ModelEqn1D - FINITE ELEMENT ANALYSIS OF A 1D MODEL PROBLEM...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
ModelEqn 1D - 1 JN Reddy FINITE ELEMENT ANALYSIS OF A 1D MODEL PROBLEM WITH A SINGLE VARIABLE Finite element model development of a linear 1D model differential equation involving a single dependent unknown (governing equations, FE model development weak form).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ModelEqn 1D - 2 JN Reddy GOVERNING EQUATION GOVERNING EQUATION point boundary a at ) ( ) ( in ) ( ) ( ) ( P u u b dx du a L , x f u x c dx du x a dx d = + = = + 0 0 E, A b=k L u ( L ) u 0 P x, u a = EA k, A u ( L ) u 0 P x, u a = kA Elastic deformation of a bar Heat transfer in a bar uninsulated bar
Background image of page 2
0 = + f cu dx du a dx d
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ModelEqn 1D - 3 JN Reddy FINITE ELEMENT APPROXIMATION FINITE ELEMENT APPROXIMATION (NEED FOR SEEKING SOLUTION ON SUB (NEED FOR SEEKING SOLUTION ON SUB -INTERVALS) INTERVALS) Approximation of the actual solution over the entire domain requires higher-order approximation 1. Actual solution may be defined by Sub-intervals because of discontiuity of the data a ( x ). 2. Approximation over sub-intervals subintervals allows lower-order Approximation of the actual solution
Background image of page 4
ModelEqn 1D - 4 JN Reddy FINITE ELEMENT DISCRETIZATION FINITE ELEMENT DISCRETIZATION Approximation over sub-intervals subintervals allows lower-order Approximation of the actual solution x x x a Q b Q a x x = b x x = A typical element (geometry and ‘forces’) length element heats or forces end b = = a b a x x h Q , Q
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ModelEqn 1D - 5 JN Reddy b b a x x x b x a x x x x x x x x Q x w Q x w dx wf cwu dx du dx dw a dx du a x w dx du a x w dx wf cwu dx du dx dw a dx du a w dx wf cwu dx du dx dw a dx x f u x c dx du x a dx d w b a b a b a b a b a b a + = + = + = + = ) ( ) ( ) ( ) ( ) ( ) ( ) ( a 0 WEAK FORM OVER AN ELEMENT WEAK FORM OVER AN ELEMENT Heat/Force input b x Q dx du a b a x Q dx du a a h e 2 1 Heat input Force out
Background image of page 6
ModelEqn 1D - 6 JN Reddy LINEAR AND BILINEAR FORMS LINEAR AND BILINEAR FORMS AND THE VARIATIONAL PROBLEM AND THE VARIATIONAL PROBLEM w w l u , w B all for holds ) ( ) ( = + + = + = b b a x x x x Q x w Q x w dx wf w l , dx cwu dx du dx dw a u , w B a b b a ) ( ) ( ) ( ) ( a Bilinear Form and Linear Form Weak Form ) ( ) ( ) ( ) ( ) ( ) ( a a w l u , w B Q x w Q x w dx wf dx cwu dx du dx dw a Q x w Q x w dx wf cwu dx du dx dw a b b a x x x x b b a x x a b b a b a = + + + = + = 0 Variational Problem : Find u such that
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ModelEqn 1D - 7 JN Reddy EQUIVALENCE BETWEEN MINIMUM OF A EQUIVALENCE BETWEEN MINIMUM OF A QUADRATIC FUNCTIONAL AND WEAK FORM QUADRATIC FUNCTIONAL AND WEAK FORM δ u u l u , δ u B δ I all for ) ( ) (0 0 = δ = which is the same as the weak form or the variational problem with δ u = w Quadratic Functional: Variational Problem : Find u such that I ( u ) is a minimum:
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

Page1 / 23

ModelEqn1D - FINITE ELEMENT ANALYSIS OF A 1D MODEL PROBLEM...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online