terms of outgoing and ingoing coordinates, which leads to outgoing and ingoing waves with different analytic properties on either side of the event horizon.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. These solutions are then expressed in terms of outgoing and ingoing coordinates, which have distinct analytic properties inside and outside the event horizon.
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
The document presents information about differential equations including:
- A definition of a differential equation as an equation containing the derivative of one or more variables.
- Classification of differential equations by type (ordinary vs. partial), order, and linearity.
- Methods for solving different types of differential equations such as variable separable form, homogeneous equations, exact equations, and linear equations.
- An example problem demonstrating how to use the cooling rate formula to calculate the time of death based on measured body temperatures.
This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
This document presents a framework for analyzing the convergence of Galerkin approximations for a class of noncoercive operators. It begins by introducing assumptions on the operators and establishing well-posedness of the continuous problem. It then analyzes a "GAP" condition on the finite element discretization that is sufficient for stability and quasi-optimal convergence. Finally, it discusses two applications of the theory: Maxwell's equations with variable coefficients, and a boundary integral formulation for electromagnetic wave propagation.
The document discusses ordinary differential equations, including exponential growth/decay models, separation of variables, numerical and hybrid numerical-symbolic solving techniques, orthogonal curves, Newton's law of heating and cooling, and medical modeling examples. Specific examples are provided to illustrate concepts like families of solutions, implicit solutions, direction fields, and determining parameter values from initial conditions.
This document discusses various methods for modeling shallow water flows and waves using numerical techniques. It covers topics like wave theories, wave modeling approaches, meshfree Lagrangian methods, smoothed particle hydrodynamics (SPH), and the use of graphics processing units (GPUs) for real-time simulations. SPH is presented as a meshfree Lagrangian technique for modeling wave breaking processes. The document outlines the governing SPH equations, kernel approximations, time stepping approaches, and submodels for viscosity and turbulence. Validation examples are shown comparing SPH simulations to experimental data.
Integrability and weak diffraction in a two-particle Bose-Hubbard model jiang-min zhang
We report a bound state, which is embedded in the continuum spectrum, of the one-dimensional two-particle (Bose or Fermion) Hubbard model with an impurity potential. The state has the Bethe-ansatz form, although this model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.
This document provides an overview of the Gauss-Seidel and Newton-Raphson power flow solution methods. It begins by describing the Gauss-Seidel iterative method for solving nonlinear power flow equations using a scalar example. It then discusses applying Gauss-Seidel to vector power flow problems and provides an example of solving a two bus system. The document next describes the Newton-Raphson method, extending it to multidimensional problems using Taylor series approximations and defining the Jacobian matrix. It concludes with brief discussions of advantages and disadvantages of each method.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko
We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
The document summarizes key points about equality constrained minimization problems and Newton's method for solving them. It discusses:
1) Equality constrained minimization problems and their equivalent forms via eliminating constraints or using the dual problem.
2) Newton's method extended to include equality constraints, where the Newton step is defined to satisfy the linearized optimality conditions and ensures feasible descent.
3) An infeasible start Newton method that computes steps to reduce the primal-dual residual norm, ensuring iterates become feasible within a finite number of steps.
This document discusses quantum mechanical concepts related to particles in one-dimensional potential wells and harmonic oscillators. It covers:
1) The wavefunctions for a free particle in a 1D infinite and finite potential well, and the boundary conditions that apply.
2) The quantized energy levels and wavefunctions of the 1D quantum harmonic oscillator.
3) Operators in quantum mechanics and their use to extract measurable quantities from wavefunctions. Eigenfunctions and eigenvalues are introduced as special cases where operators produce scaled versions of the original wavefunction.
This document contains solutions to several problems involving vector calculus and partial differential equations.
For problem 1, key points include: deriving an identity involving curl and dot products; showing that curl is self-adjoint under certain boundary conditions where the vector field is parallel to the boundary normal; and explaining how Maxwell's equations with these boundary conditions would produce oscillating electromagnetic wave solutions.
Problem 2 involves solving the eigenproblem for the Laplacian in an annular region using separation of variables. Continuity conditions at the inner and outer radii lead to a transcendental equation determining the eigenvalues.
Problem 3 examines eigenproblems for the Laplacian and curl operators, showing they are self-adjoint and obtaining matrix and finite difference
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document discusses integrals involving exponential functions. It shows that integrating the exponential function results in dividing the constant in the exponent. It evaluates the important definite integral from 0 to infinity of e^-ax, which equals 1/a. It also evaluates the double integral from -infinity to infinity of e^-a(x^2+y^2), which equals sqrt(pi/a). Taking derivatives of these integrals generates related integrals involving x and x^4 that are useful in kinetic theory of gases.
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
This document summarizes key concepts about the particle in a rigid one-dimensional box:
1. It finds the energy eigenstates and discusses the wave functions and their properties like orthogonality.
2. It calculates the probability and expected values for the particle's position and discusses the physical interpretation of the wave function and coefficients when expanding an arbitrary function in the eigenstates.
3. It addresses several questions about normalized wave functions, time-dependent wave functions, energy measurements, and the wave function after a measurement.
This document provides an introduction to elementary quantum mechanics. It begins by defining Hilbert spaces and establishing complex exponentials as an orthonormal basis for L2 spaces. It then discusses Fourier series and using a linear combination of complex exponentials to represent L2 functions. Next, it introduces the Fourier transform and Parseval's identity. It derives the one-dimensional Schrodinger equation and discusses its physical interpretation. Finally, it formally defines quantum mechanics as a Hilbert space with a Hamiltonian and presents the Heisenberg uncertainty principle.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Levelised Cost of Hydrogen (LCOH) Calculator ManualMassimo Talia
The aim of this manual is to explain the
methodology behind the Levelized Cost of
Hydrogen (LCOH) calculator. Moreover, this
manual also demonstrates how the calculator
can be used for estimating the expenses associated with hydrogen production in Europe
using low-temperature electrolysis considering different sources of electricity
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...DharmaBanothu
The Network on Chip (NoC) has emerged as an effective
solution for intercommunication infrastructure within System on
Chip (SoC) designs, overcoming the limitations of traditional
methods that face significant bottlenecks. However, the complexity
of NoC design presents numerous challenges related to
performance metrics such as scalability, latency, power
consumption, and signal integrity. This project addresses the
issues within the router's memory unit and proposes an enhanced
memory structure. To achieve efficient data transfer, FIFO buffers
are implemented in distributed RAM and virtual channels for
FPGA-based NoC. The project introduces advanced FIFO-based
memory units within the NoC router, assessing their performance
in a Bi-directional NoC (Bi-NoC) configuration. The primary
objective is to reduce the router's workload while enhancing the
FIFO internal structure. To further improve data transfer speed,
a Bi-NoC with a self-configurable intercommunication channel is
suggested. Simulation and synthesis results demonstrate
guaranteed throughput, predictable latency, and equitable
network access, showing significant improvement over previous
designs
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Ce 595 section 2
1. FEA Theory -1-
Section 2: Finite Element Analysis Theory
1. Method of Weighted Residuals
2. Calculus of Variations
Two distinct ways to develop the underlying
equations of FEA!
2. FEA Theory -2-
Section 2: FEA Theory
Some definitions:
•V = volume of object
•A = surface area
= Au + As
•Au = surface of known
displacements
•As = surface of known stresses
•b = body force
•t = surface stresses (tractions)
, ,
, , ; , , .
, ,
u x y z
x y z v x y z
w x y z
x u x
3. FEA Theory -3-
A group of methods that take governing
equations in the strong form and turn them into
(related) statements in the weak form.
Applicable to a wide class of problems
(elasticity, heat conduction, mass flow, …).
A “purely mathematical” concept.
Section 2.1: Weighted Residual Methods
4. FEA Theory -4-
2.1: Weighted residual methods (cont.)
Need to write the equilibrium equations and boundary
conditions in an abstract form as follows:
0
0
0 ,
0
on
.
ˆ on
xy
x xz
x
xy y yz
y
yz
xz z
z
u
b
x y z
b
x y z
b
x y z
A
As
s
s
s
E u x 0
u u 0
B u x 0
σ n t 0
Solve these for u(x)!
5. FEA Theory -5-
2.1: Weighted residual methods (cont.)
Let be the exact solution to the problem
(differential equation and boundary conditions)
Then, for any choice of vectors W and W’:
exact
u x
in !
on !
exact
exact
everywhere V
everywhere A
E u x 0
B u x 0
0 in !
0 on !
exact
exact
everywhere V
everywhere A
W E u x
W B u x
6. FEA Theory -6-
2.1: Weighted residual methods (cont.)
Integrate these “results” over the entire volume
and surface:
Previous expression is still true if W and W’ are
functions of x (called weighting functions):
0
exact exact
V A
dV dA
W E u x W B u x
1 1
2 2
3 3
, , , ,
, , , , ,
, , , ,
0
exact exact
V A
W x y z W x y z
W x y z W x y z
W x y z W x y z
dV dA
W x W x
W x E u x W x B u x
7. FEA Theory -7-
Now, consider an approximate solution to the
same problem:
Matrix/vector form of this:
2.1: Weighted residual methods (cont.)
11 12 1
1 21 2 22 n 2
31 32 3
, , , , , , , , , ,
, , a * , , a * , , a * , , , ,
, , , , , , , , , ,
approx n exact
approx n exact
approx n exact
u x y z N x y z N x y z N x y z u x y z
v x y z N x y z N x y z N x y z v x y z
w x y z N x y z N x y z N x y z w x y z
1
.
a
n
approx k k exact
k
u x N x u x
Known functions
Unknown constants
1
11 12 1
2
21 22 2
31 32 3
n
unknowns
a
, , , , , , , ,
a
, , , , , , , ,
, , , , , , , ,
a
known functions
approx n exact
approx n
approx n
u x y z N x y z N x y z N x y z u
v x y z N x y z N x y z N x y z
w x y z N x y z N x y z N x y z
, ,
, , .
, ,
exact
exact
approx exact
x y z
v x y z
w x y z
u x N x a u x
8. FEA Theory -8-
2.1: Weighted residual methods (cont.)
Plugging this approximate solution into the
differential equation and boundary conditions
results in some errors, called the residuals.
Repeating the previous process now gives us an
integral close to but not exactly equal to zero!
, , 0 .
E B
V A
I dV dA
a W x R x a W x R x a
, in !
, on !
E approx
B approx
V
A
R x a E u x 0
R x a B u x 0
9. FEA Theory -9-
2.1: Weighted residual methods (cont.)
Goal: Find the value of a that makes this integral
as close as possible to zero – “best approximation”.
Idea: for n different choices of the weighting
functions, derive an equation for a by requiring
that the above integral equal zero:
Solve these equations for a!
1 1 1
2 2 2
Equation #1: , , 0 .
Equation #2: , , 0 .
Equation #n:
E B
V A
E B
V A
n n E
I dV dA
I dV dA
I
a W x R x a W x R x a
a W x R x a W x R x a
a W x R x
, , 0 .
n B
V A
dV dA
a W x R x a
10. FEA Theory -10-
2.1: Weighted residual methods (cont.)
Notes on weighted residual methods:
It is typical (but not required) to assume that the
known functions satisfy the displacement boundary
conditions exactly on Au. (Essential conditions)
In some methods, one must integrate the volume
integral by parts to get “appropriate” equations.
Different methods result from different ideas about
how to choose the weighting functions.
, , 0 , 1,2, , .
k k E k B
V A
I dV dA k n
s
a W x R x a W x R x a
11. FEA Theory -11-
2.1: Weighted residual methods (cont.)
1.Collocation Method:
Assume only one PDE and one BC to solve!
Idea: pick n points in object
(at least one in V and one
on A) and require residual
to be zero at each point!
, , ; , , .
E E B B
R R
R x a x a R x a x a
, =0, 1,2, , .
, =0, 1,2, , .
.
E i V
B j A
V A
R i n
R j n
n n n
x a
x a
12. FEA Theory -12-
2.1: Weighted residual methods (cont.)
2. Subdomain Method:
Assume only one PDE and one BC to solve!
Divide object up into n distinct regions (at least one
in V and one on A).
Require integral over
each region to be zero.
, , ; , , .
E E B B
R R
R x a x a R x a x a
, 0, 1,2, ,
, 0, 1,2, , .
.
i
j
i E V
V
j B A
A
V A
I R dV i n
I R dA j n
n n n
a x a
a x a
13. FEA Theory -13-
2.1: Weighted residual methods (cont.)
Notes on collocation and subdomain methods:
Weighting functions for collocation method are the Dirac
delta functions:
Weighting functions for subdomain method are the
indicator functions:
Advantage: Simple to formulate.
Disadvantage: Used mostly for problems with only one
governing equation (axial bar, beam, heat,…).
, 1,2, , . , 1,2, , .
i i V j j A
i n j n
W x x x W x x x
1 if
1 if
, 1,2, , . , 1,2, , .
0 if
0 if
j j
i i
i V j A
i j
i i
A
V
i n j n
A
V
x
x
W x W x
x
x
14. FEA Theory -14-
2.1: Weighted residual methods (cont.)
3. Least Squares Method:
Considers magnitude of residual over the object.
Finds minimum by setting derivatives to zero
, , , , 0.
LS E E B B
V A
I dV dA
s
a R x a R x a R x a R x a
k k k
, , , , 0 .
a a a
LS E B
k E B
V A
I
I dV dA
s
a R R
a x a R x a x a R x a
W x
W x
15. FEA Theory -15-
2.1: Weighted residual methods (cont.)
4.Galerkin’s Method:
Idea: Project residual of differential equation
onto original approximating functions!
To get W’, must integrate any derivatives in
volume integral by parts!
k
, 1,2, , .
a
approx
k k k n
u x
W x N x
, ,
, ,
Let , , , ;
, , . (Assume derivative involves .)
E E deriv E noderiv
E deriv E deriv dx x
R x a R x a R x a
R x a R x a
16. FEA Theory -16-
2.1: Weighted residual methods (cont.)
Must use integrated-by-parts version of !
,
Introduces the boundary conditions!
,
,
, ,
,
,
k E k E deriv x
V A
k
E deriv
V
k E noderiv
V
k
dV n dA
dV
x
dV
I
N x R x a N x R x a
N x
R x a
N x R x a
, 0 , 1,2, , .
k E
V
dV k n
a N x R x a
k
I a
17. FEA Theory -17-
2.1: Weighted residual methods (cont.)
Notes on least square and Galerkin methods:
More widely used than collocation and subdomain,
since they are truly global methods.
For least squares method:
Equations to solve for a are always symmetric but tend
to be ill-conditioned.
Approximate solution needs to be very smooth.
For Galerkin’s method:
Equations to solve for a are usually symmetric but much
more “robust”.
Integrating by parts produces “less smooth” version of
approximate solution; more useful for FEA!
18. FEA Theory -18-
2.1: Weighted residual methods (cont.)
Example: 1D Axial Rod “dynamics”
Given: Axial rod has constant density ρ, area A, length L, and spins at
constant rate ω. It is pinned at x = 0 and has applied force -F at x = L. The
governing equation and boundary conditions for the steady-state rotation of
the rod are:
Required: Using each of the four weighted residual methods and the
approximate solution , estimate the displacement of the rod.
2
2
2
0 for 0 ;
0 0; .
d u
E x x L
dx
du F
u x E x L
dx A
2
1 2
u x a x a x
19. FEA Theory -19-
2.1: Weighted residual methods (cont.)
Some preliminaries:
Problem has an exact solution given by
Approximate solution satisfies essential boundary
condition u(x = 0) = 0.
Two unknown constants → n = 2.
Notation:
2 2
3
1 1
2 6
* ; .
x x x
o o
L L L
FL A L
u x u u
EA F
2
2
2
0 ; 0 ; .
; .
u
V x L A x A x L
d u du F
E x E
dx dx A
s
E u B u
20. FEA Theory -20-
2.1: Weighted residual methods (cont.)
Solution:
1. Collocation Method --
Since n=2, have two collocation points. One must be at
x = L (must have one on As). Assume other at x = L/3.
Equation #1: evaluate residual of E(u) at x = L/3:
Equation #2: evaluate residual of B(u) at x = L:
2
2 2 2
1 2 2
2
2
1
3
2
, a a 2 a .
Equation #1 is: 2 a 0.
E approx
d
R x E x x x E x
dx
E L
a E u
2
1 2 1 2
1 2
, a a a 2 a .
Equation #1 is: a 2 a 0.
B approx
d F F
R x E x x E E x
dx A A
F
E E L
A
a B u
21. FEA Theory -21-
2.1: Weighted residual methods (cont.)
Solution:
1. Collocation Method --
Solve simultaneous equations:
Plot results:
2 2 2
2
1 1
3 6
1 2
a + ; a * .
3 6
x x x
L L L
approx o
F L L
u x u
EA E E
22. FEA Theory -22-
2.1: Weighted residual methods (cont.)
Solution:
2. Subdomain Method --
Since n=2, have two subdomains. One must be at
x = L (= As). Other must be 0 < x < L (= V).
Equation #1: integrate residual of E(u) over V:
Equation #2: evaluate residual of B(u) at x = L:
2 2 2 2
1
2
2 1 2 2
0
2 2
1
2
2
, 2 a 2 a 2 a .
Equation #1 is: 2 a 0.
L
E
R x E x I E x dx EL L
EL L
a a
2
1 2 1 2
1 2
, a a a 2 a .
Equation #1 is: a 2 a 0.
B approx
d F F
R x E x x E E x
dx A A
F
E E L
A
a B u
23. FEA Theory -23-
2.1: Weighted residual methods (cont.)
Solution:
2. Subdomain Method --
Solve simultaneous equations:
Plot results:
2 2 2
2
1 1
2 4
1 2
a + ; a * .
2 4
x x x
L L L
approx o
F L L
u x u
EA E E
24. FEA Theory -24-
2.1: Weighted residual methods (cont.)
Solution:
3. Least Squares Method --
For dimensional equality, take in ILS(a). Once again,
the “integral” over As is just evaluation at x = L.
Equation #1: take derivative with respect to a1:
2
2 1 2
1 1 1 1
1 2 1 2
, 2 a 0; , a 2 a .
a a a a
Equation #1 is: a 2 a a 2 a 0.
E B
x L
R R F
x E x x E E x E
A
E F E F
E E x E E L
L A L A
a a
1 L
k k k
1
, , , , =0.
a a a
LS E B
k E B
V
I
I dV x L x L
L
a R R
a x a R x a a R a
25. FEA Theory -25-
2.1: Weighted residual methods (cont.)
Solution:
3. Least Squares Method --
Equation #2: take derivative with respect to a2:
Solve equations:
2
2 1 2
2 2 2 2
2 2 2 2 2
2 2 1 2 1 2
0
2 2 2 2
1 2
, 2 a 2 ; , a 2 a 2 .
a a a a
2 2
2 2 a + a 2 a 2 a 8 a .
2
So Equation #2 is 2 a 8 a
E B
L
x L
R R F
x E x E x E E x Ex
A
Ex F EF
I E E x dx E E x E E L E L
L A A
EF
E E L E L
a a
a
0.
A
2 2 2
2
1 1
2 4
1 2
a + ; a * .
2 4
x x x
L L L
approx o
F L L
u x u
EA E E
Same as subdomain method!
26. FEA Theory -26-
2.1: Weighted residual methods (cont.)
Solution:
4.Galerkin’s Method --
Weighting functions are
Integrate general expression for volume integral
by parts first:
2
1 1 2 2
; .
N x x N x x
W x W x
2
0
0
0
2
0
, *
*
+ * .
L
k E k approx
V
x L
k approx x
L
k approx
L
k
dV N x Eu x x dx
N x Eu x
N x Eu x dx
N x x dx
N x R x a
Set this equal to
zero for k = 1,2!
27. FEA Theory -27-
2.1: Weighted residual methods (cont.)
Solution:
4.Galerkin’s Method --
Equation #1 uses N1(x)=x in previous:
Equation #2 uses N2(x)=x2 in previous:
2
1 1 2
0
0 0
2 2 3
1
3
1 2
* 1* a 2a * 0.
Equation #1 is a a 0.
L L
x L
approx
x
I x Eu x E x dx x x dx
FL
E L E L L
A
a
2 2 2
2 1 2
0
0 0
2
2 3 2 4
4 1
3 4
1 2
* 2 * a 2a * 0.
Equation #2 is a a 0.
L L
x L
approx
x
I x Eu x x E x dx x x dx
FL
E L E L L
A
a
28. FEA Theory -28-
2.1: Weighted residual methods (cont.)
Solution:
4. Galerkin’s Method --
Solve simultaneous equations:
Plot results:
2 2 2
2
7 1
12 4
1 2
7
a ; a * .
12 4
x x x
L L L
approx o
F L L
u x u
EA E E
29. FEA Theory -29-
Section 2: Finite Element Analysis Theory
1. Method of Weighted Residuals
2. Calculus of Variations
Two distinct ways to develop the underlying
equations of FEA!
30. FEA Theory -30-
A formal technique for associating minimum or
maximum principles with weak form equations
that can be solved approximately.
A more physically motivated approach than
weighted residuals.
Not all problems amenable to this technique.
Section 2.2: Calculus of Variations
31. FEA Theory -31-
2.2: Calculus of Variations (cont.)
Minimum/Maximum Principles (Variational
Principles) involve the following:
A set of equations and boundary
conditions to solve for .
A scalar quantity “related” to E and B (called
a functional).
A variational principle states that solving
and is equivalent to finding the function that
gives a maximum or minimum value.
Requires -- “First variation of
must be zero (stationarity)”.
E u x 0
B u x 0
u x
J u x
E u x 0
B u x 0
J u x
0
J
u x
J u x
32. FEA Theory -32-
2.2: Calculus of Variations (cont.)
What is a functional?
A function takes a point in space as input and returns
a scalar number as output.
(Vector-valued function gives vector as output.)
A functional takes a function as input and returns a
scalar number as output.
0
2
1
E.g.,
a
f x f x dx
E.g., , , 2 3 .
u x y z x y z
u x
Arc-length of f(x) from
x=0 to x=a.
33. FEA Theory -33-
2.2: Calculus of Variations (cont.)
A few examples:
Recall that straight line is shortest distance
between two points. How do we prove that?
4
0
4
0
1
2
2
1
1 2
2 2
1
2 2
1
1 4.4721.
2
1 9.2936.
, = 4,2 ; =
, = 4,2 ; = 6
f x dx
f x x dx
a b f x x
a b f x x
1 1 .
Let = scalar #, any function such that 0 0 4 .
Should have for all and
f x g x f x
g x g g
g x
34. FEA Theory -34-
For a given function , consider a Taylor series
expansion of arc-length formula in terms of α:
2.2: Calculus of Variations (cont.)
g x
2
2
1 1 1 1
2
0 0
1
* *
2
d d
f x g x f x f x g x f x g x
d d
4
2
1 1
0 0 0
4
1
2
0
1
0
4
1
2
0
1
1
1
1
d d
f x g x f x g x dx
d d
f x g x g x
dx
f x g x
f x g x
dx
f x
= some number β;
Assume β > 0.
35. FEA Theory -35-
Suppose that α is small and negative:
Same problem if β < 0 and α small and positive.
So, must have β = 0!
2.2: Calculus of Variations (cont.)
2
2
1 1 1 1
2
0 0
!
1 1 1
1
* *
2
* !
Negligible
d d
f x g x f x f x g x f x g x
d d
f x g x f x f x
Can’t happen!!!
4
1
2
0
1
0.
1
f x g x
dx
f x
36. FEA Theory -36-
Integrate by parts:
But and
2.2: Calculus of Variations (cont.)
4
4 4
1 1 1
2 2 2
0 0
1 1 1
0
* 0.
1 1 1
x
x
f x g x f x f x
d
dx g x g x dx
dx
f x f x f x
1
1
2
1
1
1 1 2
constant, or some other constant.
1
and must pass through 0,0 and 4,2 !
f x
f x
f x
f x mx b f x x
0 0
g x
4 0.
g x
4 4
1 1
2 2
0 0
1 1
0 for any choice of .
1 1
f x g x f x
d
dx g x dx g x
dx
f x f x
Must equal zero!!!
37. FEA Theory -37-
2.2: Calculus of Variations (cont.)
Key ideas in this “proof”:
Considered an arbitrary increment of the input
function.
Derivative of the functional forced to be zero.
This implies a certain equation must equal zero.
Calculus of Variations gives you a “direct” way of
performing these calculations!
38. FEA Theory -38-
2.2: Calculus of Variations (cont.)
Some definitions:
General form of a functional is
A variation of is
Note: if must satisfy some boundary conditions,
so must .
2
2
2
2
, , , , , , ,
+ , , , , , , , .
n
n
V
m
m
A
J E dV
x y z x z
B dA
x y z x z
u u u u u
u x x u x x x x x x
u u u u u
x u x x x x x x
, 1.
u x v x
u x
u x
u x u x
40. FEA Theory -40-
2.2: Calculus of Variations (cont.)
Some properties of the variation of :
Derivatives and variations can interchange.
Integrals and variations can interchange.
Variation of sum is sum of variations.
Variation of product obeys “product rule”.
u x
.
u x u x u x u x u x u x
.
z z z z
v x u x
u x v x
.
u x u x u x u x
.
V V
dV dV
u x u x
41. FEA Theory -41-
2.2: Calculus of Variations (cont.)
Some properties of the variation of :
“Chain rule” applies to dependent variables only!
u x
, , , , , , , ,
+ , , , ,
n n
n n
n
n
x
E
E
x z x z
E
x z x
u
u u u u
x u x x x x u x x x u
u
u u u
x u x x x
+
+ , , , , .
n
n
n n
n n
z
E
x z z
u
u u u
x u x x x
42. FEA Theory -42-
2.2: Calculus of Variations (cont.)
Let’s go back to arc-length example:
1 1 1
0
*
d
f x g x f x f x g x
d
=
1
f x
=
1 .
f x
4 4
2 2
1 1 1
0 0
2
1
4 1
2
2
0
1
4 1
1 1
2
2
0
1
1 1
*
1
*2 *
1
f x f x dx f x dx
f x
dx
f x
f x f x
dx
f x
Thus, we see that
Just like before!
4
1
1 2
0
1
1
f x g x
f x dx
f x
=
g x
43. FEA Theory -43-
2.2: Calculus of Variations (cont.)
Minimum/Maximum Principles (Variational
Principles) involve the following:
A set of equations and boundary
conditions to solve for .
A scalar quantity “related” to E and B (called
a functional).
E u x 0
B u x 0
u x
J u x
What is the relation?
44. FEA Theory -44-
2.2: Calculus of Variations (cont.)
Let’s consider a 1D version of this:
Want to minimize J(u), so require δJ(u) = 0:
; ; , , , .
b
a
x
x
u x E u x J u x E x u u u dx
u x E u x
2
2
, , ,
0.
b
a
b
a
b
a
x
x
x
x
x
x
J u x E x u u u dx
E E E
u u u dx
u u u
E E d E d
u u u dx
u u dx u dx
45. FEA Theory -45-
2.2: Calculus of Variations (cont.)
Integrate 2nd term by parts:
involves the boundary conditions!
Essential BC’s: E.g.,
Natural BC’s: E.g,
Other BC’s: E.g.,
*
b
b b
a a
a
x x
x x
x x
x x
E d E d E
u dx u u dx
u dx u dx u
*
b
a
x x
x x
E
u
u
0 or 0.
a b
u x x u x x
0 or 0.
a b
E E
x x x x
u u
some number.
a
E
x x
u
46. FEA Theory -46-
2.2: Calculus of Variations (cont.)
Integrate 3rd term by parts twice:
2
2
2
2
* *
* * * .
b
b b
a a
a
b
b b
a
a a
x x
x x
x x
x x
x x
x x x
x
x x x x
E d E d d E d
u dx u u dx
u dx u dx dx u dx
E d d E d E
u u u dx
u dx dx u dx u
* and * involve BC's!
b
b
a a
x x
x x
x x x x
E d d E
u u
u dx dx u
47. FEA Theory -47-
2.2: Calculus of Variations (cont.)
Pull all of this together:
2
2
0 * *
* * * .
b
b b
a a
a
b
b b
a
a a
x x
x x
x x
x x
x x
x x x
x
x x x x
E E d E
J u x u dx u u dx
u u dx u
E d d E d E
u u u dx
u dx dx u dx u
2
2
0 *
+ boundary condition terms.
b
a
x
x
E d E d E
J u x u dx
u dx u dx u
48. FEA Theory -48-
2.2: Calculus of Variations (cont.)
Assuming all boundary conditions are either essential
or natural, end up with:
for any choice of
2
2
0!
E d E d E
u dx u u
dx
The Euler equation for
2
2
0 *
b
a
x
x
E d E d E
u dx
u dx u dx u
u
J u x
49. FEA Theory -49-
2.2: Calculus of Variations (cont.)
The “relation” between being minimum and
is as follows:
J u x
0
E u x
If you can find an operator such that
then solving is the
same as solving .
, , ,
E x u u u
2
2
0,
E d E d E
E u x
u dx u dx u
, , , 0
b
a
x
x
J u x E x u u u dx
0
E u x
50. FEA Theory -50-
2.2: Calculus of Variations (cont.)
Some notes:
If you have boundary conditions that neither essential nor natural,
then must explicitly include a “boundary term” in the functional.
As number of dependent variables increases (e.g., 2D), one
functional will produce multiple Euler equations:
, , , , , , , .
b
b
a
a
x
x x
x x
x
J u x E x u u u dx B x u u u u
, , , , , , , , , ,
0 and 0
u v u v
x x y y
Area
u u v v
x y x y
J u x y v x y E x y u v dA
E E E E E E
u x y v x y
(See Slide #10 for general statement of this idea.)
51. FEA Theory -51-
2.2: Calculus of Variations (cont.)
Notes:
There are no general procedures for finding the operator
for a given set of equations
However, is known for many of the more common finite
element analysis problems.
Special case for which can always be found:
.
E u x 0
E
E
E
2 2 1
2
;
= matrix of derivative operators such that satisfies
BC's
for all possible choices of and .
V V
dV dV
1
1
E u x M x u x b 0
M x M x
u x M x u x u x M x u x
u x u x “Self-adjoint” equations
52. FEA Theory -52-
2.2: Calculus of Variations (cont.)
Notes:
For self-adjoint equations, and can be shown
to be:
(Depending on problem details, may be necessary to
integrate by parts before taking variation.)
1
;
2
1
.
2
V
E
J dV
u x M x u x u x b
u u x M x u x u x b
E
J u
J u
53. FEA Theory -53-
2.2: Calculus of Variations (cont.)
Example: axial deformation of fixed rod with axial load –
Can re-write governing equations as:
0 0
.
1 0
0
f x
d
dx E
d
dx
u x
x
b
M u x
0; .
0 0 .
f x
d du
x x
dx E dx
u x u x L
54. FEA Theory -54-
2.2: Calculus of Variations (cont.)
Example:
Functional is then calculated as follows:
Euler equations for this functional:
2
1 1 1
2 2 2
2
1 1 1
2 2 2
0
0
1
= ;
1
2 0
.
f x
d
uf x
dx E d du
dx dx E
d
dx
L
uf x
d du
dx dx E
u u u
E u
J u dx
u
1 1
2 2
1 1
2 2
0 + 0, or + 0.
0 0, or 0.
f x f x
d d d
dx E dx dx E
du
dx
du d du
dx dx dx
d
dx
E d E
u dx
E d E
u
dx
55. FEA Theory -55-
2.2: Calculus of Variations (cont.)
So, what’s all of this have to do with finite elements?
Have a set of equations and boundary
conditions to solve for .
Have a functional
related to and via the Euler equations on .
Finite element analysis attempts to find the best
approximate solution to
E u x 0
B u x 0
u x
E B
, , , ,
x y z
V
J E dV
u u u
u x x u
E
, , , , 0.
x y z
V
J E dV
u u u
u x x u
Weak form of governing equations!
56. FEA Theory -56-
2.2: Calculus of Variations (cont.)
Look more closely at 1D version:
Suppose we make “usual” approximation –
1
1
a
a .
n
approx k k
k
n
approx k k
k
u x u x N x
u x u x N x
, , , ,
* boundary terms 0.
b
a
x
E x u u E x u u
d
u dx u
x
u dx
2 2
0 1 2 0 1 2
E.g., if a a a ,then a a a .
approx approx
u x x x u x x x
A “space” of trial functions Must belong to same “space”
57. FEA Theory -57-
2.2: Calculus of Variations (cont.)
Plug in approximations (ignoring boundary terms for
now) –
Since each ak is arbitrary, best approximation comes
from
, , , ,
* 0, 1,2, ,
b
approx approx approx approx
a
x
E x u u u u E x u u u u
d
k u dx u
x
N x dx k n
, , , ,
1
, , , ,
1
a * 0,
or a * * 0.
b
approx approx approx approx
a
b
approx approx approx approx
a
x n
E x u u u u E x u u u u
d
k k u dx u
k
x
x
n
E x u u u u E x u u u u
d
k k u dx u
k x
N x dx
N x dx
Function of a1, a2, …, an Get n equations for n constants!
58. FEA Theory -58-
Notice the following:
Galerkin’s Method and Calculus of Variations give
same equations when “proper” is used!
, , , ,
If , then
, .
* , 0, 1,2, , .
approx approx approx approx
b
a
E x u u u u E x u u u u
d
approx E
u dx u
x
k E
x
E d E
E u x
u dx u
E u u R x
N x R x dx k n
a
a
2.2: Calculus of Variations (cont.)
Galerkin’s
method!
E
59. FEA Theory -59-
2.2: Calculus of Variations (cont.)
Notice something else:
, , .
b
a
approx exact
x
approx approx exact
x
J u u J u u
E x u u u u dx J u u
a
a a
, , , ,
a a
, , , ,
, ,
* *
* *
b
k k
a
b
approx approx approx approx
approx approx
k k
a
approx approx approx approx
x
E
approx approx
x
x
E x u u u u E x u u u u
u u
u u
x
E x u u u u E x u u u u
k
u u
x u u u u dx
dx
N x N
b
a
x
k
x
x dx
60. FEA Theory -60-
2.2: Calculus of Variations (cont.)
Integrate 2nd term by parts (and ignore boundary
terms again):
Rayleigh-Ritz Method on gives same equations as
J = 0 !
, , , ,
a
, , , ,
, ,
a
* *
* .
0 *
b
approx approx approx approx
k
a
b
approx approx approx approx
a
approx approx
k
x
E x u u u u E x u u u u
d
k k
u dx u
x
x
E x u u u u E x u u u u
d
k u dx u
x
E x u u u u
k
N x N x dx
N x dx
N x
, ,
0.
b
approx approx
a
x
E x u u u u
d
u dx u
x
dx
61. FEA Theory -61-
2.2: Calculus of Variations (cont.)
Example: 1D Axial Rod “dynamics”
Given: Axial rod has constant density ρ, area A, length L, and spins at
constant rate ω. It is pinned at x = 0 and has applied force -F at x = L. The
governing equation and boundary conditions for the steady-state rotation of
the rod are:
Required: Using the calculus of variations on an appropriate variational
principle along with the approximate solution , estimate the
displacement of the rod.
2
2
2
0 for 0 ;
0 0; .
d u
E x x L
dx
du F
u x E x L
dx A
2
1 2
u x a x a x
62. FEA Theory -62-
2 2
2 2
2 2
2 2
2 2
2 2
1 1
2 2
0
is self-adjoint, with and .
* * .
L
d u d u
dx dx
d u d
E x E x
dx dx
E u E u x J u E u x dx
E u M b
2.2: Calculus of Variations (cont.)
Solution:
Find appropriate variational principle:
Problem: there is a nonzero boundary condition –
2
2
1
2
2
1 1
2 2
0
* (Work done by applied force.)
* * .
F
A
L
d u
F
A dx
B u x L
J u x L u E u x dx
Needs to be integrated by parts!
63. FEA Theory -63-
2 2
1 1 1
2 2 2
0
0 0
2 2
1
2
0 0
* *
= * .
L L
x L
du du
F
A dx dx
x
L L
du
F
A dx
J u x L u E E dx u x dx
u x L E dx u x dx
2.2: Calculus of Variations (cont.)
Solution:
Doing this gives:
Require the first variation to equal zero:
2
0
* * * 0.
L
d u
du
F
A dx dx
J u x L u x E dx
64. FEA Theory -64-
2.2: Calculus of Variations (cont.)
Solution:
Using the given approximate function:
After some integrating, result is:
2 2
1 2 1 2
2
1 2
2 2
1 2 1 2 1 2
0
a a a a .
a a *
a a * a 2a * a 2 a 0.
F
A
L
u x x x u x x x
J L L
x x x E x x dx
2
2 3 2
1
1 2 1
3
2 4 2 3
1 4
1 2 2
4 3
a a a
a a a 0.
FL
A
FL
A
J L EL EL
L EL EL
=0
=0
65. FEA Theory -65-
2.2: Calculus of Variations (cont.)
Solution:
Solve the two equations to get:
2 2 2
2
7 1
12 4
1 2
7
a ; a * .
12 4
x x x
L L L
approx o
F L L
u x u
EA E E
Same as Galerkin’s method solution!
66. FEA Theory -66-
What if we had forgotten about the BC?
Functional becomes:
So the first variation becomes:
2 2
1 1
2 2
0
0 0
2 2
1 1
2 2
0 0
*
= * .
L L
x L
du du
dx dx
x
L L
du
F
A dx
J u E E dx u x dx
u x L E dx u x dx
2.2: Calculus of Variations (cont.)
2
2
0
* * * 0.
L
d u
du
F
A dx dx
J u x L u x E dx
Force is cut in half!
67. FEA Theory -67-
2.2: Calculus of Variations (cont.)
Solution:
Solution becomes:
2 2 2
2
7 1
12 4 2
1 2
7
a ; a * .
2 12 4
x x x
L L L
approx o
F L L
u x u
EA E E