Chapter 3 - ETH Z€¦ · Chapter 3 Variational Formulation & the Galerkin Method . ......

Institute of Structural Engineering

Method of Finite Elements I

Chapter 3

Variational Formulation & the Galerkin Method



Today’s Lecture Contents:

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



The differential form of physical processes Physical processes are governed by laws (equations) most probably expressed in a differential form:

•  The Laplace equation in two dimensions:

(e.g. the heat conduction problem)

•  The isotropic slab equation:

•  The axially loaded bar equation:



The differential form of physical processes Focus: The axially loaded bar example. Consider a bar loaded with constant end load R.

Given: Length L, Section Area A, Young'ʹs modulus E Find: stresses and deformations. Assumptions: The cross-‐‑section of the bar does not change after loading. The material is linear elastic, isotropic, and homogeneous. The load is centric. End-‐‑effects are not of interest to us.

Strength of Materials Approach

Equilibrium equation

( )( ) Rf x R xA

σ= ⇒ =x

R

R

L-‐x

f(x)

Constitutive equation (Hooke’s Law)

( )( )x RxE AE

σε = =

Kinematics equation

ε x( ) = δ (x)

x

δ x( ) = Rx

AE



The differential form of physical processes Focus: The axially loaded bar example. Consider and infinitesimal element of the bar:

Given: Length L, Section Area A, Young'ʹs modulus E Find: stresses and deformations. Assumptions: The cross-‐‑section of the bar does not change after loading. The material is linear elastic, isotropic, and homogeneous. The load is centric. End-‐‑effects are not of interest to us.

The Differential Approach Equilibrium equation

Constitutive equation (Hooke’s Law) Eσ ε=

Kinematics equation dudx

ε =

Aσ = A(σ + Δσ )⇒ A lim

Δx→0!

ΔσΔx

= 0⇒ Adσdx

= 0

AEd 2udx2 = 0 Strong Form

Boundary Conditions (BC)u(0) = 0 Essential BC

σ L( ) = 0⇒

AEdudx x=L

= R Natural BC



The differential form of physical processes

Definition

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

Focus: The axially loaded bar example.

The Differential Approach


u(0) = 0 Essential BC

AEdudx x=L

= R Natural BCx

R



The differential form of physical processes

Analytical Solution:

u(x) = uh = C1x +C2 & ε(x) = du(x)

dx= C1 = const!

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

Focus: The axially loaded bar example.




AEdudx x=L

= R Natural BCx

R



The differential form of physical processes Focus: The axially loaded bar example.



u(x) = uh = C1x +C2u(0)=0

dudx x=L

= REA

⎯ →⎯⎯⎯C2 = 0

C1 =R

EA



AEdudx x=L

= R Natural BC

⇒ u x( ) = Rx

AEv  Same as in the mechanical approach!

x

R

To fully define the solution (i.e., to evaluate the values of parameters ) we have to use the given boundary conditions (BC): C1,C2



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



su : supported area with prescribed displacements Usu

s f : surface with prescribed forces f s f

f B : body forces (per unit volume)U : displacement vectorε : strain tensor (vector)σ : stress tensor (vector)

Deriving the Strong form – 3D case



Kinematic Relations

U =UVW

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥, L =

∂∂x

0 0

0 ∂∂y

0

0 0 ∂∂z

∂∂y

∂∂x

0

0 ∂∂z

∂∂y

∂∂z

0 ∂∂x

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

, ε T = ε xx ε yy ε zz 2ε xy 2ε yz 2ε xz⎡⎣⎢

⎤⎦⎥

ε = LU




2 2 2

2 2

2 2 2

2 2

2 2 2

2 2

1 2 2 2 2

2

2 2 2 2

2

2 2 2 2

2

0 2 0 0

0 0 2 0

0 0 0 2

0 0

0 0

0 0

y x x y

z y y z

z x x z

y z x z x x y

x z z y x y y

x y z x z y z

⎡ ⎤∂ ∂ ∂−⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂−⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥−∂ ∂ ∂ ∂⎢ ⎥= ⎢ ⎥∂ ∂ ∂ ∂− −⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥− −

∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥− −⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

L L1ε = 0⇒

Strain Compatibility – Saint Venant principle




2

2, =

B

T Txx yy zz xy yz zxτ τ τ τ τ τ⎡ ⎤= ⎣ ⎦

L τ + f = 0

τ L L

0 0 0on we have where 0 0 0

0 0 0

fsf

l m ns m l n

n m l

⎡ ⎤⎢ ⎥− = ⎢ ⎥⎢ ⎥⎣ ⎦

Ντ f = 0 N

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

Equilibrium Equations


Body loads:

Surface loads:



=τ Cε

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

1−=ε C τ

Constitutive Law




Differential Formulation

•  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

Summary -‐‑ General Form 3D case:



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

f f

f

S T ST T B iT iC

iV V S

dV dV dS= + +∑∫ ∫ ∫ε τ U f U f U R

Stresses in equilibrium with applied loads

Virtual strains corresponding to virtual displacements

Internal Virtual Work External Virtual Work



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



Variational Formulation

• Based on the principle of stationarity of a functional, which is usually potential or complementary energy

• Two classes of boundary conditions: essential (geometric) and natural (force) boundary conditions

• Scalar quantities (energies, potentials) are considered rather than vector quantities



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

•  is the stress potential:

/ij ijUτ ε= ∂ ∂

U = 1

2v∫ ε TCεdv

W = f BT

Uv∫ dv + f S f

s∫ Uds

U



Principle of Minimum Total Potential Energy

• Total potential energy is a sum of strain energy and potential of loads

• 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

dP = dU − dW = 0

P =U −W

P



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 a_ained 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.



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.



Variational Methods approximation is based on the minimization of a functional, as those defined in the earlier slides.

•  Rayleigh-Ritz Method

Weighted Residual Methods start with an estimate of the the solution and demand that its weighted average error is minimized

•  The Galerkin Method

•  The Least Square Method

•  The Collocation Method

•  The Subdomain Method

•  Pseudo-spectral Methods

Approximative Methods



The differential form of physical processes Focus: The axially loaded bar with distributed load example.


u(x) = uh + up = C1x +C2 −ax3

6EAu(0)=0

dudx x=L

= REA

⎯ →⎯⎯⎯C2 = 0

C1 =aL2

2EA

AEd 2udx2 = −ax Strong Form


AEdudx x=L

= 0 Natural BC

⇒ u x( ) = aL2

2EAx − ax3

6EA

x

To fully define the solution (i.e., to evaluate the values of parameters ) we have to use the given boundary conditions (BC): C1,C2

ax



The differential form of physical processes Focus: The axially loaded bar with distributed load example.

x

ax

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2Length (m)

00.5

11.5

22.5

3

0 0.5 1 1.5 2Length (m)

u x( ) = aL2

2EAx − ax3

6EA



and its corresponding strong form:

Given an arbitrary weighting function w that satisfies the essential conditions and additionally:

If then,

x

R

Weighted Residual Methods

AEd 2udx2 = −ax Strong Form

Boundary Conditions (BC)u(0) = 0 Essential BC

σ L( ) = 0⇒

AEdudx x=L

= 0 Natural BC

Focus: The axially loaded bar with distributed load example.

ax



Multiplying the strong form by w and integrating over L:

Integrating equation I by parts the following relation is derived:

x

Weighted Residual Methods Focus: The axially loaded bar with distributed load example.

ax



Elaborating a li_le bit more on the relation:

x

R


ax



Elaborating a li_le bit more on the relation:

why?

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

why?

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

x


ax



Weighted Residual Methods start with an estimate of the solution and demand that its weighted average error is minimized:

•  The Galerkin Method

•  The Least Square Method

•  The Collocation Method

•  The Subdomain Method

•  Pseudo-spectral Methods Boris Grigoryevich Galerkin – (1871-1945) mathematician/ engineer

Weighted Residual Methods



Theory – Consider the general case of a differential equation:

Example – The axially loaded bar:

Try an approximate solution to the equation of the following form

where are test functions (input) and are unknown quantities that we need to evaluate. The solution must satisfy the boundary conditions.

Since is an approximation, substituting it in the initial equation will result in an error:

Choose the following approximation

Demand that the approximation satisfies the essential conditions:

The approximation error in this case is:

The Galerkin Method



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

The Galerkin Method




Therefore once again integration by parts leads to

But that’s the Weak Form!!!!!

The Galerkin Method




Therefore once again integration by parts leads to

But that’s the Weak Form!!!!!

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,

The Galerkin Method



The following relation is retrieved:

where:

The Galerkin Method



Performing the integration, the following relations are established:

The Galerkin Method



Or in matrix form:

that’s a linear system of equations:

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

The Galerkin Method



Now lets try the following approximation:

The weight function assumes the same form:

Which again needs to satisfy the natural BCs, therefore:

Substituting now into the weak form:

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

The Galerkin Method

w2 EAu2 − x ax( )( )dx

0

L

∫ = 0⇒ u2 =αL2

3EA



0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2Length (m)

Strong Form Galerkin-Cubic order Galerkin-Linear

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 basis functions. These are by definition absolutely accurate at the boundaries of the problem. So, why not increase the boundaries?

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

1 2 3

The Galerkin Method

element: 1 element: 2



1 2 3

element: 1

The Galerkin Method element: 2



1 2 3

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

The Galerkin Method



And then an approximation was defined for the displacement field, for example

Vector of degrees of freedom

Shape Function Matrix

The weak form also involves the first derivative of the approximation

Strain Displacement Matrix

Strain field

Displacement field

The Galerkin Method



Therefore if we return to the weak form :

The following FUNDAMENTAL FEM expression is derived

and set:

or even be]er Why??

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}

Ιf [B] is a function of {d}

Material Nonlinearity

Geometrical Nonlinearity

The Galerkin Method

Chapter 3 - ETH Z€¦ · Chapter 3 Variational Formulation & the Galerkin Method . ......

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