FEA 03 Strong & Weak form PDF

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Strong & Weak form...

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Strong Form & Weak Form • One of the key steps in the FE analysis is derivation of the integral form (weak form) of the system from the governing differential equations (strong form). • This is a critical and important step in the FEM and is very common in all procedures. • Introduction of interpolation functions in the weak form will lead to the discretized FE equations (algebraic equations). • We start with one-dimensional boundary value problems (BVP) governed by ordinary differential eqs.

Strong form: elastic bar under axial loading (strong form) Considering the bar and its differential element:

F(x) is the external applied force at point x, b(x) is the body force, and A(x) is the cross-sectional area at x. Applying the equilibrium eq., we get: dx ) dx  F ( x  x)  0 2 F ( x  x)  F ( x) dx  b(x  )  0 dx 2 dF when dx  0  b( x)  0 dx  F (x )  b ( x 

We need to substitute u(x) in the balance equation :

F   . A  E . A  EA  BC :

du dx

d  du   EA   b  0 dx  dx  u (L )  u o

,

 ( 0)  E

du (0)   o dx

In case that EA is independent of x , the differential equation leads to:

d 2u EA 2  b( x)  0 dx which is the second order differential equation governing the displacement of a rod under axial loading.

Strong form: heat conduction in one-dimension (strong form) Considering the bar and its differential element:

T(x) is the temperature at x, q(x) is the heat flux at point x, b(x) is the energy of heat source, and A(x) is the cross-sectional area at x. Applying the energy balance eq., we get:

dx )dx  q ( x  x ) A( x  x )  0 2 q (x  x )A (x  x )  q (x ) A (x ) dx  b(x  )  0 dx 2 d ( qA) when dx  0  b( x)  0 dx  q( x) A( x)  b( x 

We substitute temperature, T(x) , in the balance equation. From Fourier law:

q  k BC :

dT dx



T ( L)  To

d  dT  k b  0 dx  dx  dT , q(0)   k ( 0)  q o dx

In case that k is independent of x , the differential equation leads to:

d 2T k 2  b( x )  0 dx This is the second order differential equation governing the heat flow (temperature distribution) of a one-dimensional heat conduction.

Weak form: elastic bar under axial loading To derive the weak form, multiply the ODE by an arbitrary weight function w(x) (satisfies the BC) and integrate on the domain (similarly for the natural boundary conditions). Considering the bar under loading with defined boundary conditions. d  du   EA   b  0 dx  dx  u ( L )  u o ,  (0)  E

L

d 

du 



  dx  EA dx   b .w( x )dx  0

,

w( x) in 0  x  L

0

du (0)   o dx

  du  u ( L )  uo ,  wA E (0)   o   x 0  0    dx

Integration by part from the first equation leads to: x L

L

L

du dw   du   w     EA  EA dx   bw.dx  0 dx  x0 0  dx dx   0 Based on weak form of the traction boundary condition with w(L) =0: L

L

du dw   EA  0  dx dx dx  wA 0 x0  0 bw.dx

Weak form: weak problem statement Find u(x) among the smooth functions with defined BC’s such that: L L du dw   0  EA dx dx dx  wA 0  x0  0 bw.dx

• The integration by parts leads to a symmetric form in u and w, this leads to a symmetric stiffness. • The weak form is equivalent to the strong form. In other words, if u(x) satisfies the strong form the above is also true, and if u(x) satisfies the weak form then it is also the solution of the strong form of the problem. • Smoothness in u and w is needed for the integrals in the weak form to exist, it means that trial function u and weight function w must have 1st derivative that is squared integrable.

variational principle – stationary principle – strong form - weak form All above forms can be extracted from each other as we have shown, integrating the strong form leads to weak from, and backward integration of the weak form yields to the strong form. Strong form is another form of the total potential energy and it is also derivable from the variational principle and principle of virtual work. Minimizing the total potential energy (stationary principle) leads to the governing differential equations of equilibrium and essential/natural boundary conditions or strong form. In many areas it is quite common that a functional form (as in the potential energy) can not be developed such as in fluid mechanics and advection-diffusion problems. It is important to realize that variational form only exists for selfadjoint systems. The weak form for the advection-diffusion equation is not symmetric, and it is not a self-adjoint system.

In such cases the finite element method in conjunction with Weighted Residual Method can be used to find a numerical solution for the governing differential equations....