Objectives
• In this section well be learn deriving the system of linear equations th t t f li ti that approximate the solution of an ODE using piecewise continuous functions.
The Finite Element Method
Top-Down Approach
MATH1674 Dr. Eng. Mohammad Tawfik
MATH1674 Dr. Eng. Mohammad Tawfik
Finite Element
• As a first step, we need to define points in the domain, domain including the boundary which we will boundary, call nodes • These nodes will appear to be connected by elements
Trial Functions
• Out of all the admissible functions, we will select the ones that are equal to unity at l t th th t lt it t some specific point in the domain, node, and reach zero at the neighbouring points! • The neighbouring nodes are connected by elements of the domain
MATH1674 Dr. Eng. Mohammad Tawfik
MATH1674 Dr. Eng. Mohammad Tawfik
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Bar Example
• Using the week form derived from the Galerkin method, method the admissibility conditions on the function are that it should be at least once differentiable and satisfy the geometric boundary conditions.
The Trial functions become
ψi
= = x − xi −1 xi − xi −1 xi +1 − x xi +1 − xi xi −1 ≤ x < xi xi ≤ x < xi +1
But in this case, the unknown multipliers become the physical quantity that we are looking for
u ( x ) = ∑ uiψ i ( x )
i =1
MATH1674 Dr. Eng. Mohammad Tawfik MATH1674 Dr. Eng. Mohammad Tawfik
n
Note!
• Note that the functions have intersection domains only with the neighboring functions, those sharing the same element! • This means that the integral will become zero whenever the functions are not overlapping.
Evaluating the matrix elements:
• The integral becomes: dψ j ( x ) dψ i ( x ) dx
Domain
∫
dx
dx
(xi − xi −1 ) + (xi +1 − xi ) (xi − xi −1 )2 (xi +1 − xi )2 (x − x ) = − i i −1 2 (xi − xi −1 ) (x − x ) = − i +1 i 2 (xi +1 − xi )
= =0
j =i j = i −1 j = i +1 otherwise
MATH1674 Dr. Eng. Mohammad Tawfik
MATH1674 Dr. Eng. Mohammad Tawfik
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Relating to element length
• But (xi-xi-1)=li-1
dψ j ( x ) dψ i ( x ) dx dx
In Matrix Form
For equal-length elements
∫
Domain
dx
=
1 1 + li −1 li 1 li −1 1 li
j =i j = i −1 j = i +1 otherwise
MATH1674 Dr. Eng. Mohammad Tawfik
1 lelement
=− =− =0
⎡ 1 −1 0 0 ⎢− 1 2 − 1 0 ⎢ ⎢ 0 −1 2 −1 ⎢ ⎢ 0 0 −1 2 ⎢M M M M ⎢ ⎢0 0 0 0 ⎣
... 0⎤ ⎧ u1 ⎫ ⎧ f1 ⎫ ... 0⎥ ⎪u2 ⎪ ⎪ f 2 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... 0⎥ ⎪u3 ⎪ ⎪ f 3 ⎪ ⎥⎨ ⎬ = ⎨ ⎬ ... 0⎥ ⎪u4 ⎪ ⎪ f 4 ⎪ O M ⎥⎪ M ⎪ ⎪ M ⎪ ⎥⎪ ⎪ ⎪ ⎪ ... 1⎥ ⎪u n ⎪ ⎪ f n ⎪ ⎦⎩ ⎭ ⎩ ⎭
MATH1674 Dr. Eng. Mohammad Tawfik
The element matrix becomes:
Applying Boundary Conditions
Fixed from the left; u1=0 0
[k s ]{us } = { f }
Where [ks] is the structure stiffness matrix, {us} is the structure displacement vector, and {f} is the generalized force vector. NOTE: The stiffness matrix is SINGULAR!
1 lelement
⎡ 1 −1 0 0 ⎢− 1 2 − 1 0 ⎢ ⎢ 0 −1 2 −1 ⎢ ⎢ 0 0 −1 2 ⎢M M M M ⎢ ⎢0 0 0 0 ⎣
... 0⎤ ⎧ u1 ⎫ ⎧ f1 ⎫ ... 0⎥ ⎪u2 ⎪ ⎪ f 2 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ... 0⎥ ⎪u3 ⎪ ⎪ f 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎨ ⎬ = ⎨ ⎬ ... 0⎥ ⎪u4 ⎪ ⎪ f 4 ⎪ O M ⎥⎪ M ⎪ ⎪ M ⎪ ⎥⎪ ⎪ ⎪ ⎪ ... 1⎥ ⎪u n ⎪ ⎪ f n ⎪ ⎦⎩ ⎭ ⎩ ⎭
MATH1674 Dr. Eng. Mohammad Tawfik
MATH1674 Dr. Eng. Mohammad Tawfik
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You get two sets of equations
• The first equation will relate the force (f1), support reaction, t th displacements of t ti to the di l t f the structure • The rest of the equations will relate the unknown displacements to the external forces.
Conclusion
• In this section we learned:
– Selecting piecewise continuous function provides the main distinction of the finite element method – Evaluating the system of equations that approximate the differential equation using Galerkin Method – Applying the boundary conditions to the system of equations – Recognizing the system matrix as a sparse matrix; most of its elements are zeros.
MATH1674 Dr. Eng. Mohammad Tawfik
MATH1674 Dr. Eng. Mohammad Tawfik
Homework #2
• Use the top down approach outlined in the previous slides to write down a matrix equation describing the dynamics of a bar. • Use four nodes and assume the external force to be zero. • Write down your equation in the form:
& [M ]{u&}+ [K ]{u} = 0
MATH1674 Dr. Eng. Mohammad Tawfik
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