Institute of Structural Page 1

Chapter 3

Variational Formulation & the Galerkin Method

Method of Finite Elements I Institute of Structural Engineering Page 2

Today’s Lecture Contents:

• Introduction • Differential formulation • Principle of Virtual Work • Variational formulations • Approximative methods • The Galerkin Approach

Method of Finite Elements I Institute of Structural Engineering Page 3 The differential form of physical processes Physical processes are governed by laws (equations) most probably expressed in a differential form:

• The axially loaded bar equation:

• The isotropic slab equation:

• The Laplace equation in two dimensions: (e.g. the heat conduction problem)

Method of Finite Elements I Institute of Structural Engineering Page 4 The differential form of physical processes Focus: The axially loaded bar example. Consider a bar loaded with constant end load R. Strength of Materials Approach

Equilibrium equation R R fx()= R⇒σ ( x) = A x Constitutive equation (Hooke’s Law) σ (x) R ε ()x == R E AE f(x) Kinematics equation L-x δ (x) ε x = ( ) x Given: Length L, Section Area A, Young's modulus E Find: stresses and deformations.

Rx Assumptions: δ (x) = The cross-section of the bar does not change after loading. AE The material is linear elastic, isotropic, and homogeneous. The load is centric. End-effects are not of interest to us.

Method of Finite Elements I Institute of Structural Engineering Page 5 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach Consider and infinitesimal element of the bar: Equilibrium equation Δσ dσ Aσ = A(σ + Δσ ) ⇒ A lim = 0 ⇒ A = 0 ! x dx Δx→0 Δ Constitutive equation (Hooke’s Law) σε= E Kinematics equation du ε = dx

d 2u AE = 0 Strong Form dx2 Boundary Conditions (BC) Given: Length L, Section Area A, Young's modulus E Find: stresses and deformations. u(0) = 0 Essential BC

Assumptions: σ (L) = 0 ⇒ The cross-section of the bar does not change after loading. du The material is linear elastic, isotropic, and homogeneous. AE = R Natural BC The load is centric. dx x=L End-effects are not of interest to us. Method of Finite Elements I Institute of Structural Engineering Page 6 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach

d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L

Definition

The strong form of a physical process is the well posed set of the underlying differential equation with the accompanying boundary conditions

Method of Finite Elements I Institute of Structural Engineering Page 7 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach

d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L

v This is a homogeneous 2nd order ODE with known solution:

Analytical Solution: du(x) u(x) = uh = C1x + C2 & ε(x) = = C1 = const! dx

Method of Finite Elements I Institute of Structural Engineering Page 8 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach

d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L Analytical Solution:

To fully define the solution (i.e., to evaluate the values of parameters C 1 , C 2 ) we have to use the given boundary conditions (BC):

C2 = 0 u(0)=0 u(x) = uh = C1x + C2 ⎯d⎯u ⎯R⎯→ R = dx EA C = x=L 1 EA Rx ⇒ u(x) = v Same as in the mechanical approach! AE

Method of Finite Elements I Institute of Structural Engineering Page 9 The Strong form – 2D case A generic expression of the two-dimensional strong form is:

and a generic expression of the accompanying set of boundary conditions:

: Essential or Dirichlet BCs

: Natural or von Neumann BCs

Disadvantages The analytical solution in such equations is i. In many cases difficult to be evaluated ii. In most cases CANNOT be evaluated at all. Why? • Complex geometries • Complex loading and boundary conditions

Method of Finite Elements I Institute of Structural Engineering Page 10 Deriving the Strong form – 3D case

su su : supported area with prescribed displacements U

s f s f : surface with prescribed forces f f B : body forces (per unit volume) U : displacement vector ε : strain tensor (vector) σ : stress tensor (vector)

Method of Finite Elements I Institute of Structural Engineering Page 11 Deriving the Strong form – 3D case Kinematic Relations

LU ε = ⎡ ∂ ⎤ ⎢ 0 0 ⎥ ⎢ ∂x ⎥ ⎢ ∂ ⎥ 0 0 ⎢ y ⎥ ⎢ ∂ ⎥ ⎢ ∂ ⎥ ⎡ U ⎤ ⎢ 0 0 ⎥ ⎢ ⎥ ∂z T ⎡ ⎤ U = V , L = ⎢ ⎥, ε = ε xx ε yy ε zz 2ε xy 2ε yz 2ε xz ⎢ ⎥ ⎢ ∂ ∂ ⎥ ⎣⎢ ⎦⎥ ⎢ W ⎥ 0 ⎣ ⎦ ⎢ ∂y ∂x ⎥ ⎢ ⎥ ⎢ ∂ ∂ ⎥ 0 ⎢ ∂z ∂y ⎥ ⎢ ⎥ ⎢ ∂ ∂ ⎥ 0 ⎢ z x ⎥ ⎣ ∂ ∂ ⎦

Method of Finite Elements I Institute of Structural Engineering Page 12 Deriving the Strong form – 3D case Strain Compatibility – Saint Venant principle

⎡⎤∂∂22 ∂ 2 02− 0 0 ⎢⎥yx22 xy ⎢⎥∂∂ ∂∂ ⎢⎥∂∂22 ∂ 2 ⎢⎥002022 − ⎢⎥∂∂zy ∂∂ yz ⎢⎥∂∂22 ∂ 2 ⎢⎥0002− ∂∂zx22 ∂∂ xz L = ⎢⎥ L ε = 0 ⇒ 1 2222 1 ⎢⎥∂∂∂∂ ⎢⎥00−−2 ⎢⎥∂∂yz ∂∂ xz ∂ x ∂∂ xy ⎢⎥∂∂∂∂2222 ⎢⎥00−− ⎢⎥∂∂xz ∂∂ zy ∂∂ xy ∂ y2 ⎢⎥22 2 2 ⎢⎥∂∂ ∂ ∂ 00 2 −− ⎣⎦⎢⎥∂∂xy ∂ z ∂∂ xz ∂∂ yz

Method of Finite Elements I Institute of Structural Engineering Page 13 Deriving the Strong form – 3D case Equilibrium Equations

B Body loads: L2τ+f =0

τTT, LL = = ⎣⎦⎡⎤ττττττxx yy zz xy yz zx 2

Surface loads: ⎡⎤lmn00 0 s on smln we have Ντ− ff = 0 where N =⎢⎥ 0 0 0 f ⎢⎥ ⎣⎦⎢⎥00nml 0

l, m and n are cosines of the angles between the normal on the surface and X, Y and Z

Method of Finite Elements I Institute of Structural Engineering Page 14 Deriving the Strong form – 3D case Constitutive Law

τ= Cε

C is elasticity matrix and depends on material properties E and ν (modulus of elasticity and Poisson’s ratio)

ε= C−1τ

Method of Finite Elements I Institute of Structural Engineering Page 15 Differential Formulation

Summary - General Form 3D case:

• Boundary problem of the linear theory of elasticity: differential equations and boundary conditions • 15 unknowns: 6 stress components, 6 strain components and 3 displacement components • 3 equilibrium equations, 6 relationships between displacements and strains, material law (6 equations) • Together with boundary conditions, the state of stress and deformation is completely defined

Method of Finite Elements I Institute of Structural Engineering Page 16 Principle of Virtual Work

• The principle of virtual displacements: the virtual work of a system of equilibrium forces vanishes on compatible virtual displacements; the virtual displacements are taken in the form of variations of the real displacements • Equilibrium is a consequence of vanishing of a virtual work

Internal Virtual Work External Virtual Work

ετTdV Uf T B dV USTff f S dS UR iT i =+ +∑ C ∫∫ ∫ i VV Sf

Stresses in equilibrium with applied loads

Virtual strains corresponding to virtual displacements

Method of Finite Elements I Institute of Structural Engineering Page 17 Principle of Complementary Virtual Work

• The principle of virtual forces: virtual work of equilibrium variations of the stresses and the forces on the strains and displacements vanishes; the stress field considered is a statically admissible field of variation • Equilibrium is assumed to hold a priori and the compatibility of deformations is a consequence of vanishing of a virtual work • Both principles do not depend on a constitutive law

Method of Finite Elements I Institute of Structural Engineering Page 18 Variational Formulation

• Based on the principle of stationarity of a functional, which is usually potential or complementary • Two classes of boundary conditions: essential (geometric) and natural (force) boundary conditions • Scalar quantities (, potentials) are considered rather than vector quantities

Method of Finite Elements I Institute of Structural Engineering Page 19 Principle of Minimum Total Potential Energy • Conservation of energy: Work = change in potential, kinetic and thermal energy • For elastic problems (linear and non-linear) a special case of the Principle of Virtual Work – Principle of minimum total potential energy can be applied

• U is the stress potential: τεij=∂U / ∂ ij 1 U = ε T Cεdv ∫ 2 v

T S W = ∫ f B Udv + ∫ f f Uds v s

Method of Finite Elements I Institute of Structural Engineering Page 20 Principle of Minimum Total Potential Energy

• Total potential energy is a sum of strain energy and potential of loads P = U −W • This equation, which gives P as a function of deformation components, together with compatibility relations within the solid and geometric boundary conditions, defines the so called Lagrange functional • Applying the variation we invoke the stationary condition of the functional P

dP = dU − dW = 0

Method of Finite Elements I Institute of Structural Engineering Page 21 Variational Formulation

• By utilizing the variational formulation, it is possible to obtain a formulation of the problem, which is of lower complexity than the original differential form (strong form). • This is also known as the weak form, which however can also be aained by following an alternate path (see Galerkin formulation). • For approximate solutions, a larger class of trial functions than in the differential formulation can be employed; for example, the trial functions need not satisfy the natural boundary conditions because these boundary conditions are implicitly contained in the functional – this is extensively used in MFE.

Method of Finite Elements I Institute of Structural Engineering Page 22 Approximative Methods Instead of trying to find the exact solution of the continuous system, i.e., of the strong form, try to derive an estimate of what the solution should be at specific points within the system.

The procedure of reducing the physical process to its discrete counterpart is the discretisation process.

Method of Finite Elements I Institute of Structural Engineering Page 23 Approximative Methods Variational Methods Weighted Residual Methods approximation is based on the start with an estimate of the the solution and minimization of a functional, as those demand that its weighted average error is defned in the earlier slides. minimized • Rayleigh-Ritz Method • Te Galerkin Method • Te Least Square Method • Te • Te Subdomain Method • Pseudo-spectral Methods

Method of Finite Elements I Institute of Structural Engineering Page 24 The differential form of physical processes Focus: The axially loaded bar with distributed load example.

2 ax d u AE = −ax Strong Form dx2 u(0) = 0 Essential BC du x AE = 0 Natural BC dx x=L Analytical Solution:

To fully define the solution (i.e., to evaluate the values of parameters C 1 , C 2 ) we have to use the given boundary conditions (BC):

3 C2 = 0 ax u(0)=0 2 u(x) = uh + up = C1x + C2 − ⎯d⎯u ⎯R⎯→ aL 6EA = dx EA C = x=L 1 2EA aL2 ax3 ⇒ u(x) = x − 2EA 6EA

Method of Finite Elements I Institute of Structural Engineering Page 25 The differential form of physical processes Focus: The axially loaded bar with distributed load example.

ax aL2 ax3 u(x) = x − 2EA 6EA x

3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Length (m) Length (m)

Method of Finite Elements I Institute of Structural Engineering Page 26 Weighted Residual Methods Focus: The axially loaded bar with distributed load example.

ax

R and its corresponding strong form:

d 2u x AE = −ax Strong Form dx2 Boundary Conditions (BC) Given an arbitrary weighting function w that satisfies the essential conditions and u(0) = 0 Essential BC additionally: σ (L) = 0 ⇒ du AE = 0 Natural BC If then, dx x=L

Method of Finite Elements I Institute of Structural Engineering Page 27 Weighted Residual Methods Focus: The axially loaded bar with distributed load example.

ax

x Multiplying the strong form by w and integrating over L:

Integrating equation I by parts the following relation is derived:

Method of Finite Elements I Institute of Structural Engineering Page 28 Weighted Residual Methods Focus: The axially loaded bar with distributed load example.

ax

R

x

Elaborating a lile bit more on the relation:

Method of Finite Elements I Institute of Structural Engineering Page 29 Weighted Residual Methods Focus: The axially loaded bar with distributed load example.

ax Elaborating a lile bit more on the relation:

why? why? x

Therefore, the weak form of the problem is defined as Find such that:

Observe that the weak form involves derivatives of a lesser order than the original strong form.

Method of Finite Elements I Institute of Structural Engineering Page 30 Weighted Residual Methods

Weighted Residual Methods start with an estimate of the solution and demand that its weighted average error is minimized: • Te Galerkin Method • Te Least Square Method • Te Collocation Method • Te Subdomain Method • Boris Grigoryevich Galerkin – (1871-1945) Pseudo-spectral Methods mathematician/ engineer

Method of Finite Elements I Institute of Structural Engineering Page 31 The Galerkin Method Theory – Consider the general case of a Example – The axially loaded bar: differential equation: Try an approximate solution to the equation of the following form Choose the following approximation

Demand that the approximation where are test functions (input) and satisfies the essential conditions: are unknown quantities that we need to evaluate. The solution must satisfy the boundary conditions. The approximation error in this case is: Since is an approximation, substituting it in the initial equation will result in an error:

Method of Finite Elements I Institute of Structural Engineering Page 32 The Galerkin Method

Assumption 1: The weighted average error of the approximation should be zero

Method of Finite Elements I Institute of Structural Engineering Page 33 The Galerkin Method

Assumption 1: The weighted average error of the approximation should be zero

But that’s the Weak Form!!!!!

Therefore once again integration by parts leads to

Method of Finite Elements I Institute of Structural Engineering Page 34 The Galerkin Method

Assumption 1: The weighted average error of the approximation should be zero

But that’s the Weak Form!!!!!

Therefore once again integration by parts leads to

Assumption 2: The weight function is approximated using the same scheme as for the solution Remember that the weight function must also satisfy the BCs

Substituting the approximations for both and in the weak form,

Method of Finite Elements I Institute of Structural Engineering Page 35 The Galerkin Method The following relation is retrieved:

where:

Method of Finite Elements I Institute of Structural Engineering Page 36 The Galerkin Method

Performing the integration, the following relations are established:

Method of Finite Elements I Institute of Structural Engineering Page 37

Or in matrix form: The Galerkin Method

that’s a linear system of equations:

and that’s of course the exact solution. Why?

Method of Finite Elements I Institute of Structural Engineering Page 38 The Galerkin Method

Now lets try the following approximation:

Which again needs to satisfy the natural BCs, therefore: The weight function assumes the same form:

Substituting now into the weak form:

2 L L w EAu − x ax dx = 0 ⇒ u = α 2 0 ( 2 ( )) 2 ∫ 3EA

Wasn’t that much easier? But….is it correct?

Method of Finite Elements I Institute of Structural Engineering Page 39 The Galerkin Method

Strong Form Galerkin-Cubic order Galerkin-Linear 3

2.5

2

1.5

1

0.5

0 0 0.5 1 1.5 2 Length (m)

Method of Finite Elements I Institute of Structural Engineering Page 40 The Galerkin Method We saw in the previous example that the Galerkin method is based on the approximation of the strong form solution using a set of functions. These are by definition absolutely accurate at the boundaries of the problem. So, why not increase the boundaries?

element: 1 element: 2

1 2 3

Instead of seeking the solution of a single bar we chose to divide it into three interconnected and not overlapping elements

Method of Finite Elements I Institute of Structural Engineering Page 41 The Galerkin Method element: 1 element: 2

1 2 3

Method of Finite Elements I Institute of Structural Engineering Page 42 The Galerkin Method

1 2 3

Method of Finite Elements I Institute of Structural Engineering Page 43 The Galerkin Method The weak form of a continuous problem was derived in a systematic way: This part involves only the solution approximation

This part only involves the essential boundary conditions a.k.a. loading

Method of Finite Elements I Institute of Structural Engineering Page 44 The Galerkin Method And then an approximation was defined for the displacement field, for example

Vector of degrees of freedom Displacement field Shape Function Matrix The weak form also involves the first derivative of the approximation

Strain field Strain Displacement Matrix

Method of Finite Elements I Institute of Structural Engineering Page 45 The Galerkin Method Therefore if we return to the weak form :

and set:

The following FUNDAMENTAL FEM expression is derived

Why?? or even beer

Method of Finite Elements I Institute of Structural Engineering Page 46 The Galerkin Method

EA has to do only with material and cross-sectional properties

We call The Finite Element stiffness Matrix

Ιf E is a function of {d} Material Nonlinearity

Ιf [B] is a function of {d} Geometrical Nonlinearity

Method of Finite Elements I