Calculus of Variations

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1 Structural Engineering Research Group (SERG) Summer Seminar Series #9 July, 4 Calculus of Variations Tzuyang Yu Associate Professor, Ph.D. Department of Civil and Environmental Engineering The University of Massachusetts Lowell Lowell, Massachusetts

2 Calculus of Variations Background Maimum and Minimum of Functions Maimum and Minimum of Functionals The Variational Notation Constraints and Lagrange Multiplier Applications Approimate Methods Variational Methods Eamples Summary References

3 Background Definition A function is a mapping of single values to single values. A functional is a mapping of function values to single or function values. It usually contains single or multiple variables and their derivatives. Dirichlet Principle: There eists one stationary ground state for energy. Euler s Equation defines the condition for finding the etrema of functionals. à An etremal is the maimum or minimum integral curves of Euler s equation of a functional. Calculus of Functionals: Determining the properties of functionals. Calculus of Variations: Finding the etremals of functionals. 3

4 Background Single value calculus: Functions take etreme values on bounded domain. Necessary condition for etremum at, if f is differentiable: fʹ ( ) = Calculus of variations Test function v(), which vanishes at endpoints, used to find etremal: w ( ) = u ( ) +εv ( ) (,, ) Necessary condition for etremal: di dε = I b ε = F ww d a 4

5 Maimum and Minimum of Functions Maimum and minimum (a) If f() is twice continuously differentiable on [, ] i.e. Nec. condition for a ma. (min.) of f() at [, ] is that Fʹ ( ) = Suff. condition for a ma (min.) of f() at are that and Fʹ ( ) = or Fʹ ʹ ( ) Fʹ ʹ ( ) (b) If f() over closed domain D. Then nec. and suff. condition for a ma. (min.) of f() at that is a negative infinite. i j [, ] f D D are that = i f = = i =, n and also 5

6 Maimum and Minimum of Functions (c) If f() on closed domain D If we want to etremize f() subject to the constraints i=,, k (k < n) g (, K, ) = i n EX: Find the etrema of f(,y) subject to g(,y) = i) Approach One: Direct differentiation of g(, y) To etremize f dg = gd + g ydy = g dy = d g df = fd + f ydy = g ( f fy ) d= g y y 6

7 Maimum and Minimum of Functions We have fg fg = and g = y y to find (,y ) which is to etremize f subject to g = ii) Approach Two: Lagrange Multiplier Let vy (,, λ) = f( y, ) + λgy (, ) etrema of v without any constraint etrema of f subject to g = To etremize v v = f + λg = v = f + y λg = y y fg fg = y y v = g = λ We obtain the same equations by etremizing v. where the Lagrange Multiplier. λ is called 7

8 Maimum and Minimum of Functionals Functionals are function s function. y y ( ) y y y ( ) +η( ) η( ) The basic problem in calculus of variations. Determine y ( ) c [, ] such that the functional: I y( ) where F c points. ( ),! ( ) ( ) = F (, y y ) d as an etrema over its entire domain, subject to y( ) = y, y( ) = y at the end 8

9 Maimum and Minimum of Functionals Using integrating by parts of the nd term, it leads to [F y! (, y, y!)η] [ d d F y!(, y, y! ) F (, y, y!)] ηd = y () = = η( ) η( ) η( ) Since and is arbitrary, d F yʹ (, y, y ʹ ) F y(, y, y ʹ ) = d () (Euler s Equation) Natural B.C s F y ʹ = or/and F ʹ y = = The above requirements are called national b.c s. 9

10 The Variational Notation Variations Imbed u() in a parameter family of function variation of u() is defined as δu = εη( ) The corresponding variation of F, δf to the order in ε is, φ(, ε) = u( ) + εη( ) the since and Then δf = F( + y+ εη, yʹ + εηʹ ) F(, y, yʹ ) F F = δy+ δyʹ y yʹ Iu Fu uʹ ʹ d G ( + εη ) = (, + εη, + εη ) = ( ε ) δi = δ F(, y, yʹ ) d = δ Fyyd (,, ʹ ) F y = δy + F δyʹ d yʹ

11 The Variational Notation F d F F δy y d yʹ yʹ = δyd + Thus, a stationary function for a functional is one for which the first variation. F y = For more general cases (a) Several dependent variables EX: = Euler s Eq. I F(, y,z ; y,z ) d F d F ( ) = y d yʹ ʹ ʹ, F d F ( ) = z d zʹ (b) Several Independent variables EX: I F(, y, u, u, uy) ddy Euler s Eq. = R F F F ( ) ( ) = u u y u y

12 The Variational Notation (c) High Orders EX: I = F(, y, yʹ, yʹ ʹ ) d Euler s Eq. F d F d F ( ) + ( ) = y d yʹ d yʹ ʹ Variables Causing more equations Order Causing longer equations

13 Constraints and Lagrange Multiplier Lagrange multiplier Lagrange multiplier can be used to find the etreme value of a multivariate function f subjected to the constraints. EX: (a) Find the etreme value of I = F(,,, u v u, v ) d where u ( ) v ( ) u ( ) = u = u = v v ( ) = v and subject to the constraints Guv (,, ) = (3) From δ F d F F d F = + = v d u v d v I δu v d (4) 3

14 Constraints and Lagrange Multiplier Because of the constraints, we don t get two Euler s equations. From δ G G G u v u δ = + v δ = Gv v δ u G δ = u Therefore, Eq. (4) becomes G F d F F d F δi Gu u d u v d v G F d F G F d F = v u d u u v d v v = + δvd = (5) (6) The above equations, together with Eq. (3), are used to solve for u, v. 4

15 (b) Simple Isoparametric Problem To etremize i) Constraints and Lagrange Multiplier ii) y ( ) = y, y ( ) = y I = ( ʹ ) (,, ʹ ) F y y d J = G y y d= const,,., subject to the constraint: Take the variation of two-parameter family: (where and η are some equations which satisfy η ( ) ( ) ( ) ( ) ( ) ( ) η η η η = = = = ) ( ) ( ) y+ δy= y+ εη + ε η Then, I F y yʹ ʹ ʹ d ( ε, ε ) = (, + εη + ε η, + εη + ε η ) J( ε, ε ) = G(, y+ εη + ε η, yʹ + εηʹ + ε η ʹ ) d To base on the Lagrange Multiplier Method, we can get : 5

16 ( I + λj) ε = ε = = ε ( I + λj) ε = ε = = ε F d F G d G λ ηid + = y d yʹ y d yʹ i =, The Euler s equation becomes d ( F + λg) ( F+ λg) = y d y ʹ G d G when =, λ is arbitrary numbers. y d yʹ Constraints and Lagrange Multiplier The constraint is trivial, we can ignore. λ 6

17 Applications Helmholtz Equation EX: Force vibration of a membrane u u u c ( + ) = f(, y, t) t y if the forcing function f is of the form f(, y, t) = P(, y)sin( ωt+ α) (7) we may write the steady state displacement u in the form u = v(, y)sin( ωt+ α) v v c ( + ) + ω v+ p= y (8) 7

18 Consider R c v R δ vddy v v c ( + ) + ω v+ p δvddy = y = c [( v δv) v δv ] ddy R Applications c v R yyδvddy = c [( v δv) v δv ] ddy R y y y y Note that ( v δv) = v δv+ v δv ( v δv) = v δv+ v δv y y yy y y V = V i+ V j y V = vδv V = v δ v n = cosθi + sinθ j iv = V + V y y = (ν δv) y y + (ν y δv) y 8

19 Applications ( iv ) da = R γ V in ds = (v δvcosθ + v y δvsinθ)ds c v δ vddy + c v δ vddy R R +c v y δvsinθds = c v δvcosθds γ γ yy R c δ(v ) ddy c δ(v y ) ddy R c (v γ cosθ + v y sinθ)δvds + ωδ ( v ) ddy+ Pδvddy= R R R c δ[(v ) + (v y ) ]ddy (9) v c vds [ c ( v) v Pv] ddy= n δ δ ω γ R 9

20 Applications Hence, i) if v= f(, y) is given on i.e. on δ v = γ γ then the variational problem δ c [ ( v) ω v pvddy ] = R ii) if v = is given on n the variation problem is same as Eq. () v iii) if = () s is given on n ψ γ δ ω ψ γ [ c ( v) v pv ddy c vd] = R τ () ()

21 Applications Diffusion Equation EX: Steady State Heat Condition ( k T) = f(, T) in D B.C s : T = T kn T = q kn T = h( T T ) δt 3 on B on B on B 3 Multiply the equation by, and integrate over the domain D. After integrating by parts, we find the variational problem as follow. δ k T f T dtʹ dτ + qtdσ + h( T T3) dσ] B = B3 T [ ( ) + (, ʹ ) D T with T = T on B.

22 Applications Poisson s Equation EX: Torsion of a Prismatic Bar ψ = ψ = in on R γ where ψ with ψ = on γ. is the Prandtl stress function and ψ στz = Gα y, σzy = σ ψ α The variation problem becomes { } D ddy δ [( ψ) 4 ψ] = R

23 Approimate Methods I) Method of Weighted Residuals (MWR) Lu [ ] = in D with homogeneous b.c s in B. Assume an approimate solution. where each trial function u u Cφ n = = n i i i= φ i satisfies the b.c s. The residual is R n = L[ u ] n In this method (MWR), C i are chosen such that R n is forced to be zero in an average sense. i.e. < w j, R n > =, j =,,,n where w j are the weighting functions.. 3

24 Approimate Methods II) Galerkin Method w j are chosen to be the trial functions φ j as members of a complete set of functions. hence the trial functions is chosen Galerkin method force the residual to be zero w.r.t. an orthogonal complete set. EX: Torsion of a Square Shaft ψ = ψ = on =± a, y =± a i) One term approimation ψ = c ( a )( y a ) R = ψ + = c[( a) + ( y a) ] + i φ = ( a )( y a ) 4

25 a a From φ = Therefore, R ddy a a c = 5 8 a 5 ( a )( y a ) ψ = 8a The torsional rigidity is determined by D = G ψ ddy=.388 G( a) The eact value of D is R D =.46 G( a) a The approimation error is -.%. Approimate Methods 4 4 5

26 Approimate Methods ii) Two term approimation ψ = ( a )( y a )[ c + c ( + y )] From φ ddy= R R and φ ddy= We obtain c Therefore R R =, c = 6 a 443 a By symmetry R = ψ + φ = ( a )( y a ) ( a )( y a )( y ) φ = + 4 = ψ =.44 ( ) à The error is -.4%. R D G ddy G a 6

27 Variational Methods I) Kantorovich Method [Kantorovich (948)] Assuming the approimate solution as : u = n where U i is a known function decided by b.c. condition. i= C i ( n )U i C i is a unknown function decided by minimal I. Euler Equation of C i EX : The Torsional Problem with a Functional I. u u y a b ( ) = [( ) ( ) 4 ] a + b Iu uddy 7

28 Variational Methods Assuming the one-term approimate solution as : Then, uy b y C (, ) = ( ) () a b (C) = {( ) [C ʹ ( )] + 4 C ( ) 4( )C( )} a b I b y y b y ddy Integrate by y a I(C) = [ b Cʹ + b C b C] d a Euler s equation is 5 5 C( ) C( ) b b ʹ ʹ = where b.c. condition is C( ± a ) = General solution is 5 5 C( ) = Acosh( ) Asinh( ) b + b + 8

29 Variational Methods where A =, A = 5 a cosh( ) b C( ) = cosh( ) b 5 a cosh( ) b and 5 Therefore, the one-term approimate solution is 5 cosh( ) b 5 a cosh( ) b u = ( b y ) 9

30 Variational Methods II) Rayleigh-Ritz Method This is used when the eact solution is impossible or difficult to obtain. First, we assume the approimate solution as : where U i are some approimate function which satisfy the b.c s. Then, we can calculate etreme I. I = I c K c n (,, ) Choose c ~ c n i.e. yʹ ʹ + y = y() = y() = EX:, Its solution can be obtained from ( ) u n = CU i I c yʹ ʹ + y+ δ yd= ( ʹ ) i= I = K = = c I = y y y d i n 3

31 Assuming that i) One-term approimation Then, I = c ii) Two-term approimation ( )( 3 ) y= c + c + c K ( ) ( ) = = y ʹ = c ( ) y c c 3 4 I( c ) = c ( 4 4 ) c ( ) c( ) d + + c 4 c 9 c = + + c = c Variational Methods 9 6 c = c =.63 y() =.63 ( ) y= ( )( c + c ) = c ( ) + c ( ) 3 3

32 Then Variational Methods yʹ = c + c ( ) ( 3 ) { ( ) ( 3 Ic ( ), c) = [ c cc c3 (4 + 9 ) c + + cc + } { ( ) ( ) { } + c ( + ) c + c ] d } ( ) ( ) c = cc c c c c c = c + cc + c

33 Variational Methods I = c 9 + = 6 7 c c I = c.37 c +.7 c =.5 c =.77, c = = 7 84 c c y () = ( )( ) 33

34 Eamples I) The Brachistochrone (fastest descent) Curve Problem Proposed by Johann Bernoulli (696) II) Structural Dynamics Formulation of Governing Equation of Motion 34

35 Summary Many governing equations in physics and chemistry can be formulated by functionals. Finding the etremals of functionals can lead to the solutions in many problems in science and engineering. Euler s Equations provide the necessary condition for evaluating problems involving functionals. However, analyzing Euler s equations sometimes can be difficult. Calculus of Variations determines the etremals of functionals, even though the solution is only approimated. Calculus of Variations provides the theoretical basis for many methods in engineering, such as the Principle of Virtual Displacement (PVD) and the Finite Element Method (FEM). 35

36 References O. Bolza (973), Lectures on the Calculus of Variations, Chelsea Publishing Company, New York, NY. I.M. Gelfand, S.V. Fomin (963), Calculus of Variations, New York, NY. H.J. Greenberg (949), On the Variational Principles of Plasticity, Brown University Press, Providence, RI. C. Lanezos (949), The Variational Principles of Mechanics, University of Toronto Press, Toronto, Canada. L. Kantorovich (948), Functional analysis and applied mathematics, (Russian) UMN3, 6 (8), pp G. Bliss (946), Lectures on the Calculus of Variations, University of Chicago Press, Chicago, IL. 36

VARIATIONAL PRINCIPLES

CHAPTER - II VARIATIONAL PRINCIPLES Unit : Euler-Lagranges s Differential Equations: Introduction: We have seen that co-ordinates are the tools in the hands of a mathematician. With the help of these co-ordinates