Introduction to Variational CalculusandVariational Formulations in FEMJayadeep U. B.M.E.D., NIT CalicutRef.: Forray, Marvin J., Variational Calculus in Science and Engineering, McGraw Hill International Edn.Reddy, J. N., An Introduction to the Finite Element Method, McGraw Hill International Edition.Zienkiewicz, O. C., and Morgan, K., Finite Elements and Approximation, John Wiley & Sons.

Department of Mechanical Engineering, National Institute of Technology Calicut

IntroductionConsider the problem of the bending of a cantilever under loads as in figure below:

The Fundamental Idea:The displacement function w(x) at equilibrium will be the one, corresponding to which, the internal forces, generated due to resistance of beam to deform, and the external forces are equal.

From an energy perspective, the equilibrium of the system corresponds to minimum potential energy.

Lecture - 01

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Introduction contd. Assuming that the potential energy (P.E.) of the initial system to be zero, P.E. of the final system has two parts: Strain energy stored in the cantilever beam Reduction in P.E. of the force

If the stiffness of the beam and the force are constant and known, the P.E. becomes a function of only the displacement, which by itself is a function.

For example, in this problem of the cantilever, the admissible displacements, satisfying the specified B.C. are the functions with:

However, the P.E. corresponding to these different functions will be different. The displacement of the physical system, will correspond to minimum P.E.

Lecture - 01

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Introduction contd. Hence, the analysis problem is solved, if we can find the displacement function, which minimizes the P.E. of the system, from a space of admissible functions.

To be more precise, P.E. is a scalar valued function, in the form of an integral over the complete domain, of the displacement function. 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 variable , one dependent variable and its first order derivative w.r.t. the independent variable:

Lecture - 01

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Minimization & MaximizationBefore delving into the details of variational calculus, let us have a brief review of the minimization/maximization problems in ordinary calculus (extremum value of a function).

A general problem:

Necessary and Sufficient conditions for a minimum at x = x0:

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Minimization & Maximization contd.We know that at the minimum value of a function:

Sufficiency condition:

The Basics:Lecture - 01

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Minimization & Maximization contd.In case of a problem with multiple independent variables:

Necessary conditions for a minimum at x = c, y = d:

Sufficiency condition is more complicated. The Hessian Matrix:

These rules can be generalized for any number of independent variables.

Lecture - 01

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Minimization & Maximization contd.A symmetric matrix [K] is said to be positive definite, if:

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).

Positive Definiteness: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.

Lecture - 01

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Variational CalculusThe word Brachistochrone means shortest-time. The problem was formulated by John Bernoulli in 1696.

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.

The Beginnings Brachistochrone Problem:Ans.: Cycloidal Curve

Lecture - 02

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Brachistochrone Problem contd. The time take for the travel, is a scalar-valued function of the path (a function) followed a Functional.

The time taken for any small segment (length = ds) is depends on the instantaneous velocity (v).

Assuming the path to be frictionless & zero initial velocity, the instantaneous velocity at any point depends on the vertical distance (of fall) from that point to starting point.

If the path is expressed by the function y y(x):

Lecture - 02

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Brachistochrone Problem contd. To solve the problem, we need to find the function y(x), which 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.

Lecture - 02

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Other Classical ProblemsShortest-Length Problem: Perhaps the most basic problem in variational calculus is to find the curve with minimum length, connecting two points, say (x0, y0) & (x1, y1).

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. The objective is to maximize the area, while the perimeter acts like a constraint.

The answer is a Circle.

Lecture - 02

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The Basic ProblemWe have to find the stationary condition of a general functional, depending on the independent variable (x), dependent variable (y) and its first order derivative (y), with prescribed end values y0 & y1 the simplest case.

Let u u(x) be the function, which minimizes the functional I. Any small change in this function, satisfying the B.C. called a Weak Variation can be written as:

Lecture - 02

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The Basic Problem contd. Substituting into the expression for the functional:

Note that, in this equation, I() is not a functional, it is function of .

Hence we can use the usual methods of calculus to find the extremum.

Lecture - 02

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The Basic Problem contd. Since 0, we can simplify this equation as:

The second term inside the integral can be evaluated using integration by parts:

Substituting:

Lecture - 02

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The Basic Problem contd. The Fundamental Lemma of Variational Calculus:Lecture - 02

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The Basic Problem contd. Using this fundamental lemma:

This equation is called the Euler Equation or Euler-Lagrange Equation in Variational Calculus.

For the stationarity of a functional, Euler equation must be satisfied, along with the B.C.:

Note that these conditions are only the necessary conditions. The sufficient condition for a minimum in general case is difficult to obtain. One option is to compute value of the functional for solutions of Euler equation and few other functions and decide whether it is minimum/maximum.

Another significant point is: there exists an Euler equation for any functional, while it is not true the other way.

Lecture - 02

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Example Shortest-Length ProblemThe functional corresponding to the length of curve connecting two points is obtained as:

Corresponding Euler equation (assuming u(x) minimizes I) is:

Therefore, we get:

Lecture - 02

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Euler Equations for General CasesMore than one Dependent Variable:

Corresponding Euler equations:

Integrals with Higher Derivatives:

Corresponding Euler equation:

Lecture - 03

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Euler Equations for General Cases contd.More than one Independent Variable:

Corresponding Euler equations:

In these cases, the boundary conditions will have to be suitably modified.

In the last case, Greens Lemma will have to be used instead of the integration by parts.

Lecture - 03

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Natural Boundary ConditionsLet us consider the basic problem again, but with the difference in B.C. as shown below:

considering a Weak Variation from the function minimizing the functional:

Substituting into the expression for the functional:

Lecture - 03

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Natural Boundary Condition contd. Differentiating I w.r.t. :

The second term inside the integral can be evaluated using integration by parts:

Substituting:

Lecture - 03

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Natural Boundary Condition contd. Since (x) is an arbitrary function, these two terms should separately be equal to zero. Therefore we have the conditions:

The second requirement above is called the natural boundary condition (This is same as the natural B.C., we have seen in the W.R. formulation).

Lecture - 03

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Example: Hanging RodThe objective is to formulate the functional, derive Euler equation and natural B.C. for the 1D problem below:

Lecture - 03

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Hanging Rod Example contd. Euler Equation:

This is the same as the governing D.E. of the problem. Natural B.C. at x = L:

This is same as the natural B.C. that the force acting at the lower end of the rod is zero.

Lecture - 03

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The Variational NotationConsider a general functional:

For a fixed value of x, the integrand depends on y(x) and its derivative y(x). The change in y(x) for a fixed value of x:

It may be noted that:

In other words, the derivative of variation with respect to an independent variable is same as variation of the derivative.

Lecture - 04

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The Variational Notation contd. Ignoring the higher order terms in , the variation in F:

The change in F, caused by the change y:

Similar to derivative, variation and integration commute. Hence, the First Variation in the functional give:

Lecture - 04

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The Variational Notation contd. If the B.C. are specified at both x = a and x = b, the boundary terms will be identically zero.

Integration by parts, of the second term gives:

The integrand in the above equation is identically equal to zero, form Eulers equation. Hence we get, another form of the necessary condition for a functional to be stationary, which is given by the equation:

If the boundaries are not fully constrained, we need to satisfy the natural B.C., in addition to the above equation.

Lecture - 04

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Rayleigh-Ritz MethodLet the solution be approximated as:

If there is a functional corresponding to a problem, the solution is the function, which makes the functional stationary. If we have the functional:

Substituting the approximation into the functional, the functional becomes an ordinary function of the unknown parameters am. Hence, the necessary condition for stationarity becomes:

This is a set of M equations for determining the M unknown parameters (Ritz Coefficients).

Lecture - 04

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Variational Formulation in Stress AnalysisHence the solution is the displacement function, which minimizes the P.E.:

We have the potential energy of the system, as the functional for stress analysis problems.

Strain Energy (3D):

(-ve of) Change in Potential of loads:

Lecture - 04

Department of Mechanical Engineering, National Institute of Technology Calicut

Variational Formulation in Stress Analysis contd.The elemental contribution to the strain energy:

Using a finite element discretization, net P.E. can be considered as sum of elemental contributions:

Elemental contribution to potential of loads:

Lecture - 04

Department of Mechanical Engineering, National Institute of Technology Calicut

Variational Formulation in Stress Analysis contd.Assembling these elemental systems, we get the global system, given as:

Using the final step in Rayleigh-Ritz method, i.e. taking partial derivatives w.r.t. the nodal displacements:

Lecture - 04

Department of Mechanical Engineering, National Institute of Technology Calicut

Variational Formulation in Heat TransferHence the solution is the temperature function, which makes this functional stationary, which is given by:

The functional corresponding to heat transfer problems is given by:

Using the finite element method:

Rayleigh-Ritz method is to make:

Lecture - 04

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Variational Formulation in Heat Transfer contd.Substitution gives:

Considering the elemental contribution & the finite element approximation over any given element:

Rewriting the functional as:

Lecture - 04

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Variational Formulation in Heat Transfer contd.We get the elemental contribution:

Equating the partial derivatives to zero, the elemental system is obtained as:

Lecture - 04

Department of Mechanical Engineering, National Institute of Technology Calicut

Assignment No. 3Exercise Problem No: 3.2, 3.5, 3.14, 6.2 & 6.3 in the book: Finite Elements and Approximation by Zienkiewicz, O. C., and Morgan, K.

Due Date: As announced in the class.

Any suitable assumptions can be made, but clearly state the assumptions and their justifications along with the answers.

Lecture - 04

Department of Mechanical Engineering, National Institute of Technology Calicut

Isoperimetric Problems & Essential B.C.Problem of Dido: The requirement is to find the closed curve of given length, with maximum enclosed area. The answer: circle.

Another similar problem: Of all the rectangles of same perimeter, we have to find the one with maximum area. The answer: Square.

In both the above problems, we can express the perimeter & enclosed area as functionals.

Hence the objective is to find the maximum or minimum of a functional, while keeping another functional constant.

Such problems are generally called Isoperimetric Problems, even when the functional involved is not a perimeter.

Hence, we can call even the inverse problem of Dido, i.e., to find the curve minimizing the perimeter, for a given area as an isoperimetric problem.

Lecture - 05

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Isoperimetric Problems & Essential B.C. contd.For a general isoperimetric problem:

We can use the Lagrange Multiplier method for solving such problems. We shall formulate a new problem:

The function, which makes this functional stationary will be the answer to the initial problem.

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Isoperimetric Problems & Essential B.C. contd.Coming back to problem of FE formulation:

Using Lagrange Multiplier method, the problem is re-formulated as:

The Lagrange Multiplier is a function of space coordinates. Hence we have the stationarity condition:

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Isoperimetric Problems & Essential B.C. contd.The first variation in the new functional:

Since this should be true for any arbitrary variation, all the terms in R.H.S. above must be zero. The first term gives:

Using the Fundamental Lemma of variational calculus, the second term gives:

And since the Lagrange Multiplier is an arbitrary parameter, third term gives:

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Isoperimetric Problems & Essential B.C. contd.This last equation needs some explanation. For the essential B.C., the variation:

In general:

The same result can be arrived by noting that: since r is independent of , it should remain unaffected due to the variation in .

We can apply the Rayleigh-Ritz method to solve this new functional.

Generally, we will have to enforce the essential B.C. in this manner, since the original functional could be incorporating the effect of natural B.C.

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Essential B.C. using Lagrange MultipliersConsidering a stress analysis problem, without surface & body forces, the functional to be minimized is:

Using FE formulation, the functional (P.E.) becomes a function of nodal displacements:

Re-writing:

Let us assume that we have to enforce the essential B.C.:

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Essential B.C. & Lagrange Multipliers contd.Rewriting the original functional:

To enforce the essential B.C., a new functional (or function of nodal displacements) can be formulated:

Lecture - 05

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Essential B.C. & Lagrange Multipliers contd.The first equation is obtained by taking partial derivatives:

Simplifying:

Lecture - 05

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Essential B.C. & Lagrange Multipliers contd.We can re-write this equation as:

The last equation can be obtained by taking the partial derivative:

Writing in a more convenient form:

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Essential B.C. & Lagrange Multipliers contd.Combining all the equations:

Thus we have the same system, as was obtained in the discrete system analysis.

All the arguments, for & against the use of this method, are still valid.

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Essential B.C. using Penalty MethodThis method also can be thought as a modification to the functional to be minimized. However, there are some major differences. The modified functional is:

Taking partial derivatives, we get the first equation:

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Essential B.C. using Penalty Method contd.This equation can be re-written as:

The complete system is:

This is the Penalty Method, we have seen earlier!!!

Lecture - 05

Department of Mechanical Engineering, National Institute of Technology Calicut

Concluding RemarksThe variational calculus gives an alternate method to arrive at the finite element stiffness matrices, if a functional corresponding to the problem could be found.

The scope of Galerkin WR method is clearly wider, but there are some specific merits for variational formulation:

Variational formulation is physically more meaningful as compared to WR statements (e.g.: minimization of P.E.)

Many methods used in FEA, which were known to work, were given proper mathematical reasoning by the use of variational calculus.

Other methods, like Least-square method, can also be used for the finite element formulation. However, they are not widely used like the Galerkin and Variational methods.

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