Introduction to Finite Element Method

Judy P. Yang (楊子儀)

Dec. 1, 2016

Department of Civil Engineering National Chiao Tung University (NCTU)

Chapter 11 Strong and Weak Formulations

• 11.1 Strong Form of One-Dimensional Heat Equation • 11.2 Strong Form of Axially Loaded Elastic Bar • 11.3 Strong Form of Flexible String

Chapter 11 Strong and Weak Formulations

• For the problems considered in the previous chapters, the FE equations could be formulated directly

• In general, however, the FE method is a numerical method to solve arbitrary differential equations

• To achieve this objective, it is a characteristic feature of the FE approach that the differential equations in question are first reformulated into an equivalent form, the so-called weak formulation

• This chapter introduces the formulation in terms of so-called strong and weak forms

Overview of the Finite Element Method

( ) ( ) ( ) ( )MGWS ⇔≈⇔Strong

form

Weak

form

Galerkin

approx.

Matrix

form

11.1 Strong Form of One-Dimensional Heat Equation

• Consider a fin with an inhomogeneous temperature distribution 𝑇(𝑥) – Assume a 1D problem, in which heat only flows along the

fin

𝐴 𝑥 : cross-sectional area of the fin 𝑄: [𝐽 𝑠⁄ 𝑚], heat input per unit time & per unit length of the fin 𝑄 may be transferred to the fin across its outer surface or created internally 𝑄 is measured as positive if heat is supplied to the fin

11.1 Strong Form of One-Dimensional Heat Equation

• Consider an infinitely small part 𝑑𝑥 of the fin

• Conservation equation – Assume the problem is time independent

• 𝐻 + 𝑄𝑑𝑥 = 𝐻 + 𝑑𝐻 • 𝑑𝑑

𝑑𝑑= 𝑄

𝐻: heat flow per unit time at position 𝑥 𝐻 + 𝑑𝐻: heat flow per unit time at position 𝑥 + 𝑑𝑥 𝐻 and 𝐻 + 𝑑𝐻: positive when directed along the 𝑥 axis 𝑄𝑑𝑥: heat supply per unit time

11.1 Strong Form of One-Dimensional Heat Equation

• Define 𝐻 𝑥 = 𝐴 𝑥 𝑞 𝑥 – 𝑞: [𝐽 𝑚2⁄ 𝑠], flux

• The energy passing through a unit area per unit time • positive when heat flows in the 𝑥 direction

– 𝑑 𝐴𝐴 𝑑𝑑

= 𝑄 (1)

• Fourier’s law of heat conduction – 𝑞 = −𝑘 𝑑𝑑

𝑑𝑑 (2)

• A constitutive relation describes how heat flows within the material • 𝑇: temperature • 𝑘: thermal conductivity, [𝐽 ℃⁄ 𝑚𝑠]> 0

– If 𝑑𝑑𝑑𝑑

< 0, then 𝑞 > 0 is in accordance with the physical fact that heat flows from hotter to cooler regions

11.1 Strong Form of One-Dimensional Heat Equation

• One-dimensional heat equation

• Boundary conditions (BCs)

– Neumann BC: 𝑞 𝑥 = 0 = − 𝑘 𝑑𝑑𝑑𝑑 𝑑=0

= ℎ

– Dirichlet BC: 𝑇 𝑥 = 𝐿 = 𝑔 • ℎ,𝑔: given

(2) (1): 0, 0d dTAk Q x Ldx dx

⇒ + = ≤ ≤

2

2constant 0, 0d TAk Ak Q x Ldx

= ⇒ + = ≤ ≤

11.1 Strong Form of One-Dimensional Heat Equation

• Strong form for one-dimensional heat flow

( )

( )0

0, 0

0x

d dTAk Q x Ldx dx

dTq x k hdx

T x L g=

+ = ≤ ≤

= = − =

= =

11.2 Strong Form of Axially Loaded Elastic Bar

• Consider an axially loaded elastic bar

– 𝑏: 𝑁 𝑚⁄ , body force per unit axial length – The body force is measured as positive if pointing to

the right – 𝐴(𝑥): the cross-sectional area of the bar

11.2 Strong Form of Axially Loaded Elastic Bar

• Consider the infinitely small part 𝑑𝑥 of the bar

– Equilibrium condition • −𝑁 + 𝑏𝑑𝑥 + 𝑁 + 𝑑𝑁 = 0

• 𝑑𝑑𝑑𝑑

+ 𝑏 = 0

𝑁: the tensile force at the left end 𝑁 + 𝑑𝑁: the tensile force at the right end 𝑏𝑑𝑥 : total body force acting on small body

11.2 Strong Form of Axially Loaded Elastic Bar

– Recall 𝑁 = 𝐴(𝑥)𝜎(𝑥)

• 𝑑 𝐴𝜎𝑑𝑑

+ 𝑏 = 0

– Constitutive relation 𝜎 = 𝐸𝜀

• 𝑑 𝐴𝐴𝜀𝑑𝑑

+ 𝑏 = 0

– Strain-displacement (kinematic) relation 𝜀 = 𝑑𝑑𝑑𝑑

• 𝑑𝑑𝑑

𝐴𝐸 𝑑𝑑𝑑𝑑

+ 𝑏 = 0

11.2 Strong Form of Axially Loaded Elastic Bar

• Strong form for an axially loaded elastic bar – Boundary conditions

• ℎ,𝑔: given

2

2

0, 0

constant, 0, 0

d duAE b x Ldx dx

d uAE AE b x Ldx

+ = ≤ ≤

= + = ≤ ≤

( ) ( )

( )

00

0x

x

duN x A AE hdx

u x L g

σ=

=

= = = =

= =

11.2 Strong Form of Axially Loaded Elastic Bar

• One-dimensional heat flow

• Axially Loaded Elastic Bar

The behavior of an axially loaded elastic bar is controlled by the same equations as for one-dimensional heat flow

( )

( )0

0, 0

0x

d dTAk Q x Ldx dx

dTq x k hdx

T x L g=

+ = ≤ ≤

= = − =

= =

( )

( )0

0, 0

0x

d duAE b x Ldx dx

duN x AE hdx

u x L g=

+ = ≤ ≤

= = =

= =

11.3 Strong Form of Flexible String • Consider a laterally loaded flexible string

– 𝑤 𝑥 : the lateral deflection in the 𝑧 direction – 𝑝 𝑥 : 𝑁 𝑚⁄ , lateral loading per unit axial length, positive in

the 𝑧 direction – 𝑆: the tensile force in the string

11.3 Strong Form of Flexible String • Consider an infinitely small part 𝑑𝑥 of the string

– At position 𝑥, the deflection is 𝑤 • 𝜃: the angle that the string makes with the 𝑥 axis • 𝑆: the string force • 𝐻, 𝑉: the horizontal & vertical components of the tensile force 𝑆 in the

string

11.3 Strong Form of Flexible String • Consider an infinitely small part 𝑑𝑥 of the string

– At position 𝑥 + 𝑑𝑥 • 𝑆 + 𝑑𝑆: the string force • 𝐻 + 𝑑𝐻, 𝑉 + 𝑑𝑉: the horizontal & vertical components of 𝑆 + 𝑑𝑆 • 𝑝𝑑𝑥 : the total external force acting in the 𝑧 direction on the part 𝑑𝑥

11.3 Strong Form of Flexible String • Equilibrium condition

– Horizontal equilibrium

– Vertical equilibrium

– As 𝑆 is tangential to the flexible string,

0 constantH H dH H− + + = ⇒ =

0 0 (1)dVV pdx V dV pdx

− + + + = ⇒ + =

tan & tan (2)V dw dwV HH dx dx

θ θ= = ⇒ =

2

2(2) (1) : 0d wH pdx

⇒ + =

11.3 Strong Form of Flexible String

– From geometry, 𝐻 = 𝑆cos𝜃 – For small deflection, the slope 𝜃 is small and cos𝜃 ≈ 1

• 𝐻 = 𝑆 = 𝑐𝑐𝑐𝑠𝑐𝑐𝑐𝑐 • which means that the tensile force in the string is independent of the

loading

11.3 Strong Form of Flexible String • Strong form for a flexible string

– Boundary conditions

2

2 0, 0d wS p x Ldx

+ = ≤ ≤

( )( )

0 0

0

w x

w x L

= =

= =

Examples of Second-Order Differential Equations • Adopted from N. Ottosen & H. Petersson, Introduction to the Finite Element

Method, Prentice Hall, 1992.

Examples of Second-Order Differential Equations • Adopted from N. Ottosen & H. Petersson, Introduction to the Finite Element

Method, Prentice Hall, 1992.