finite element methods - cairo university •reddy, an introduction to the finite element method...
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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