Finite Element Methods - Cairo University •Reddy, An introduction to the finite element method •Hughes, The finite element method-linear static and dynamic finite element analysis

Finite Element Methods

Instructor: Mohamed Abdou Mahran Kasem, Ph.D.

Aerospace Engineering Department

Cairo University

Contact details

Email: [email protected]

Office hours: Monday

Site link: https://scholar.cu.edu.eg/?q=mohamedabdou/classes

Course details

Textbooks:• Reddy, An introduction to the finite element method

• Hughes, The finite element method-linear static and dynamic finite elementanalysis

Grades:

➢10% Assignments

➢5% Attendance

➢15% Projects

➢70% Final exam

Discussion question

Why do you study the finite element method ?

CAD Model Finite Element Mesh

Discussion question – Why FEM?

Physical Problem

Mathematical Model

PDF + BC’s

Analytical Solution Numerical Solution Experimentally

Based on PDE

i.e. Finite Difference

Based on Integral Form

i.e. FE, BE, Meshless

Methods, Finite Volume

(ANSYS CFD).

Learning Outcomes

• Understand the concepts behind the finite element methods

• Solve 1-D trusses, 1-D beams, and 2-D plates using FEM

• Conduct modal analysis using FEM.

• Perform heat transfer analysis using FEM.

• Advanced topics

Idea of FEM

➢ The finite element method is based on dividing the whole structureinto subdomains that called elements.

➢The government equation is approximated over each element usingvariational methods.

➢This approximation is based on the idea of representing a complexdomain (or governing equation) using a set of simple functions.

➢The points that connect the elements together are called nodes.

Idea of FEM

History of FEM

➢The development of the finite element method starts in the 1940s in structuralengineering by Hrennikoff in 1941 and McHenry in 1943.

They used one-dimensional elements (bars and beams) to solve stresses incontinuous solids.

➢In 1956, Turner et al. derived stiffness matrices for truss elements, beam elements,and two-dimensional triangular and rectangular elements in plane stress andoutlined the procedure.

➢The term finite element was presented by Clough in 1960 in plane stress problemthat was solved using rectangular and triangular elements.

History of FEM

➢ Thermal analysis, and large deflection were first considered byTurner et al. in 1960.

➢ Material nonlinearities was considered by Gallagher et al. in 1962.

➢ Buckling problems were initially treated by Gallagher and Padlog in1963.

➢ The method was extended to solve visco-elasticity problems byZienkiewicz et al. in 1968.

Finite Element MethodBasic concepts

Remember, “ the purpose of analysis is to understand the problem and

gain insight – not generate numbers.” Thomass P. Sarafin.

A mathematical model

• We call the real system or a structure “the physical model”.

• Usually we cannot solve the real system, instead we solves an approximaterepresentation to this real system that we call “Mathematical Model/Idealizedmodel”.

• The mathematical model for most systems is represented by a differentialequation that we call the government equation .

• In FEM, we do not solve the DE, instead we solve an equivalent form to it thatwe call the weak form.

A mathematical model

Mathematical models

➢ The finite element method is used to provide an approximatesolution to a mathematical model.

➢A mathematical model is a set of equations that presents theessential features of the physical system.

➢ This mathematical model can be derived based on thefundamental scientific lows of physics such as theconservation of mass and momentum.

➢An example is the equilibrium equation that represents theproblems of linear structure analysis

Types of boundary conditions

There are mainly two types of boundary conditions:

1. Conditions applied to the primary or independentvariables itself (i.e. 𝒖(𝒙𝟎) = 𝟎) that are referred toas Dirichlet or essential boundary conditions.

2. Conditions applied to the derivatives of primary orindependent variables (i.e. 𝒖,𝒙 (𝒙𝟎) = 𝟎) that arereferred to as Neumann or natural boundaryconditions.

Domain,Ω

Notation

Notation

Notation

Notation

As we discussed earlier, in finite element method we approximate

the solution of the differential equation u by an approximate

function 𝑢ℎ.

Weighting Function

Integration by parts

𝑤𝑣 𝑥 = 𝑤𝑥𝑣 + 𝑤𝑣𝑥

𝑤𝑣 = න𝑤𝑥𝑣 𝑑𝑥 +න𝑤𝑣𝑥 𝑑𝑥

Integrate both sides w.r.t x

Rearrange

න𝑎

𝑏

𝑤𝑣𝑥 𝑑𝑥 = ቚ𝑤𝑣𝑎

𝑏−න

𝑎

𝑏

𝑤𝑥𝑣 𝑑𝑥

Index notation

𝑢,𝑖𝑖

Index notation

𝐱 =

𝑥1𝑥2𝑥3

𝑥𝑖 , 𝑖 = 1: 3

Vector

Tensor

𝛔 =

𝜎11 𝜎12 𝜎13𝜎21 𝜎22 𝜎23𝜎31 𝜎32 𝜎33

, 𝜎𝑖𝑗 , 𝑖 = 1: 3 𝑎𝑛𝑑 𝑗 = 1: 3

Functional

• Roughly speaking, a functional is an operator which maps a

function into a scalar or numbers.

• We can assume it as function of functions.

• It is usually integral statement of functions that measure their

overall performance.

Functional

Variation

Objective of numerical analysis

The numerical analysis objective is tosolve the government equations anddetermine the dependent variablefunction in certain domain 𝛀 andsome boundaries of the domain 𝚪. Domain,Ω

Objective of numerical analysis

➢ The approximate functions are known as shape functions

➢ They are defined by class 𝑪𝒎 𝛀 , if the function derivatives can beobtained up to the m derivative and continuous in 𝛀.

Objective of numerical analysis

➢ For example, 𝑪𝟎 function means that the dependent function iscontinuous, but its derivatives are not continuous within the domain 𝛀.

➢ Similarly, 𝑪𝟏 function means that the function first derivatives areexist and continuous, but its second derivative are not continuous in 𝛀.

➢ The domain dimension is defined based on the number ofindependent variables.

Weak formulation of BVPs

Consider the governing equation:

−𝑑

𝑑𝑥𝑎 𝑥 𝑢,𝑥 = 𝑓, 𝑥 𝜖 Ω = 0, 𝐿

𝑢 0 = 𝑢0, ቚ𝑎𝑢,𝑥𝑥=𝐿

= 𝐹0

This equation formulation called a strong form equation in the solution domain 𝒮 .

• Assume a weighting function 𝓌, then multiply 𝓌 by the governing equation and integrate.

නΩ

𝓌 −𝑑

𝑑𝑥𝑎 𝑥 𝑢,𝑥 − 𝑓 𝑑Ω = 0

This form is known as the weighted residual form.

Using integration by parts

න𝑢 𝑑𝑣 = 𝑢𝑣 − 𝑣 𝑑𝑢

Weak formulation of BVPs

Using integration by parts

න𝑢 𝑑𝑣 = 𝑢𝑣 − 𝑣 𝑑𝑢

නΩ

−𝓌𝑑

𝑑𝑥𝑎𝑢,𝑥 −𝓌𝑓 𝑑Ω = 0

නΩ

𝓌,𝑥𝑎𝑢,𝑥𝑑Ω − ቚ𝓌𝑎𝑢,𝑥0

𝐿− න

Ω

𝓌𝑓 𝑑Ω = 0

• Which takes the form

නΩ

𝓌,𝑥𝑎𝑢,𝑥𝑑Ω = නΩ

𝓌𝑓 𝑑Ω + ቚ𝓌𝑎𝑢,𝑥0

𝐿

All approximations of u should satisfy the essential BC’s, and the weight function should be zero at essential boundaries. Then 𝓌 0 = 0

By applying the boundary conditionsΩ 𝓌,𝑥𝑎𝑢,𝑥𝑑Ω = Ω 𝓌𝑓 𝑑Ω +𝓌 𝐿 𝐹0

Weak formulation of BVPs

The weak form statement is

𝐹𝑖𝑛𝑑 𝑢 𝜖 𝒰 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡

නΩ

𝓌,𝑥𝑎𝑢,𝑥𝑑Ω = නΩ

𝓌𝑓 𝑑Ω +𝓌 𝐿 𝐹0

𝑓𝑜𝑟 𝑎𝑙𝑙 𝓌 𝜖 𝕍

Where 𝒰 is the solution space and 𝕍 is the weighting space. Both the strong and weak forms are equivalent.

Weak formulation of BVPs

We can write the weak form in terms of the following functionals

𝐵 𝓌, 𝑢 = 𝑙 𝓌

Where

𝐵 𝓌, 𝑢 = නΩ

𝓌,𝑥𝑎𝑢,𝑥𝑑Ω

𝑙 𝓌 = නΩ

𝓌𝑓 𝑑Ω +𝓌 𝐿 𝐹0

➢𝐵 𝓌, 𝑢 𝑖𝑠 𝑏𝑖𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙, 𝑎𝑛𝑑 𝑙 𝓌 𝑖𝑠 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙.

➢𝐵 𝓌, 𝑢 is symmetric, i.e. 𝐵 𝓌, 𝑢 = 𝐵 𝑢,𝓌 .

The Blackbox

Mathematical Model +

SolutionInputs Outputs

If you do not understand what’s under the Blackbox

Mathematical Model +

SolutionGarbage in Garbage out

The objective of this course is to teach you what’s under the Blackbox

Model validation and verification

Verification

- Did I solve the model right?

- Results consistent with mathematical model.

- Level of numerical error is acceptable.

- Done through comparison with hand calculations

Validation

- Did I solve the right model?

- Does the mathematical model represent the physical problem?

- Check with experimental data