Rayleigh ritz method in calculus of variations ppt

Forray, Marvin J. Reddy, J. Zienkiewicz, O. Introduction The Fundamental Idea: Consider the problem of the bending of a cantilever under loads as in figure below:.

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From an energy perspective, the equilibrium of the system corresponds to minimum potential energy. Lecture - Introduction contd. Assuming that the potential energy P. For example, in this problem of the cantilever, the admissible displacements, satisfying the specified B. However, the P.

Rayleigh–Ritz method explained

The displacement of the physical system, will correspond to minimum P. Hence, the analysis problem is solved, if we can find the displacement function, which minimizes the P. To be more precise, P. Such a quantity is called as a Functional. Variational Calculus is the calculus of Functionals mainly concerned with extremization problems.

General form of a functional, with only one independent variableone dependent variable and its first order derivative w. A general problem:. If the second order derivative also is zero, higher order derivatives need to be considered. The Basics: We know that at the minimum value of a function:. At a minimum point, the slope has to become positive for x 0 and negative, in case of x 0. In case of a problem with multiple independent variables:. These rules can be generalized for any number of independent variables.

An example is the stiffness matrix in linear elastic structural systems, where the quadratic form above is twice the strain energy, which is positive for any displacement vector and zero only if the displacement is identically zero. It is assumed that the system is properly constrained; otherwise, the stiffness matrix is positive semi-definite. All the Eigen values of a positive definite matrix are positive think of buckling loads and natural frequencies.

A symmetric matrix [A] is negative definite, if [A] is positive definite. Sufficiency condition for maximum is that the Hessian Matrix should be negative definite.

The problem was formulated by John Bernoulli in The problem is to find the path corresponding to shortest-time for a particle sliding from point x0, y0 to x1, y1 in the vertical plane, under the action of gravity. Brachistochrone Problem contd.The Rayleigh Ritz method is a classical approximate method to find the displacement function of an object such that the it is in equilibrium with the externally applied loads.

The Rayleigh Ritz method relies on the principle of minimum potential energy for conservative systems. The method involves assuming a form or a shape for the unknown displacement functions, and thus, the displacement functions would have a few unknown parameters.

These assumed shape functions are termed Trial Functions. Afterwards, the potential energy function of the system is written in terms of those few parameters, and the values of those parameters that would minimize the potential energy function of the system are calculated.

Mathematically speaking, we assume that the unknown displacement function is a member of a certain space of functions; for example, we can assume that where is the set of all the possible vector valued linear functions defined on the body represented by the set.

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With this assumption, we restrict the possible solutions to those in the formand thus, the unknown parameters are restricted to the matrixwhich in this particular example, has nine components. Finally, the nine components that would minimize the potential-energy function are obtained.

The method will be illustrated using various one-dimensional examples. The Rayleigh Ritz method seeks to find an approximate solution to minimize the potential energy of the system. For plane bars under axial loading, the unknown displacement function is. In this case, the potential energy of the system has the following form See beams under axial loads and energy expressions :.

To obtain an approximate solution fora trial function of a finite number of unknowns is first selected. In the case of bars under axial loads, the trial function has to satisfy the displacement boundary conditions.

The following examples show the method for different external loads, and bar configurations. Let a bar with a length and a constant cross sectional area be aligned with the coordinate axis. Also, assume that the bar has a constant force of value applied at the end.

If the bar is fixed at the endfind the displacement of the bar by directly solving the differential equation of equilibrium. Also, find the displacement function using the Rayleigh Ritz method assuming a polynomial function for the displacement of the degrees 0, 1, 2, 3, and 4.

Exact Solution: First, the exact solution that would satisfy the equilibrium equations can be obtained. The equilibrium equation as shown in the beams under axial loading section when and are constant is:. Therefore, the exact solution for the displacement and the stress are:. The Rayleigh Ritz Method: The first step in the Rayleigh Ritz method finds the minimizer of the potential energy of the system which can be written as:.

Notice that the potential energy lost by the action of the end force is equal to the product of and the displacement evaluated at while the potential energy lost by the action of the distributed body forces is an integral since it acts on each point along the beam length. Further, we will use the constitutive equation to rewrite the potential energy function in terms of the function :. To find an approximate solution, an assumption for the shape or the form of has to be introduced. Polynomial of the Zero Degree: First, we can assume that is a constant function a polynomial of the zero degree :. Polynomial of the First Degree: As a more refined approximation, we can assume that is a linear function a polynomial of the first degree :.

To satisfy the essential boundary condition the coefficient is equal to zero.We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions.

In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical.

Rayleigh–Ritz method explained

In variational problems linear boundary conditions are often divided into essential geometric and natural dynamic [ 1II. More generally, one calls the boundary conditions essential if they involve derivatives of order less than half of the order of the differential equation and natural otherwise [ 3I.

In the standard exposition of the Ritz method the trial functions may violate the natural conditions but must satisfy all the essential ones [ 24. The reason is that the variational equations force the natural conditions on the trial solutions anyway, even if the trial functions themselves do not satisfy them. But what if we wish to use trial functions that violate the essential conditions as well? For instance, in problems involving parametric asymptotics, the trial functions are preimposed with no regard for boundary conditions [ 56 ] and in initial-boundary problems with time-dependent boundary conditions the time independent trial functions can not satisfy them in principle.

One may also wish to use such violating trial functions because they are simpler; see [ 7 ] for other possible reasons. Thus, there is abundant motivation to generalize the Ritz method to trial functions that do not satisfy the essential conditions. From a theoretical viewpoint this is a particular case of approximating solutions by nonconforming functions, the nonconformity here being at the boundary [ 8 ].

A natural idea is to treat the essential boundary conditions as variational constraints and to remove them as any other constraints using the Lagrange multipliers. Such approach is naively taken in some applied works at least since [ 9 ]; see also [ 10 ] where the authors explicitly cite the simplicity of the trial functions as a reason for using them. This approach however is very far from the intuitive reasoning behind the naive application of the Lagrange multipliers in [ 9 ] or in [ 10 ].

The standard approach applies to problems with quadratic functionals [ 13II. It applies to general convex functionals and is much closer to the standard theory of the Ritz method [ 15 — 17 ]. We avoid the saddle point reformulation altogether and work directly with the original variational problem. A system of boundary weights is used to approximate the Lagrange multipliers, and our assumptions are stated in terms of completeness of the trial functions and of the boundary weights.

On the other hand, completeness is much more intuitive and straightforward to verify than an inf-sup condition. As a trade-off, we only prove convergence of the method rather than obtain an explicit error estimate, so the relation between spaces of the trial functions and of the boundary weights is less precise.

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But hopefully a more intuitive approach provides a better understanding of analytic and numerical issues involved.I have just discover you site about a month ago and I really do like what you are doing with sharing all these.

Keep it going. Shape function is step 8?? Shouldnt that be linear?? U : Strain energy. F : Pulling Force. Hi Pruthvi, Shape function is chosen in step 2. Step 8 is the resulting horizontal displacement calculated on the basis of that particular shape function.

If a different shape function had been chosen in step 2, a different equation would have arisen out of Rayleigh-Ritz method. However, as you pointed out well, hey are linear when e and F are independent. But then the value of K doesn't apply anymore and we are in the plastic range. I suggest you to make a very simple Excel exercise where e, F and U are pictured according to the Strain-stress curve to realize this. Thanks again for commenting!

Dude, thanks a lot for your explanations. My teacher sucked at teaching this, and the available literature is indeed not for humans, it seems to be destined at people that already know these things. It's very hard to get a mental picture of stuff, when it's pages after pages of rambling about mathematical formulas and derivations, without getting to the "why is it done this way, and not another".

Thanks a lot for your insights again. How could I assess the precision of the equation? Publicado por Rabindranath Andujar Moreno en Unknown 21 de febrero de Rabindranath Andujar Moreno 21 de febrero de Rafael 6 de febrero de Unknown 14 de mayo de Mbegin 14 de noviembre de Suscribirse a: Enviar comentarios Atom.

Displacement e vs Force F on a spring of elastic material. Shaded area represents stored strain energy.

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When pulled with a force F its tip deforms an amount e.The Rayleigh—Ritz method is a numerical method of finding approximations to eigenvalue equations that are difficult to solve analytically, particularly in the context of solving physical boundary value problems that can be expressed as matrix differential equations. It is used in mechanical engineering to approximate the eigenmodes of a physical system, such as finding the resonant frequencies of a structure to guide appropriate damping.

The name is a common misnomer used to describe the method that is more appropriately termed the Ritz method or the Galerkin method. This method was invented by Walther Ritz inbut it bears some similarity to the Rayleigh quotient and so the misnomer persists.

If a Krylov subspace is used and A is a general matrix, then this is the Arnoldi algorithm. Convergence of the procedure means that as i tends to infinity, the approximation will tend towards the exact function.

In many cases one uses a complete set of functions e. A set of functions. The above outlined procedure can be extended to cases with more than one independent variable. The Rayleigh—Ritz method is often used in mechanical engineering for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheel s on a shaft with varying cross section.

It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns. Consider the case whereby we want to find the resonant frequency of oscillation of a system. First, write the oscillation in the form.

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By conservation of energy, the average kinetic energy must be equal to the average potential energy. Thus, if we knew the mode shape. For a non-trivial solution of c, we require determinant of the matrix coefficient of c to be zero.

This gives a solution for the first N eigenfrequencies and eigenmodes of the system, with N being the number of approximating functions. The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. We also know that without damping, the maximal KE equals the maximal PE. Note that the overall amplitude of the mode shape cancels out from each side, always.Forray, Marvin J. Reddy, J. Zienkiewicz, O. Introduction The Fundamental Idea: Consider the problem of the bending of a cantilever under loads as in figure below:.

From an energy perspective, the equilibrium of the system corresponds to minimum potential energy. Lecture - Introduction contd. Assuming that the potential energy P. For example, in this problem of the cantilever, the admissible displacements, satisfying the specified B. However, the P. The displacement of the physical system, will correspond to minimum P. Hence, the analysis problem is solved, if we can find the displacement function, which minimizes the P.

To be more precise, P. Such a quantity is called as a Functional.

Rayleigh–Ritz method

Variational Calculus is the calculus of Functionals mainly concerned with extremization problems. General form of a functional, with only one independent variableone dependent variable and its first order derivative w.

A general problem:. If the second order derivative also is zero, higher order derivatives need to be considered. The Basics: We know that at the minimum value of a function:. At a minimum point, the slope has to become positive for x 0 and negative, in case of x 0. In case of a problem with multiple independent variables:. These rules can be generalized for any number of independent variables. An example is the stiffness matrix in linear elastic structural systems, where the quadratic form above is twice the strain energy, which is positive for any displacement vector and zero only if the displacement is identically zero.

It is assumed that the system is properly constrained; otherwise, the stiffness matrix is positive semi-definite. All the Eigen values of a positive definite matrix are positive think of buckling loads and natural frequencies.

A symmetric matrix [A] is negative definite, if [A] is positive definite. Sufficiency condition for maximum is that the Hessian Matrix should be negative definite. The problem was formulated by John Bernoulli in The problem is to find the path corresponding to shortest-time for a particle sliding from point x0, y0 to x1, y1 in the vertical plane, under the action of gravity.

Brachistochrone Problem contd. The time take for the travel, is a scalar-valued function of the path a function followed a Functional. If the path is expressed by the function y y x :. To solve the problem, we need to find the function y xwhich minimizes the functional, which is the total time taken by the particle to travel from x0, y0 to x1, y1.

Similar to the ordinary calculus, the methods to solve this problem is based on finding the value of functional such that for small changes in the path, the functional is not affected the functional becomes Stationary. The major difference is that in finding the extremum of a function, we find the point at which it happens, while in checking the stationary character, the requirement is to find the corresponding function. The answer is obviously a straight line.

Minimum Surface of Revolution: The aim is to find the curve connecting two points, which when rotated about the x-axis, gives minimum surface area. The answer is a Catenary Curve. Isoperimetric Problem: In this case, the objective is to find the curve with a specified length, which encloses the maximum area. We have two functionals here the length of curve perimeter and the area enclosed.He has been a visiting researcher at the University of Manchester and University College London, and is a visiting professor at the University of Palermo, Italy.

Professor Cassel's research utilizes computational fluid dynamics in conjunction with advanced analytical methods to address problems in bio-fluids, unsteady aerodynamics, multiphase flow, and cryogenic fluid flow and heat transfer. Cambridge University Press Bolero Ozon. Variational Methods with Applications in Science and Engineering. Kevin W. There is a resurgence of applications in which the calculus of variations has direct relevance.

In addition to application to solid mechanics and dynamics, it is now being applied in a variety of numerical methods, numerical grid generation, modern physics, various optimization settings, and fluid dynamics, for example.

Many of these applications, such as nonlinear optimal control theory applied to continuous systems, have only recently become tractable computationally, with the advent of advanced algorithms and large computer systems.

The content of the text reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. Most readers will be pleased to note that the mathematical fundamentals of calculus of variations at least those necessary to pursue applications is rather compact and is contained in a single chapter of the book. Therefore, the majority of the text consists of applications of variational calculus for a variety of fields.

Hamiltons Principle.  